Optimal. Leaf size=316 \[ \frac{x^5 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{5 b^4}-\frac{a x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{2 b^5}+\frac{a^{5/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 b^{17/3}}-\frac{a^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 b^{17/3}}-\frac{a^{5/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{\sqrt{3} b^{17/3}}+\frac{x^8 \left (a^2 f-a b e+b^2 d\right )}{8 b^3}+\frac{x^{11} (b e-a f)}{11 b^2}+\frac{f x^{14}}{14 b} \]
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Rubi [A] time = 0.306078, antiderivative size = 316, 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} \[ \frac{x^5 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{5 b^4}-\frac{a x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{2 b^5}+\frac{a^{5/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 b^{17/3}}-\frac{a^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 b^{17/3}}-\frac{a^{5/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{\sqrt{3} b^{17/3}}+\frac{x^8 \left (a^2 f-a b e+b^2 d\right )}{8 b^3}+\frac{x^{11} (b e-a f)}{11 b^2}+\frac{f x^{14}}{14 b} \]
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^7 \left (c+d x^3+e x^6+f x^9\right )}{a+b x^3} \, dx &=\frac{f x^{14}}{14 b}+\frac{\int \frac{x^7 \left (14 b c+14 b d x^3+14 (b e-a f) x^6\right )}{a+b x^3} \, dx}{14 b}\\ &=\frac{f x^{14}}{14 b}+\frac{\int \left (-\frac{14 a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{b^4}+\frac{14 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^4}{b^3}+\frac{14 \left (b^2 d-a b e+a^2 f\right ) x^7}{b^2}+\frac{14 (b e-a f) x^{10}}{b}-\frac{14 \left (-a^2 b^3 c+a^3 b^2 d-a^4 b e+a^5 f\right ) x}{b^4 \left (a+b x^3\right )}\right ) \, dx}{14 b}\\ &=-\frac{a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{2 b^5}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^5}{5 b^4}+\frac{\left (b^2 d-a b e+a^2 f\right ) x^8}{8 b^3}+\frac{(b e-a f) x^{11}}{11 b^2}+\frac{f x^{14}}{14 b}+\frac{\left (a^2 \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^5}\\ &=-\frac{a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{2 b^5}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^5}{5 b^4}+\frac{\left (b^2 d-a b e+a^2 f\right ) x^8}{8 b^3}+\frac{(b e-a f) x^{11}}{11 b^2}+\frac{f x^{14}}{14 b}-\frac{\left (a^{5/3} \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^{16/3}}+\frac{\left (a^{5/3} \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^{16/3}}\\ &=-\frac{a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{2 b^5}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^5}{5 b^4}+\frac{\left (b^2 d-a b e+a^2 f\right ) x^8}{8 b^3}+\frac{(b e-a f) x^{11}}{11 b^2}+\frac{f x^{14}}{14 b}-\frac{a^{5/3} \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^{17/3}}+\frac{\left (a^{5/3} \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^{17/3}}+\frac{\left (a^2 \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^{16/3}}\\ &=-\frac{a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{2 b^5}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^5}{5 b^4}+\frac{\left (b^2 d-a b e+a^2 f\right ) x^8}{8 b^3}+\frac{(b e-a f) x^{11}}{11 b^2}+\frac{f x^{14}}{14 b}-\frac{a^{5/3} \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^{17/3}}+\frac{a^{5/3} \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^{17/3}}+\frac{\left (a^{5/3} \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^{17/3}}\\ &=-\frac{a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{2 b^5}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^5}{5 b^4}+\frac{\left (b^2 d-a b e+a^2 f\right ) x^8}{8 b^3}+\frac{(b e-a f) x^{11}}{11 b^2}+\frac{f x^{14}}{14 b}-\frac{a^{5/3} \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^{17/3}}-\frac{a^{5/3} \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^{17/3}}+\frac{a^{5/3} \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^{17/3}}\\ \end{align*}
