Optimal. Leaf size=174 \[ -\frac{a^2 \sqrt [3]{a+b x^3}}{b^3 d}+\frac{a^{7/3} \log \left (a-b x^3\right )}{3\ 2^{2/3} b^3 d}-\frac{a^{7/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2^{2/3} b^3 d}+\frac{\sqrt [3]{2} a^{7/3} \tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^3 d}-\frac{\left (a+b x^3\right )^{7/3}}{7 b^3 d} \]
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Rubi [A] time = 0.17849, antiderivative size = 174, 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, 57, 617, 204, 31} \[ -\frac{a^2 \sqrt [3]{a+b x^3}}{b^3 d}+\frac{a^{7/3} \log \left (a-b x^3\right )}{3\ 2^{2/3} b^3 d}-\frac{a^{7/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2^{2/3} b^3 d}+\frac{\sqrt [3]{2} a^{7/3} \tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^3 d}-\frac{\left (a+b x^3\right )^{7/3}}{7 b^3 d} \]
Antiderivative was successfully verified.
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Rule 446
Rule 88
Rule 50
Rule 57
Rule 617
Rule 204
Rule 31
Rubi steps
\begin{align*} \int \frac{x^8 \sqrt [3]{a+b x^3}}{a d-b d x^3} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^2 \sqrt [3]{a+b x}}{a d-b d x} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (-\frac{(a+b x)^{4/3}}{b^2 d}+\frac{a^2 \sqrt [3]{a+b x}}{b^2 (a d-b d x)}\right ) \, dx,x,x^3\right )\\ &=-\frac{\left (a+b x^3\right )^{7/3}}{7 b^3 d}+\frac{a^2 \operatorname{Subst}\left (\int \frac{\sqrt [3]{a+b x}}{a d-b d x} \, dx,x,x^3\right )}{3 b^2}\\ &=-\frac{a^2 \sqrt [3]{a+b x^3}}{b^3 d}-\frac{\left (a+b x^3\right )^{7/3}}{7 b^3 d}+\frac{\left (2 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{(a+b x)^{2/3} (a d-b d x)} \, dx,x,x^3\right )}{3 b^2}\\ &=-\frac{a^2 \sqrt [3]{a+b x^3}}{b^3 d}-\frac{\left (a+b x^3\right )^{7/3}}{7 b^3 d}+\frac{a^{7/3} \log \left (a-b x^3\right )}{3\ 2^{2/3} b^3 d}+\frac{a^{7/3} \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{2} \sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x^3}\right )}{2^{2/3} b^3 d}+\frac{a^{8/3} \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 )}{\sqrt [3]{2} b^3 d}\\ &=-\frac{a^2 \sqrt [3]{a+b x^3}}{b^3 d}-\frac{\left (a+b x^3\right )^{7/3}}{7 b^3 d}+\frac{a^{7/3} \log \left (a-b x^3\right )}{3\ 2^{2/3} b^3 d}-\frac{a^{7/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2^{2/3} b^3 d}-\frac{\left (\sqrt [3]{2} a^{7/3}\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 \sqrt [3]{a+b x^3}}{b^3 d}-\frac{\left (a+b x^3\right )^{7/3}}{7 b^3 d}+\frac{\sqrt [3]{2} a^{7/3} \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^{7/3} \log \left (a-b x^3\right )}{3\ 2^{2/3} b^3 d}-\frac{a^{7/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2^{2/3} b^3 d}\\ \end{align*}
Mathematica [A] time = 0.124325, size = 207, normalized size = 1.19 \[ \frac{1}{3} \left (-\frac{3 a^2 \sqrt [3]{a+b x^3}}{b^3 d}-\frac{\frac{2 \sqrt [3]{2} a^{7/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{b d}-\frac{\sqrt [3]{2} a^{7/3} \left (\log \left (2^{2/3} a^{2/3}+\sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )\right )}{b d}}{2 b^2}-\frac{3 \left (a+b x^3\right )^{7/3}}{7 b^3 d}\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.043, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{8}}{-bd{x}^{3}+ad}\sqrt [3]{b{x}^{3}+a}}\, 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 = 1.33823, size = 483, normalized size = 2.78 \begin{align*} -\frac{14 \, \sqrt{3} 2^{\frac{1}{3}} \left (-a\right )^{\frac{1}{3}} a^{2} \arctan \left (\frac{\sqrt{3} 2^{\frac{2}{3}}{\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (-a\right )^{\frac{2}{3}} + \sqrt{3} a}{3 \, a}\right ) + 7 \cdot 2^{\frac{1}{3}} \left (-a\right )^{\frac{1}{3}} a^{2} \log \left (2^{\frac{2}{3}} \left (-a\right )^{\frac{2}{3}} - 2^{\frac{1}{3}}{\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (-a\right )^{\frac{1}{3}} +{\left (b x^{3} + a\right )}^{\frac{2}{3}}\right ) - 14 \cdot 2^{\frac{1}{3}} \left (-a\right )^{\frac{1}{3}} a^{2} \log \left (2^{\frac{1}{3}} \left (-a\right )^{\frac{1}{3}} +{\left (b x^{3} + a\right )}^{\frac{1}{3}}\right ) + 6 \,{\left (b^{2} x^{6} + 2 \, a b x^{3} + 8 \, a^{2}\right )}{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{42 \, 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} \sqrt [3]{a + b x^{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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