Optimal. Leaf size=227 \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 a^{7/3} b^{5/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^{7/3} b^{5/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{\sqrt{3} a^{7/3} b^{5/3}}+\frac{b c-a d}{a^2 x}-\frac{c}{4 a x^4}+\frac{f x^2}{2 b} \]
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Rubi [A] time = 0.185703, antiderivative size = 227, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.233, Rules used = {1834, 292, 31, 634, 617, 204, 628} \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 a^{7/3} b^{5/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^{7/3} b^{5/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{\sqrt{3} a^{7/3} b^{5/3}}+\frac{b c-a d}{a^2 x}-\frac{c}{4 a x^4}+\frac{f x^2}{2 b} \]
Antiderivative was successfully verified.
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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^5 \left (a+b x^3\right )} \, dx &=\int \left (\frac{c}{a x^5}+\frac{-b c+a d}{a^2 x^2}+\frac{f x}{b}-\frac{\left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) x}{a^2 b \left (a+b x^3\right )}\right ) \, dx\\ &=-\frac{c}{4 a x^4}+\frac{b c-a d}{a^2 x}+\frac{f x^2}{2 b}-\frac{\left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) \int \frac{x}{a+b x^3} \, dx}{a^2 b}\\ &=-\frac{c}{4 a x^4}+\frac{b c-a d}{a^2 x}+\frac{f x^2}{2 b}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^{7/3} b^{4/3}}+\frac{\left (b^3 c-a b^2 d+a^2 b e-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}{3 a^{7/3} b^{4/3}}\\ &=-\frac{c}{4 a x^4}+\frac{b c-a d}{a^2 x}+\frac{f x^2}{2 b}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{7/3} b^{5/3}}+\frac{\left (b^3 c-a b^2 d+a^2 b e-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}{6 a^{7/3} b^{5/3}}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 a^2 b^{4/3}}\\ &=-\frac{c}{4 a x^4}+\frac{b c-a d}{a^2 x}+\frac{f x^2}{2 b}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{7/3} b^{5/3}}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{7/3} b^{5/3}}+\frac{\left (b^3 c-a b^2 d+a^2 b e-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 )}{a^{7/3} b^{5/3}}\\ &=-\frac{c}{4 a x^4}+\frac{b c-a d}{a^2 x}+\frac{f x^2}{2 b}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{7/3} b^{5/3}}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{7/3} b^{5/3}}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{7/3} b^{5/3}}\\ \end{align*}
Mathematica [A] time = 0.111304, size = 220, normalized size = 0.97 \[ \frac{1}{12} \left (\frac{2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{a^{7/3} b^{5/3}}+\frac{4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{a^{7/3} b^{5/3}}+\frac{4 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{a^{7/3} b^{5/3}}+\frac{12 (b c-a d)}{a^2 x}-\frac{3 c}{a x^4}+\frac{6 f x^2}{b}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.007, size = 412, normalized size = 1.8 \begin{align*}{\frac{f{x}^{2}}{2\,b}}+{\frac{af}{3\,{b}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{e}{3\,b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{d}{3\,a}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{bc}{3\,{a}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{af}{6\,{b}^{2}}\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{e}{6\,b}\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{d}{6\,a}\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{bc}{6\,{a}^{2}}\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{a\sqrt{3}f}{3\,{b}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{\sqrt{3}e}{3\,b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{d\sqrt{3}}{3\,a}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{b\sqrt{3}c}{3\,{a}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{c}{4\,a{x}^{4}}}-{\frac{d}{ax}}+{\frac{bc}{{a}^{2}x}} \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 = 1.38581, size = 1251, normalized size = 5.51 \begin{align*} \left [\frac{6 \, a^{3} b^{2} f x^{6} - 6 \, \sqrt{\frac{1}{3}}{\left (a b^{4} c - a^{2} b^{3} d + a^{3} b^{2} e - a^{4} b f\right )} x^{4} \sqrt{-\frac{\left (a b^{2}\right )^{\frac{1}{3}}}{a}} \log \left (\frac{2 \, b^{2} x^{3} - a b - 3 \, \sqrt{\frac{1}{3}}{\left (a b x + 2 \, \left (a b^{2}\right )^{\frac{2}{3}} x^{2} - \left (a b^{2}\right )^{\frac{1}{3}} a\right )} \sqrt{-\frac{\left (a b^{2}\right )^{\frac{1}{3}}}{a}} - 3 \, \left (a b^{2}\right )^{\frac{2}{3}} x}{b x^{3} + a}\right ) + 2 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (a b^{2}\right )^{\frac{2}{3}} x^{4} \log \left (b^{2} x^{2} - \left (a b^{2}\right )^{\frac{1}{3}} b x + \left (a b^{2}\right )^{\frac{2}{3}}\right ) - 4 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (a b^{2}\right )^{\frac{2}{3}} x^{4} \log \left (b x + \left (a b^{2}\right )^{\frac{1}{3}}\right ) - 3 \, a^{2} b^{3} c + 12 \,{\left (a b^{4} c - a^{2} b^{3} d\right )} x^{3}}{12 \, a^{3} b^{3} x^{4}}, \frac{6 \, a^{3} b^{2} f x^{6} - 12 \, \sqrt{\frac{1}{3}}{\left (a b^{4} c - a^{2} b^{3} d + a^{3} b^{2} e - a^{4} b f\right )} x^{4} \sqrt{\frac{\left (a b^{2}\right )^{\frac{1}{3}}}{a}} \arctan \left (-\frac{\sqrt{\frac{1}{3}}{\left (2 \, b x - \left (a b^{2}\right )^{\frac{1}{3}}\right )} \sqrt{\frac{\left (a b^{2}\right )^{\frac{1}{3}}}{a}}}{b}\right ) + 2 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (a b^{2}\right )^{\frac{2}{3}} x^{4} \log \left (b^{2} x^{2} - \left (a b^{2}\right )^{\frac{1}{3}} b x + \left (a b^{2}\right )^{\frac{2}{3}}\right ) - 4 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (a b^{2}\right )^{\frac{2}{3}} x^{4} \log \left (b x + \left (a b^{2}\right )^{\frac{1}{3}}\right ) - 3 \, a^{2} b^{3} c + 12 \,{\left (a b^{4} c - a^{2} b^{3} d\right )} x^{3}}{12 \, a^{3} b^{3} x^{4}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 8.04656, size = 411, normalized size = 1.81 \begin{align*} \operatorname{RootSum}{\left (27 t^{3} a^{7} b^{5} - a^{9} f^{3} + 3 a^{8} b e f^{2} - 3 a^{7} b^{2} d f^{2} - 3 a^{7} b^{2} e^{2} f + 3 a^{6} b^{3} c f^{2} + 6 a^{6} b^{3} d e f + a^{6} b^{3} e^{3} - 6 a^{5} b^{4} c e f - 3 a^{5} b^{4} d^{2} f - 3 a^{5} b^{4} d e^{2} + 6 a^{4} b^{5} c d f + 3 a^{4} b^{5} c e^{2} + 3 a^{4} b^{5} d^{2} e - 3 a^{3} b^{6} c^{2} f - 6 a^{3} b^{6} c d e - a^{3} b^{6} d^{3} + 3 a^{2} b^{7} c^{2} e + 3 a^{2} b^{7} c d^{2} - 3 a b^{8} c^{2} d + b^{9} c^{3}, \left ( t \mapsto t \log{\left (\frac{9 t^{2} a^{5} b^{3}}{a^{6} f^{2} - 2 a^{5} b e f + 2 a^{4} b^{2} d f + a^{4} b^{2} e^{2} - 2 a^{3} b^{3} c f - 2 a^{3} b^{3} d e + 2 a^{2} b^{4} c e + a^{2} b^{4} d^{2} - 2 a b^{5} c d + b^{6} c^{2}} + x \right )} \right )\right )} + \frac{f x^{2}}{2 b} - \frac{a c + x^{3} \left (4 a d - 4 b c\right )}{4 a^{2} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08082, size = 417, normalized size = 1.84 \begin{align*} \frac{f x^{2}}{2 \, b} - \frac{{\left (b^{3} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} - a b^{2} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 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 )}{3 \, a^{3} b} - \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - \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 )}{3 \, a^{3} b^{3}} + \frac{{\left (\left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - \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 )}{6 \, a^{3} b^{3}} + \frac{4 \, b c x^{3} - 4 \, a d x^{3} - a c}{4 \, a^{2} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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