Optimal. Leaf size=335 \[ \frac{x^2 \left (3 a^2 b e-4 a^3 f-2 a b^2 d+b^3 c\right )}{2 b^5}+\frac{a x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 b^5 \left (a+b x^3\right )}-\frac{a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (11 a^2 b e-14 a^3 f-8 a b^2 d+5 b^3 c\right )}{18 b^{17/3}}+\frac{a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (11 a^2 b e-14 a^3 f-8 a b^2 d+5 b^3 c\right )}{9 b^{17/3}}+\frac{a^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (11 a^2 b e-14 a^3 f-8 a b^2 d+5 b^3 c\right )}{3 \sqrt{3} b^{17/3}}+\frac{x^5 \left (3 a^2 f-2 a b e+b^2 d\right )}{5 b^4}+\frac{x^8 (b e-2 a f)}{8 b^3}+\frac{f x^{11}}{11 b^2} \]
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Rubi [A] time = 0.705022, antiderivative size = 335, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 10, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {1828, 1851, 1836, 1488, 292, 31, 634, 617, 204, 628} \[ \frac{x^2 \left (3 a^2 b e-4 a^3 f-2 a b^2 d+b^3 c\right )}{2 b^5}+\frac{a x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 b^5 \left (a+b x^3\right )}-\frac{a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (11 a^2 b e-14 a^3 f-8 a b^2 d+5 b^3 c\right )}{18 b^{17/3}}+\frac{a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (11 a^2 b e-14 a^3 f-8 a b^2 d+5 b^3 c\right )}{9 b^{17/3}}+\frac{a^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (11 a^2 b e-14 a^3 f-8 a b^2 d+5 b^3 c\right )}{3 \sqrt{3} b^{17/3}}+\frac{x^5 \left (3 a^2 f-2 a b e+b^2 d\right )}{5 b^4}+\frac{x^8 (b e-2 a f)}{8 b^3}+\frac{f x^{11}}{11 b^2} \]
Antiderivative was successfully verified.
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Rule 1828
Rule 1851
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 )}{\left (a+b x^3\right )^2} \, dx &=\frac{a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 b^5 \left (a+b x^3\right )}-\frac{\int \frac{2 a^2 b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x-3 a b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^4-3 a b^3 \left (b^2 d-a b e+a^2 f\right ) x^7-3 a b^4 (b e-a f) x^{10}-3 a b^5 f x^{13}}{a+b x^3} \, dx}{3 a b^6}\\ &=\frac{a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 b^5 \left (a+b x^3\right )}-\frac{\int \frac{x \left (2 a^2 b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )-3 a b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^3-3 a b^3 \left (b^2 d-a b e+a^2 f\right ) x^6-3 a b^4 (b e-a f) x^9-3 a b^5 f x^{12}\right )}{a+b x^3} \, dx}{3 a b^6}\\ &=\frac{f x^{11}}{11 b^2}+\frac{a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 b^5 \left (a+b x^3\right )}-\frac{\int \frac{x \left (22 a^2 b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )-33 a b^3 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^3-33 a b^4 \left (b^2 d-a b e+a^2 f\right ) x^6-33 a b^5 (b e-2 a f) x^9\right )}{a+b x^3} \, dx}{33 a b^7}\\ &=\frac{(b e-2 a f) x^8}{8 b^3}+\frac{f x^{11}}{11 b^2}+\frac{a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 b^5 \left (a+b x^3\right )}-\frac{\int \frac{x \left (176 a^2 b^3 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )-264 a b^4 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^3-264 a b^5 \left (b^2 d-2 a b e+3 a^2 f\right ) x^6\right )}{a+b x^3} \, dx}{264 a b^8}\\ &=\frac{(b e-2 a f) x^8}{8 b^3}+\frac{f x^{11}}{11 b^2}+\frac{a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 b^5 \left (a+b x^3\right )}-\frac{\int \left (-264 a b^3 \left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x-264 a b^4 \left (b^2 d-2 a b e+3 a^2 f\right ) x^4-\frac{88 \left (-5 a^2 b^6 c+8 a^3 b^5 d-11 a^4 b^4 e+14 a^5 b^3 f\right ) x}{a+b x^3}\right ) \, dx}{264 a b^8}\\ &=\frac{\left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x^2}{2 b^5}+\frac{\left (b^2 d-2 a b e+3 a^2 f\right ) x^5}{5 b^4}+\frac{(b e-2 a f) x^8}{8 b^3}+\frac{f x^{11}}{11 b^2}+\frac{a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 b^5 \left (a+b x^3\right )}-\frac{\left (a \left (5 b^3 c-8 a b^2 d+11 a^2 b e-14 a^3 f\right )\right ) \int \frac{x}{a+b x^3} \, dx}{3 b^5}\\ &=\frac{\left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x^2}{2 b^5}+\frac{\left (b^2 d-2 a b e+3 a^2 f\right ) x^5}{5 b^4}+\frac{(b e-2 a f) x^8}{8 b^3}+\frac{f x^{11}}{11 b^2}+\frac{a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 b^5 \left (a+b x^3\right )}+\frac{\left (a^{2/3} \left (5 b^3 c-8 a b^2 d+11 a^2 b e-14 a^3 f\right )\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 b^{16/3}}-\frac{\left (a^{2/3} \left (5 b^3 c-8 a b^2 d+11 a^2 b e-14 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}{9 b^{16/3}}\\ &=\frac{\left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x^2}{2 b^5}+\frac{\left (b^2 d-2 a b e+3 a^2 f\right ) x^5}{5 b^4}+\frac{(b e-2 a f) x^8}{8 b^3}+\frac{f x^{11}}{11 b^2}+\frac{a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 b^5 \left (a+b x^3\right )}+\frac{a^{2/3} \left (5 b^3 c-8 a b^2 d+11 a^2 b e-14 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{17/3}}-\frac{\left (a^{2/3} \left (5 b^3 c-8 a b^2 d+11 a^2 b e-14 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}{18 b^{17/3}}-\frac{\left (a \left (5 b^3 c-8 a b^2 d+11 a^2 b e-14 a^3 f\right )\right ) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 b^{16/3}}\\ &=\frac{\left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x^2}{2 b^5}+\frac{\left (b^2 d-2 a b e+3 a^2 f\right ) x^5}{5 b^4}+\frac{(b e-2 a f) x^8}{8 b^3}+\frac{f x^{11}}{11 b^2}+\frac{a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 b^5 \left (a+b x^3\right )}+\frac{a^{2/3} \left (5 b^3 c-8 a b^2 d+11 a^2 b e-14 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{17/3}}-\frac{a^{2/3} \left (5 b^3 c-8 a b^2 d+11 a^2 b e-14 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 b^{17/3}}-\frac{\left (a^{2/3} \left (5 b^3 c-8 a b^2 d+11 a^2 b e-14 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 )}{3 b^{17/3}}\\ &=\frac{\left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x^2}{2 b^5}+\frac{\left (b^2 d-2 a b e+3 a^2 f\right ) x^5}{5 b^4}+\frac{(b e-2 a f) x^8}{8 b^3}+\frac{f x^{11}}{11 b^2}+\frac{a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 b^5 \left (a+b x^3\right )}+\frac{a^{2/3} \left (5 b^3 c-8 a b^2 d+11 a^2 b e-14 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} b^{17/3}}+\frac{a^{2/3} \left (5 b^3 c-8 a b^2 d+11 a^2 b e-14 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{17/3}}-\frac{a^{2/3} \left (5 b^3 c-8 a b^2 d+11 a^2 b e-14 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 b^{17/3}}\\ \end{align*}
Mathematica [A] time = 0.165613, size = 319, normalized size = 0.95 \[ \frac{1980 b^{2/3} x^2 \left (3 a^2 b e-4 a^3 f-2 a b^2 d+b^3 c\right )+\frac{1320 a b^{2/3} x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{a+b x^3}+220 a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-11 a^2 b e+14 a^3 f+8 a b^2 d-5 b^3 c\right )-440 a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-11 a^2 b e+14 a^3 f+8 a b^2 d-5 b^3 c\right )-440 \sqrt{3} a^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (-11 a^2 b e+14 a^3 f+8 a b^2 d-5 b^3 c\right )+792 b^{5/3} x^5 \left (3 a^2 f-2 a b e+b^2 d\right )+495 b^{8/3} x^8 (b e-2 a f)+360 b^{11/3} f x^{11}}{3960 b^{17/3}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.01, size = 584, normalized size = 1.7 \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 = 1.29785, size = 1057, normalized size = 3.16 \begin{align*} \frac{360 \, b^{4} f x^{14} + 45 \,{\left (11 \, b^{4} e - 14 \, a b^{3} f\right )} x^{11} + 