Optimal. Leaf size=242 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 a^{10/3} b^{2/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^{10/3} b^{2/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{\sqrt{3} a^{10/3} b^{2/3}}-\frac{a^2 e-a b d+b^2 c}{a^3 x}+\frac{b c-a d}{4 a^2 x^4}-\frac{c}{7 a x^7} \]
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Rubi [A] time = 0.187964, antiderivative size = 242, 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} \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 a^{10/3} b^{2/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^{10/3} b^{2/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{\sqrt{3} a^{10/3} b^{2/3}}-\frac{a^2 e-a b d+b^2 c}{a^3 x}+\frac{b c-a d}{4 a^2 x^4}-\frac{c}{7 a x^7} \]
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^8 \left (a+b x^3\right )} \, dx &=\int \left (\frac{c}{a x^8}+\frac{-b c+a d}{a^2 x^5}+\frac{b^2 c-a b d+a^2 e}{a^3 x^2}+\frac{\left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) x}{a^3 \left (a+b x^3\right )}\right ) \, dx\\ &=-\frac{c}{7 a x^7}+\frac{b c-a d}{4 a^2 x^4}-\frac{b^2 c-a b d+a^2 e}{a^3 x}+\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^3}\\ &=-\frac{c}{7 a x^7}+\frac{b c-a d}{4 a^2 x^4}-\frac{b^2 c-a b d+a^2 e}{a^3 x}+\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^{10/3} \sqrt [3]{b}}-\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^{10/3} \sqrt [3]{b}}\\ &=-\frac{c}{7 a x^7}+\frac{b c-a d}{4 a^2 x^4}-\frac{b^2 c-a b d+a^2 e}{a^3 x}+\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^{10/3} b^{2/3}}-\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^{10/3} b^{2/3}}-\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^3 \sqrt [3]{b}}\\ &=-\frac{c}{7 a x^7}+\frac{b c-a d}{4 a^2 x^4}-\frac{b^2 c-a b d+a^2 e}{a^3 x}+\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^{10/3} b^{2/3}}-\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^{10/3} b^{2/3}}-\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^{10/3} b^{2/3}}\\ &=-\frac{c}{7 a x^7}+\frac{b c-a d}{4 a^2 x^4}-\frac{b^2 c-a b d+a^2 e}{a^3 x}+\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^{10/3} b^{2/3}}+\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^{10/3} b^{2/3}}-\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^{10/3} b^{2/3}}\\ \end{align*}
Mathematica [A] time = 0.111245, size = 231, normalized size = 0.95 \[ \frac{\frac{14 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{b^{2/3}}+\frac{28 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{b^{2/3}}+\frac{28 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{b^{2/3}}-\frac{84 \sqrt [3]{a} \left (a^2 e-a b d+b^2 c\right )}{x}+\frac{21 a^{4/3} (b c-a d)}{x^4}-\frac{12 a^{7/3} c}{x^7}}{84 a^{10/3}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.007, size = 440, normalized size = 1.8 \begin{align*} -{\frac{f}{3\,b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{e}{3\,a}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{bd}{3\,{a}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{{b}^{2}c}{3\,{a}^{3}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{f}{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{e}{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{bd}{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{{b}^{2}c}{6\,{a}^{3}}\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{\sqrt{3}f}{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{\sqrt{3}e}{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{\sqrt{3}bd}{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{\sqrt{3}{b}^{2}c}{3\,{a}^{3}}\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}{7\,a{x}^{7}}}-{\frac{d}{4\,a{x}^{4}}}+{\frac{bc}{4\,{a}^{2}{x}^{4}}}-{\frac{e}{ax}}+{\frac{bd}{{a}^{2}x}}-{\frac{{b}^{2}c}{{a}^{3}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 = 1.43548, size = 1359, normalized size = 5.62 \begin{align*} \left [-\frac{42 \, \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^{7} \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 ) + 14 \,{\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^{7} \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 ) - 28 \,{\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^{7} \log \left (b x - \left (-a b^{2}\right )^{\frac{1}{3}}\right ) + 84 \,{\left (a b^{4} c - a^{2} b^{3} d + a^{3} b^{2} e\right )} x^{6} + 12 \, a^{3} b^{2} c - 21 \,{\left (a^{2} b^{3} c - a^{3} b^{2} d\right )} x^{3}}{84 \, a^{4} b^{2} x^{7}}, -\frac{84 \, \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^{7} \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 ) + 14 \,{\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^{7} \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 ) - 28 \,{\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^{7} \log \left (b x - \left (-a b^{2}\right )^{\frac{1}{3}}\right ) + 84 \,{\left (a b^{4} c - a^{2} b^{3} d + a^{3} b^{2} e\right )} x^{6} + 12 \, a^{3} b^{2} c - 21 \,{\left (a^{2} b^{3} c - a^{3} b^{2} d\right )} x^{3}}{84 \, a^{4} b^{2} x^{7}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 30.4187, size = 432, normalized size = 1.79 \begin{align*} \operatorname{RootSum}{\left (27 t^{3} a^{10} b^{2} + 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^{7} b}{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{4 a^{2} c + x^{6} \left (28 a^{2} e - 28 a b d + 28 b^{2} c\right ) + x^{3} \left (7 a^{2} d - 7 a b c\right )}{28 a^{3} x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07466, size = 444, normalized size = 1.83 \begin{align*} \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^{4}} + \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^{4} b^{2}} - \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^{4} b^{2}} - \frac{28 \, b^{2} c x^{6} - 28 \, a b d x^{6} + 28 \, a^{2} x^{6} e - 7 \, a b c x^{3} + 7 \, a^{2} d x^{3} + 4 \, a^{2} c}{28 \, a^{3} x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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