Optimal. Leaf size=375 \[ \frac{a x \left (25 a^2 b e-31 a^3 f-19 a b^2 d+13 b^3 c\right )}{18 b^6 \left (a+b x^3\right )}-\frac{a^2 x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 b^6 \left (a+b x^3\right )^2}+\frac{\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (65 a^2 b e-104 a^3 f-35 a b^2 d+14 b^3 c\right )}{54 b^{19/3}}+\frac{x \left (6 a^2 b e-10 a^3 f-3 a b^2 d+b^3 c\right )}{b^6}-\frac{\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (65 a^2 b e-104 a^3 f-35 a b^2 d+14 b^3 c\right )}{27 b^{19/3}}+\frac{\sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (65 a^2 b e-104 a^3 f-35 a b^2 d+14 b^3 c\right )}{9 \sqrt{3} b^{19/3}}+\frac{x^4 \left (6 a^2 f-3 a b e+b^2 d\right )}{4 b^5}+\frac{x^7 (b e-3 a f)}{7 b^4}+\frac{f x^{10}}{10 b^3} \]
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Rubi [A] time = 0.606352, antiderivative size = 375, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {1828, 1858, 1887, 200, 31, 634, 617, 204, 628} \[ \frac{a x \left (25 a^2 b e-31 a^3 f-19 a b^2 d+13 b^3 c\right )}{18 b^6 \left (a+b x^3\right )}-\frac{a^2 x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 b^6 \left (a+b x^3\right )^2}+\frac{\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (65 a^2 b e-104 a^3 f-35 a b^2 d+14 b^3 c\right )}{54 b^{19/3}}+\frac{x \left (6 a^2 b e-10 a^3 f-3 a b^2 d+b^3 c\right )}{b^6}-\frac{\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (65 a^2 b e-104 a^3 f-35 a b^2 d+14 b^3 c\right )}{27 b^{19/3}}+\frac{\sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (65 a^2 b e-104 a^3 f-35 a b^2 d+14 b^3 c\right )}{9 \sqrt{3} b^{19/3}}+\frac{x^4 \left (6 a^2 f-3 a b e+b^2 d\right )}{4 b^5}+\frac{x^7 (b e-3 a f)}{7 b^4}+\frac{f x^{10}}{10 b^3} \]
Antiderivative was successfully verified.
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Rule 1828
Rule 1858
Rule 1887
Rule 200
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{x^9 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx &=-\frac{a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 b^6 \left (a+b x^3\right )^2}-\frac{\int \frac{-a^3 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )+6 a^2 b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^3-6 a b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^6-6 a b^3 \left (b^2 d-a b e+a^2 f\right ) x^9-6 a b^4 (b e-a f) x^{12}-6 a b^5 f x^{15}}{\left (a+b x^3\right )^2} \, dx}{6 a b^6}\\ &=-\frac{a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 b^6 \left (a+b x^3\right )^2}+\frac{a \left (13 b^3 c-19 a b^2 d+25 a^2 b e-31 a^3 f\right ) x}{18 b^6 \left (a+b x^3\right )}+\frac{\int \frac{-2 a^3 b^5 \left (5 b^3 c-8 a b^2 d+11 a^2 b e-14 a^3 f\right )+18 a^2 b^6 \left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x^3+18 a^2 b^7 \left (b^2 d-2 a b e+3 a^2 f\right ) x^6+18 a^2 b^8 (b e-2 a f) x^9+18 a^2 b^9 f x^{12}}{a+b x^3} \, dx}{18 a^2 b^{11}}\\ &=-\frac{a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 b^6 \left (a+b x^3\right )^2}+\frac{a \left (13 b^3 c-19 a b^2 d+25 a^2 b e-31 a^3 f\right ) x}{18 b^6 \left (a+b x^3\right )}+\frac{\int \left (18 a^2 b^5 \left (b^3 c-3 a b^2 d+6 a^2 b e-10 a^3 f\right )+18 a^2 b^6 \left (b^2 d-3 a b e+6 a^2 f\right ) x^3+18 a^2 b^7 (b e-3 a f) x^6+18 a^2 b^8 f x^9+\frac{2 \left (-14 a^3 b^8 c+35 a^4 b^7 d-65 a^5 b^6 e+104 a^6 b^5 f\right )}{a+b x^3}\right ) \, dx}{18 a^2 b^{11}}\\ &=\frac{\left (b^3 c-3 a b^2 d+6 a^2 b e-10 a^3 f\right ) x}{b^6}+\frac{\left (b^2 d-3 a b e+6 a^2 f\right ) x^4}{4 b^5}+\frac{(b e-3 a f) x^7}{7 b^4}+\frac{f x^{10}}{10 