99 \,{\left (8 \, b^{4} d - 11 \, a b^{3} e + 14 \, a^{2} b^{2} f\right )} x^{8} + 396 \,{\left (5 \, b^{4} c - 8 \, a b^{3} d + 11 \, a^{2} b^{2} e - 14 \, a^{3} b f\right )} x^{5} + 660 \,{\left (5 \, a b^{3} c - 8 \, a^{2} b^{2} d + 11 \, a^{3} b e - 14 \, a^{4} f\right )} x^{2} - 440 \, \sqrt{3}{\left (5 \, a b^{3} c - 8 \, a^{2} b^{2} d + 11 \, a^{3} b e - 14 \, a^{4} f +{\left (5 \, b^{4} c - 8 \, a b^{3} d + 11 \, a^{2} b^{2} e - 14 \, a^{3} b f\right )} x^{3}\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 (5 \, a b^{3} c - 8 \, a^{2} b^{2} d + 11 \, a^{3} b e - 14 \, a^{4} f +{\left (5 \, b^{4} c - 8 \, a b^{3} d + 11 \, a^{2} b^{2} e - 14 \, a^{3} b f\right )} x^{3}\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 (5 \, a b^{3} c - 8 \, a^{2} b^{2} d + 11 \, a^{3} b e - 14 \, a^{4} f +{\left (5 \, b^{4} c - 8 \, a b^{3} d + 11 \, a^{2} b^{2} e - 14 \, a^{3} b f\right )} x^{3}\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 )}{3960 \,{\left (b^{6} x^{3} + a b^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 37.4541, size = 530, normalized size = 1.58 \begin{align*} - \frac{x^{2} \left (a^{4} f - a^{3} b e + a^{2} b^{2} d - a b^{3} c\right )}{3 a b^{5} + 3 b^{6} x^{3}} + \operatorname{RootSum}{\left (729 t^{3} b^{17} + 2744 a^{11} f^{3} - 6468 a^{10} b e f^{2} + 4704 a^{9} b^{2} d f^{2} + 5082 a^{9} b^{2} e^{2} f - 2940 a^{8} b^{3} c f^{2} - 7392 a^{8} b^{3} d e f - 1331 a^{8} b^{3} e^{3} + 4620 a^{7} b^{4} c e f + 2688 a^{7} b^{4} d^{2} f + 2904 a^{7} b^{4} d e^{2} - 3360 a^{6} b^{5} c d f - 1815 a^{6} b^{5} c e^{2} - 2112 a^{6} b^{5} d^{2} e + 1050 a^{5} b^{6} c^{2} f + 2640 a^{5} b^{6} c d e + 512 a^{5} b^{6} d^{3} - 825 a^{4} b^{7} c^{2} e - 960 a^{4} b^{7} c d^{2} + 600 a^{3} b^{8} c^{2} d - 125 a^{2} b^{9} c^{3}, \left ( t \mapsto t \log{\left (\frac{81 t^{2} b^{11}}{196 a^{7} f^{2} - 308 a^{6} b e f + 224 a^{5} b^{2} d f + 121 a^{5} b^{2} e^{2} - 140 a^{4} b^{3} c f - 176 a^{4} b^{3} d e + 110 a^{3} b^{4} c e + 64 a^{3} b^{4} d^{2} - 80 a^{2} b^{5} c d + 25 a b^{6} c^{2}} + x \right )} \right )\right )} + \frac{f x^{11}}{11 b^{2}} - \frac{x^{8} \left (2 a f - b e\right )}{8 b^{3}} + \frac{x^{5} \left (3 a^{2} f - 2 a b e + b^{2} d\right )}{5 b^{4}} - \frac{x^{2} \left (4 a^{3} f - 3 a^{2} b e + 2 a b^{2} d - 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 = 1.08749, size = 597, normalized size = 1.78 \begin{align*} \frac{{\left (5 \, a b^{3} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 8 \, a^{2} b^{2} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 14 \, a^{4} f \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 11 \, a^{3} 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 )}{9 \, a b^{5}} + \frac{\sqrt{3}{\left (5 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - 8 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - 14 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + 11 \, \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 )}{9 \, b^{7}} + \frac{a b^{3} c x^{2} - a^{2} b^{2} d x^{2} - a^{4} f x^{2} + a^{3} b x^{2} e}{3 \,{\left (b x^{3} + a\right )} b^{5}} - \frac{{\left (5 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - 8 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - 14 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + 11 \, \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 )}{18 \, b^{7}} + \frac{40 \, b^{20} f x^{11} - 110 \, a b^{19} f x^{8} + 55 \, b^{20} x^{8} e + 88 \, b^{20} d x^{5} + 264 \, a^{2} b^{18} f x^{5} - 176 \, a b^{19} x^{5} e + 220 \, b^{20} c x^{2} - 440 \, a b^{19} d x^{2} - 880 \, a^{3} b^{17} f x^{2} + 660 \, a^{2} b^{18} x^{2} e}{440 \, b^{22}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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