Optimal. Leaf size=297 \[ -\frac{x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^4 \left (a+b x^3\right )}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (4 a^2 b e+a^3 (-f)-7 a b^2 d+10 b^3 c\right )}{18 a^{13/3} b^{2/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (4 a^2 b e+a^3 (-f)-7 a b^2 d+10 b^3 c\right )}{9 a^{13/3} b^{2/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (4 a^2 b e+a^3 (-f)-7 a b^2 d+10 b^3 c\right )}{3 \sqrt{3} a^{13/3} b^{2/3}}-\frac{a^2 e-2 a b d+3 b^2 c}{a^4 x}+\frac{2 b c-a d}{4 a^3 x^4}-\frac{c}{7 a^2 x^7} \]
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Rubi [A] time = 0.383572, 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, 292, 31, 634, 617, 204, 628} \[ -\frac{x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^4 \left (a+b x^3\right )}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (4 a^2 b e+a^3 (-f)-7 a b^2 d+10 b^3 c\right )}{18 a^{13/3} b^{2/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (4 a^2 b e+a^3 (-f)-7 a b^2 d+10 b^3 c\right )}{9 a^{13/3} b^{2/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (4 a^2 b e+a^3 (-f)-7 a b^2 d+10 b^3 c\right )}{3 \sqrt{3} a^{13/3} b^{2/3}}-\frac{a^2 e-2 a b d+3 b^2 c}{a^4 x}+\frac{2 b c-a d}{4 a^3 x^4}-\frac{c}{7 a^2 x^7} \]
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^8 \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^2}{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{b^3 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^9}{a^3}}{x^8 \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^2}{3 a^4 \left (a+b x^3\right )}-\frac{\int \left (-\frac{3 b^3 c}{a x^8}-\frac{3 b^3 (-2 b c+a d)}{a^2 x^5}-\frac{3 b^3 \left (3 b^2 c-2 a b d+a^2 e\right )}{a^3 x^2}-\frac{b^3 \left (-10 b^3 c+7 a b^2 d-4 a^2 b e+a^3 f\right ) x}{a^3 \left (a+b x^3\right )}\right ) \, dx}{3 a b^3}\\ &=-\frac{c}{7 a^2 x^7}+\frac{2 b c-a d}{4 a^3 x^4}-\frac{3 b^2 c-2 a b d+a^2 e}{a^4 x}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^4 \left (a+b x^3\right )}-\frac{\left (10 b^3 c-7 a b^2 d+4 a^2 b e-a^3 f\right ) \int \frac{x}{a+b x^3} \, dx}{3 a^4}\\ &=-\frac{c}{7 a^2 x^7}+\frac{2 b c-a d}{4 a^3 x^4}-\frac{3 b^2 c-2 a b d+a^2 e}{a^4 x}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^4 \left (a+b x^3\right )}+\frac{\left (10 b^3 c-7 a b^2 d+4 a^2 b e-a^3 f\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a^{13/3} \sqrt [3]{b}}-\frac{\left (10 b^3 c-7 a b^2 d+4 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}{9 a^{13/3} \sqrt [3]{b}}\\ &=-\frac{c}{7 a^2 x^7}+\frac{2 b c-a d}{4 a^3 x^4}-\frac{3 b^2 c-2 a b d+a^2 e}{a^4 x}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^4 \left (a+b x^3\right )}+\frac{\left (10 b^3 c-7 a b^2 d+4 a^2 b e-a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{13/3} b^{2/3}}-\frac{\left (10 b^3 c-7 a b^2 d+4 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}{18 a^{13/3} b^{2/3}}-\frac{\left (10 b^3 c-7 a b^2 d+4 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}{6 a^4 \sqrt [3]{b}}\\ &=-\frac{c}{7 a^2 x^7}+\frac{2 b c-a d}{4 a^3 x^4}-\frac{3 b^2 c-2 a b d+a^2 e}{a^4 x}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^4 \left (a+b x^3\right )}+\frac{\left (10 b^3 c-7 a b^2 d+4 a^2 b e-a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{13/3} b^{2/3}}-\frac{\left (10 b^3 c-7 a b^2 d+4 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 )}{18 a^{13/3} b^{2/3}}-\frac{\left (10 b^3 c-7 a b^2 d+4 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 )}{3 a^{13/3} b^{2/3}}\\ &=-\frac{c}{7 a^2 x^7}+\frac{2 b c-a d}{4 a^3 x^4}-\frac{3 b^2 c-2 a b d+a^2 e}{a^4 x}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^4 \left (a+b x^3\right )}+\frac{\left (10 b^3 c-7 a b^2 d+4 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 )}{3 \sqrt{3} a^{13/3} b^{2/3}}+\frac{\left (10 b^3 c-7 a b^2 d+4 a^2 b e-a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{13/3} b^{2/3}}-\frac{\left (10 b^3 c-7 a b^2 d+4 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 )}{18 a^{13/3} b^{2/3}}\\ \end{align*}
Mathematica [A] time = 0.179237, size = 281, normalized size = 0.95 \[ \frac{\frac{84 \sqrt [3]{a} x^2 \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{a+b x^3}+\frac{14 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-4 a^2 b e+a^3 f+7 a b^2 d-10 b^3 c\right )}{b^{2/3}}+\frac{28 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (4 a^2 b e+a^3 (-f)-7 a b^2 d+10 b^3 c\right )}{b^{2/3}}+\frac{28 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (4 a^2 b e+a^3 (-f)-7 a b^2 d+10 b^3 c\right )}{b^{2/3}}-\frac{252 \sqrt [3]{a} \left (a^2 e-2 a b d+3 b^2 c\right )}{x}-\frac{63 a^{4/3} (a d-2 b c)}{x^4}-\frac{36 a^{7/3} c}{x^7}}{252 a^{13/3}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.015, size = 529, 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 = 1.46461, size = 2159, normalized size = 7.27 \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 = 1.11736, size = 522, normalized size = 1.76 \begin{align*} \frac{{\left (10 \, b^{3} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 7 \, a b^{2} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} - a^{3} f \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 4 \, 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 )}{9 \, a^{5}} - \frac{b^{3} c x^{2} - a b^{2} d x^{2} - a^{3} f x^{2} + a^{2} b x^{2} e}{3 \,{\left (b x^{3} + a\right )} a^{4}} + \frac{\sqrt{3}{\left (10 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - 7 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + 4 \, \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^{5} b^{2}} - \frac{{\left (10 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - 7 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + 4 \, \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^{5} b^{2}} - \frac{84 \, b^{2} c x^{6} - 56 \, a b d x^{6} + 28 \, a^{2} x^{6} e - 14 \, a b c x^{3} + 7 \, a^{2} d x^{3} + 4 \, a^{2} c}{28 \, a^{4} x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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