b^3}-\frac{a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 b^6 \left (a+b x^3\right )^2}+\frac{a \left (13 b^3 c-19 a b^2 d+25 a^2 b e-31 a^3 f\right ) x}{18 b^6 \left (a+b x^3\right )}-\frac{\left (a \left (14 b^3 c-35 a b^2 d+65 a^2 b e-104 a^3 f\right )\right ) \int \frac{1}{a+b x^3} \, dx}{9 b^6}\\ &=\frac{\left (b^3 c-3 a b^2 d+6 a^2 b e-10 a^3 f\right ) x}{b^6}+\frac{\left (b^2 d-3 a b e+6 a^2 f\right ) x^4}{4 b^5}+\frac{(b e-3 a f) x^7}{7 b^4}+\frac{f x^{10}}{10 b^3}-\frac{a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 b^6 \left (a+b x^3\right )^2}+\frac{a \left (13 b^3 c-19 a b^2 d+25 a^2 b e-31 a^3 f\right ) x}{18 b^6 \left (a+b x^3\right )}-\frac{\left (\sqrt [3]{a} \left (14 b^3 c-35 a b^2 d+65 a^2 b e-104 a^3 f\right )\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 b^6}-\frac{\left (\sqrt [3]{a} \left (14 b^3 c-35 a b^2 d+65 a^2 b e-104 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}{27 b^6}\\ &=\frac{\left (b^3 c-3 a b^2 d+6 a^2 b e-10 a^3 f\right ) x}{b^6}+\frac{\left (b^2 d-3 a b e+6 a^2 f\right ) x^4}{4 b^5}+\frac{(b e-3 a f) x^7}{7 b^4}+\frac{f x^{10}}{10 b^3}-\frac{a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 b^6 \left (a+b x^3\right )^2}+\frac{a \left (13 b^3 c-19 a b^2 d+25 a^2 b e-31 a^3 f\right ) x}{18 b^6 \left (a+b x^3\right )}-\frac{\sqrt [3]{a} \left (14 b^3 c-35 a b^2 d+65 a^2 b e-104 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 b^{19/3}}+\frac{\left (\sqrt [3]{a} \left (14 b^3 c-35 a b^2 d+65 a^2 b e-104 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}{54 b^{19/3}}-\frac{\left (a^{2/3} \left (14 b^3 c-35 a b^2 d+65 a^2 b e-104 a^3 f\right )\right ) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{18 b^6}\\ &=\frac{\left (b^3 c-3 a b^2 d+6 a^2 b e-10 a^3 f\right ) x}{b^6}+\frac{\left (b^2 d-3 a b e+6 a^2 f\right ) x^4}{4 b^5}+\frac{(b e-3 a f) x^7}{7 b^4}+\frac{f x^{10}}{10 b^3}-\frac{a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 b^6 \left (a+b x^3\right )^2}+\frac{a \left (13 b^3 c-19 a b^2 d+25 a^2 b e-31 a^3 f\right ) x}{18 b^6 \left (a+b x^3\right )}-\frac{\sqrt [3]{a} \left (14 b^3 c-35 a b^2 d+65 a^2 b e-104 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 b^{19/3}}+\frac{\sqrt [3]{a} \left (14 b^3 c-35 a b^2 d+65 a^2 b e-104 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 b^{19/3}}-\frac{\left (\sqrt [3]{a} \left (14 b^3 c-35 a b^2 d+65 a^2 b e-104 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 )}{9 b^{19/3}}\\ &=\frac{\left (b^3 c-3 a b^2 d+6 a^2 b e-10 a^3 f\right ) x}{b^6}+\frac{\left (b^2 d-3 a b e+6 a^2 f\right ) x^4}{4 b^5}+\frac{(b e-3 a f) x^7}{7 b^4}+\frac{f x^{10}}{10 b^3}-\frac{a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 b^6 \left (a+b x^3\right )^2}+\frac{a \left (13 b^3 c-19 a b^2 d+25 a^2 b e-31 a^3 f\right ) x}{18 b^6 \left (a+b x^3\right )}+\frac{\sqrt [3]{a} \left (14 b^3 c-35 a b^2 d+65 a^2 b e-104 a^3 f\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} b^{19/3}}-\frac{\sqrt [3]{a} \left (14 b^3 c-35 a b^2 d+65 a^2 b e-104 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 b^{19/3}}+\frac{\sqrt [3]{a} \left (14 b^3 c-35 a b^2 d+65 a^2 b e-104 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 b^{19/3}}\\ \end{align*}
