Optimal. Leaf size=288 \[ -\frac{x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 b^4 \left (a+b x^3\right )}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (7 a^2 b e-10 a^3 f-4 a b^2 d+b^3 c\right )}{18 a^{2/3} b^{13/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (7 a^2 b e-10 a^3 f-4 a b^2 d+b^3 c\right )}{9 a^{2/3} b^{13/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (7 a^2 b e-10 a^3 f-4 a b^2 d+b^3 c\right )}{3 \sqrt{3} a^{2/3} b^{13/3}}+\frac{x \left (3 a^2 f-2 a b e+b^2 d\right )}{b^4}+\frac{x^4 (b e-2 a f)}{4 b^3}+\frac{f x^7}{7 b^2} \]
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Rubi [A] time = 0.32561, antiderivative size = 288, 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 = {1828, 1887, 200, 31, 634, 617, 204, 628} \[ -\frac{x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 b^4 \left (a+b x^3\right )}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (7 a^2 b e-10 a^3 f-4 a b^2 d+b^3 c\right )}{18 a^{2/3} b^{13/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (7 a^2 b e-10 a^3 f-4 a b^2 d+b^3 c\right )}{9 a^{2/3} b^{13/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (7 a^2 b e-10 a^3 f-4 a b^2 d+b^3 c\right )}{3 \sqrt{3} a^{2/3} b^{13/3}}+\frac{x \left (3 a^2 f-2 a b e+b^2 d\right )}{b^4}+\frac{x^4 (b e-2 a f)}{4 b^3}+\frac{f x^7}{7 b^2} \]
Antiderivative was successfully verified.
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Rule 1828
Rule 1887
Rule 200
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{x^3 \left (c+d x^3+e x^6+f x^9\right )}{\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}{3 b^4 \left (a+b x^3\right )}-\frac{\int \frac{-a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )-3 a b \left (b^2 d-a b e+a^2 f\right ) x^3-3 a b^2 (b e-a f) x^6-3 a b^3 f x^9}{a+b x^3} \, dx}{3 a b^4}\\ &=-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 b^4 \left (a+b x^3\right )}-\frac{\int \left (-3 a \left (b^2 d-2 a b e+3 a^2 f\right )-3 a b (b e-2 a f) x^3-3 a b^2 f x^6+\frac{-a b^3 c+4 a^2 b^2 d-7 a^3 b e+10 a^4 f}{a+b x^3}\right ) \, dx}{3 a b^4}\\ &=\frac{\left (b^2 d-2 a b e+3 a^2 f\right ) x}{b^4}+\frac{(b e-2 a f) x^4}{4 b^3}+\frac{f x^7}{7 b^2}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 b^4 \left (a+b x^3\right )}+\frac{\left (b^3 c-4 a b^2 d+7 a^2 b e-10 a^3 f\right ) \int \frac{1}{a+b x^3} \, dx}{3 b^4}\\ &=\frac{\left (b^2 d-2 a b e+3 a^2 f\right ) x}{b^4}+\frac{(b e-2 a f) x^4}{4 b^3}+\frac{f x^7}{7 b^2}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 b^4 \left (a+b x^3\right )}+\frac{\left (b^3 c-4 a b^2 d+7 a^2 b e-10 a^3 f\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a^{2/3} b^4}+\frac{\left (b^3 c-4 a b^2 d+7 a^2 b e-10 a^3 f\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^{2/3} b^4}\\ &=\frac{\left (b^2 d-2 a b e+3 a^2 f\right ) x}{b^4}+\frac{(b e-2 a f) x^4}{4 b^3}+\frac{f x^7}{7 b^2}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 b^4 \left (a+b x^3\right )}+\frac{\left (b^3 c-4 a b^2 d+7 a^2 b e-10 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{2/3} b^{13/3}}-\frac{\left (b^3 c-4 a b^2 d+7 a^2 b e-10 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^{2/3} b^{13/3}}+\frac{\left (b^3 c-4 a b^2 d+7 a^2 b e-10 a^3 f\right ) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 \sqrt [3]{a} b^4}\\ &=\frac{\left (b^2 d-2 a b e+3 a^2 f\right ) x}{b^4}+\frac{(b e-2 a f) x^4}{4 b^3}+\frac{f x^7}{7 b^2}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 b^4 \left (a+b x^3\right )}+\frac{\left (b^3 c-4 a b^2 d+7 a^2 b e-10 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{2/3} b^{13/3}}-\frac{\left (b^3 c-4 a b^2 