Optimal. Leaf size=317 \[ \frac{x^2 \left (2 a^2 b e+a^3 f-5 a b^2 d+8 b^3 c\right )}{9 a^4 b \left (a+b x^3\right )}+\frac{x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 a^3 b \left (a+b x^3\right )^2}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (2 a^2 b e+a^3 f-14 a b^2 d+35 b^3 c\right )}{54 a^{13/3} b^{5/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (2 a^2 b e+a^3 f-14 a b^2 d+35 b^3 c\right )}{27 a^{13/3} b^{5/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (2 a^2 b e+a^3 f-14 a b^2 d+35 b^3 c\right )}{9 \sqrt{3} a^{13/3} b^{5/3}}+\frac{3 b c-a d}{a^4 x}-\frac{c}{4 a^3 x^4} \]
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Rubi [A] time = 0.369523, antiderivative size = 317, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {1829, 1484, 1488, 292, 31, 634, 617, 204, 628} \[ \frac{x^2 \left (2 a^2 b e+a^3 f-5 a b^2 d+8 b^3 c\right )}{9 a^4 b \left (a+b x^3\right )}+\frac{x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 a^3 b \left (a+b x^3\right )^2}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (2 a^2 b e+a^3 f-14 a b^2 d+35 b^3 c\right )}{54 a^{13/3} b^{5/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (2 a^2 b e+a^3 f-14 a b^2 d+35 b^3 c\right )}{27 a^{13/3} b^{5/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (2 a^2 b e+a^3 f-14 a b^2 d+35 b^3 c\right )}{9 \sqrt{3} a^{13/3} b^{5/3}}+\frac{3 b c-a d}{a^4 x}-\frac{c}{4 a^3 x^4} \]
Antiderivative was successfully verified.
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Rule 1829
Rule 1484
Rule 1488
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 )^3} \, dx &=\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^3 b \left (a+b x^3\right )^2}-\frac{\int \frac{-6 b^3 c+6 b^3 \left (\frac{b c}{a}-d\right ) x^3-2 b^2 \left (\frac{2 b^3 c}{a^2}-\frac{2 b^2 d}{a}+2 b e+a f\right ) x^6}{x^5 \left (a+b x^3\right )^2} \, dx}{6 a b^3}\\ &=\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^3 b \left (a+b x^3\right )^2}+\frac{\left (8 b^3 c-5 a b^2 d+2 a^2 b e+a^3 f\right ) x^2}{9 a^4 b \left (a+b x^3\right )}-\frac{\int \frac{-18 a^2 b^5 c+18 a b^5 (2 b c-a d) x^3-2 b^4 \left (8 b^3 c-5 a b^2 d+2 a^2 b e+a^3 f\right ) x^6}{x^5 \left (a+b x^3\right )} \, dx}{18 a^4 b^5}\\ &=\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^3 b \left (a+b x^3\right )^2}+\frac{\left (8 b^3 c-5 a b^2 d+2 a^2 b e+a^3 f\right ) x^2}{9 a^4 b \left (a+b x^3\right )}-\frac{\int \left (-\frac{18 a b^5 c}{x^5}+\frac{18 b^5 (3 b c-a d)}{x^2}-\frac{2 b^4 \left (35 b^3 c-14 a b^2 d+2 a^2 b e+a^3 f\right ) x}{a+b x^3}\right ) \, dx}{18 a^4 b^5}\\ &=-\frac{c}{4 a^3 x^4}+\frac{3 b c-a d}{a^4 x}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^3 b \left (a+b x^3\right )^2}+\frac{\left (8 b^3 c-5 a b^2 d+2 a^2 b e+a^3 f\right ) x^2}{9 a^4 b \left (a+b x^3\right )}+\frac{\left (35 b^3 c-14 a b^2 d+2 a^2 b e+a^3 f\right ) \int \frac{x}{a+b x^3} \, dx}{9 a^4 b}\\ &=-\frac{c}{4 a^3 x^4}+\frac{3 b c-a d}{a^4 x}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^3 b \left (a+b x^3\right )^2}+\frac{\left (8 b^3 c-5 a b^2 d+2 a^2 b e+a^3 f\right ) x^2}{9 a^4 b \left (a+b x^3\right )}-\frac{\left (35 b^3 c-14 a b^2 d+2 a^2 b e+a^3 f\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{13/3} b^{4/3}}+\frac{\left (35 b^3 c-14 a b^2 d+2 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}{27 a^{13/3} b^{4/3}}\\ &=-\frac{c}{4 a^3 x^4}+\frac{3 b c-a d}{a^4 x}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^3 b \left (a+b x^3\right )^2}+\frac{\left (8 b^3 c-5 a b^2 d+2 a^2 b e+a^3 f\right ) x^2}{9 a^4 b \left (a+b x^3\right )}-\frac{\left (35 b^3 c-14 a b^2 d+2 a^2 b