Optimal. Leaf size=162 \[ -\frac{(b c-a d)^{2/3} \log \left (c+d x^3\right )}{6 d^{5/3}}+\frac{(b c-a d)^{2/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{5/3}}+\frac{(b c-a d)^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt{3}}\right )}{\sqrt{3} d^{5/3}}+\frac{\left (a+b x^3\right )^{2/3}}{2 d} \]
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Rubi [A] time = 0.170246, antiderivative size = 162, 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 = {444, 50, 56, 617, 204, 31} \[ -\frac{(b c-a d)^{2/3} \log \left (c+d x^3\right )}{6 d^{5/3}}+\frac{(b c-a d)^{2/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{5/3}}+\frac{(b c-a d)^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt{3}}\right )}{\sqrt{3} d^{5/3}}+\frac{\left (a+b x^3\right )^{2/3}}{2 d} \]
Antiderivative was successfully verified.
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Rule 444
Rule 50
Rule 56
Rule 617
Rule 204
Rule 31
Rubi steps
\begin{align*} \int \frac{x^2 \left (a+b x^3\right )^{2/3}}{c+d x^3} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{(a+b x)^{2/3}}{c+d x} \, dx,x,x^3\right )\\ &=\frac{\left (a+b x^3\right )^{2/3}}{2 d}-\frac{(b c-a d) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a+b x} (c+d x)} \, dx,x,x^3\right )}{3 d}\\ &=\frac{\left (a+b x^3\right )^{2/3}}{2 d}-\frac{(b c-a d)^{2/3} \log \left (c+d x^3\right )}{6 d^{5/3}}+\frac{(b c-a d)^{2/3} \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^{5/3}}-\frac{(b c-a d) \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^2}\\ &=\frac{\left (a+b x^3\right )^{2/3}}{2 d}-\frac{(b c-a d)^{2/3} \log \left (c+d x^3\right )}{6 d^{5/3}}+\frac{(b c-a d)^{2/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{5/3}}-\frac{(b c-a d)^{2/3} \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^{5/3}}\\ &=\frac{\left (a+b x^3\right )^{2/3}}{2 d}+\frac{(b c-a d)^{2/3} \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^{5/3}}-\frac{(b c-a d)^{2/3} \log \left (c+d x^3\right )}{6 d^{5/3}}+\frac{(b c-a d)^{2/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{5/3}}\\ \end{align*}
Mathematica [C] time = 0.0155015, size = 47, normalized size = 0.29 \[ -\frac{\left (a+b x^3\right )^{2/3} \left (\, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{d \left (b x^3+a\right )}{a d-b c}\right )-1\right )}{2 d} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.039, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{2}}{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 = 1.53618, size = 736, normalized size = 4.54 \begin{align*} \frac{2 \, \sqrt{3} \left (\frac{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{d^{2}}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3}{\left (b x^{3} + a\right )}^{\frac{1}{3}} d \left (\frac{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{d^{2}}\right )^{\frac{1}{3}} - \sqrt{3}{\left (b c - a d\right )}}{3 \,{\left (b c - a d\right )}}\right ) - \left (\frac{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{d^{2}}\right )^{\frac{1}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac{1}{3}} d \left (\frac{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{d^{2}}\right )^{\frac{2}{3}} -{\left (b x^{3} + a\right )}^{\frac{2}{3}}{\left (b c - a d\right )} -{\left (b c - a d\right )} \left (\frac{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{d^{2}}\right )^{\frac{1}{3}}\right ) + 2 \, \left (\frac{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{d^{2}}\right )^{\frac{1}{3}} \log \left (-d \left (\frac{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{d^{2}}\right )^{\frac{2}{3}} -{\left (b x^{3} + a\right )}^{\frac{1}{3}}{\left (b c - a d\right )}\right ) + 3 \,{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{6 \, d} \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^{2} \left (a + b x^{3}\right )^{\frac{2}{3}}}{c + d x^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.23453, size = 350, normalized size = 2.16 \begin{align*} \frac{{\left (b c d \left (-\frac{b c - a d}{d}\right )^{\frac{1}{3}} - a d^{2} \left (-\frac{b c - a d}{d}\right )^{\frac{1}{3}}\right )} \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 )}{3 \,{\left (b c d^{2} - a d^{3}\right )}} + \frac{{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{2 \, d} + \frac{\sqrt{3}{\left (-b c d^{2} + a d^{3}\right )}^{\frac{2}{3}} \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 )}{3 \, d^{3}} - \frac{{\left (-b c d^{2} + a d^{3}\right )}^{\frac{2}{3}} \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 \, d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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