Optimal. Leaf size=301 \[ \frac{x^2 \left (-4 a^2 b e+7 a^3 f+a b^2 d+2 b^3 c\right )}{9 a^2 b^3 \left (a+b x^3\right )}+\frac{x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 a b^3 \left (a+b x^3\right )^2}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (5 a^2 b e-20 a^3 f+a b^2 d+2 b^3 c\right )}{54 a^{7/3} b^{11/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (5 a^2 b e-20 a^3 f+a b^2 d+2 b^3 c\right )}{27 a^{7/3} b^{11/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (5 a^2 b e-20 a^3 f+a b^2 d+2 b^3 c\right )}{9 \sqrt{3} a^{7/3} b^{11/3}}+\frac{f x^2}{2 b^3} \]
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Rubi [A] time = 0.368046, antiderivative size = 301, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 10, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {1828, 1594, 1482, 459, 292, 31, 634, 617, 204, 628} \[ \frac{x^2 \left (-4 a^2 b e+7 a^3 f+a b^2 d+2 b^3 c\right )}{9 a^2 b^3 \left (a+b x^3\right )}+\frac{x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 a b^3 \left (a+b x^3\right )^2}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (5 a^2 b e-20 a^3 f+a b^2 d+2 b^3 c\right )}{54 a^{7/3} b^{11/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (5 a^2 b e-20 a^3 f+a b^2 d+2 b^3 c\right )}{27 a^{7/3} b^{11/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (5 a^2 b e-20 a^3 f+a b^2 d+2 b^3 c\right )}{9 \sqrt{3} a^{7/3} b^{11/3}}+\frac{f x^2}{2 b^3} \]
Antiderivative was successfully verified.
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Rule 1828
Rule 1594
Rule 1482
Rule 459
Rule 292
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{x \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx &=\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a b^3 \left (a+b x^3\right )^2}-\frac{\int \frac{-2 b \left (2 b^3 c+a b^2 d-a^2 b e+a^3 f\right ) x-6 a b^2 (b e-a f) x^4-6 a b^3 f x^7}{\left (a+b x^3\right )^2} \, dx}{6 a b^4}\\ &=\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a b^3 \left (a+b x^3\right )^2}-\frac{\int \frac{x \left (-2 b \left (2 b^3 c+a b^2 d-a^2 b e+a^3 f\right )-6 a b^2 (b e-a f) x^3-6 a b^3 f x^6\right )}{\left (a+b x^3\right )^2} \, dx}{6 a b^4}\\ &=\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a b^3 \left (a+b x^3\right )^2}+\frac{\left (2 b^3 c+a b^2 d-4 a^2 b e+7 a^3 f\right ) x^2}{9 a^2 b^3 \left (a+b x^3\right )}+\frac{\int \frac{x \left (2 b^3 \left (\frac{2 b^3 c}{a}+b^2 d+5 a b e-11 a^2 f\right )+18 a b^4 f x^3\right )}{a+b x^3} \, dx}{18 a b^6}\\ &=\frac{f x^2}{2 b^3}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a b^3 \left (a+b x^3\right )^2}+\frac{\left (2 b^3 c+a b^2 d-4 a^2 b e+7 a^3 f\right ) x^2}{9 a^2 b^3 \left (a+b x^3\right )}+\frac{\left (2 b^3 c+a b^2 d+5 a^2 b e-20 a^3 f\right ) \int \frac{x}{a+b x^3} \, dx}{9 a^2 b^3}\\ &=\frac{f x^2}{2 b^3}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a b^3 \left (a+b x^3\right )^2}+\frac{\left (2 b^3 c+a b^2 d-4 a^2 b e+7 a^3 f\right ) x^2}{9 a^2 b^3 \left (a+b x^3\right )}-\frac{\left (2 b^3 c+a b^2 d+5 a^2 b e-20 a^3 f\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{7/3} b^{10/3}}+\frac{\left (2 b^3 c+a b^2 d+5 a^2 b e-20 a^3 f\right ) \int \frac{\sqrt [3]{a}+\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{27 a^{7/3} b^{10/3}}\\ &=\frac{f x^2}{2 b^3}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a b^3 \left (a+b x^3\right )^2}+\frac{\left (2 b^3 c+a b^2 d-4 a^2 b e+7 a^3 f\right ) x^2}{9 a^2 b^3 \left (a+b x^3\right )}-\frac{\left (2 b^3 c+a b^2 d+5 a^2 b e-20 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{7/3} b^{11/3}}+\frac{\left (2 b^3 c+a b^2 d+5 a^2 b e-20 a^3 f\right ) \int \frac{-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{54 a^{7/3} b^{11/3}}+\frac{\left (2 b^3 c+a b^2 d+5 a^2 b e-20 a^3 f\right ) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{18 a^2 b^{10/3}}\\ &=\frac{f x^2}{2 b^3}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a b^3 \left (a+b x^3\right )^2}+\frac{\left (2 b^3 c+a b^2 d-4 a^2 b e+7 a^3 f\right ) x^2}{9 a^2 b^3 \left (a+b x^3\right )}-\frac{\left (2 b^3 c+a b^2 d+5 a^2 b e-20 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{7/3} b^{11/3}}+\frac{\left (2 b^3 c+a b^2 d+5 a^2 b e-20 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{7/3} b^{11/3}}+\frac{\left (2 b^3 c+a b^2 d+5 a^2 b e-20 a^3 f\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{9 a^{7/3} b^{11/3}}\\ &=\frac{f x^2}{2 b^3}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a b^3 \left (a+b x^3\right )^2}+\frac{\left (2 b^3 c+a b^2 d-4 a^2 b e+7 a^3 f\right ) x^2}{9 a^2 b^3 \left (a+b x^3\right )}-\frac{\left (2 b^3 c+a b^2 d+5 a^2 b e-20 a^3 f\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{7/3} b^{11/3}}-\frac{\left (2 b^3 c+a b^2 d+5 a^2 b e-20 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{7/3} b^{11/3}}+\frac{\left (2 b^3 c+a b^2 d+5 a^2 b e-20 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{7/3} b^{11/3}}\\ \end{align*}
Mathematica [A] time = 0.200122, size = 284, normalized size = 0.94 \[ \frac{\frac{6 b^{2/3} x^2 \left (-4 a^2 b e+7 a^3 f+a b^2 d+2 b^3 c\right )}{a^2 \left (a+b x^3\right )}+\frac{9 b^{2/3} x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{a \left (a+b x^3\right )^2}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (5 a^2 b e-20 a^3 f+a b^2 d+2 b^3 c\right )}{a^{7/3}}-\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (5 a^2 b e-20 a^3 f+a b^2 d+2 b^3 c\right )}{a^{7/3}}-\frac{2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (5 a^2 b e-20 a^3 f+a b^2 d+2 b^3 c\right )}{a^{7/3}}+27 b^{2/3} f x^2}{54 b^{11/3}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.011, size = 550, normalized size = 1.8 \begin{align*} \text{result too large to display} \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 [B] time = 1.49314, size = 2538, normalized size = 8.43 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09405, size = 522, normalized size = 1.73 \begin{align*} \frac{f x^{2}}{2 \, b^{3}} - \frac{{\left (2 \, b^{3} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} + a b^{2} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 20 \, a^{3} f \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 5 \, a^{2} b \left (-\frac{a}{b}\right )^{\frac{1}{3}} e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a^{3} b^{3}} - \frac{\sqrt{3}{\left (2 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c + \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - 20 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + 5 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{2} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, a^{3} b^{5}} + \frac{4 \, b^{4} c x^{5} + 2 \, a b^{3} d x^{5} + 14 \, a^{3} b f x^{5} - 8 \, a^{2} b^{2} x^{5} e + 7 \, a b^{3} c x^{2} - a^{2} b^{2} d x^{2} + 11 \, a^{4} f x^{2} - 5 \, a^{3} b x^{2} e}{18 \,{\left (b x^{3} + a\right )}^{2} a^{2} b^{3}} + \frac{{\left (2 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c + \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - 20 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + 5 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{2} b e\right )} \log \left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{54 \, a^{3} b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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