Optimal. Leaf size=64 \[ \frac{x^8 \sqrt [3]{\frac{b x^3}{a}+1} F_1\left (\frac{8}{3};\frac{1}{3},1;\frac{11}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{8 c \sqrt [3]{a+b x^3}} \]
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Rubi [A] time = 0.0526605, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {511, 510} \[ \frac{x^8 \sqrt [3]{\frac{b x^3}{a}+1} F_1\left (\frac{8}{3};\frac{1}{3},1;\frac{11}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{8 c \sqrt [3]{a+b x^3}} \]
Antiderivative was successfully verified.
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Rule 511
Rule 510
Rubi steps
\begin{align*} \int \frac{x^7}{\sqrt [3]{a+b x^3} \left (c+d x^3\right )} \, dx &=\frac{\sqrt [3]{1+\frac{b x^3}{a}} \int \frac{x^7}{\sqrt [3]{1+\frac{b x^3}{a}} \left (c+d x^3\right )} \, dx}{\sqrt [3]{a+b x^3}}\\ &=\frac{x^8 \sqrt [3]{1+\frac{b x^3}{a}} F_1\left (\frac{8}{3};\frac{1}{3},1;\frac{11}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{8 c \sqrt [3]{a+b x^3}}\\ \end{align*}
Mathematica [B] time = 0.113332, size = 144, normalized size = 2.25 \[ \frac{-2 x^5 \sqrt [3]{\frac{b x^3}{a}+1} (a d+2 b c) F_1\left (\frac{5}{3};\frac{1}{3},1;\frac{8}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )-5 a c x^2 \sqrt [3]{\frac{b x^3}{a}+1} F_1\left (\frac{2}{3};\frac{1}{3},1;\frac{5}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )+5 c x^2 \left (a+b x^3\right )}{20 b c d \sqrt [3]{a+b x^3}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.055, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{7}}{d{x}^{3}+c}{\frac{1}{\sqrt [3]{b{x}^{3}+a}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{7}}{{\left (b x^{3} + a\right )}^{\frac{1}{3}}{\left (d x^{3} + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{7}}{\sqrt [3]{a + b x^{3}} \left (c + d x^{3}\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{7}}{{\left (b x^{3} + a\right )}^{\frac{1}{3}}{\left (d x^{3} + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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