Optimal. Leaf size=175 \[ \frac{a^{5/3} \log \left (a-b x^3\right )}{3 \sqrt [3]{2} b^2 d}-\frac{a^{5/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{2} b^2 d}-\frac{2^{2/3} a^{5/3} \tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^2 d}-\frac{a \left (a+b x^3\right )^{2/3}}{2 b^2 d}-\frac{\left (a+b x^3\right )^{5/3}}{5 b^2 d} \]
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Rubi [A] time = 0.156465, antiderivative size = 175, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {446, 80, 50, 55, 617, 204, 31} \[ \frac{a^{5/3} \log \left (a-b x^3\right )}{3 \sqrt [3]{2} b^2 d}-\frac{a^{5/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{2} b^2 d}-\frac{2^{2/3} a^{5/3} \tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^2 d}-\frac{a \left (a+b x^3\right )^{2/3}}{2 b^2 d}-\frac{\left (a+b x^3\right )^{5/3}}{5 b^2 d} \]
Antiderivative was successfully verified.
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Rule 446
Rule 80
Rule 50
Rule 55
Rule 617
Rule 204
Rule 31
Rubi steps
\begin{align*} \int \frac{x^5 \left (a+b x^3\right )^{2/3}}{a d-b d x^3} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x (a+b x)^{2/3}}{a d-b d x} \, dx,x,x^3\right )\\ &=-\frac{\left (a+b x^3\right )^{5/3}}{5 b^2 d}+\frac{a \operatorname{Subst}\left (\int \frac{(a+b x)^{2/3}}{a d-b d x} \, dx,x,x^3\right )}{3 b}\\ &=-\frac{a \left (a+b x^3\right )^{2/3}}{2 b^2 d}-\frac{\left (a+b x^3\right )^{5/3}}{5 b^2 d}+\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a+b x} (a d-b d x)} \, dx,x,x^3\right )}{3 b}\\ &=-\frac{a \left (a+b x^3\right )^{2/3}}{2 b^2 d}-\frac{\left (a+b x^3\right )^{5/3}}{5 b^2 d}+\frac{a^{5/3} \log \left (a-b x^3\right )}{3 \sqrt [3]{2} b^2 d}+\frac{a^{5/3} \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{2} \sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{2} b^2 d}-\frac{a^2 \operatorname{Subst}\left (\int \frac{1}{2^{2/3} a^{2/3}+\sqrt [3]{2} \sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+b x^3}\right )}{b^2 d}\\ &=-\frac{a \left (a+b x^3\right )^{2/3}}{2 b^2 d}-\frac{\left (a+b x^3\right )^{5/3}}{5 b^2 d}+\frac{a^{5/3} \log \left (a-b x^3\right )}{3 \sqrt [3]{2} b^2 d}-\frac{a^{5/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{2} b^2 d}+\frac{\left (2^{2/3} a^{5/3}\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2^{2/3} \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}\right )}{b^2 d}\\ &=-\frac{a \left (a+b x^3\right )^{2/3}}{2 b^2 d}-\frac{\left (a+b x^3\right )^{5/3}}{5 b^2 d}-\frac{2^{2/3} a^{5/3} \tan ^{-1}\left (\frac{1+\frac{2^{2/3} \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{\sqrt{3} b^2 d}+\frac{a^{5/3} \log \left (a-b x^3\right )}{3 \sqrt [3]{2} b^2 d}-\frac{a^{5/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{2} b^2 d}\\ \end{align*}
Mathematica [A] time = 0.129372, size = 143, normalized size = 0.82 \[ \frac{5\ 2^{2/3} a^{5/3} \log \left (a-b x^3\right )-3 \left (5\ 2^{2/3} a^{5/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )+\left (a+b x^3\right )^{2/3} \left (7 a+2 b x^3\right )\right )-10\ 2^{2/3} \sqrt{3} a^{5/3} \tan ^{-1}\left (\frac{\frac{2^{2/3} \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}+1}{\sqrt{3}}\right )}{30 b^2 d} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.04, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{5}}{-bd{x}^{3}+ad} \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 [A] time = 1.84128, size = 493, normalized size = 2.82 \begin{align*} -\frac{10 \cdot 4^{\frac{1}{3}} \sqrt{3} \left (-a^{2}\right )^{\frac{1}{3}} a \arctan \left (\frac{4^{\frac{1}{3}} \sqrt{3}{\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (-a^{2}\right )^{\frac{1}{3}} - \sqrt{3} a}{3 \, a}\right ) + 5 \cdot 4^{\frac{1}{3}} \left (-a^{2}\right )^{\frac{1}{3}} a \log \left (4^{\frac{2}{3}}{\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (-a^{2}\right )^{\frac{2}{3}} + 2 \,{\left (b x^{3} + a\right )}^{\frac{2}{3}} a - 2 \cdot 4^{\frac{1}{3}} \left (-a^{2}\right )^{\frac{1}{3}} a\right ) - 10 \cdot 4^{\frac{1}{3}} \left (-a^{2}\right )^{\frac{1}{3}} a \log \left (-4^{\frac{2}{3}} \left (-a^{2}\right )^{\frac{2}{3}} + 2 \,{\left (b x^{3} + a\right )}^{\frac{1}{3}} a\right ) + 3 \,{\left (2 \, b x^{3} + 7 \, a\right )}{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{30 \, b^{2} 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*} - \frac{\int \frac{x^{5} \left (a + b x^{3}\right )^{\frac{2}{3}}}{- a + b x^{3}}\, dx}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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