Optimal. Leaf size=328 \[ \frac{a x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 b^5 \left (a+b x^3\right )}+\frac{\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (10 a^2 b e-13 a^3 f-7 a b^2 d+4 b^3 c\right )}{18 b^{16/3}}+\frac{x \left (3 a^2 b e-4 a^3 f-2 a b^2 d+b^3 c\right )}{b^5}-\frac{\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (10 a^2 b e-13 a^3 f-7 a b^2 d+4 b^3 c\right )}{9 b^{16/3}}+\frac{\sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (10 a^2 b e-13 a^3 f-7 a b^2 d+4 b^3 c\right )}{3 \sqrt{3} b^{16/3}}+\frac{x^4 \left (3 a^2 f-2 a b e+b^2 d\right )}{4 b^4}+\frac{x^7 (b e-2 a f)}{7 b^3}+\frac{f x^{10}}{10 b^2} \]
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Rubi [A] time = 0.368865, antiderivative size = 328, 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} \[ \frac{a x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 b^5 \left (a+b x^3\right )}+\frac{\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (10 a^2 b e-13 a^3 f-7 a b^2 d+4 b^3 c\right )}{18 b^{16/3}}+\frac{x \left (3 a^2 b e-4 a^3 f-2 a b^2 d+b^3 c\right )}{b^5}-\frac{\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (10 a^2 b e-13 a^3 f-7 a b^2 d+4 b^3 c\right )}{9 b^{16/3}}+\frac{\sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (10 a^2 b e-13 a^3 f-7 a b^2 d+4 b^3 c\right )}{3 \sqrt{3} b^{16/3}}+\frac{x^4 \left (3 a^2 f-2 a b e+b^2 d\right )}{4 b^4}+\frac{x^7 (b e-2 a f)}{7 b^3}+\frac{f x^{10}}{10 b^2} \]
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^6 \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}{3 b^5 \left (a+b x^3\right )}-\frac{\int \frac{a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )-3 a b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^3-3 a b^2 \left (b^2 d-a b e+a^2 f\right ) x^6-3 a b^3 (b e-a f) x^9-3 a b^4 f x^{12}}{a+b x^3} \, dx}{3 a b^5}\\ &=\frac{a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 b^5 \left (a+b x^3\right )}-\frac{\int \left (-3 a \left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right )-3 a b \left (b^2 d-2 a b e+3 a^2 f\right ) x^3-3 a b^2 (b e-2 a f) x^6-3 a b^3 f x^9+\frac{4 a^2 b^3 c-7 a^3 b^2 d+10 a^4 b e-13 a^5 f}{a+b x^3}\right ) \, dx}{3 a b^5}\\ &=\frac{\left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x}{b^5}+\frac{\left (b^2 d-2 a b e+3 a^2 f\right ) x^4}{4 b^4}+\frac{(b e-2 a f) x^7}{7 b^3}+\frac{f x^{10}}{10 b^2}+\frac{a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 b^5 \left (a+b x^3\right )}-\frac{\left (a \left (4 b^3 c-7 a b^2 d+10 a^2 b e-13 a^3 f\right )\right ) \int \frac{1}{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}{b^5}+\frac{\left (b^2 d-2 a b e+3 a^2 f\right ) x^4}{4 b^4}+\frac{(b e-2 a f) x^7}{7 b^3}+\frac{f x^{10}}{10 b^2}+\frac{a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 b^5 \left (a+b x^3\right )}-\frac{\left (\sqrt [3]{a} \left (4 b^3 c-7 a b^2 d+10 a^2 b e-13 a^3 f\right )\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 b^5}-\frac{\left (\sqrt [3]{a} \left (4 b^3 c-7 a b^2 d+10 a^2 b e-13 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}{9 b^5}\\ &=\frac{\left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x}{b^5}+\frac{\left (b^2 d-2 a b e+3 a^2 f\right ) x^4}{4 b^4}+\frac{(b e-2 a f) x^7}{7 b^3}+\frac{f x^{10}}{10 b^2}+\frac{a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 b^5 \left (a+b x^3\right )}-\frac{\sqrt [3]{a} \left (4 b^3 c-7 a b^2 d+10 a^2 b e-13 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{16/3}}+\frac{\left (\sqrt [3]{a} \left (4 b^3 c-7 a b^2 d+10 a^2 b e-13 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^{16/3}}-\frac{\left (a^{2/3} \left (4 b^3 c-7 a b^2 d+10 a^2 b e-13 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^5}\\ &=\frac{\left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x}{b^5}+\frac{\left (b^2 d-2 a b e+3 a^2 f\right ) x^4}{4 b^4}+\frac{(b e-2 a f) x^7}{7 b^3}+\frac{f x^{10}}{10 b^2}+\frac{a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 b^5 \left (a+b x^3\right )}-\frac{\sqrt [3]{a} \left (4 b^3 c-7 a b^2 d+10 a^2 b e-13 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{16/3}}+\frac{\sqrt [3]{a} \left (4 b^3 c-7 a b^2 d+10 a^2 b e-13 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 b^{16/3}}-\frac{\left (\sqrt [3]{a} \left (4 b^3 c-7 a b^2 d+10 a^2 b e-13 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^{16/3}}\\ &=\frac{\left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x}{b^5}+\frac{\left (b^2 d-2 a b e+3 a^2 f\right ) x^4}{4 b^4}+\frac{(b e-2 a f) x^7}{7 b^3}+\frac{f x^{10}}{10 b^2}+\frac{a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 b^5 \left (a+b x^3\right )}+\frac{\sqrt [3]{a} \left (4 b^3 c-7 a b^2 d+10 a^2 b e-13 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^{16/3}}-\frac{\sqrt [3]{a} \left (4 b^3 c-7 a b^2 d+10 a^2 b e-13 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{16/3}}+\frac{\sqrt [3]{a} \left (4 b^3 c-7 a b^2 d+10 a^2 b e-13 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 b^{16/3}}\\ \end{align*}