Mathematica [A] time = 0.0975826, size = 311, normalized size = 0.98 \[ \frac{x^5 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{5 b^4}+\frac{a x^2 \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{2 b^5}-\frac{a^{5/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{6 b^{17/3}}+\frac{a^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{3 b^{17/3}}+\frac{a^{5/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{\sqrt{3} b^{17/3}}+\frac{x^8 \left (a^2 f-a b e+b^2 d\right )}{8 b^3}+\frac{x^{11} (b e-a f)}{11 b^2}+\frac{f x^{14}}{14 b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.003, size = 554, 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 = 1.44431, size = 726, normalized size = 2.3 \begin{align*} \frac{660 \, b^{4} f x^{14} + 840 \,{\left (b^{4} e - a b^{3} f\right )} x^{11} + 1155 \,{\left (b^{4} d - a b^{3} e + a^{2} b^{2} f\right )} x^{8} + 1848 \,{\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} x^{5} - 4620 \,{\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} x^{2} + 3080 \, \sqrt{3}{\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} 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 ) + 1540 \,{\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} 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 ) - 3080 \,{\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} 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 )}{9240 \, b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.45072, size = 496, normalized size = 1.57 \begin{align*} \operatorname{RootSum}{\left (27 t^{3} b^{17} - a^{14} f^{3} + 3 a^{13} b e f^{2} - 3 a^{12} b^{2} d f^{2} - 3 a^{12} b^{2} e^{2} f + 3 a^{11} b^{3} c f^{2} + 6 a^{11} b^{3} d e f + a^{11} b^{3} e^{3} - 6 a^{10} b^{4} c e f - 3 a^{10} b^{4} d^{2} f - 3 a^{10} b^{4} d e^{2} + 6 a^{9} b^{5} c d f + 3 a^{9} b^{5} c e^{2} + 3 a^{9} b^{5} d^{2} e - 3 a^{8} b^{6} c^{2} f - 6 a^{8} b^{6} c d e - a^{8} b^{6} d^{3} + 3 a^{7} b^{7} c^{2} e + 3 a^{7} b^{7} c d^{2} - 3 a^{6} b^{8} c^{2} d + a^{5} b^{9} c^{3}, \left ( t \mapsto t \log{\left (\frac{9 t^{2} b^{11}}{a^{9} f^{2} - 2 a^{8} b e f + 2 a^{7} b^{2} d f + a^{7} b^{2} e^{2} - 2 a^{6} b^{3} c f - 2 a^{6} b^{3} d e + 2 a^{5} b^{4} c e + a^{5} b^{4} d^{2} - 2 a^{4} b^{5} c d + a^{3} b^{6} c^{2}} + x \right )} \right )\right )} + \frac{f x^{14}}{14 b} - \frac{x^{11} \left (a f - b e\right )}{11 b^{2}} + \frac{x^{8} \left (a^{2} f - a b e + b^{2} d\right )}{8 b^{3}} - \frac{x^{5} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{5 b^{4}} + \frac{x^{2} \left (a^{4} f - a^{3} b e + a^{2} b^{2} d - a b^{3} c\right )}{2 b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.0825, size = 595, normalized size = 1.88 \begin{align*} -\frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{2}{3}} a b^{3} c - \left (-a b^{2}\right )^{\frac{2}{3}} a^{2} b^{2} d - \left (-a b^{2}\right )^{\frac{2}{3}} a^{4} f + \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} 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^{7}} + \frac{{\left (\left (-a b^{2}\right )^{\frac{2}{3}} a b^{3} c - \left (-a b^{2}\right )^{\frac{2}{3}} a^{2} b^{2} d - \left (-a b^{2}\right )^{\frac{2}{3}} a^{4} f + \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} 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^{7}} - \frac{{\left (a^{2} b^{12} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} - a^{3} b^{11} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} - a^{5} b^{9} f \left (-\frac{a}{b}\right )^{\frac{1}{3}} + a^{4} b^{10} \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^{14}} + \frac{220 \, b^{13} f x^{14} - 280 \, a b^{12} f x^{11} + 280 \, b^{13} x^{11} e + 385 \, b^{13} d x^{8} + 385 \, a^{2} b^{11} f x^{8} - 385 \, a b^{12} x^{8} e + 616 \, b^{13} c x^{5} - 616 \, a b^{12} d x^{5} - 616 \, a^{3} b^{10} f x^{5} + 616 \, a^{2} b^{11} x^{5} e - 1540 \, a b^{12} c x^{2} + 1540 \, a^{2} b^{11} d x^{2} + 1540 \, a^{4} b^{9} f x^{2} - 1540 \, a^{3} b^{10} x^{2} e}{3080 \, b^{14}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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