Optimal. Leaf size=165 \[ \frac{c \log \left (c+d x^3\right )}{6 d^{4/3} (b c-a d)^{2/3}}-\frac{c \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{4/3} (b c-a d)^{2/3}}+\frac{c \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt{3}}\right )}{\sqrt{3} d^{4/3} (b c-a d)^{2/3}}+\frac{\sqrt [3]{a+b x^3}}{b d} \]
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Rubi [A] time = 0.155348, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {446, 80, 58, 617, 204, 31} \[ \frac{c \log \left (c+d x^3\right )}{6 d^{4/3} (b c-a d)^{2/3}}-\frac{c \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{4/3} (b c-a d)^{2/3}}+\frac{c \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt{3}}\right )}{\sqrt{3} d^{4/3} (b c-a d)^{2/3}}+\frac{\sqrt [3]{a+b x^3}}{b d} \]
Antiderivative was successfully verified.
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Rule 446
Rule 80
Rule 58
Rule 617
Rule 204
Rule 31
Rubi steps
\begin{align*} \int \frac{x^5}{\left (a+b x^3\right )^{2/3} \left (c+d x^3\right )} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x}{(a+b x)^{2/3} (c+d x)} \, dx,x,x^3\right )\\ &=\frac{\sqrt [3]{a+b x^3}}{b d}-\frac{c \operatorname{Subst}\left (\int \frac{1}{(a+b x)^{2/3} (c+d x)} \, dx,x,x^3\right )}{3 d}\\ &=\frac{\sqrt [3]{a+b x^3}}{b d}+\frac{c \log \left (c+d x^3\right )}{6 d^{4/3} (b c-a d)^{2/3}}-\frac{c \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^{4/3} (b c-a d)^{2/3}}-\frac{c \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^{5/3} \sqrt [3]{b c-a d}}\\ &=\frac{\sqrt [3]{a+b x^3}}{b d}+\frac{c \log \left (c+d x^3\right )}{6 d^{4/3} (b c-a d)^{2/3}}-\frac{c \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{4/3} (b c-a d)^{2/3}}-\frac{c \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^{4/3} (b c-a d)^{2/3}}\\ &=\frac{\sqrt [3]{a+b x^3}}{b d}+\frac{c \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^{4/3} (b c-a d)^{2/3}}+\frac{c \log \left (c+d x^3\right )}{6 d^{4/3} (b c-a d)^{2/3}}-\frac{c \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{4/3} (b c-a d)^{2/3}}\\ \end{align*}
Mathematica [A] time = 0.103148, size = 202, normalized size = 1.22 \[ \frac{b c \log \left (-\sqrt [3]{d} \sqrt [3]{a+b x^3} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+d^{2/3} \left (a+b x^3\right )^{2/3}\right )+6 \sqrt [3]{d} \sqrt [3]{a+b x^3} (b c-a d)^{2/3}-2 b c \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )-2 \sqrt{3} b c \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}-1}{\sqrt{3}}\right )}{6 b d^{4/3} (b c-a d)^{2/3}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.04, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{5}}{d{x}^{3}+c} \left ( b{x}^{3}+a \right ) ^{-{\frac{2}{3}}}}\, 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 = 1.52908, size = 2276, normalized size = 13.79 \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^{5}}{\left (a + b x^{3}\right )^{\frac{2}{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 = 1.18109, size = 342, normalized size = 2.07 \begin{align*} -\frac{\frac{6 \,{\left (-b c d^{2} + a d^{3}\right )}^{\frac{1}{3}} b c \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^{2} - \sqrt{3} a d^{3}} + \frac{{\left (-b c d^{2} + a d^{3}\right )}^{\frac{1}{3}} b c \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 )}{b c d^{2} - a d^{3}} - \frac{2 \, b c \left (-\frac{b c - a d}{d}\right )^{\frac{1}{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 )}{b c d - a d^{2}} - \frac{6 \,{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{d}}{6 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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