Optimal. Leaf size=187 \[ -\frac{\sqrt [3]{a+b x^3} (b c-a d)}{d^2}-\frac{(b c-a d)^{4/3} \log \left (c+d x^3\right )}{6 d^{7/3}}+\frac{(b c-a d)^{4/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{7/3}}-\frac{(b c-a d)^{4/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt{3}}\right )}{\sqrt{3} d^{7/3}}+\frac{\left (a+b x^3\right )^{4/3}}{4 d} \]
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Rubi [A] time = 0.207714, antiderivative size = 187, 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 = {444, 50, 58, 617, 204, 31} \[ -\frac{\sqrt [3]{a+b x^3} (b c-a d)}{d^2}-\frac{(b c-a d)^{4/3} \log \left (c+d x^3\right )}{6 d^{7/3}}+\frac{(b c-a d)^{4/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{7/3}}-\frac{(b c-a d)^{4/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt{3}}\right )}{\sqrt{3} d^{7/3}}+\frac{\left (a+b x^3\right )^{4/3}}{4 d} \]
Antiderivative was successfully verified.
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Rule 444
Rule 50
Rule 58
Rule 617
Rule 204
Rule 31
Rubi steps
\begin{align*} \int \frac{x^2 \left (a+b x^3\right )^{4/3}}{c+d x^3} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{(a+b x)^{4/3}}{c+d x} \, dx,x,x^3\right )\\ &=\frac{\left (a+b x^3\right )^{4/3}}{4 d}-\frac{(b c-a d) \operatorname{Subst}\left (\int \frac{\sqrt [3]{a+b x}}{c+d x} \, dx,x,x^3\right )}{3 d}\\ &=-\frac{(b c-a d) \sqrt [3]{a+b x^3}}{d^2}+\frac{\left (a+b x^3\right )^{4/3}}{4 d}+\frac{(b c-a d)^2 \operatorname{Subst}\left (\int \frac{1}{(a+b x)^{2/3} (c+d x)} \, dx,x,x^3\right )}{3 d^2}\\ &=-\frac{(b c-a d) \sqrt [3]{a+b x^3}}{d^2}+\frac{\left (a+b x^3\right )^{4/3}}{4 d}-\frac{(b c-a d)^{4/3} \log \left (c+d x^3\right )}{6 d^{7/3}}+\frac{(b c-a d)^{4/3} \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^{7/3}}+\frac{(b c-a d)^{5/3} \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^{8/3}}\\ &=-\frac{(b c-a d) \sqrt [3]{a+b x^3}}{d^2}+\frac{\left (a+b x^3\right )^{4/3}}{4 d}-\frac{(b c-a d)^{4/3} \log \left (c+d x^3\right )}{6 d^{7/3}}+\frac{(b c-a d)^{4/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{7/3}}+\frac{(b c-a d)^{4/3} \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^{7/3}}\\ &=-\frac{(b c-a d) \sqrt [3]{a+b x^3}}{d^2}+\frac{\left (a+b x^3\right )^{4/3}}{4 d}-\frac{(b c-a d)^{4/3} \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^{7/3}}-\frac{(b c-a d)^{4/3} \log \left (c+d x^3\right )}{6 d^{7/3}}+\frac{(b c-a d)^{4/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{7/3}}\\ \end{align*}
Mathematica [A] time = 0.295421, size = 232, normalized size = 1.24 \[ \frac{(a d-b c) \left (\sqrt [3]{b c-a d} \log \left (-\sqrt [3]{d} \sqrt [3]{a+b x^3} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+d^{2/3} \left (a+b x^3\right )^{2/3}\right )-2 \sqrt [3]{b c-a d} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )-2 \sqrt{3} \sqrt [3]{b c-a d} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}-1}{\sqrt{3}}\right )+6 \sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{6 d^{7/3}}+\frac{\left (a+b x^3\right )^{4/3}}{4 d} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.042, 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 = 2.57733, size = 572, normalized size = 3.06 \begin{align*} \frac{4 \, \sqrt{3}{\left (b c - a d\right )} \left (-\frac{b c - a d}{d}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3}{\left (b x^{3} + a\right )}^{\frac{1}{3}} d \left (-\frac{b c - a d}{d}\right )^{\frac{2}{3}} - \sqrt{3}{\left (b c - a d\right )}}{3 \,{\left (b c - a d\right )}}\right ) + 2 \,{\left (b c - a d\right )} \left (-\frac{b c - a d}{d}\right )^{\frac{1}{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 ) - 4 \,{\left (b c - a d\right )} \left (-\frac{b c - a d}{d}\right )^{\frac{1}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac{1}{3}} - \left (-\frac{b c - a d}{d}\right )^{\frac{1}{3}}\right ) + 3 \,{\left (b d x^{3} - 4 \, b c + 5 \, a d\right )}{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{12 \, d^{2}} \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}}}{c + d x^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24612, size = 401, normalized size = 2.14 \begin{align*} -\frac{{\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\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^{4} - a d^{5}\right )}} + \frac{\sqrt{3}{\left (-b c d^{2} + a d^{3}\right )}^{\frac{1}{3}}{\left (b c - a d\right )} \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{1}{3}}{\left (b c - a d\right )} \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}} - \frac{4 \,{\left (b x^{3} + a\right )}^{\frac{1}{3}} b c d^{2} -{\left (b x^{3} + a\right )}^{\frac{4}{3}} d^{3} - 4 \,{\left (b x^{3} + a\right )}^{\frac{1}{3}} a d^{3}}{4 \, d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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