e+a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{13/3} b^{5/3}}+\frac{\left (35 b^3 c-14 a b^2 d+2 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}{54 a^{13/3} b^{5/3}}+\frac{\left (35 b^3 c-14 a b^2 d+2 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}{18 a^4 b^{4/3}}\\ &=-\frac{c}{4 a^3 x^4}+\frac{3 b c-a d}{a^4 x}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^3 b \left (a+b x^3\right )^2}+\frac{\left (8 b^3 c-5 a b^2 d+2 a^2 b e+a^3 f\right ) x^2}{9 a^4 b \left (a+b x^3\right )}-\frac{\left (35 b^3 c-14 a b^2 d+2 a^2 b e+a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{13/3} b^{5/3}}+\frac{\left (35 b^3 c-14 a b^2 d+2 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 )}{54 a^{13/3} b^{5/3}}+\frac{\left (35 b^3 c-14 a b^2 d+2 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 )}{9 a^{13/3} b^{5/3}}\\ &=-\frac{c}{4 a^3 x^4}+\frac{3 b c-a d}{a^4 x}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 a^3 b \left (a+b x^3\right )^2}+\frac{\left (8 b^3 c-5 a b^2 d+2 a^2 b e+a^3 f\right ) x^2}{9 a^4 b \left (a+b x^3\right )}-\frac{\left (35 b^3 c-14 a b^2 d+2 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 )}{9 \sqrt{3} a^{13/3} b^{5/3}}-\frac{\left (35 b^3 c-14 a b^2 d+2 a^2 b e+a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{13/3} b^{5/3}}+\frac{\left (35 b^3 c-14 a b^2 d+2 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 )}{54 a^{13/3} b^{5/3}}\\ \end{align*}
Mathematica [A] time = 0.232386, size = 303, normalized size = 0.96 \[ \frac{\frac{12 \sqrt [3]{a} x^2 \left (2 a^2 b e+a^3 f-5 a b^2 d+8 b^3 c\right )}{b \left (a+b x^3\right )}-\frac{18 a^{4/3} x^2 \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{b \left (a+b x^3\right )^2}+\frac{2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (2 a^2 b e+a^3 f-14 a b^2 d+35 b^3 c\right )}{b^{5/3}}-\frac{4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (2 a^2 b e+a^3 f-14 a b^2 d+35 b^3 c\right )}{b^{5/3}}-\frac{4 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (2 a^2 b e+a^3 f-14 a b^2 d+35 b^3 c\right )}{b^{5/3}}-\frac{27 a^{4/3} c}{x^4}-\frac{108 \sqrt [3]{a} (a d-3 b c)}{x}}{108 a^{13/3}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.013, size = 574, 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 [B] time = 2.07189, size = 2773, normalized size = 8.75 \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 = 1.10135, size = 547, normalized size = 1.73 \begin{align*} -\frac{{\left (35 \, b^{3} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 14 \, a b^{2} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} + a^{3} f \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 2 \, 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 )}{27 \, a^{5} b} - \frac{\sqrt{3}{\left (35 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - 14 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d + \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + 2 \, \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 )}{27 \, a^{5} b^{3}} + \frac{16 \, b^{4} c x^{5} - 10 \, a b^{3} d x^{5} + 2 \, a^{3} b f x^{5} + 4 \, a^{2} b^{2} x^{5} e + 19 \, a b^{3} c x^{2} - 13 \, a^{2} b^{2} d x^{2} - a^{4} f x^{2} + 7 \, a^{3} b x^{2} e}{18 \,{\left (b x^{3} + a\right )}^{2} a^{4} b} + \frac{{\left (35 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - 14 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d + \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + 2 \, \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 )}{54 \, a^{5} b^{3}} + \frac{12 \, b c x^{3} - 4 \, a d x^{3} - a c}{4 \, a^{4} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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