Optimal. Leaf size=335 \[ \frac{b x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^5 \left (a+b x^3\right )}+\frac{2 a^2 b e+a^3 (-f)-3 a b^2 d+4 b^3 c}{2 a^5 x^2}-\frac{b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (8 a^2 b e-5 a^3 f-11 a b^2 d+14 b^3 c\right )}{18 a^{17/3}}+\frac{b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (8 a^2 b e-5 a^3 f-11 a b^2 d+14 b^3 c\right )}{9 a^{17/3}}-\frac{b^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (8 a^2 b e-5 a^3 f-11 a b^2 d+14 b^3 c\right )}{3 \sqrt{3} a^{17/3}}-\frac{a^2 e-2 a b d+3 b^2 c}{5 a^4 x^5}+\frac{2 b c-a d}{8 a^3 x^8}-\frac{c}{11 a^2 x^{11}} \]
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Rubi [A] time = 0.434195, antiderivative size = 335, 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} \[ \frac{b x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^5 \left (a+b x^3\right )}+\frac{2 a^2 b e+a^3 (-f)-3 a b^2 d+4 b^3 c}{2 a^5 x^2}-\frac{b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (8 a^2 b e-5 a^3 f-11 a b^2 d+14 b^3 c\right )}{18 a^{17/3}}+\frac{b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (8 a^2 b e-5 a^3 f-11 a b^2 d+14 b^3 c\right )}{9 a^{17/3}}-\frac{b^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (8 a^2 b e-5 a^3 f-11 a b^2 d+14 b^3 c\right )}{3 \sqrt{3} a^{17/3}}-\frac{a^2 e-2 a b d+3 b^2 c}{5 a^4 x^5}+\frac{2 b c-a d}{8 a^3 x^8}-\frac{c}{11 a^2 x^{11}} \]
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^{12} \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}{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{2 b^4 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^{12}}{a^4}}{x^{12} \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}{3 a^5 \left (a+b x^3\right )}-\frac{\int \left (-\frac{3 b^3 c}{a x^{12}}-\frac{3 b^3 (-2 b c+a d)}{a^2 x^9}-\frac{3 b^3 \left (3 b^2 c-2 a b d+a^2 e\right )}{a^3 x^6}-\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^3}+\frac{b^4 \left (-14 b^3 c+11 a b^2 d-8 a^2 b e+5 a^3 f\right )}{a^4 \left (a+b x^3\right )}\right ) \, dx}{3 a b^3}\\ &=-\frac{c}{11 a^2 x^{11}}+\frac{2 b c-a d}{8 a^3 x^8}-\frac{3 b^2 c-2 a b d+a^2 e}{5 a^4 x^5}+\frac{4 b^3 c-3 a b^2 d+2 a^2 b e-a^3 f}{2 a^5 x^2}+\frac{b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 a^5 \left (a+b x^3\right )}+\frac{\left (b \left (14 b^3 c-11 a b^2 d+8 a^2 b e-5 a^3 f\right )\right ) \int \frac{1}{a+b x^3} \, dx}{3 a^5}\\ &=-\frac{c}{11 a^2 x^{11}}+\frac{2 b c-a d}{8 a^3 x^8}-\frac{3 b^2 c-2 a b d+a^2 e}{5 a^4 x^5}+\frac{4 b^3 c-3 a b^2 d+2 a^2 b e-a^3 f}{2 a^5 x^2}+\frac{b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 a^5 \left (a+b x^3\right )}+\frac{\left (b \left (14 b^3 c-11 a b^2 d+8 a^2 b e-5 a^3 f\right )\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a^{17/3}}+\frac{\left (b \left (14 b^3 c-11 a b^2 d+8 a^2 b e-5 a^3 f\right )\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^{17/3}}\\ &=-\frac{c}{11 a^2 x^{11}}+\frac{2 b c-a d}{8 a^3 x^8}-\frac{3 b^2 c-2 a b d+a^2 e}{5 a^4 x^5}+\frac{4 b^3 c-3 a b^2 d+2 a^2 b e-a^3 f}{2 a^5 x^2}+\frac{b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 a^5 \left (a+b x^3\right )}+\frac{b^{2/3} \left (14 b^3 c-11 a b^2 d+8 a^2 b e-5 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{17/3}}-\frac{\left (b^{2/3} \left (14 b^3 c-11 a b^2 d+8 a^2 b e-5 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^{17/3}}+\frac{\left (b \left (14 b^3 c-11 a b^2 d+8 a^2 b e-5 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^{16/3}}\\ &=-\frac{c}{11 a^2 x^{11}}+\frac{2 b c-a d}{8 a^3 x^8}-\frac{3 b^2 c-2 a b d+a^2 e}{5 a^4 x^5}+\frac{4 b^3 c-3 a b^2 d+2 a^2 b e-a^3 f}{2 a^5 x^2}+\frac{b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 a^5 \left (a+b x^3\right )}+\frac{b^{2/3} \left (14 b^3 c-11 a b^2 d+8 a^2 b e-5 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{17/3}}-\frac{b^{2/3} \left (14 b^3 c-11 a b^2 d+8 a^2 b e-5 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{17/3}}+\frac{\left (b^{2/3} \left (14 b^3 c-11 a b^2 d+8 a^2 b e-5 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^{17/3}}\\ &=-\frac{c}{11 a^2 x^{11}}+\frac{2 b c-a d}{8 a^3 x^8}-\frac{3 b^2 c-2 a b d+a^2 e}{5 a^4 x^5}+\frac{4 b^3 c-3 a b^2 d+2 a^2 b e-a^3 f}{2 a^5 x^2}+\frac{b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 a^5 \left (a+b x^3\right )}-\frac{b^{2/3} \left (14 b^3 c-11 a b^2 d+8 a^2 b e-5 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^{17/3}}+\frac{b^{2/3} \left (14 b^3 c-11 a b^2 d+8 a^2 b e-5 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{17/3}}-\frac{b^{2/3} \left (14 b^3 c-11 a b^2 d+8 a^2 b e-5 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{17/3}}\\ \end{align*}
