Optimal. Leaf size=279 \[ \frac{x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{2 b^4}-\frac{a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 b^{14/3}}+\frac{a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 b^{14/3}}+\frac{a^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{\sqrt{3} b^{14/3}}+\frac{x^5 \left (a^2 f-a b e+b^2 d\right )}{5 b^3}+\frac{x^8 (b e-a f)}{8 b^2}+\frac{f x^{11}}{11 b} \]
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Rubi [A] time = 0.273338, antiderivative size = 279, 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, 292, 31, 634, 617, 204, 628} \[ \frac{x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{2 b^4}-\frac{a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 b^{14/3}}+\frac{a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 b^{14/3}}+\frac{a^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{\sqrt{3} b^{14/3}}+\frac{x^5 \left (a^2 f-a b e+b^2 d\right )}{5 b^3}+\frac{x^8 (b e-a f)}{8 b^2}+\frac{f x^{11}}{11 b} \]
Antiderivative was successfully verified.
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Rule 1836
Rule 1488
Rule 292
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{x^4 \left (c+d x^3+e x^6+f x^9\right )}{a+b x^3} \, dx &=\frac{f x^{11}}{11 b}+\frac{\int \frac{x^4 \left (11 b c+11 b d x^3+11 (b e-a f) x^6\right )}{a+b x^3} \, dx}{11 b}\\ &=\frac{f x^{11}}{11 b}+\frac{\int \left (\frac{11 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{b^3}+\frac{11 \left (b^2 d-a b e+a^2 f\right ) x^4}{b^2}+\frac{11 (b e-a f) x^7}{b}+\frac{11 \left (-a b^3 c+a^2 b^2 d-a^3 b e+a^4 f\right ) x}{b^3 \left (a+b x^3\right )}\right ) \, dx}{11 b}\\ &=\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{2 b^4}+\frac{\left (b^2 d-a b e+a^2 f\right ) x^5}{5 b^3}+\frac{(b e-a f) x^8}{8 b^2}+\frac{f x^{11}}{11 b}-\frac{\left (a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )\right ) \int \frac{x}{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^2}{2 b^4}+\frac{\left (b^2 d-a b e+a^2 f\right ) x^5}{5 b^3}+\frac{(b e-a f) x^8}{8 b^2}+\frac{f x^{11}}{11 b}+\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}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 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{\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^{13/3}}\\ &=\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{2 b^4}+\frac{\left (b^2 d-a b e+a^2 f\right ) x^5}{5 b^3}+\frac{(b e-a f) x^8}{8 b^2}+\frac{f x^{11}}{11 b}+\frac{a^{2/3} \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^{14/3}}-\frac{\left (a^{2/3} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )\right ) \int \frac{-\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^{14/3}}-\frac{\left (a \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^{13/3}}\\ &=\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{2 b^4}+\frac{\left (b^2 d-a b e+a^2 f\right ) x^5}{5 b^3}+\frac{(b e-a f) x^8}{8 b^2}+\frac{f x^{11}}{11 b}+\frac{a^{2/3} \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^{14/3}}-\frac{a^{2/3} \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^{14/3}}-\frac{\left (a^{2/3} \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^{14/3}}\\ &=\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{2 b^4}+\frac{\left (b^2 d-a b e+a^2 f\right ) x^5}{5 b^3}+\frac{(b e-a f) x^8}{8 b^2}+\frac{f x^{11}}{11 b}+\frac{a^{2/3} \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^{14/3}}+\frac{a^{2/3} \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^{14/3}}-\frac{a^{2/3} \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^{14/3}}\\ \end{align*}
Mathematica [A] time = 0.0915794, size = 266, normalized size = 0.95 \[ \frac{660 b^{2/3} x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )+220 a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )-440 a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )-440 \sqrt{3} a^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )+264 b^{5/3} x^5 \left (a^2 f-a b e+b^2 d\right )+165 b^{8/3} x^8 (b e-a f)+120 b^{11/3} f x^{11}}{1320 b^{14/3}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.004, size = 502, 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 = 1.40658, size = 635, normalized size = 2.28 \begin{align*} \frac{120 \, b^{3} f x^{11} + 165 \,{\left (b^{3} e - a b^{2} f\right )} x^{8} + 264 \,{\left (b^{3} d - a b^{2} e + a^{2} b f\right )} x^{5} + 660 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{2} - 440 \, \sqrt{3}{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} b x \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} + \sqrt{3} a}{3 \, a}\right ) + 220 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \log \left (a x^{2} - b x \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}} - a \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}}\right ) - 440 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \log \left (a x + b \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}}\right )}{1320 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.38275, size = 459, normalized size = 1.65 \begin{align*} \operatorname{RootSum}{\left (27 t^{3} b^{14} + a^{11} f^{3} - 3 a^{10} b e f^{2} + 3 a^{9} b^{2} d f^{2} + 3 a^{9} b^{2} e^{2} f - 3 a^{8} b^{3} c f^{2} - 6 a^{8} b^{3} d e f - a^{8} b^{3} e^{3} + 6 a^{7} b^{4} c e f + 3 a^{7} b^{4} d^{2} f + 3 a^{7} b^{4} d e^{2} - 6 a^{6} b^{5} c d f - 3 a^{6} b^{5} c e^{2} - 3 a^{6} b^{5} d^{2} e + 3 a^{5} b^{6} c^{2} f + 6 a^{5} b^{6} c d e + a^{5} b^{6} d^{3} - 3 a^{4} b^{7} c^{2} e - 3 a^{4} b^{7} c d^{2} + 3 a^{3} b^{8} c^{2} d - a^{2} b^{9} c^{3}, \left ( t \mapsto t \log{\left (\frac{9 t^{2} b^{9}}{a^{7} f^{2} - 2 a^{6} b e f + 2 a^{5} b^{2} d f + a^{5} b^{2} e^{2} - 2 a^{4} b^{3} c f - 2 a^{4} b^{3} d e + 2 a^{3} b^{4} c e + a^{3} b^{4} d^{2} - 2 a^{2} b^{5} c d + a b^{6} c^{2}} + x \right )} \right )\right )} + \frac{f x^{11}}{11 b} - \frac{x^{8} \left (a f - b e\right )}{8 b^{2}} + \frac{x^{5} \left (a^{2} f - a b e + b^{2} d\right )}{5 b^{3}} - \frac{x^{2} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{2 b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07492, size = 521, normalized size = 1.87 \begin{align*} \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 \, b^{6}} - \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 \, b^{6}} + \frac{{\left (a b^{10} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} - a^{2} b^{9} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} - a^{4} b^{7} f \left (-\frac{a}{b}\right )^{\frac{1}{3}} + a^{3} b^{8} \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 b^{11}} + \frac{40 \, b^{10} f x^{11} - 55 \, a b^{9} f x^{8} + 55 \, b^{10} x^{8} e + 88 \, b^{10} d x^{5} + 88 \, a^{2} b^{8} f x^{5} - 88 \, a b^{9} x^{5} e + 220 \, b^{10} c x^{2} - 220 \, a b^{9} d x^{2} - 220 \, a^{3} b^{7} f x^{2} + 220 \, a^{2} b^{8} x^{2} e}{440 \, b^{11}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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