Optimal. Leaf size=269 \[ \frac{x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^3 b \left (a+b x^3\right )}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^2 b e+2 a^3 f-4 a b^2 d+7 b^3 c\right )}{18 a^{10/3} b^{5/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^2 b e+2 a^3 f-4 a b^2 d+7 b^3 c\right )}{9 a^{10/3} b^{5/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^2 b e+2 a^3 f-4 a b^2 d+7 b^3 c\right )}{3 \sqrt{3} a^{10/3} b^{5/3}}+\frac{2 b c-a d}{a^3 x}-\frac{c}{4 a^2 x^4} \]
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Rubi [A] time = 0.288055, antiderivative size = 269, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {1829, 1488, 292, 31, 634, 617, 204, 628} \[ \frac{x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^3 b \left (a+b x^3\right )}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^2 b e+2 a^3 f-4 a b^2 d+7 b^3 c\right )}{18 a^{10/3} b^{5/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^2 b e+2 a^3 f-4 a b^2 d+7 b^3 c\right )}{9 a^{10/3} b^{5/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^2 b e+2 a^3 f-4 a b^2 d+7 b^3 c\right )}{3 \sqrt{3} a^{10/3} b^{5/3}}+\frac{2 b c-a d}{a^3 x}-\frac{c}{4 a^2 x^4} \]
Antiderivative was successfully verified.
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Rule 1829
Rule 1488
Rule 292
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{c+d x^3+e x^6+f x^9}{x^5 \left (a+b x^3\right )^2} \, dx &=\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^3 b \left (a+b x^3\right )}-\frac{\int \frac{-3 b^3 c+3 b^3 \left (\frac{b c}{a}-d\right ) x^3-b^2 \left (\frac{b^3 c}{a^2}-\frac{b^2 d}{a}+b e+2 a f\right ) x^6}{x^5 \left (a+b x^3\right )} \, dx}{3 a b^3}\\ &=\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^3 b \left (a+b x^3\right )}-\frac{\int \left (-\frac{3 b^3 c}{a x^5}-\frac{3 b^3 (-2 b c+a d)}{a^2 x^2}-\frac{b^2 \left (7 b^3 c-4 a b^2 d+a^2 b e+2 a^3 f\right ) x}{a^2 \left (a+b x^3\right )}\right ) \, dx}{3 a b^3}\\ &=-\frac{c}{4 a^2 x^4}+\frac{2 b c-a d}{a^3 x}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^3 b \left (a+b x^3\right )}+\frac{\left (7 b^3 c-4 a b^2 d+a^2 b e+2 a^3 f\right ) \int \frac{x}{a+b x^3} \, dx}{3 a^3 b}\\ &=-\frac{c}{4 a^2 x^4}+\frac{2 b c-a d}{a^3 x}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^3 b \left (a+b x^3\right )}-\frac{\left (7 b^3 c-4 a b^2 d+a^2 b e+2 a^3 f\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a^{10/3} b^{4/3}}+\frac{\left (7 b^3 c-4 a b^2 d+a^2 b e+2 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}{9 a^{10/3} b^{4/3}}\\ &=-\frac{c}{4 a^2 x^4}+\frac{2 b c-a d}{a^3 x}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^3 b \left (a+b x^3\right )}-\frac{\left (7 b^3 c-4 a b^2 d+a^2 b e+2 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{10/3} b^{5/3}}+\frac{\left (7 b^3 c-4 a b^2 d+a^2 b e+2 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}{18 a^{10/3} b^{5/3}}+\frac{\left (7 b^3 c-4 a b^2 d+a^2 b e+2 a^3 f\right ) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 a^3 b^{4/3}}\\ &=-\frac{c}{4 a^2 x^4}+\frac{2 b c-a d}{a^3 x}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^3 b \left (a+b x^3\right )}-\frac{\left (7 b^3 c-4 a b^2 d+a^2 b e+2 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{10/3} b^{5/3}}+\frac{\left (7 b^3 c-4 a b^2 d+a^2 b e+2 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{10/3} b^{5/3}}+\frac{\left (7 b^3 c-4 a b^2 d+a^2 b e+2 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 )}{3 a^{10/3} b^{5/3}}\\ &=-\frac{c}{4 a^2 x^4}+\frac{2 b c-a d}{a^3 x}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^3 b \left (a+b x^3\right )}-\frac{\left (7 b^3 c-4 a b^2 d+a^2 b e+2 a^3 f\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{10/3} b^{5/3}}-\frac{\left (7 b^3 c-4 a b^2 d+a^2 b e+2 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{10/3} b^{5/3}}+\frac{\left (7 b^3 c-4 a b^2 d+a^2 b e+2 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{10/3} b^{5/3}}\\ \end{align*}
