Optimal. Leaf size=384 \[ \frac {x^5 \left (6 a^2 f-3 a b e+b^2 d\right )}{5 b^5}+\frac {x^2 \left (-10 a^3 f+6 a^2 b e-3 a b^2 d+b^3 c\right )}{2 b^6}+\frac {a x^2 \left (-16 a^3 f+13 a^2 b e-10 a b^2 d+7 b^3 c\right )}{9 b^6 \left (a+b x^3\right )}-\frac {a^2 x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^6 \left (a+b x^3\right )^2}-\frac {a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-119 a^3 f+77 a^2 b e-44 a b^2 d+20 b^3 c\right )}{54 b^{20/3}}+\frac {a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-119 a^3 f+77 a^2 b e-44 a b^2 d+20 b^3 c\right )}{27 b^{20/3}}+\frac {a^{2/3} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (-119 a^3 f+77 a^2 b e-44 a b^2 d+20 b^3 c\right )}{9 \sqrt {3} b^{20/3}}+\frac {x^8 (b e-3 a f)}{8 b^4}+\frac {f x^{11}}{11 b^3} \]
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Rubi [A] time = 1.05, antiderivative size = 384, normalized size of antiderivative = 1.00, number of steps used = 14, 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} \[ \frac {x^2 \left (6 a^2 b e-10 a^3 f-3 a b^2 d+b^3 c\right )}{2 b^6}+\frac {a x^2 \left (13 a^2 b e-16 a^3 f-10 a b^2 d+7 b^3 c\right )}{9 b^6 \left (a+b x^3\right )}-\frac {a^2 x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 b^6 \left (a+b x^3\right )^2}-\frac {a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (77 a^2 b e-119 a^3 f-44 a b^2 d+20 b^3 c\right )}{54 b^{20/3}}+\frac {a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (77 a^2 b e-119 a^3 f-44 a b^2 d+20 b^3 c\right )}{27 b^{20/3}}+\frac {a^{2/3} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (77 a^2 b e-119 a^3 f-44 a b^2 d+20 b^3 c\right )}{9 \sqrt {3} b^{20/3}}+\frac {x^5 \left (6 a^2 f-3 a b e+b^2 d\right )}{5 b^5}+\frac {x^8 (b e-3 a f)}{8 b^4}+\frac {f x^{11}}{11 b^3} \]
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 292
Rule 617
Rule 628
Rule 634
Rule 1488
Rule 1828
Rule 1836
Rule 1851
Rubi steps
\begin {align*} \int \frac {x^{10} \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^2}{6 b^6 \left (a+b x^3\right )^2}-\frac {\int \frac {-2 a^3 b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x+6 a^2 b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^4-6 a b^3 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^7-6 a b^4 \left (b^2 d-a b e+a^2 f\right ) x^{10}-6 a b^5 (b e-a f) x^{13}-6 a b^6 f x^{16}}{\left (a+b x^3\right )^2} \, dx}{6 a b^7}\\ &=-\frac {a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 b^6 \left (a+b x^3\right )^2}-\frac {\int \frac {x \left (-2 a^3 b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )+6 a^2 b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^3-6 a b^3 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^6-6 a b^4 \left (b^2 d-a b e+a^2 f\right ) x^9-6 a b^5 (b e-a f) x^{12}-6 a b^6 f x^{15}\right )}{\left (a+b x^3\right )^2} \, dx}{6 a b^7}\\ &=-\frac {a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 