Mathematica [A] time = 0.37398, size = 362, normalized size = 0.97 \[ \frac{\frac{210 a \sqrt [3]{b} x \left (25 a^2 b e-31 a^3 f-19 a b^2 d+13 b^3 c\right )}{a+b x^3}+\frac{630 a^2 \sqrt [3]{b} x \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{\left (a+b x^3\right )^2}-70 \sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-65 a^2 b e+104 a^3 f+35 a b^2 d-14 b^3 c\right )+3780 \sqrt [3]{b} x \left (6 a^2 b e-10 a^3 f-3 a b^2 d+b^3 c\right )+140 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-65 a^2 b e+104 a^3 f+35 a b^2 d-14 b^3 c\right )-140 \sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (-65 a^2 b e+104 a^3 f+35 a b^2 d-14 b^3 c\right )+945 b^{4/3} x^4 \left (6 a^2 f-3 a b e+b^2 d\right )+540 b^{7/3} x^7 (b e-3 a f)+378 b^{10/3} f x^{10}}{3780 b^{19/3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 651, 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 = 1.44258, size = 1424, normalized size = 3.8 \begin{align*} \frac{378 \, b^{5} f x^{16} + 108 \,{\left (5 \, b^{5} e - 8 \, a b^{4} f\right )} x^{13} + 27 \,{\left (35 \, b^{5} d - 65 \, a b^{4} e + 104 \, a^{2} b^{3} f\right )} x^{10} + 270 \,{\left (14 \, b^{5} c - 35 \, a b^{4} d + 65 \, a^{2} b^{3} e - 104 \, a^{3} b^{2} f\right )} x^{7} + 735 \,{\left (14 \, a b^{4} c - 35 \, a^{2} b^{3} d + 65 \, a^{3} b^{2} e - 104 \, a^{4} b f\right )} x^{4} - 140 \, \sqrt{3}{\left ({\left (14 \, b^{5} c - 35 \, a b^{4} d + 65 \, a^{2} b^{3} e - 104 \, a^{3} b^{2} f\right )} x^{6} + 14 \, a^{2} b^{3} c - 35 \, a^{3} b^{2} d + 65 \, a^{4} b e - 104 \, a^{5} f + 2 \,{\left (14 \, a b^{4} c - 35 \, a^{2} b^{3} d + 65 \, a^{3} b^{2} e - 104 \, a^{4} 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 ({\left (14 \, b^{5} c - 35 \, a b^{4} d + 65 \, a^{2} b^{3} e - 104 \, a^{3} b^{2} f\right )} x^{6} + 14 \, a^{2} b^{3} c - 35 \, a^{3} b^{2} d + 65 \, a^{4} b e - 104 \, a^{5} f + 2 \,{\left (14 \, a b^{4} c - 35 \, a^{2} b^{3} d + 65 \, a^{3} b^{2} e - 104 \, a^{4} 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 ({\left (14 \, b^{5} c - 35 \, a b^{4} d + 65 \, a^{2} b^{3} e - 104 \, a^{3} b^{2} f\right )} x^{6} + 14 \, a^{2} b^{3} c - 35 \, a^{3} b^{2} d + 65 \, a^{4} b e - 104 \, a^{5} f + 2 \,{\left (14 \, a b^{4} c - 35 \, a^{2} b^{3} d + 65 \, a^{3} b^{2} e - 104 \, a^{4} 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 (14 \, a^{2} b^{3} c - 35 \, a^{3} b^{2} d + 65 \, a^{4} b e - 104 \, a^{5} f\right )} x}{3780 \,{\left (b^{8} x^{6} + 2 \, a b^{7} x^{3} + a^{2} b^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08754, size = 598, normalized size = 1.59 \begin{align*} -\frac{\sqrt{3}{\left (14 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - 35 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - 104 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + 65 \, \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 )}{27 \, b^{7}} + \frac{{\left (14 \, a b^{3} c - 35 \, a^{2} b^{2} d - 104 \, a^{4} f + 65 \, 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 )}{27 \, a b^{6}} - \frac{{\left (14 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - 35 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - 104 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + 65 \, \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 )}{54 \, b^{7}} + \frac{13 \, a b^{4} c x^{4} - 19 \, a^{2} b^{3} d x^{4} - 31 \, a^{4} b f x^{4} + 25 \, a^{3} b^{2} x^{4} e + 10 \, a^{2} b^{3} c x - 16 \, a^{3} b^{2} d x - 28 \, a^{5} f x + 22 \, a^{4} b x e}{18 \,{\left (b x^{3} + a\right )}^{2} b^{6}} + \frac{14 \, b^{27} f x^{10} - 60 \, a b^{26} f x^{7} + 20 \, b^{27} x^{7} e + 35 \, b^{27} d x^{4} + 210 \, a^{2} b^{25} f x^{4} - 105 \, a b^{26} x^{4} e + 140 \, b^{27} c x - 420 \, a b^{26} d x - 1400 \, a^{3} b^{24} f x + 840 \, a^{2} b^{25} x e}{140 \, b^{30}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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