Optimal. Leaf size=274 \[ \frac{\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 b^{13/3}}+\frac{x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{b^4}-\frac{\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 b^{13/3}}+\frac{\sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{\sqrt{3} b^{13/3}}+\frac{x^4 \left (a^2 f-a b e+b^2 d\right )}{4 b^3}+\frac{x^7 (b e-a f)}{7 b^2}+\frac{f x^{10}}{10 b} \]
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Rubi [A] time = 0.26722, antiderivative size = 274, 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 = {1836, 1488, 200, 31, 634, 617, 204, 628} \[ \frac{\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 b^{13/3}}+\frac{x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{b^4}-\frac{\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 b^{13/3}}+\frac{\sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{\sqrt{3} b^{13/3}}+\frac{x^4 \left (a^2 f-a b e+b^2 d\right )}{4 b^3}+\frac{x^7 (b e-a f)}{7 b^2}+\frac{f x^{10}}{10 b} \]
Antiderivative was successfully verified.
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Rule 1836
Rule 1488
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 )}{a+b x^3} \, dx &=\frac{f x^{10}}{10 b}+\frac{\int \frac{x^3 \left (10 b c+10 b d x^3+10 (b e-a f) x^6\right )}{a+b x^3} \, dx}{10 b}\\ &=\frac{f x^{10}}{10 b}+\frac{\int \left (\frac{10 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{b^3}+\frac{10 \left (b^2 d-a b e+a^2 f\right ) x^3}{b^2}+\frac{10 (b e-a f) x^6}{b}+\frac{10 \left (-a b^3 c+a^2 b^2 d-a^3 b e+a^4 f\right )}{b^3 \left (a+b x^3\right )}\right ) \, dx}{10 b}\\ &=\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{b^4}+\frac{\left (b^2 d-a b e+a^2 f\right ) x^4}{4 b^3}+\frac{(b e-a f) x^7}{7 b^2}+\frac{f x^{10}}{10 b}-\frac{\left (a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )\right ) \int \frac{1}{a+b x^3} \, dx}{b^4}\\ &=\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{b^4}+\frac{\left (b^2 d-a b e+a^2 f\right ) x^4}{4 b^3}+\frac{(b e-a f) x^7}{7 b^2}+\frac{f x^{10}}{10 b}-\frac{\left (\sqrt [3]{a} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 b^4}-\frac{\left (\sqrt [3]{a} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )\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}{3 b^4}\\ &=\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{b^4}+\frac{\left (b^2 d-a b e+a^2 f\right ) x^4}{4 b^3}+\frac{(b e-a f) x^7}{7 b^2}+\frac{f x^{10}}{10 b}-\frac{\sqrt [3]{a} \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 b^{13/3}}+\frac{\left (\sqrt [3]{a} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )\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 b^{13/3}}-\frac{\left (a^{2/3} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )\right ) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 b^4}\\ &=\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{b^4}+\frac{\left (b^2 d-a b e+a^2 f\right ) x^4}{4 b^3}+\frac{(b e-a f) x^7}{7 b^2}+\frac{f x^{10}}{10 b}-\frac{\sqrt [3]{a} \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 b^{13/3}}+\frac{\sqrt [3]{a} \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 b^{13/3}}-\frac{\left (\sqrt [3]{a} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{b^{13/3}}\\ &=\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{b^4}+\frac{\left (b^2 d-a b e+a^2 f\right ) x^4}{4 b^3}+\frac{(b e-a f) x^7}{7 b^2}+\frac{f x^{10}}{10 b}+\frac{\sqrt [3]{a} \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} b^{13/3}}-\frac{\sqrt [3]{a} \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 b^{13/3}}+\frac{\sqrt [3]{a} \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 b^{13/3}}\\ \end{align*}
Mathematica [A] time = 0.0911862, size = 264, normalized size = 0.96 \[ \frac{-70 \sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )+420 \sqrt [3]{b} x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )+140 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )-140 \sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )+105 b^{4/3} x^4 \left (a^2 f-a b e+b^2 d\right )+60 b^{7/3} x^7 (b e-a f)+42 b^{10/3} f x^{10}}{420 b^{13/3}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.003, size = 492, 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 = 1.29716, size = 566, normalized size = 2.07 \begin{align*} \frac{42 \, b^{3} f x^{10} + 60 \,{\left (b^{3} e - a b^{2} f\right )} x^{7} + 105 \,{\left (b^{3} d - a b^{2} e + a^{2} b f\right )} x^{4} - 140 \, \sqrt{3}{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (\frac{a}{b}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} b x \left (\frac{a}{b}\right )^{\frac{2}{3}} - \sqrt{3} a}{3 \, a}\right ) + 70 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x^{2} - x \left (\frac{a}{b}\right )^{\frac{1}{3}} + \left (\frac{a}{b}\right )^{\frac{2}{3}}\right ) - 140 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x + \left (\frac{a}{b}\right )^{\frac{1}{3}}\right ) + 420 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x}{420 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.71881, size = 371, normalized size = 1.35 \begin{align*} \operatorname{RootSum}{\left (27 t^{3} b^{13} - a^{10} f^{3} + 3 a^{9} b e f^{2} - 3 a^{8} b^{2} d f^{2} - 3 a^{8} b^{2} e^{2} f + 3 a^{7} b^{3} c f^{2} + 6 a^{7} b^{3} d e f + a^{7} b^{3} e^{3} - 6 a^{6} b^{4} c e f - 3 a^{6} b^{4} d^{2} f - 3 a^{6} b^{4} d e^{2} + 6 a^{5} b^{5} c d f + 3 a^{5} b^{5} c e^{2} + 3 a^{5} b^{5} d^{2} e - 3 a^{4} b^{6} c^{2} f - 6 a^{4} b^{6} c d e - a^{4} b^{6} d^{3} + 3 a^{3} b^{7} c^{2} e + 3 a^{3} b^{7} c d^{2} - 3 a^{2} b^{8} c^{2} d + a b^{9} c^{3}, \left ( t \mapsto t \log{\left (\frac{3 t b^{4}}{a^{3} f - a^{2} b e + a b^{2} d - b^{3} c} + x \right )} \right )\right )} + \frac{f x^{10}}{10 b} - \frac{x^{7} \left (a f - b e\right )}{7 b^{2}} + \frac{x^{4} \left (a^{2} f - a b e + b^{2} d\right )}{4 b^{3}} - \frac{x \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08937, size = 467, normalized size = 1.7 \begin{align*} -\frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + \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 )}{3 \, b^{5}} - \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + \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 )}{6 \, b^{5}} + \frac{{\left (a b^{9} c - a^{2} b^{8} d - a^{4} b^{6} f + a^{3} b^{7} 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 b^{10}} + \frac{14 \, b^{9} f x^{10} - 20 \, a b^{8} f x^{7} + 20 \, b^{9} x^{7} e + 35 \, b^{9} d x^{4} + 35 \, a^{2} b^{7} f x^{4} - 35 \, a b^{8} x^{4} e + 140 \, b^{9} c x - 140 \, a b^{8} d x - 140 \, a^{3} b^{6} f x + 140 \, a^{2} b^{7} x e}{140 \, b^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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