Integrand size = 15, antiderivative size = 38 \[ \int \frac {x^6}{\left (a+b x^3\right )^{2/3}} \, dx=\frac {x^7 \sqrt [3]{a+b x^3} \operatorname {Hypergeometric2F1}\left (1,\frac {8}{3},\frac {10}{3},-\frac {b x^3}{a}\right )}{7 a} \]
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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} \[ \int \frac {x^6}{\left (a+b x^3\right )^{2/3}} \, dx=\frac {x^7 \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {7}{3},\frac {10}{3},-\frac {b x^3}{a}\right )}{7 \left (a+b x^3\right )^{2/3}} \]
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Rule 371
Rule 372
Rubi steps \begin{align*} \text {integral}& = \frac {\left (1+\frac {b x^3}{a}\right )^{2/3} \int \frac {x^6}{\left (1+\frac {b x^3}{a}\right )^{2/3}} \, dx}{\left (a+b x^3\right )^{2/3}} \\ & = \frac {x^7 \left (1+\frac {b x^3}{a}\right )^{2/3} \, _2F_1\left (\frac {2}{3},\frac {7}{3};\frac {10}{3};-\frac {b x^3}{a}\right )}{7 \left (a+b x^3\right )^{2/3}} \\ \end{align*}
Time = 4.87 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.34 \[ \int \frac {x^6}{\left (a+b x^3\right )^{2/3}} \, dx=\frac {x^7 \left (1+\frac {b x^3}{a}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {7}{3},\frac {10}{3},-\frac {b x^3}{a}\right )}{7 \left (a+b x^3\right )^{2/3}} \]
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\[\int \frac {x^{6}}{\left (b \,x^{3}+a \right )^{\frac {2}{3}}}d x\]
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\[ \int \frac {x^6}{\left (a+b x^3\right )^{2/3}} \, dx=\int { \frac {x^{6}}{{\left (b x^{3} + a\right )}^{\frac {2}{3}}} \,d x } \]
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Result contains complex when optimal does not.
Time = 0.51 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.97 \[ \int \frac {x^6}{\left (a+b x^3\right )^{2/3}} \, dx=\frac {x^{7} \Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {2}{3}, \frac {7}{3} \\ \frac {10}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {2}{3}} \Gamma \left (\frac {10}{3}\right )} \]
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\[ \int \frac {x^6}{\left (a+b x^3\right )^{2/3}} \, dx=\int { \frac {x^{6}}{{\left (b x^{3} + a\right )}^{\frac {2}{3}}} \,d x } \]
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\[ \int \frac {x^6}{\left (a+b x^3\right )^{2/3}} \, dx=\int { \frac {x^{6}}{{\left (b x^{3} + a\right )}^{\frac {2}{3}}} \,d x } \]
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Timed out. \[ \int \frac {x^6}{\left (a+b x^3\right )^{2/3}} \, dx=\int \frac {x^6}{{\left (b\,x^3+a\right )}^{2/3}} \,d x \]
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