Evaluate the integral. $$ \int 10 x^{2} \sqrt[4]{8+2 x^{3}} d x $$

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Calculate the integral $$ \int 10x^{2}\sqrt[4]{8+2x^{3}}dx $$.
Evaluate the integral by following steps:
- step0: Evaluate using substitution:
  $$\int 10x^{2}\sqrt[4]{8+2x^{3}} dx$$
- step1: Evaluate the power:
  $$\int 10x^{2}\left(8+2x^{3}ight)^{\frac{1}{4}} dx$$
- step2: Use properties of integrals:
  $$10	imes \int x^{2}\left(8+2x^{3}ight)^{\frac{1}{4}} dx$$
- step3: Use the substitution $$dx=\frac{1}{6x^{2}} dt$$ to transform the integral$$:$$
  $$10	imes \int x^{2}\left(8+2x^{3}ight)^{\frac{1}{4}}	imes \frac{1}{6x^{2}} dt$$
- step4: Simplify:
  $$10	imes \int \frac{\left(8+2x^{3}ight)^{\frac{1}{4}}}{6} dt$$
- step5: Use the substitution $$t=2x^{3}$$ to transform the integral$$:$$
  $$10	imes \int \frac{\left(8+tight)^{\frac{1}{4}}}{6} dt$$
- step6: Rewrite the expression:
  $$10	imes \int \frac{1}{6}\left(8+tight)^{\frac{1}{4}} dt$$
- step7: Use properties of integrals:
  $$10	imes \frac{1}{6}	imes \int \left(8+tight)^{\frac{1}{4}} dt$$
- step8: Multiply the numbers:
  $$\frac{5}{3}	imes \int \left(8+tight)^{\frac{1}{4}} dt$$
- step9: Use the substitution $$dt=1 dv$$ to transform the integral$$:$$
  $$\frac{5}{3}	imes \int \left(8+tight)^{\frac{1}{4}}	imes 1 dv$$
- step10: Simplify:
  $$\frac{5}{3}	imes \int \left(8+tight)^{\frac{1}{4}} dv$$
- step11: Use the substitution $$v=8+t$$ to transform the integral$$:$$
  $$\frac{5}{3}	imes \int v^{\frac{1}{4}} dv$$
- step12: Evaluate the integral:
  $$\frac{5}{3}	imes \frac{v^{\frac{1}{4}+1}}{\frac{1}{4}+1}$$
- step13: Simplify:
  $$\frac{5}{3}	imes \frac{v^{\frac{5}{4}}}{\frac{5}{4}}$$
- step14: Multiply the terms:
  $$\frac{5v^{\frac{5}{4}}}{3	imes \frac{5}{4}}$$
- step15: Multiply the terms:
  $$\frac{5v^{\frac{5}{4}}}{\frac{15}{4}}$$
- step16: Reduce the fraction:
  $$\frac{4v^{\frac{5}{4}}}{3}$$
- step17: Substitute back:
  $$\frac{4\left(8+tight)^{\frac{5}{4}}}{3}$$
- step18: Substitute back:
  $$\frac{4\left(8+2x^{3}ight)^{\frac{5}{4}}}{3}$$
- step19: Simplify:
  $$\frac{4\sqrt[4]{\left(8+2x^{3}ight)^{5}}}{3}$$
- step20: Add the constant of integral C:
  $$\frac{4\sqrt[4]{\left(8+2x^{3}ight)^{5}}}{3} + C, C \in \mathbb{R}$$
The integral of $$10x^{2}\sqrt[4]{8+2x^{3}}$$ with respect to $$x$$ is $$\frac{4\sqrt[4]{(8+2x^{3})^{5}}}{3} + C$$, where $$C$$ is an arbitrary constant.
The integral of $$10x^{2}\sqrt[4]{8+2x^{3}}$$ with respect to $$x$$ is $$\frac{4\sqrt[4]{(8+2x^{3})^{5}}}{3} + C$$.

Evaluate the integral.

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