b^6 \left (a+b x^3\right )^2}+\frac {a \left (7 b^3 c-10 a b^2 d+13 a^2 b e-16 a^3 f\right ) x^2}{9 b^6 \left (a+b x^3\right )}+\frac {\int \frac {-2 a^3 b^7 \left (11 b^3 c-17 a b^2 d+23 a^2 b e-29 a^3 f\right ) x+18 a^2 b^8 \left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x^4+18 a^2 b^9 \left (b^2 d-2 a b e+3 a^2 f\right ) x^7+18 a^2 b^{10} (b e-2 a f) x^{10}+18 a^2 b^{11} f x^{13}}{a+b x^3} \, dx}{18 a^2 b^{13}}\\ &=-\frac {a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 b^6 \left (a+b x^3\right )^2}+\frac {a \left (7 b^3 c-10 a b^2 d+13 a^2 b e-16 a^3 f\right ) x^2}{9 b^6 \left (a+b x^3\right )}+\frac {\int \frac {x \left (-2 a^3 b^7 \left (11 b^3 c-17 a b^2 d+23 a^2 b e-29 a^3 f\right )+18 a^2 b^8 \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^9 \left (b^2 d-2 a b e+3 a^2 f\right ) x^6+18 a^2 b^{10} (b e-2 a f) x^9+18 a^2 b^{11} f x^{12}\right )}{a+b x^3} \, dx}{18 a^2 b^{13}}\\ &=\frac {f x^{11}}{11 b^3}-\frac {a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 b^6 \left (a+b x^3\right )^2}+\frac {a \left (7 b^3 c-10 a b^2 d+13 a^2 b e-16 a^3 f\right ) x^2}{9 b^6 \left (a+b x^3\right )}+\frac {\int \frac {x \left (-22 a^3 b^8 \left (11 b^3 c-17 a b^2 d+23 a^2 b e-29 a^3 f\right )+198 a^2 b^9 \left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x^3+198 a^2 b^{10} \left (b^2 d-2 a b e+3 a^2 f\right ) x^6+198 a^2 b^{11} (b e-3 a f) x^9\right )}{a+b x^3} \, dx}{198 a^2 b^{14}}\\ &=\frac {(b e-3 a f) x^8}{8 b^4}+\frac {f x^{11}}{11 b^3}-\frac {a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 b^6 \left (a+b x^3\right )^2}+\frac {a \left (7 b^3 c-10 a b^2 d+13 a^2 b e-16 a^3 f\right ) x^2}{9 b^6 \left (a+b x^3\right )}+\frac {\int \frac {x \left (-176 a^3 b^9 \left (11 b^3 c-17 a b^2 d+23 a^2 b e-29 a^3 f\right )+1584 a^2 b^{10} \left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x^3+1584 a^2 b^{11} \left (b^2 d-3 a b e+6 a^2 f\right ) x^6\right )}{a+b x^3} \, dx}{1584 a^2 b^{15}}\\ &=\frac {(b e-3 a f) x^8}{8 b^4}+\frac {f x^{11}}{11 b^3}-\frac {a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 b^6 \left (a+b x^3\right )^2}+\frac {a \left (7 b^3 c-10 a b^2 d+13 a^2 b e-16 a^3 f\right ) x^2}{9 b^6 \left (a+b x^3\right )}+\frac {\int \left (1584 a^2 b^9 \left (b^3 c-3 a b^2 d+6 a^2 b e-10 a^3 f\right ) x+1584 a^2 b^{10} \left (b^2 d-3 a b e+6 a^2 f\right ) x^4+\frac {176 \left (-20 a^3 b^{12} c+44 a^4 b^{11} d-77 a^5 b^{10} e+119 a^6 b^9 f\right ) x}{a+b x^3}\right ) \, dx}{1584 a^2 b^{15}}\\ &=\frac {\left (b^3 c-3 a b^2 d+6 a^2 b e-10 a^3 f\right ) x^2}{2 b^6}+\frac {\left (b^2 d-3 a b e+6 a^2 f\right ) x^5}{5 b^5}+\frac {(b e-3 a f) x^8}{8 b^4}+\frac {f x^{11}}{11 b^3}-\frac {a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 b^6 \left (a+b x^3\right )^2}+\frac {a \left (7 b^3 c-10 a b^2 d+13 a^2 b e-16 a^3 f\right ) x^2}{9 b^6 \left (a+b x^3\right )}-\frac {\left (a \left (20 b^3 c-44 a b^2 