Optimal. Leaf size=203 \[ -\frac{\left (a+b x^3\right )^{2/3} (a d+b c)}{2 b^2 d^2}+\frac{\left (a+b x^3\right )^{5/3}}{5 b^2 d}+\frac{c^2 \log \left (c+d x^3\right )}{6 d^{8/3} \sqrt [3]{b c-a d}}-\frac{c^2 \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{8/3} \sqrt [3]{b c-a d}}-\frac{c^2 \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt{3}}\right )}{\sqrt{3} d^{8/3} \sqrt [3]{b c-a d}} \]
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Rubi [A] time = 0.21365, antiderivative size = 203, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {446, 88, 56, 617, 204, 31} \[ -\frac{\left (a+b x^3\right )^{2/3} (a d+b c)}{2 b^2 d^2}+\frac{\left (a+b x^3\right )^{5/3}}{5 b^2 d}+\frac{c^2 \log \left (c+d x^3\right )}{6 d^{8/3} \sqrt [3]{b c-a d}}-\frac{c^2 \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{8/3} \sqrt [3]{b c-a d}}-\frac{c^2 \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt{3}}\right )}{\sqrt{3} d^{8/3} \sqrt [3]{b c-a d}} \]
Antiderivative was successfully verified.
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Rule 446
Rule 88
Rule 56
Rule 617
Rule 204
Rule 31
Rubi steps
\begin{align*} \int \frac{x^8}{\sqrt [3]{a+b x^3} \left (c+d x^3\right )} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^2}{\sqrt [3]{a+b x} (c+d x)} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{-b c-a d}{b d^2 \sqrt [3]{a+b x}}+\frac{(a+b x)^{2/3}}{b d}+\frac{c^2}{d^2 \sqrt [3]{a+b x} (c+d x)}\right ) \, dx,x,x^3\right )\\ &=-\frac{(b c+a d) \left (a+b x^3\right )^{2/3}}{2 b^2 d^2}+\frac{\left (a+b x^3\right )^{5/3}}{5 b^2 d}+\frac{c^2 \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a+b x} (c+d x)} \, dx,x,x^3\right )}{3 d^2}\\ &=-\frac{(b c+a d) \left (a+b x^3\right )^{2/3}}{2 b^2 d^2}+\frac{\left (a+b x^3\right )^{5/3}}{5 b^2 d}+\frac{c^2 \log \left (c+d x^3\right )}{6 d^{8/3} \sqrt [3]{b c-a d}}+\frac{c^2 \operatorname{Subst}\left (\int \frac{1}{\frac{(b c-a d)^{2/3}}{d^{2/3}}-\frac{\sqrt [3]{b c-a d} x}{\sqrt [3]{d}}+x^2} \, dx,x,\sqrt [3]{a+b x^3}\right )}{2 d^3}-\frac{c^2 \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt [3]{b c-a d}}{\sqrt [3]{d}}+x} \, dx,x,\sqrt [3]{a+b x^3}\right )}{2 d^{8/3} \sqrt [3]{b c-a d}}\\ &=-\frac{(b c+a d) \left (a+b x^3\right )^{2/3}}{2 b^2 d^2}+\frac{\left (a+b x^3\right )^{5/3}}{5 b^2 d}+\frac{c^2 \log \left (c+d x^3\right )}{6 d^{8/3} \sqrt [3]{b c-a d}}-\frac{c^2 \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{8/3} \sqrt [3]{b c-a d}}+\frac{c^2 \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}\right )}{d^{8/3} \sqrt [3]{b c-a d}}\\ &=-\frac{(b c+a d) \left (a+b x^3\right )^{2/3}}{2 b^2 d^2}+\frac{\left (a+b x^3\right )^{5/3}}{5 b^2 d}-\frac{c^2 \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt{3}}\right )}{\sqrt{3} d^{8/3} \sqrt [3]{b c-a d}}+\frac{c^2 \log \left (c+d x^3\right )}{6 d^{8/3} \sqrt [3]{b c-a d}}-\frac{c^2 \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{8/3} \sqrt [3]{b c-a d}}\\ \end{align*}
Mathematica [C] time = 0.0608934, size = 103, normalized size = 0.51 \[ \frac{\left (a+b x^3\right )^{2/3} \left (3 a^2 d^2+5 b^2 c^2 \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{d \left (b x^3+a\right )}{a d-b c}\right )+2 a b d \left (c-d x^3\right )+b^2 c \left (2 d x^3-5 c\right )\right )}{10 b^2 d^2 (b c-a d)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.043, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{8}}{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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.96938, size = 1716, normalized size = 8.45 \begin{align*} \text{result too large to display} \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^{8}}{\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 [A] time = 1.22217, size = 423, normalized size = 2.08 \begin{align*} -\frac{b^{12} c^{2} d^{3} \left (-\frac{b c - a d}{d}\right )^{\frac{2}{3}} \log \left ({\left |{\left (b x^{3} + a\right )}^{\frac{1}{3}} - \left (-\frac{b c - a d}{d}\right )^{\frac{1}{3}} \right |}\right )}{3 \,{\left (b^{13} c d^{5} - a b^{12} d^{6}\right )}} - \frac{{\left (-b c d^{2} + a d^{3}\right )}^{\frac{2}{3}} c^{2} \arctan \left (\frac{\sqrt{3}{\left (2 \,{\left (b x^{3} + a\right )}^{\frac{1}{3}} + \left (-\frac{b c - a d}{d}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{b c - a d}{d}\right )^{\frac{1}{3}}}\right )}{\sqrt{3} b c d^{4} - \sqrt{3} a d^{5}} + \frac{{\left (-b c d^{2} + a d^{3}\right )}^{\frac{2}{3}} c^{2} \log \left ({\left (b x^{3} + a\right )}^{\frac{2}{3}} +{\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (-\frac{b c - a d}{d}\right )^{\frac{1}{3}} + \left (-\frac{b c - a d}{d}\right )^{\frac{2}{3}}\right )}{6 \,{\left (b c d^{4} - a d^{5}\right )}} - \frac{5 \,{\left (b x^{3} + a\right )}^{\frac{2}{3}} b^{9} c d^{3} - 2 \,{\left (b x^{3} + a\right )}^{\frac{5}{3}} b^{8} d^{4} + 5 \,{\left (b x^{3} + a\right )}^{\frac{2}{3}} a b^{8} d^{4}}{10 \, b^{10} d^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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