Optimal. Leaf size=177 \[ -\frac{a^2 \left (a+b x^3\right )^{2/3}}{2 b^3 d}+\frac{a^{8/3} \log \left (a-b x^3\right )}{3 \sqrt [3]{2} b^3 d}-\frac{a^{8/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{2} b^3 d}-\frac{2^{2/3} a^{8/3} \tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^3 d}-\frac{\left (a+b x^3\right )^{8/3}}{8 b^3 d} \]
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Rubi [A] time = 0.199533, antiderivative size = 177, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {446, 88, 50, 55, 617, 204, 31} \[ -\frac{a^2 \left (a+b x^3\right )^{2/3}}{2 b^3 d}+\frac{a^{8/3} \log \left (a-b x^3\right )}{3 \sqrt [3]{2} b^3 d}-\frac{a^{8/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{2} b^3 d}-\frac{2^{2/3} a^{8/3} \tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^3 d}-\frac{\left (a+b x^3\right )^{8/3}}{8 b^3 d} \]
Antiderivative was successfully verified.
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Rule 446
Rule 88
Rule 50
Rule 55
Rule 617
Rule 204
Rule 31
Rubi steps
\begin{align*} \int \frac{x^8 \left (a+b x^3\right )^{2/3}}{a d-b d x^3} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^2 (a+b x)^{2/3}}{a d-b d x} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (-\frac{(a+b x)^{5/3}}{b^2 d}+\frac{a^2 (a+b x)^{2/3}}{b^2 (a d-b d x)}\right ) \, dx,x,x^3\right )\\ &=-\frac{\left (a+b x^3\right )^{8/3}}{8 b^3 d}+\frac{a^2 \operatorname{Subst}\left (\int \frac{(a+b x)^{2/3}}{a d-b d x} \, dx,x,x^3\right )}{3 b^2}\\ &=-\frac{a^2 \left (a+b x^3\right )^{2/3}}{2 b^3 d}-\frac{\left (a+b x^3\right )^{8/3}}{8 b^3 d}+\frac{\left (2 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a+b x} (a d-b d x)} \, dx,x,x^3\right )}{3 b^2}\\ &=-\frac{a^2 \left (a+b x^3\right )^{2/3}}{2 b^3 d}-\frac{\left (a+b x^3\right )^{8/3}}{8 b^3 d}+\frac{a^{8/3} \log \left (a-b x^3\right )}{3 \sqrt [3]{2} b^3 d}+\frac{a^{8/3} \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^3 d}-\frac{a^3 \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^3 d}\\ &=-\frac{a^2 \left (a+b x^3\right )^{2/3}}{2 b^3 d}-\frac{\left (a+b x^3\right )^{8/3}}{8 b^3 d}+\frac{a^{8/3} \log \left (a-b x^3\right )}{3 \sqrt [3]{2} b^3 d}-\frac{a^{8/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{2} b^3 d}+\frac{\left (2^{2/3} a^{8/3}\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^3 d}\\ &=-\frac{a^2 \left (a+b x^3\right )^{2/3}}{2 b^3 d}-\frac{\left (a+b x^3\right )^{8/3}}{8 b^3 d}-\frac{2^{2/3} a^{8/3} \tan ^{-1}\left (\frac{1+\frac{2^{2/3} \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{\sqrt{3} b^3 d}+\frac{a^{8/3} \log \left (a-b x^3\right )}{3 \sqrt [3]{2} b^3 d}-\frac{a^{8/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{2} b^3 d}\\ \end{align*}
Mathematica [A] time = 0.163851, size = 153, normalized size = 0.86 \[ \frac{-3 \left (\left (a+b x^3\right )^{2/3} \left (5 a^2+2 a b x^3+b^2 x^6\right )+4\ 2^{2/3} a^{8/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )\right )+4\ 2^{2/3} a^{8/3} \log \left (a-b x^3\right )-8\ 2^{2/3} \sqrt{3} a^{8/3} \tan ^{-1}\left (\frac{\frac{2^{2/3} \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}+1}{\sqrt{3}}\right )}{24 b^3 d} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.042, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{8}}{-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 = 2.65444, size = 517, normalized size = 2.92 \begin{align*} -\frac{8 \cdot 4^{\frac{1}{3}} \sqrt{3} \left (-a^{2}\right )^{\frac{1}{3}} a^{2} \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 ) + 4 \cdot 4^{\frac{1}{3}} \left (-a^{2}\right )^{\frac{1}{3}} a^{2} \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 ) - 8 \cdot 4^{\frac{1}{3}} \left (-a^{2}\right )^{\frac{1}{3}} a^{2} \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 (b^{2} x^{6} + 2 \, a b x^{3} + 5 \, a^{2}\right )}{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{24 \, b^{3} 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^{8} \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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