d+77 a^2 b e-119 a^3 f\right )\right ) \int \frac {x}{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^2}{2 b^6}+\frac {\left (b^2 d-3 a b e+6 a^2 f\right ) x^5}{5 b^5}+\frac {(b e-3 a f) x^8}{8 b^4}+\frac {f x^{11}}{11 b^3}-\frac {a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 b^6 \left (a+b x^3\right )^2}+\frac {a \left (7 b^3 c-10 a b^2 d+13 a^2 b e-16 a^3 f\right ) x^2}{9 b^6 \left (a+b x^3\right )}+\frac {\left (a^{2/3} \left (20 b^3 c-44 a b^2 d+77 a^2 b e-119 a^3 f\right )\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 b^{19/3}}-\frac {\left (a^{2/3} \left (20 b^3 c-44 a b^2 d+77 a^2 b e-119 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}{27 b^{19/3}}\\ &=\frac {\left (b^3 c-3 a b^2 d+6 a^2 b e-10 a^3 f\right ) x^2}{2 b^6}+\frac {\left (b^2 d-3 a b e+6 a^2 f\right ) x^5}{5 b^5}+\frac {(b e-3 a f) x^8}{8 b^4}+\frac {f x^{11}}{11 b^3}-\frac {a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 b^6 \left (a+b x^3\right )^2}+\frac {a \left (7 b^3 c-10 a b^2 d+13 a^2 b e-16 a^3 f\right ) x^2}{9 b^6 \left (a+b x^3\right )}+\frac {a^{2/3} \left (20 b^3 c-44 a b^2 d+77 a^2 b e-119 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 b^{20/3}}-\frac {\left (a^{2/3} \left (20 b^3 c-44 a b^2 d+77 a^2 b e-119 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^{20/3}}-\frac {\left (a \left (20 b^3 c-44 a b^2 d+77 a^2 b e-119 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^{19/3}}\\ &=\frac {\left (b^3 c-3 a b^2 d+6 a^2 b e-10 a^3 f\right ) x^2}{2 b^6}+\frac {\left (b^2 d-3 a b e+6 a^2 f\right ) x^5}{5 b^5}+\frac {(b e-3 a f) x^8}{8 b^4}+\frac {f x^{11}}{11 b^3}-\frac {a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 b^6 \left (a+b x^3\right )^2}+\frac {a \left (7 b^3 c-10 a b^2 d+13 a^2 b e-16 a^3 f\right ) x^2}{9 b^6 \left (a+b x^3\right )}+\frac {a^{2/3} \left (20 b^3 c-44 a b^2 d+77 a^2 b e-119 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 b^{20/3}}-\frac {a^{2/3} \left (20 b^3 c-44 a b^2 d+77 a^2 b e-119 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 b^{20/3}}-\frac {\left (a^{2/3} \left (20 b^3 c-44 a b^2 d+77 a^2 b e-119 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^{20/3}}\\ &=\frac {\left (b^3 c-3 a b^2 d+6 a^2 b e-10 a^3 f\right ) x^2}{2 b^6}+\frac {\left (b^2 d-3 a b e+6 a^2 f\right ) x^5}{5 b^5}+\frac {(b e-3 a f) x^8}{8 b^4}+\frac {f x^{11}}{11 b^3}-\frac {a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{6 b^6 \left (a+b x^3\right )^2}+\frac {a \left (7 b^3 c-10 a b^2 d+13 a^2 b e-16 a^3 f\right ) x^2}{9 b^6 \left (a+b x^3\right )}+\frac {a^{2/3} \left (20 b^3 c-44 a b^2 d+77 a^2 b e-119 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^{20/3}}+\frac {a^{2/3} \left (20 b^3 c-44 a b^2 d+77 a^2 b e-119 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 b^{20/3}}-\frac {a^{2/3} \left (20 b^3 c-44 a b^2 d+77 a^2 b e-119 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 b^{20/3}}\\ \end {align*}
