Optimal. Leaf size=334 \[ \frac{b x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^5 \left (a+b x^3\right )}+\frac{\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (7 a^2 b e-4 a^3 f-10 a b^2 d+13 b^3 c\right )}{18 a^{16/3}}+\frac{2 a^2 b e+a^3 (-f)-3 a b^2 d+4 b^3 c}{a^5 x}-\frac{\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (7 a^2 b e-4 a^3 f-10 a b^2 d+13 b^3 c\right )}{9 a^{16/3}}-\frac{\sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (7 a^2 b e-4 a^3 f-10 a b^2 d+13 b^3 c\right )}{3 \sqrt{3} a^{16/3}}-\frac{a^2 e-2 a b d+3 b^2 c}{4 a^4 x^4}+\frac{2 b c-a d}{7 a^3 x^7}-\frac{c}{10 a^2 x^{10}} \]
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Rubi [A] time = 0.456739, antiderivative size = 334, 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, 292, 31, 634, 617, 204, 628} \[ \frac{b x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^5 \left (a+b x^3\right )}+\frac{\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (7 a^2 b e-4 a^3 f-10 a b^2 d+13 b^3 c\right )}{18 a^{16/3}}+\frac{2 a^2 b e+a^3 (-f)-3 a b^2 d+4 b^3 c}{a^5 x}-\frac{\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (7 a^2 b e-4 a^3 f-10 a b^2 d+13 b^3 c\right )}{9 a^{16/3}}-\frac{\sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (7 a^2 b e-4 a^3 f-10 a b^2 d+13 b^3 c\right )}{3 \sqrt{3} a^{16/3}}-\frac{a^2 e-2 a b d+3 b^2 c}{4 a^4 x^4}+\frac{2 b c-a d}{7 a^3 x^7}-\frac{c}{10 a^2 x^{10}} \]
Antiderivative was successfully verified.
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Rule 1829
Rule 1834
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^{11} \left (a+b x^3\right )^2} \, dx &=\frac{b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^5 \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{3 b^3 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^9}{a^3}-\frac{b^4 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^{12}}{a^4}}{x^{11} \left (a+b x^3\right )} \, dx}{3 a b^3}\\ &=\frac{b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^5 \left (a+b x^3\right )}-\frac{\int \left (-\frac{3 b^3 c}{a x^{11}}-\frac{3 b^3 (-2 b c+a d)}{a^2 x^8}-\frac{3 b^3 \left (3 b^2 c-2 a b d+a^2 e\right )}{a^3 x^5}-\frac{3 b^3 \left (-4 b^3 c+3 a b^2 d-2 a^2 b e+a^3 f\right )}{a^4 x^2}+\frac{b^4 \left (-13 b^3 c+10 a b^2 d-7 a^2 b e+4 a^3 f\right ) x}{a^4 \left (a+b x^3\right )}\right ) \, dx}{3 a b^3}\\ &=-\frac{c}{10 a^2 x^{10}}+\frac{2 b c-a d}{7 a^3 x^7}-\frac{3 b^2 c-2 a b d+a^2 e}{4 a^4 x^4}+\frac{4 b^3 c-3 a b^2 d+2 a^2 b e-a^3 f}{a^5 x}+\frac{b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^5 \left (a+b x^3\right )}+\frac{\left (b \left (13 b^3 c-10 a b^2 d+7 a^2 b e-4 a^3 f\right )\right ) \int \frac{x}{a+b x^3} \, dx}{3 a^5}\\ &=-\frac{c}{10 a^2 x^{10}}+\frac{2 b c-a d}{7 a^3 x^7}-\frac{3 b^2 c-2 a b d+a^2 e}{4 a^4 x^4}+\frac{4 b^3 c-3 a b^2 d+2 a^2 b e-a^3 f}{a^5 x}+\frac{b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^5 \left (a+b x^3\right )}-\frac{\left (b^{2/3} \left (13 b^3 c-10 a b^2 d+7 a^2 b e-4 a^3 f\right )\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a^{16/3}}+\frac{\left (b^{2/3} \left (13 b^3 c-10 a b^2 d+7 a^2 b e-4 a^3 f\right )\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^{16/3}}\\ &=-\frac{c}{10 a^2 x^{10}}+\frac{2 b c-a d}{7 a^3 x^7}-\frac{3 b^2 c-2 a b d+a^2 e}{4 a^4 x^4}+\frac{4 b^3 c-3 a b^2 d+2 a^2 b e-a^3 f}{a^5 x}+\frac{b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^5 \left (a+b x^3\right )}-\frac{\sqrt [3]{b} \left (13 b^3 c-10 a b^2 d+7 a^2 b e-4 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{16/3}}+\frac{\left (\sqrt [3]{b} \left (13 b^3 c-10 a b^2 d+7 a^2 b e-4 a^3 f\right )\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^{16/3}}+\frac{\left (b^{2/3} \left (13 b^3 c-10 a b^2 d+7 a^2 b e-4 a^3 f\right )\right ) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 a^5}\\ &=-\frac{c}{10 a^2 x^{10}}+\frac{2 b c-a d}{7 a^3 x^7}-\frac{3 b^2 c-2 a b d+a^2 e}{4 a^4 x^4}+\frac{4 b^3 c-3 a b^2 d+2 a^2 b e-a^3 f}{a^5 x}+\frac{b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^5 \left (a+b x^3\right )}-\frac{\sqrt [3]{b} \left (13 b^3 c-10 a b^2 d+7 a^2 b e-4 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{16/3}}+\frac{\sqrt [3]{b} \left (13 b^3 c-10 a b^2 d+7 a^2 b e-4 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{16/3}}+\frac{\left (\sqrt [3]{b} \left (13 b^3 c-10 a b^2 d+7 a^2 b e-4 a^3 f\right )\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^{16/3}}\\ &=-\frac{c}{10 a^2 x^{10}}+\frac{2 b c-a d}{7 a^3 x^7}-\frac{3 b^2 c-2 a b d+a^2 e}{4 a^4 x^4}+\frac{4 b^3 c-3 a b^2 d+2 a^2 b e-a^3 f}{a^5 x}+\frac{b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^5 \left (a+b x^3\right )}-\frac{\sqrt [3]{b} \left (13 b^3 c-10 a b^2 d+7 a^2 b e-4 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^{16/3}}-\frac{\sqrt [3]{b} \left (13 b^3 c-10 a b^2 d+7 a^2 b e-4 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{16/3}}+\frac{\sqrt [3]{b} \left (13 b^3 c-10 a b^2 d+7 a^2 b e-4 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{16/3}}\\ \end{align*}
