Optimal. Leaf size=211 \[ -\frac{c \left (a+b x^3\right )^{4/3}}{4 d^2}+\frac{c \sqrt [3]{a+b x^3} (b c-a d)}{d^3}+\frac{c (b c-a d)^{4/3} \log \left (c+d x^3\right )}{6 d^{10/3}}-\frac{c (b c-a d)^{4/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{10/3}}+\frac{c (b c-a d)^{4/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt{3}}\right )}{\sqrt{3} d^{10/3}}+\frac{\left (a+b x^3\right )^{7/3}}{7 b d} \]
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Rubi [A] time = 0.244996, antiderivative size = 211, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {446, 80, 50, 58, 617, 204, 31} \[ -\frac{c \left (a+b x^3\right )^{4/3}}{4 d^2}+\frac{c \sqrt [3]{a+b x^3} (b c-a d)}{d^3}+\frac{c (b c-a d)^{4/3} \log \left (c+d x^3\right )}{6 d^{10/3}}-\frac{c (b c-a d)^{4/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{10/3}}+\frac{c (b c-a d)^{4/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt{3}}\right )}{\sqrt{3} d^{10/3}}+\frac{\left (a+b x^3\right )^{7/3}}{7 b d} \]
Antiderivative was successfully verified.
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Rule 446
Rule 80
Rule 50
Rule 58
Rule 617
Rule 204
Rule 31
Rubi steps
\begin{align*} \int \frac{x^5 \left (a+b x^3\right )^{4/3}}{c+d x^3} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x (a+b x)^{4/3}}{c+d x} \, dx,x,x^3\right )\\ &=\frac{\left (a+b x^3\right )^{7/3}}{7 b d}-\frac{c \operatorname{Subst}\left (\int \frac{(a+b x)^{4/3}}{c+d x} \, dx,x,x^3\right )}{3 d}\\ &=-\frac{c \left (a+b x^3\right )^{4/3}}{4 d^2}+\frac{\left (a+b x^3\right )^{7/3}}{7 b d}+\frac{(c (b c-a d)) \operatorname{Subst}\left (\int \frac{\sqrt [3]{a+b x}}{c+d x} \, dx,x,x^3\right )}{3 d^2}\\ &=\frac{c (b c-a d) \sqrt [3]{a+b x^3}}{d^3}-\frac{c \left (a+b x^3\right )^{4/3}}{4 d^2}+\frac{\left (a+b x^3\right )^{7/3}}{7 b d}-\frac{\left (c (b c-a d)^2\right ) \operatorname{Subst}\left (\int \frac{1}{(a+b x)^{2/3} (c+d x)} \, dx,x,x^3\right )}{3 d^3}\\ &=\frac{c (b c-a d) \sqrt [3]{a+b x^3}}{d^3}-\frac{c \left (a+b x^3\right )^{4/3}}{4 d^2}+\frac{\left (a+b x^3\right )^{7/3}}{7 b d}+\frac{c (b c-a d)^{4/3} \log \left (c+d x^3\right )}{6 d^{10/3}}-\frac{\left (c (b c-a d)^{4/3}\right ) \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^{10/3}}-\frac{\left (c (b c-a d)^{5/3}\right ) \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^{11/3}}\\ &=\frac{c (b c-a d) \sqrt [3]{a+b x^3}}{d^3}-\frac{c \left (a+b x^3\right )^{4/3}}{4 d^2}+\frac{\left (a+b x^3\right )^{7/3}}{7 b d}+\frac{c (b c-a d)^{4/3} \log \left (c+d x^3\right )}{6 d^{10/3}}-\frac{c (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^{10/3}}-\frac{\left (c (b c-a d)^{4/3}\right ) \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^{10/3}}\\ &=\frac{c (b c-a d) \sqrt [3]{a+b x^3}}{d^3}-\frac{c \left (a+b x^3\right )^{4/3}}{4 d^2}+\frac{\left (a+b x^3\right )^{7/3}}{7 b d}+\frac{c (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^{10/3}}+\frac{c (b c-a d)^{4/3} \log \left (c+d x^3\right )}{6 d^{10/3}}-\frac{c (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^{10/3}}\\ \end{align*}
Mathematica [A] time = 0.256264, size = 255, normalized size = 1.21 \[ -\frac{c \left (a+b x^3\right )^{4/3}}{4 d^2}+\frac{c (b c-a d) \left (\sqrt [3]{b c-a d} \log \left (-\sqrt [3]{d} \sqrt [3]{a+b x^3} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+d^{2/3} \left (a+b x^3\right )^{2/3}\right )-2 \sqrt [3]{b c-a d} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )-2 \sqrt{3} \sqrt [3]{b c-a d} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}-1}{\sqrt{3}}\right )+6 \sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{6 d^{10/3}}+\frac{\left (a+b x^3\right )^{7/3}}{7 b d} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.043, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{5}}{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 = 2.54, size = 683, normalized size = 3.24 \begin{align*} \frac{28 \, \sqrt{3}{\left (b^{2} c^{2} - a b c 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 ) + 14 \,{\left (b^{2} c^{2} - a b c 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 ) - 28 \,{\left (b^{2} c^{2} - a b c 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 (4 \, b^{2} d^{2} x^{6} + 28 \, b^{2} c^{2} - 35 \, a b c d + 4 \, a^{2} d^{2} -{\left (7 \, b^{2} c d - 8 \, a b d^{2}\right )} x^{3}\right )}{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{84 \, b d^{3}} \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{4}{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 = 1.21291, size = 470, normalized size = 2.23 \begin{align*} \frac{{\left (b^{10} c^{3} d^{4} - 2 \, a b^{9} c^{2} d^{5} + a^{2} b^{8} c d^{6}\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^{9} c d^{7} - a b^{8} d^{8}\right )}} - \frac{\sqrt{3}{\left (-b c d^{2} + a d^{3}\right )}^{\frac{1}{3}}{\left (b c^{2} - a c 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^{4}} - \frac{{\left (-b c d^{2} + a d^{3}\right )}^{\frac{1}{3}}{\left (b c^{2} - a c 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^{4}} + \frac{28 \,{\left (b x^{3} + a\right )}^{\frac{1}{3}} b^{8} c^{2} d^{4} - 7 \,{\left (b x^{3} + a\right )}^{\frac{4}{3}} b^{7} c d^{5} - 28 \,{\left (b x^{3} + a\right )}^{\frac{1}{3}} a b^{7} c d^{5} + 4 \,{\left (b x^{3} + a\right )}^{\frac{7}{3}} b^{6} d^{6}}{28 \, b^{7} d^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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