Optimal. Leaf size=380 \[ \frac {3 b c-a d}{8 a^4 x^8}-\frac {c}{11 a^3 x^{11}}-\frac {a^2 e-3 a b d+6 b^2 c}{5 a^5 x^5}-\frac {b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-20 a^3 f+44 a^2 b e-77 a b^2 d+119 b^3 c\right )}{54 a^{20/3}}+\frac {b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-20 a^3 f+44 a^2 b e-77 a b^2 d+119 b^3 c\right )}{27 a^{20/3}}-\frac {b^{2/3} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (-20 a^3 f+44 a^2 b e-77 a b^2 d+119 b^3 c\right )}{9 \sqrt {3} a^{20/3}}+\frac {b x \left (-11 a^3 f+17 a^2 b e-23 a b^2 d+29 b^3 c\right )}{18 a^6 \left (a+b x^3\right )}+\frac {a^3 (-f)+3 a^2 b e-6 a b^2 d+10 b^3 c}{2 a^6 x^2}+\frac {b x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^5 \left (a+b x^3\right )^2} \]
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Rubi [A] time = 0.67, antiderivative size = 380, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {1829, 1834, 200, 31, 634, 617, 204, 628} \[ \frac {b x \left (17 a^2 b e-11 a^3 f-23 a b^2 d+29 b^3 c\right )}{18 a^6 \left (a+b x^3\right )}+\frac {3 a^2 b e+a^3 (-f)-6 a b^2 d+10 b^3 c}{2 a^6 x^2}+\frac {b x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 a^5 \left (a+b x^3\right )^2}-\frac {b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (44 a^2 b e-20 a^3 f-77 a b^2 d+119 b^3 c\right )}{54 a^{20/3}}+\frac {b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (44 a^2 b e-20 a^3 f-77 a b^2 d+119 b^3 c\right )}{27 a^{20/3}}-\frac {b^{2/3} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (44 a^2 b e-20 a^3 f-77 a b^2 d+119 b^3 c\right )}{9 \sqrt {3} a^{20/3}}-\frac {a^2 e-3 a b d+6 b^2 c}{5 a^5 x^5}+\frac {3 b c-a d}{8 a^4 x^8}-\frac {c}{11 a^3 x^{11}} \]
Antiderivative was successfully verified.
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Rule 31
Rule 200
Rule 204
Rule 617
Rule 628
Rule 634
Rule 1829
Rule 1834
Rubi steps
\begin {align*} \int \frac {c+d x^3+e x^6+f x^9}{x^{12} \left (a+b x^3\right )^3} \, dx &=\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^5 \left (a+b x^3\right )^2}-\frac {\int \frac {-6 b^3 c+6 b^3 \left (\frac {b c}{a}-d\right ) x^3-\frac {6 b^3 \left (b^2 c-a b d+a^2 e\right ) x^6}{a^2}+\frac {6 b^3 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^9}{a^3}-\frac {5 b^4 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^{12}}{a^4}}{x^{12} \left (a+b x^3\right )^2} \, dx}{6 a b^3}\\ &=\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^5 \left (a+b x^3\right )^2}+\frac {b \left (29 b^3 c-23 a b^2 d+17 a^2 b e-11 a^3 f\right ) x}{18 a^6 \left (a+b x^3\right )}+\frac {\int \frac {18 b^7 c-18 b^7 \left (\frac {2 b c}{a}-d\right ) x^3+18 b^7 \left (\frac {3 b^2 c}{a^2}-\frac {2 b d}{a}+e\right ) x^6-18 b^7 \left (\frac {4 b^3 c}{a^3}-\frac {3 b^2 d}{a^2}+\frac {2 b e}{a}-f\right ) x^9+\frac {2 b^8 \left (29 b^3 c-23 a b^2 d+17 a^2 b e-11 a^3 f\right ) x^{12}}{a^4}}{x^{12} \left (a+b x^3\right )} \, dx}{18 a^2 b^7}\\ &=\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^5 \left (a+b x^3\right )^2}+\frac {b \left (29 b^3 c-23 a b^2 d+17 a^2 b e-11 a^3 f\right ) x}{18 a^6 \left (a+b x^3\right )}+\frac {\int \left (\frac {18 b^7 c}{a x^{12}}+\frac {18 b^7 (-3 b c+a d)}{a^2 x^9}+\frac {18 b^7 \left (6 b^2 c-3 a b d+a^2 e\right )}{a^3 x^6}+\frac {18 b^7 \left (-10 b^3 c+6 a b^2 d-3 a^2 b e+a^3 f\right )}{a^4 x^3}-\frac {2 b^8 \left (-119 b^3 c+77 a b^2 d-44 a^2 b e+20 a^3 f\right )}{a^4 \left (a+b x^3\right )}\right ) \, dx}{18 a^2 b^7}\\ &=-\frac {c}{11 a^3 x^{11}}+\frac {3 b c-a d}{8 a^4 x^8}-\frac {6 b^2 c-3 a b d+a^2 e}{5 a^5 x^5}+\frac {10 b^3 c-6 a b^2 d+3 a^2 b e-a^3 f}{2 a^6 x^2}+\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^5 \left (a+b x^3\right )^2}+\frac {b \left (29 b^3 c-23 a b^2 d+17 a^2 b e-11 a^3 f\right ) x}{18 a^6 \left (a+b x^3\right )}+\frac {\left (b \left (119 b^3 c-77 a b^2 d+44 a^2 b e-20 a^3 f\right )\right ) \int \frac {1}{a+b x^3} \, dx}{9 a^6}\\ &=-\frac {c}{11 a^3 x^{11}}+\frac {3 b c-a d}{8 a^4 x^8}-\frac {6 b^2 c-3 a b d+a^2 e}{5 a^5 x^5}+\frac {10 b^3 c-6 a b^2 d+3 a^2 b e-a^3 f}{2 a^6 x^2}+\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^5 \left (a+b x^3\right )^2}+\frac {b \left (29 b^3 c-23 a b^2 d+17 a^2 b e-11 a^3 f\right ) x}{18 a^6 \left (a+b x^3\right )}+\frac {\left (b \left (119 b^3 c-77 a b^2 d+44 a^2 b e-20 a^3 f\right )\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{20/3}}+\frac {\left (b \left (119 b^3 c-77 a b^2 d+44 a^2 b e-20 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 a^{20/3}}\\ &=-\frac {c}{11 a^3 x^{11}}+\frac {3 b c-a d}{8 a^4 x^8}-\frac {6 b^2 c-3 a b d+a^2 e}{5 a^5 x^5}+\frac {10 b^3 c-6 a b^2 d+3 a^2 b e-a^3 f}{2 a^6 x^2}+\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^5 \left (a+b x^3\right )^2}+\frac {b \left (29 b^3 c-23 a b^2 d+17 a^2 b e-11 a^3 f\right ) x}{18 a^6 \left (a+b x^3\right )}+\frac {b^{2/3} \left (119 b^3 c-77 a b^2 d+44 a^2 b e-20 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{20/3}}-\frac {\left (b^{2/3} \left (119 b^3 c-77 a b^2 d+44 a^2 b e-20 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 a^{20/3}}+\frac {\left (b \left (119 b^3 c-77 a b^2 d+44 a^2 b e-20 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 a^{19/3}}\\ &=-\frac {c}{11 a^3 x^{11}}+\frac {3 b c-a d}{8 a^4 x^8}-\frac {6 b^2 c-3 a b d+a^2 e}{5 a^5 x^5}+\frac {10 b^3 c-6 a b^2 d+3 a^2 b e-a^3 f}{2 a^6 x^2}+\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^5 \left (a+b x^3\right )^2}+\frac {b \left (29 b^3 c-23 a b^2 d+17 a^2 b e-11 a^3 f\right ) x}{18 a^6 \left (a+b x^3\right )}+\frac {b^{2/3} \left (119 b^3 c-77 a b^2 d+44 a^2 b e-20 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{20/3}}-\frac {b^{2/3} \left (119 b^3 c-77 a b^2 d+44 a^2 b e-20 