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Mathematica [A] time = 0.55, size = 380, normalized size = 0.99 \[ \frac {x^5 \left (6 a^2 f-3 a b e+b^2 d\right )}{5 b^5}+\frac {x^2 \left (-10 a^3 f+6 a^2 b e-3 a b^2 d+b^3 c\right )}{2 b^6}+\frac {a x^2 \left (-16 a^3 f+13 a^2 b e-10 a b^2 d+7 b^3 c\right )}{9 b^6 \left (a+b x^3\right )}+\frac {a^2 x^2 \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{6 b^6 \left (a+b x^3\right )^2}+\frac {a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (119 a^3 f-77 a^2 b e+44 a b^2 d-20 b^3 c\right )}{54 b^{20/3}}-\frac {a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (119 a^3 f-77 a^2 b e+44 a b^2 d-20 b^3 c\right )}{27 b^{20/3}}-\frac {a^{2/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right ) \left (119 a^3 f-77 a^2 b e+44 a b^2 d-20 b^3 c\right )}{9 \sqrt {3} b^{20/3}}+\frac {x^8 (b e-3 a f)}{8 b^4}+\frac {f x^{11}}{11 b^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 634, normalized size = 1.65 \[ \frac {1080 \, b^{5} f x^{17} + 135 \, {\left (11 \, b^{5} e - 17 \, a b^{4} f\right )} x^{14} + 54 \, {\left (44 \, b^{5} d - 77 \, a b^{4} e + 119 \, a^{2} b^{3} f\right )} x^{11} + 297 \, {\left (20 \, b^{5} c - 44 \, a b^{4} d + 77 \, a^{2} b^{3} e - 119 \, a^{3} b^{2} f\right )} x^{8} + 1056 \, {\left (20 \, a b^{4} c - 44 \, a^{2} b^{3} d + 77 \, a^{3} b^{2} e - 119 \, a^{4} b f\right )} x^{5} + 660 \, {\left (20 \, a^{2} b^{3} c - 44 \, a^{3} b^{2} d + 77 \, a^{4} b e - 119 \, a^{5} f\right )} x^{2} - 440 \, \sqrt {3} {\left ({\left (20 \, b^{5} c - 44 \, a b^{4} d + 77 \, a^{2} b^{3} e - 119 \, a^{3} b^{2} f\right )} x^{6} + 20 \, a^{2} b^{3} c - 44 \, a^{3} b^{2} d + 77 \, a^{4} b e - 119 \, a^{5} f + 2 \, {\left (20 \, a b^{4} c - 44 \, a^{2} b^{3} d + 77 \, a^{3} b^{2} e - 119 \, a^{4} 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 ({\left (20 \, b^{5} c - 44 \, a b^{4} d + 77 \, a^{2} b^{3} e - 119 \, a^{3} b^{2} f\right )} x^{6} + 20 \, a^{2} b^{3} c - 44 \, a^{3} b^{2} d + 77 \, a^{4} b e - 119 \, a^{5} f + 2 \, {\left (20 \, a b^{4} c - 44 \, a^{2} b^{3} d + 77 \, a^{3} b^{2} e - 119 \, a^{4} 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 ({\left (20 \, b^{5} c - 44 \, a b^{4} d + 77 \, a^{2} b^{3} e - 119 \, a^{3} b^{2} f\right )} x^{6} + 20 \, a^{2} b^{3} c - 44 \, a^{3} b^{2} d + 77 \, a^{4} b e - 119 \, a^{5} f + 2 \, {\left (20 \, a b^{4} c - 44 \, a^{2} b^{3} d + 77 \, a^{3} b^{2} e - 119 \, a^{4} 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 )}{11880 \, {\left (b^{8} x^{6} + 2 \, a b^{7} x^{3} + a^{2} b^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 491, normalized size = 1.28 \[ \frac {{\left (20 \, a b^{3} c \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 44 \, a^{2} b^{2} d \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 119 \, a^{4} f \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 77 \, a^{3} b \left (-\frac {a}{b}\right )^{\frac {1}{3}} e\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a b^{6}} + \frac {\sqrt {3} {\left (20 \, \left (-a b^{2}\right )^{\frac {2}{3}} b^{3} c - 44 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b^{2} d - 119 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{3} f + 77 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{2} b e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, b^{8}} - \frac {{\left (20 \, \left (-a b^{2}\right )^{\frac {2}{3}} b^{3} c - 44 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b^{2} d - 119 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{3} f + 77 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{2} b e\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, b^{8}} + \frac {14 \, a b^{4} c x^{5} - 20 \, a^{2} b^{3} d x^{5} - 32 \, a^{4} b f x^{5} + 26 \, a^{3} b^{2} x^{5} e + 11 \, a^{2} b^{3} c x^{2} - 17 \, a^{3} b^{2} d x^{2} - 29 \, a^{5} f x^{2} + 23 \, a^{4} b x^{2} e}{18 \, {\left (b x^{3} + a\right )}^{2} b^{6}} + \frac {40 \, b^{30} f x^{11} - 165 \, a b^{29} f x^{8} + 55 \, b^{30} x^{8} e + 88 \, b^{30} d x^{5} + 528 \, a^{2} b^{28} f x^{5} - 264 \, a b^{29} x^{5} e + 220 \, b^{30} c x^{2} - 660 \, a b^{29} d x^{2} - 2200 \, a^{3} b^{27} f x^{2} + 1320 \, a^{2} b^{28} x^{2} e}{440 \, b^{33}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 668, normalized size = 1.74 \[ \frac {f \,x^{11}}{11 b^{3}}-\frac {3 a f \,x^{8}}{8 b^{4}}+\frac {e \,x^{8}}{8 b^{3}}-\frac {16 a^{4} f \,x^{5}}{9 \left (b \,x^{3}+a \right )^{2} b^{5}}+\frac {13 a^{3} e \,x^{5}}{9 \left (b \,x^{3}+a \right )^{2} b^{4}}-\frac {10 a^{2} d \,x^{5}}{9 \left (b \,x^{3}+a \right )^{2} b^{3}}+\frac {7 a c \,x^{5}}{9 \left (b \,x^{3}+a \right )^{2} b^{2}}+\frac {6 a^{2} f \,x^{5}}{5 b^{5}}-\frac {3 a e \,x^{5}}{5 b^{4}}+\frac {d \,x^{5}}{5 b^{3}}-\frac {29 a^{5} f \,x^{2}}{18 \left (b \,x^{3}+a \right )^{2} b^{6}}+\frac {23 a^{4} e \,x^{2}}{18 \left (b \,x^{3}+a \right )^{2} b^{5}}-\frac {17 a^{3} d \,x^{2}}{18 \left (b \,x^{3}+a \right )^{2} b^{4}}+\frac {11 a^{2} c \,x^{2}}{18 \left (b \,x^{3}+a \right )^{2} b^{3}}-\frac {5 a^{3} f \,x^{2}}{b^{6}}+\frac {3 a^{2} e \,x^{2}}{b^{5}}-\frac {3 a d \,x^{2}}{2 b^{4}}+\frac {c \,x^{2}}{2 b^{3}}+\frac {119 \sqrt {3}\, a^{4} f \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{7}}-\frac {119 a^{4} f \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{7}}+\frac {119 a^{4} f \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{7}}-\frac {77 \sqrt {3}\, a^{3} e \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{6}}+\frac {77 a^{3} e \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{6}}-\frac {77 a^{3} e \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{6}}+\frac {44 \sqrt {3}\, a^{2} d \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{5}}-\frac {44 a^{2} d \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{5}}+\frac {22 a^{2} d \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{5}}-\frac {20 \sqrt {3}\, a c \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{4}}+\frac {20 a c \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{4}}-\frac {10 a c \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.00, size = 380, normalized size = 0.99 \[ \frac {2 \, {\left (7 \, a b^{4} c - 10 \, a^{2} b^{3} d + 13 \, a^{3} b^{2} e - 16 \, a^{4} b f\right )} x^{5} + {\left (11 \, a^{2} b^{3} c - 17 \, a^{3} b^{2} d + 23 \, a^{4} b e - 29 \, a^{5} f\right )} x^{2}}{18 \, {\left (b^{8} x^{6} + 2 \, a b^{7} x^{3} + a^{2} b^{6}\right )}} - \frac {\sqrt {3} {\left (20 \, a b^{3} c - 44 \, a^{2} b^{2} d + 77 \, a^{3} b e - 119 \, a^{4} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, b^{7} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {40 \, b^{3} f x^{11} + 55 \, {\left (b^{3} e - 3 \, a b^{2} f\right )} x^{8} + 88 \, {\left (b^{3} d - 3 \, a b^{2} e + 6 \, a^{2} b f\right )} x^{5} + 220 \, {\left (b^{3} c - 3 \, a b^{2} d + 6 \, a^{2} b e - 10 \, a^{3} f\right )} x^{2}}{440 \, b^{6}} - \frac {{\left (20 \, a b^{3} c - 44 \, a^{2} b^{2} d + 77 \, a^{3} b e - 119 \, a^{4} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, b^{7} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {{\left (20 \, a b^{3} c - 44 \, a^{2} b^{2} d + 77 \, a^{3} b e - 119 \, a^{4} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, b^{7} \left (\frac {a}{b}\right )^{\frac {1}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.34, size = 425, normalized size = 1.11 \[ x^8\,\left (\frac {e}{8\,b^3}-\frac {3\,a\,f}{8\,b^4}\right )+x^2\,\left (\frac {c}{2\,b^3}-\frac {a^3\,f}{2\,b^6}-\frac {3\,a^2\,\left (\frac {e}{b^3}-\frac {3\,a\,f}{b^4}\right )}{2\,b^2}+\frac {3\,a\,\left (\frac {3\,a^2\,f}{b^5}-\frac {d}{b^3}+\frac {3\,a\,\left (\frac {e}{b^3}-\frac {3\,a\,f}{b^4}\right )}{b}\right )}{2\,b}\right )-\frac {\left (\frac {16\,f\,a^4\,b}{9}-\frac {13\,e\,a^3\,b^2}{9}+\frac {10\,d\,a^2\,b^3}{9}-\frac {7\,c\,a\,b^4}{9}\right )\,x^5+\left (\frac {29\,f\,a^5}{18}-\frac {23\,e\,a^4\,b}{18}+\frac {17\,d\,a^3\,b^2}{18}-\frac {11\,c\,a^2\,b^3}{18}\right )\,x^2}{a^2\,b^6+2\,a\,b^7\,x^3+b^8\,x^6}-x^5\,\left (\frac {3\,a^2\,f}{5\,b^5}-\frac {d}{5\,b^3}+\frac {3\,a\,\left (\frac {e}{b^3}-\frac {3\,a\,f}{b^4}\right )}{5\,b}\right )+\frac {f\,x^{11}}{11\,b^3}+\frac {a^{2/3}\,\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (-119\,f\,a^3+77\,e\,a^2\,b-44\,d\,a\,b^2+20\,c\,b^3\right )}{27\,b^{20/3}}-\frac {a^{2/3}\,\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-119\,f\,a^3+77\,e\,a^2\,b-44\,d\,a\,b^2+20\,c\,b^3\right )}{27\,b^{20/3}}+\frac {a^{2/3}\,\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-119\,f\,a^3+77\,e\,a^2\,b-44\,d\,a\,b^2+20\,c\,b^3\right )}{27\,b^{20/3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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