Mathematica [A] time = 0.168302, size = 255, normalized size = 0.95 \[ \frac{-\frac{12 \sqrt [3]{a} x^2 \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{b \left (a+b x^3\right )}+\frac{2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^2 b e+2 a^3 f-4 a b^2 d+7 b^3 c\right )}{b^{5/3}}-\frac{4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^2 b e+2 a^3 f-4 a b^2 d+7 b^3 c\right )}{b^{5/3}}-\frac{4 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (a^2 b e+2 a^3 f-4 a b^2 d+7 b^3 c\right )}{b^{5/3}}-\frac{9 a^{4/3} c}{x^4}-\frac{36 \sqrt [3]{a} (a d-2 b c)}{x}}{36 a^{10/3}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.013, size = 486, 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 [A] time = 1.52471, size = 1968, normalized size = 7.32 \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 [A] time = 138.391, size = 473, normalized size = 1.76 \begin{align*} \operatorname{RootSum}{\left (729 t^{3} a^{10} b^{5} + 8 a^{9} f^{3} + 12 a^{8} b e f^{2} - 48 a^{7} b^{2} d f^{2} + 6 a^{7} b^{2} e^{2} f + 84 a^{6} b^{3} c f^{2} - 48 a^{6} b^{3} d e f + a^{6} b^{3} e^{3} + 84 a^{5} b^{4} c e f + 96 a^{5} b^{4} d^{2} f - 12 a^{5} b^{4} d e^{2} - 336 a^{4} b^{5} c d f + 21 a^{4} b^{5} c e^{2} + 48 a^{4} b^{5} d^{2} e + 294 a^{3} b^{6} c^{2} f - 168 a^{3} b^{6} c d e - 64 a^{3} b^{6} d^{3} + 147 a^{2} b^{7} c^{2} e + 336 a^{2} b^{7} c d^{2} - 588 a b^{8} c^{2} d + 343 b^{9} c^{3}, \left ( t \mapsto t \log{\left (\frac{81 t^{2} a^{7} b^{3}}{4 a^{6} f^{2} + 4 a^{5} b e f - 16 a^{4} b^{2} d f + a^{4} b^{2} e^{2} + 28 a^{3} b^{3} c f - 8 a^{3} b^{3} d e + 14 a^{2} b^{4} c e + 16 a^{2} b^{4} d^{2} - 56 a b^{5} c d + 49 b^{6} c^{2}} + x \right )} \right )\right )} - \frac{3 a^{2} b c + x^{6} \left (4 a^{3} f - 4 a^{2} b e + 16 a b^{2} d - 28 b^{3} c\right ) + x^{3} \left (12 a^{2} b d - 21 a b^{2} c\right )}{12 a^{4} b x^{4} + 12 a^{3} b^{2} x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08189, size = 483, normalized size = 1.8 \begin{align*} -\frac{{\left (7 \, b^{3} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 4 \, a b^{2} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 2 \, a^{3} f \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 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 )}{9 \, a^{4} b} + \frac{b^{3} c x^{2} - a b^{2} d x^{2} - a^{3} f x^{2} + a^{2} b x^{2} e}{3 \,{\left (b x^{3} + a\right )} a^{3} b} - \frac{\sqrt{3}{\left (7 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - 4 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d + 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + \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 )}{9 \, a^{4} b^{3}} + \frac{{\left (7 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - 4 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d + 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + \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 )}{18 \, a^{4} b^{3}} + \frac{8 \, b c x^{3} - 4 \, a d x^{3} - a c}{4 \, a^{3} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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