Mathematica [A] time = 0.248407, size = 315, normalized size = 0.96 \[ \frac{\frac{420 a \sqrt [3]{b} x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{a+b x^3}-70 \sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-10 a^2 b e+13 a^3 f+7 a b^2 d-4 b^3 c\right )+1260 \sqrt [3]{b} x \left (3 a^2 b e-4 a^3 f-2 a b^2 d+b^3 c\right )+140 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-10 a^2 b e+13 a^3 f+7 a b^2 d-4 b^3 c\right )-140 \sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (-10 a^2 b e+13 a^3 f+7 a b^2 d-4 b^3 c\right )+315 b^{4/3} x^4 \left (3 a^2 f-2 a b e+b^2 d\right )+180 b^{7/3} x^7 (b e-2 a f)+126 b^{10/3} f x^{10}}{1260 b^{16/3}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.01, size = 567, 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 = 1.36585, size = 992, normalized size = 3.02 \begin{align*} \frac{126 \, b^{4} f x^{13} + 18 \,{\left (10 \, b^{4} e - 13 \, a b^{3} f\right )} x^{10} + 45 \,{\left (7 \, b^{4} d - 10 \, a b^{3} e + 13 \, a^{2} b^{2} f\right )} x^{7} + 315 \,{\left (4 \, b^{4} c - 7 \, a b^{3} d + 10 \, a^{2} b^{2} e - 13 \, a^{3} b f\right )} x^{4} - 140 \, \sqrt{3}{\left (4 \, a b^{3} c - 7 \, a^{2} b^{2} d + 10 \, a^{3} b e - 13 \, a^{4} f +{\left (4 \, b^{4} c - 7 \, a b^{3} d + 10 \, a^{2} b^{2} e - 13 \, a^{3} b f\right )} x^{3}\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 (4 \, a b^{3} c - 7 \, a^{2} b^{2} d + 10 \, a^{3} b e - 13 \, a^{4} f +{\left (4 \, b^{4} c - 7 \, a b^{3} d + 10 \, a^{2} b^{2} e - 13 \, a^{3} b f\right )} x^{3}\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 (4 \, a b^{3} c - 7 \, a^{2} b^{2} d + 10 \, a^{3} b e - 13 \, a^{4} f +{\left (4 \, b^{4} c - 7 \, a b^{3} d + 10 \, a^{2} b^{2} e - 13 \, a^{3} b f\right )} x^{3}\right )} \left (\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x + \left (\frac{a}{b}\right )^{\frac{1}{3}}\right ) + 420 \,{\left (4 \, a b^{3} c - 7 \, a^{2} b^{2} d + 10 \, a^{3} b e - 13 \, a^{4} f\right )} x}{1260 \,{\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 = 10.9149, size = 440, normalized size = 1.34 \begin{align*} - \frac{x \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^{16} - 2197 a^{10} f^{3} + 5070 a^{9} b e f^{2} - 3549 a^{8} b^{2} d f^{2} - 3900 a^{8} b^{2} e^{2} f + 2028 a^{7} b^{3} c f^{2} + 5460 a^{7} b^{3} d e f + 1000 a^{7} b^{3} e^{3} - 3120 a^{6} b^{4} c e f - 1911 a^{6} b^{4} d^{2} f - 2100 a^{6} b^{4} d e^{2} + 2184 a^{5} b^{5} c d f + 1200 a^{5} b^{5} c e^{2} + 1470 a^{5} b^{5} d^{2} e - 624 a^{4} b^{6} c^{2} f - 1680 a^{4} b^{6} c d e - 343 a^{4} b^{6} d^{3} + 480 a^{3} b^{7} c^{2} e + 588 a^{3} b^{7} c d^{2} - 336 a^{2} b^{8} c^{2} d + 64 a b^{9} c^{3}, \left ( t \mapsto t \log{\left (\frac{9 t b^{5}}{13 a^{3} f - 10 a^{2} b e + 7 a b^{2} d - 4 b^{3} c} + x \right )} \right )\right )} + \frac{f x^{10}}{10 b^{2}} - \frac{x^{7} \left (2 a f - b e\right )}{7 b^{3}} + \frac{x^{4} \left (3 a^{2} f - 2 a b e + b^{2} d\right )}{4 b^{4}} - \frac{x \left (4 a^{3} f - 3 a^{2} b e + 2 a b^{2} d - b^{3} c\right )}{b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.0802, size = 532, normalized size = 1.62 \begin{align*} -\frac{\sqrt{3}{\left (4 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - 7 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - 13 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + 10 \, \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 \, b^{6}} + \frac{{\left (4 \, a b^{3} c - 7 \, a^{2} b^{2} d - 13 \, a^{4} f + 10 \, a^{3} 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^{5}} - \frac{{\left (4 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - 7 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - 13 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + 10 \, \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 \, b^{6}} + \frac{a b^{3} c x - a^{2} b^{2} d x - a^{4} f x + a^{3} b x e}{3 \,{\left (b x^{3} + a\right )} b^{5}} + \frac{14 \, b^{18} f x^{10} - 40 \, a b^{17} f x^{7} + 20 \, b^{18} x^{7} e + 35 \, b^{18} d x^{4} + 105 \, a^{2} b^{16} f x^{4} - 70 \, a b^{17} x^{4} e + 140 \, b^{18} c x - 280 \, a b^{17} d x - 560 \, a^{3} b^{15} f x + 420 \, a^{2} b^{16} x e}{140 \, b^{20}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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