d+7 a^2 b e-10 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{2/3} b^{13/3}}+\frac{\left (b^3 c-4 a b^2 d+7 a^2 b e-10 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^{2/3} b^{13/3}}\\ &=\frac{\left (b^2 d-2 a b e+3 a^2 f\right ) x}{b^4}+\frac{(b e-2 a f) x^4}{4 b^3}+\frac{f x^7}{7 b^2}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 b^4 \left (a+b x^3\right )}-\frac{\left (b^3 c-4 a b^2 d+7 a^2 b e-10 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^{2/3} b^{13/3}}+\frac{\left (b^3 c-4 a b^2 d+7 a^2 b e-10 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{2/3} b^{13/3}}-\frac{\left (b^3 c-4 a b^2 d+7 a^2 b e-10 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{2/3} b^{13/3}}\\ \end{align*}
Mathematica [A] time = 0.157155, size = 277, normalized size = 0.96 \[ \frac{-\frac{84 \sqrt [3]{b} x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{a+b x^3}+\frac{14 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-7 a^2 b e+10 a^3 f+4 a b^2 d-b^3 c\right )}{a^{2/3}}+\frac{28 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (7 a^2 b e-10 a^3 f-4 a b^2 d+b^3 c\right )}{a^{2/3}}+\frac{28 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (-7 a^2 b e+10 a^3 f+4 a b^2 d-b^3 c\right )}{a^{2/3}}+252 \sqrt [3]{b} x \left (3 a^2 f-2 a b e+b^2 d\right )+63 b^{4/3} x^4 (b e-2 a f)+36 b^{7/3} f x^7}{252 b^{13/3}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.01, size = 514, 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 = 1.44706, size = 2125, normalized size = 7.38 \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 [A] time = 10.1333, size = 398, normalized size = 1.38 \begin{align*} \frac{x \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{3 a b^{4} + 3 b^{5} x^{3}} + \operatorname{RootSum}{\left (729 t^{3} a^{2} b^{13} + 1000 a^{9} f^{3} - 2100 a^{8} b e f^{2} + 1200 a^{7} b^{2} d f^{2} + 1470 a^{7} b^{2} e^{2} f - 300 a^{6} b^{3} c f^{2} - 1680 a^{6} b^{3} d e f - 343 a^{6} b^{3} e^{3} + 420 a^{5} b^{4} c e f + 480 a^{5} b^{4} d^{2} f + 588 a^{5} b^{4} d e^{2} - 240 a^{4} b^{5} c d f - 147 a^{4} b^{5} c e^{2} - 336 a^{4} b^{5} d^{2} e + 30 a^{3} b^{6} c^{2} f + 168 a^{3} b^{6} c d e + 64 a^{3} b^{6} d^{3} - 21 a^{2} b^{7} c^{2} e - 48 a^{2} b^{7} c d^{2} + 12 a b^{8} c^{2} d - b^{9} c^{3}, \left ( t \mapsto t \log{\left (- \frac{9 t a b^{4}}{10 a^{3} f - 7 a^{2} b e + 4 a b^{2} d - b^{3} c} + x \right )} \right )\right )} + \frac{f x^{7}}{7 b^{2}} - \frac{x^{4} \left (2 a f - b e\right )}{4 b^{3}} + \frac{x \left (3 a^{2} f - 2 a b e + b^{2} d\right )}{b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10032, size = 471, normalized size = 1.64 \begin{align*} -\frac{{\left (b^{3} c - 4 \, a b^{2} d - 10 \, a^{3} f + 7 \, a^{2} b 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 b^{4}} + \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - 4 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - 10 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + 7 \, \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 b^{5}} - \frac{b^{3} c x - a b^{2} d x - a^{3} f x + a^{2} b x e}{3 \,{\left (b x^{3} + a\right )} b^{4}} + \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - 4 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - 10 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + 7 \, \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 b^{5}} + \frac{4 \, b^{12} f x^{7} - 14 \, a b^{11} f x^{4} + 7 \, b^{12} x^{4} e + 28 \, b^{12} d x + 84 \, a^{2} b^{10} f x - 56 \, a b^{11} x e}{28 \, b^{14}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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