Optimal. Leaf size=167 \[ -\frac{1}{\sqrt [3]{a+b x^3} (b c-a d)}-\frac{\sqrt [3]{d} \log \left (c+d x^3\right )}{6 (b c-a d)^{4/3}}+\frac{\sqrt [3]{d} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 (b c-a d)^{4/3}}+\frac{\sqrt [3]{d} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt{3}}\right )}{\sqrt{3} (b c-a d)^{4/3}} \]
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Rubi [A] time = 0.165478, antiderivative size = 167, 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, 51, 56, 617, 204, 31} \[ -\frac{1}{\sqrt [3]{a+b x^3} (b c-a d)}-\frac{\sqrt [3]{d} \log \left (c+d x^3\right )}{6 (b c-a d)^{4/3}}+\frac{\sqrt [3]{d} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 (b c-a d)^{4/3}}+\frac{\sqrt [3]{d} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt{3}}\right )}{\sqrt{3} (b c-a d)^{4/3}} \]
Antiderivative was successfully verified.
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Rule 444
Rule 51
Rule 56
Rule 617
Rule 204
Rule 31
Rubi steps
\begin{align*} \int \frac{x^2}{\left (a+b x^3\right )^{4/3} \left (c+d x^3\right )} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{(a+b x)^{4/3} (c+d x)} \, dx,x,x^3\right )\\ &=-\frac{1}{(b c-a d) \sqrt [3]{a+b x^3}}-\frac{d \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a+b x} (c+d x)} \, dx,x,x^3\right )}{3 (b c-a d)}\\ &=-\frac{1}{(b c-a d) \sqrt [3]{a+b x^3}}-\frac{\sqrt [3]{d} \log \left (c+d x^3\right )}{6 (b c-a d)^{4/3}}+\frac{\sqrt [3]{d} \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 (b c-a d)^{4/3}}-\frac{\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 (b c-a d)}\\ &=-\frac{1}{(b c-a d) \sqrt [3]{a+b x^3}}-\frac{\sqrt [3]{d} \log \left (c+d x^3\right )}{6 (b c-a d)^{4/3}}+\frac{\sqrt [3]{d} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 (b c-a d)^{4/3}}-\frac{\sqrt [3]{d} \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 )}{(b c-a d)^{4/3}}\\ &=-\frac{1}{(b c-a d) \sqrt [3]{a+b x^3}}+\frac{\sqrt [3]{d} \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} (b c-a d)^{4/3}}-\frac{\sqrt [3]{d} \log \left (c+d x^3\right )}{6 (b c-a d)^{4/3}}+\frac{\sqrt [3]{d} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 (b c-a d)^{4/3}}\\ \end{align*}
Mathematica [C] time = 0.0145839, size = 50, normalized size = 0.3 \[ -\frac{\, _2F_1\left (-\frac{1}{3},1;\frac{2}{3};\frac{d \left (b x^3+a\right )}{a d-b c}\right )}{\sqrt [3]{a+b x^3} (b c-a d)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.034, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{2}}{d{x}^{3}+c} \left ( b{x}^{3}+a \right ) ^{-{\frac{4}{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 [A] time = 1.33494, size = 599, normalized size = 3.59 \begin{align*} -\frac{2 \, \sqrt{3}{\left (b x^{3} + a\right )} \left (-\frac{d}{b c - a d}\right )^{\frac{1}{3}} \arctan \left (\frac{2}{3} \, \sqrt{3}{\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (-\frac{d}{b c - a d}\right )^{\frac{1}{3}} + \frac{1}{3} \, \sqrt{3}\right ) -{\left (b x^{3} + a\right )} \left (-\frac{d}{b c - a d}\right )^{\frac{1}{3}} \log \left (-{\left (b x^{3} + a\right )}^{\frac{1}{3}}{\left (b c - a d\right )} \left (-\frac{d}{b c - a d}\right )^{\frac{2}{3}} +{\left (b x^{3} + a\right )}^{\frac{2}{3}} d -{\left (b c - a d\right )} \left (-\frac{d}{b c - a d}\right )^{\frac{1}{3}}\right ) + 2 \,{\left (b x^{3} + a\right )} \left (-\frac{d}{b c - a d}\right )^{\frac{1}{3}} \log \left ({\left (b c - a d\right )} \left (-\frac{d}{b c - a d}\right )^{\frac{2}{3}} +{\left (b x^{3} + a\right )}^{\frac{1}{3}} d\right ) + 6 \,{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{6 \,{\left ({\left (b^{2} c - a b d\right )} x^{3} + a b c - a^{2} d\right )}} \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{4}{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 [B] time = 1.17892, size = 385, normalized size = 2.31 \begin{align*} \frac{d \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^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}} + \frac{{\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 )}{\sqrt{3} b^{2} c^{2} d - 2 \, \sqrt{3} a b c d^{2} + \sqrt{3} a^{2} 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 \,{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )}} - \frac{1}{{\left (b x^{3} + a\right )}^{\frac{1}{3}}{\left (b c - a d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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