Optimal. Leaf size=297 \[ -\frac{x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^4 \left (a+b x^3\right )}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (5 a^2 b e-2 a^3 f-8 a b^2 d+11 b^3 c\right )}{18 a^{14/3} \sqrt [3]{b}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (5 a^2 b e-2 a^3 f-8 a b^2 d+11 b^3 c\right )}{9 a^{14/3} \sqrt [3]{b}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (5 a^2 b e-2 a^3 f-8 a b^2 d+11 b^3 c\right )}{3 \sqrt{3} a^{14/3} \sqrt [3]{b}}-\frac{a^2 e-2 a b d+3 b^2 c}{2 a^4 x^2}+\frac{2 b c-a d}{5 a^3 x^5}-\frac{c}{8 a^2 x^8} \]
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Rubi [A] time = 0.369952, antiderivative size = 297, 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, 1834, 200, 31, 634, 617, 204, 628} \[ -\frac{x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^4 \left (a+b x^3\right )}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (5 a^2 b e-2 a^3 f-8 a b^2 d+11 b^3 c\right )}{18 a^{14/3} \sqrt [3]{b}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (5 a^2 b e-2 a^3 f-8 a b^2 d+11 b^3 c\right )}{9 a^{14/3} \sqrt [3]{b}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (5 a^2 b e-2 a^3 f-8 a b^2 d+11 b^3 c\right )}{3 \sqrt{3} a^{14/3} \sqrt [3]{b}}-\frac{a^2 e-2 a b d+3 b^2 c}{2 a^4 x^2}+\frac{2 b c-a d}{5 a^3 x^5}-\frac{c}{8 a^2 x^8} \]
Antiderivative was successfully verified.
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Rule 1829
Rule 1834
Rule 200
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^9 \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}{3 a^4 \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-\frac{3 b^3 \left (b^2 c-a b d+a^2 e\right ) x^6}{a^2}+\frac{2 b^3 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^9}{a^3}}{x^9 \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}{3 a^4 \left (a+b x^3\right )}-\frac{\int \left (-\frac{3 b^3 c}{a x^9}-\frac{3 b^3 (-2 b c+a d)}{a^2 x^6}-\frac{3 b^3 \left (3 b^2 c-2 a b d+a^2 e\right )}{a^3 x^3}-\frac{b^3 \left (-11 b^3 c+8 a b^2 d-5 a^2 b e+2 a^3 f\right )}{a^3 \left (a+b x^3\right )}\right ) \, dx}{3 a b^3}\\ &=-\frac{c}{8 a^2 x^8}+\frac{2 b c-a d}{5 a^3 x^5}-\frac{3 b^2 c-2 a b d+a^2 e}{2 a^4 x^2}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 a^4 \left (a+b x^3\right )}-\frac{\left (11 b^3 c-8 a b^2 d+5 a^2 b e-2 a^3 f\right ) \int \frac{1}{a+b x^3} \, dx}{3 a^4}\\ &=-\frac{c}{8 a^2 x^8}+\frac{2 b c-a d}{5 a^3 x^5}-\frac{3 b^2 c-2 a b d+a^2 e}{2 a^4 x^2}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 a^4 \left (a+b x^3\right )}-\frac{\left (11 b^3 c-8 a b^2 d+5 a^2 b e-2 a^3 f\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a^{14/3}}-\frac{\left (11 b^3 c-8 a b^2 d+5 a^2 b e-2 a^3 f\right ) \int \frac{2 \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^{14/3}}\\ &=-\frac{c}{8 a^2 x^8}+\frac{2 b c-a d}{5 a^3 x^5}-\frac{3 b^2 c-2 a b d+a^2 e}{2 a^4 x^2}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 a^4 \left (a+b x^3\right )}-\frac{\left (11 b^3 c-8 a b^2 d+5 a^2 b e-2 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{14/3} \sqrt [3]{b}}-\frac{\left (11 b^3 c-8 a b^2 d+5 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^{13/3}}+\frac{\left (11 b^3 c-8 a b^2 d+5 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^{14/3} \sqrt [3]{b}}\\ &=-\frac{c}{8 a^2 x^8}+\frac{2 b c-a d}{5 a^3 x^5}-\frac{3 b^2 c-2 a b d+a^2 e}{2 a^4 x^2}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 a^4 \left (a+b x^3\right )}-\frac{\left (11 b^3 c-8 a b^2 d+5 a^2 b e-2 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{14/3} \sqrt [3]{b}}+\frac{\left (11 b^3 c-8 a b^2 d+5 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^{14/3} \sqrt [3]{b}}-\frac{\left (11 b^3 c-8 a b^2 d+5 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^{14/3} \sqrt [3]{b}}\\ &=-\frac{c}{8 a^2 x^8}+\frac{2 b c-a d}{5 a^3 x^5}-\frac{3 b^2 c-2 a b d+a^2 e}{2 a^4 x^2}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 a^4 \left (a+b x^3\right )}+\frac{\left (11 b^3 c-8 a b^2 d+5 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^{14/3} \sqrt [3]{b}}-\frac{\left (11 b^3 c-8 a b^2 d+5 a^2 b e-2 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{14/3} \sqrt [3]{b}}+\frac{\left (11 b^3 c-8 a b^2 d+5 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^{14/3} \sqrt [3]{b}}\\ \end{align*}
Mathematica [A] time = 0.178107, size = 280, normalized size = 0.94 \[ \frac{\frac{120 a^{2/3} x \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{a+b x^3}+\frac{20 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (5 a^2 b e-2 a^3 f-8 a b^2 d+11 b^3 c\right )}{\sqrt [3]{b}}+\frac{40 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-5 a^2 b e+2 a^3 f+8 a b^2 d-11 b^3 c\right )}{\sqrt [3]{b}}+\frac{40 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (5 a^2 b e-2 a^3 f-8 a b^2 d+11 b^3 c\right )}{\sqrt [3]{b}}-\frac{180 a^{2/3} \left (a^2 e-2 a b d+3 b^2 c\right )}{x^2}-\frac{72 a^{5/3} (a d-2 b c)}{x^5}-\frac{45 a^{8/3} c}{x^8}}{360 a^{14/3}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.014, size = 520, 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 = 1.55454, size = 2182, normalized size = 7.35 \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 = 1.0871, size = 468, normalized size = 1.58 \begin{align*} \frac{{\left (11 \, b^{3} c - 8 \, a b^{2} d - 2 \, a^{3} f + 5 \, a^{2} b 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^{5}} - \frac{\sqrt{3}{\left (11 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - 8 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + 5 \, \left (-a b^{2}\right )^{\frac{1}{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^{5} b} - \frac{b^{3} c x - a b^{2} d x - a^{3} f x + a^{2} b x e}{3 \,{\left (b x^{3} + a\right )} a^{4}} - \frac{{\left (11 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - 8 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + 5 \, \left (-a b^{2}\right )^{\frac{1}{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^{5} b} - \frac{60 \, b^{2} c x^{6} - 40 \, a b d x^{6} + 20 \, a^{2} x^{6} e - 16 \, a b c x^{3} + 8 \, a^{2} d x^{3} + 5 \, a^{2} c}{40 \, a^{4} x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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