Mathematica [A] time = 0.187121, size = 317, normalized size = 0.95 \[ \frac{-\frac{1320 a^{2/3} b x \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{a+b x^3}-\frac{1980 a^{2/3} \left (-2 a^2 b e+a^3 f+3 a b^2 d-4 b^3 c\right )}{x^2}+220 b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-8 a^2 b e+5 a^3 f+11 a b^2 d-14 b^3 c\right )+440 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (8 a^2 b e-5 a^3 f-11 a b^2 d+14 b^3 c\right )-440 \sqrt{3} b^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (8 a^2 b e-5 a^3 f-11 a b^2 d+14 b^3 c\right )-\frac{792 a^{5/3} \left (a^2 e-2 a b d+3 b^2 c\right )}{x^5}-\frac{495 a^{8/3} (a d-2 b c)}{x^8}-\frac{360 a^{11/3} c}{x^{11}}}{3960 a^{17/3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 566, 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 = 1.38418, size = 1099, normalized size = 3.28 \begin{align*} \frac{660 \,{\left (14 \, b^{4} c - 11 \, a b^{3} d + 8 \, a^{2} b^{2} e - 5 \, a^{3} b f\right )} x^{12} + 396 \,{\left (14 \, a b^{3} c - 11 \, a^{2} b^{2} d + 8 \, a^{3} b e - 5 \, a^{4} f\right )} x^{9} - 99 \,{\left (14 \, a^{2} b^{2} c - 11 \, a^{3} b d + 8 \, a^{4} e\right )} x^{6} - 360 \, a^{4} c + 45 \,{\left (14 \, a^{3} b c - 11 \, a^{4} d\right )} x^{3} - 440 \, \sqrt{3}{\left ({\left (14 \, b^{4} c - 11 \, a b^{3} d + 8 \, a^{2} b^{2} e - 5 \, a^{3} b f\right )} x^{14} +{\left (14 \, a b^{3} c - 11 \, a^{2} b^{2} d + 8 \, a^{3} b e - 5 \, a^{4} f\right )} x^{11}\right )} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} a x \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}} - \sqrt{3} b}{3 \, b}\right ) + 220 \,{\left ({\left (14 \, b^{4} c - 11 \, a b^{3} d + 8 \, a^{2} b^{2} e - 5 \, a^{3} b f\right )} x^{14} +{\left (14 \, a b^{3} c - 11 \, a^{2} b^{2} d + 8 \, a^{3} b e - 5 \, a^{4} f\right )} x^{11}\right )} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b^{2} x^{2} + a b x \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} + a^{2} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}}\right ) - 440 \,{\left ({\left (14 \, b^{4} c - 11 \, a b^{3} d + 8 \, a^{2} b^{2} e - 5 \, a^{3} b f\right )} x^{14} +{\left (14 \, a b^{3} c - 11 \, a^{2} b^{2} d + 8 \, a^{3} b e - 5 \, a^{4} f\right )} x^{11}\right )} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b x - a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}\right )}{3960 \,{\left (a^{5} b x^{14} + a^{6} x^{11}\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 = 1.08584, size = 528, normalized size = 1.58 \begin{align*} \frac{\sqrt{3}{\left (14 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - 11 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - 5 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + 8 \, \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^{6}} - \frac{{\left (14 \, b^{4} c - 11 \, a b^{3} d - 5 \, a^{3} b f + 8 \, a^{2} b^{2} 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{{\left (14 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - 11 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - 5 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + 8 \, \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^{6}} + \frac{b^{4} c x - a b^{3} d x - a^{3} b f x + a^{2} b^{2} x e}{3 \,{\left (b x^{3} + a\right )} a^{5}} + \frac{880 \, b^{3} c x^{9} - 660 \, a b^{2} d x^{9} - 220 \, a^{3} f x^{9} + 440 \, a^{2} b x^{9} e - 264 \, a b^{2} c x^{6} + 176 \, a^{2} b d x^{6} - 88 \, a^{3} x^{6} e + 110 \, a^{2} b c x^{3} - 55 \, a^{3} d x^{3} - 40 \, a^{3} c}{440 \, a^{5} x^{11}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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