Mathematica [A] time = 0.181351, size = 319, normalized size = 0.96 \[ \frac{-\frac{420 \sqrt [3]{a} b x^2 \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{a+b x^3}+70 \sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (7 a^2 b e-4 a^3 f-10 a b^2 d+13 b^3 c\right )-\frac{1260 \sqrt [3]{a} \left (-2 a^2 b e+a^3 f+3 a b^2 d-4 b^3 c\right )}{x}+140 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-7 a^2 b e+4 a^3 f+10 a b^2 d-13 b^3 c\right )-140 \sqrt{3} \sqrt [3]{b} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (7 a^2 b e-4 a^3 f-10 a b^2 d+13 b^3 c\right )-\frac{315 a^{4/3} \left (a^2 e-2 a b d+3 b^2 c\right )}{x^4}-\frac{180 a^{7/3} (a d-2 b c)}{x^7}-\frac{126 a^{10/3} c}{x^{10}}}{1260 a^{16/3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 575, normalized size = 1.7 \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 = 1.37247, size = 1034, normalized size = 3.1 \begin{align*} \frac{420 \,{\left (13 \, b^{4} c - 10 \, a b^{3} d + 7 \, a^{2} b^{2} e - 4 \, a^{3} b f\right )} x^{12} + 315 \,{\left (13 \, a b^{3} c - 10 \, a^{2} b^{2} d + 7 \, a^{3} b e - 4 \, a^{4} f\right )} x^{9} - 45 \,{\left (13 \, a^{2} b^{2} c - 10 \, a^{3} b d + 7 \, a^{4} e\right )} x^{6} - 126 \, a^{4} c + 18 \,{\left (13 \, a^{3} b c - 10 \, a^{4} d\right )} x^{3} + 140 \, \sqrt{3}{\left ({\left (13 \, b^{4} c - 10 \, a b^{3} d + 7 \, a^{2} b^{2} e - 4 \, a^{3} b f\right )} x^{13} +{\left (13 \, a b^{3} c - 10 \, a^{2} b^{2} d + 7 \, a^{3} b e - 4 \, a^{4} f\right )} x^{10}\right )} \left (\frac{b}{a}\right )^{\frac{1}{3}} \arctan \left (\frac{2}{3} \, \sqrt{3} x \left (\frac{b}{a}\right )^{\frac{1}{3}} - \frac{1}{3} \, \sqrt{3}\right ) + 70 \,{\left ({\left (13 \, b^{4} c - 10 \, a b^{3} d + 7 \, a^{2} b^{2} e - 4 \, a^{3} b f\right )} x^{13} +{\left (13 \, a b^{3} c - 10 \, a^{2} b^{2} d + 7 \, a^{3} b e - 4 \, a^{4} f\right )} x^{10}\right )} \left (\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x^{2} - a x \left (\frac{b}{a}\right )^{\frac{2}{3}} + a \left (\frac{b}{a}\right )^{\frac{1}{3}}\right ) - 140 \,{\left ({\left (13 \, b^{4} c - 10 \, a b^{3} d + 7 \, a^{2} b^{2} e - 4 \, a^{3} b f\right )} x^{13} +{\left (13 \, a b^{3} c - 10 \, a^{2} b^{2} d + 7 \, a^{3} b e - 4 \, a^{4} f\right )} x^{10}\right )} \left (\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x + a \left (\frac{b}{a}\right )^{\frac{2}{3}}\right )}{1260 \,{\left (a^{5} b x^{13} + a^{6} x^{10}\right )}} \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 = 1.1012, size = 590, normalized size = 1.77 \begin{align*} -\frac{{\left (13 \, b^{4} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 10 \, a b^{3} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 4 \, a^{3} b f \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 7 \, a^{2} b^{2} \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^{6}} - \frac{\sqrt{3}{\left (13 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - 10 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - 4 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + 7 \, \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^{6} b} + \frac{b^{4} c x^{2} - a b^{3} d x^{2} - a^{3} b f x^{2} + a^{2} b^{2} x^{2} e}{3 \,{\left (b x^{3} + a\right )} a^{5}} + \frac{{\left (13 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - 10 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - 4 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + 7 \, \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^{6} b} + \frac{560 \, b^{3} c x^{9} - 420 \, a b^{2} d x^{9} - 140 \, a^{3} f x^{9} + 280 \, a^{2} b x^{9} e - 105 \, a b^{2} c x^{6} + 70 \, a^{2} b d x^{6} - 35 \, a^{3} x^{6} e + 40 \, a^{2} b c x^{3} - 20 \, a^{3} d x^{3} - 14 \, a^{3} c}{140 \, a^{5} x^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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