Integrand size = 15, antiderivative size = 38 \[ \int x^3 \sqrt [3]{a+b x^3} \, dx=\frac {x^4 \left (a+b x^3\right )^{4/3} \operatorname {Hypergeometric2F1}\left (1,\frac {8}{3},\frac {7}{3},-\frac {b x^3}{a}\right )}{4 a} \]
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Time = 0.01 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.34, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {372, 371} \[ \int x^3 \sqrt [3]{a+b x^3} \, dx=\frac {x^4 \sqrt [3]{a+b x^3} \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},\frac {4}{3},\frac {7}{3},-\frac {b x^3}{a}\right )}{4 \sqrt [3]{\frac {b x^3}{a}+1}} \]
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Rule 371
Rule 372
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [3]{a+b x^3} \int x^3 \sqrt [3]{1+\frac {b x^3}{a}} \, dx}{\sqrt [3]{1+\frac {b x^3}{a}}} \\ & = \frac {x^4 \sqrt [3]{a+b x^3} \, _2F_1\left (-\frac {1}{3},\frac {4}{3};\frac {7}{3};-\frac {b x^3}{a}\right )}{4 \sqrt [3]{1+\frac {b x^3}{a}}} \\ \end{align*}
Time = 4.37 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.34 \[ \int x^3 \sqrt [3]{a+b x^3} \, dx=\frac {x^4 \sqrt [3]{a+b x^3} \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},\frac {4}{3},\frac {7}{3},-\frac {b x^3}{a}\right )}{4 \sqrt [3]{1+\frac {b x^3}{a}}} \]
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\[\int x^{3} \left (b \,x^{3}+a \right )^{\frac {1}{3}}d x\]
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\[ \int x^3 \sqrt [3]{a+b x^3} \, dx=\int { {\left (b x^{3} + a\right )}^{\frac {1}{3}} x^{3} \,d x } \]
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Result contains complex when optimal does not.
Time = 0.51 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.03 \[ \int x^3 \sqrt [3]{a+b x^3} \, dx=\frac {\sqrt [3]{a} x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {7}{3}\right )} \]
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\[ \int x^3 \sqrt [3]{a+b x^3} \, dx=\int { {\left (b x^{3} + a\right )}^{\frac {1}{3}} x^{3} \,d x } \]
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\[ \int x^3 \sqrt [3]{a+b x^3} \, dx=\int { {\left (b x^{3} + a\right )}^{\frac {1}{3}} x^{3} \,d x } \]
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Timed out. \[ \int x^3 \sqrt [3]{a+b x^3} \, dx=\int x^3\,{\left (b\,x^3+a\right )}^{1/3} \,d x \]
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