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{20/3}}+\frac {\left (b^{2/3} \left (119 b^3 c-77 a b^2 d+44 a^2 b e-20 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 a^{20/3}}\\ &=-\frac {c}{11 a^3 x^{11}}+\frac {3 b c-a d}{8 a^4 x^8}-\frac {6 b^2 c-3 a b d+a^2 e}{5 a^5 x^5}+\frac {10 b^3 c-6 a b^2 d+3 a^2 b e-a^3 f}{2 a^6 x^2}+\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^5 \left (a+b x^3\right )^2}+\frac {b \left (29 b^3 c-23 a b^2 d+17 a^2 b e-11 a^3 f\right ) x}{18 a^6 \left (a+b x^3\right )}-\frac {b^{2/3} \left (119 b^3 c-77 a b^2 d+44 a^2 b e-20 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} a^{20/3}}+\frac {b^{2/3} \left (119 b^3 c-77 a b^2 d+44 a^2 b e-20 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{20/3}}-\frac {b^{2/3} \left (119 b^3 c-77 a b^2 d+44 a^2 b e-20 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{20/3}}\\ \end {align*}
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Mathematica [A] time = 0.58, size = 376, normalized size = 0.99 \[ \frac {3 b c-a d}{8 a^4 x^8}-\frac {c}{11 a^3 x^{11}}-\frac {a^2 e-3 a b d+6 b^2 c}{5 a^5 x^5}+\frac {b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (20 a^3 f-44 a^2 b e+77 a b^2 d-119 b^3 c\right )}{54 a^{20/3}}+\frac {b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-20 a^3 f+44 a^2 b e-77 a b^2 d+119 b^3 c\right )}{27 a^{20/3}}+\frac {b^{2/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right ) \left (20 a^3 f-44 a^2 b e+77 a b^2 d-119 b^3 c\right )}{9 \sqrt {3} a^{20/3}}+\frac {b x \left (-11 a^3 f+17 a^2 b e-23 a b^2 d+29 b^3 c\right )}{18 a^6 \left (a+b x^3\right )}+\frac {a^3 (-f)+3 a^2 b e-6 a b^2 d+10 b^3 c}{2 a^6 x^2}+\frac {b x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^5 \left (a+b x^3\right )^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 654, normalized size = 1.72 \[ \frac {660 \, {\left (119 \, b^{5} c - 77 \, a b^{4} d + 44 \, a^{2} b^{3} e - 20 \, a^{3} b^{2} f\right )} x^{15} + 1056 \, {\left (119 \, a b^{4} c - 77 \, a^{2} b^{3} d + 44 \, a^{3} b^{2} e - 20 \, a^{4} b f\right )} x^{12} + 297 \, {\left (119 \, a^{2} b^{3} c - 77 \, a^{3} b^{2} d + 44 \, a^{4} b e - 20 \, a^{5} f\right )} x^{9} - 54 \, {\left (119 \, a^{3} b^{2} c - 77 \, a^{4} b d + 44 \, a^{5} e\right )} x^{6} - 1080 \, a^{5} c + 135 \, {\left (17 \, a^{4} b c - 11 \, a^{5} d\right )} x^{3} - 440 \, \sqrt {3} {\left ({\left (119 \, b^{5} c - 77 \, a b^{4} d + 44 \, a^{2} b^{3} e - 20 \, a^{3} b^{2} f\right )} x^{17} + 2 \, {\left (119 \, a b^{4} c - 77 \, a^{2} b^{3} d + 44 \, a^{3} b^{2} e - 20 \, a^{4} b f\right )} x^{14} + {\left (119 \, a^{2} b^{3} c - 77 \, a^{3} b^{2} d + 44 \, a^{4} b e - 20 \, a^{5} f\right )} x^{11}\right )} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} a x \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}} - \sqrt {3} b}{3 \, b}\right ) + 220 \, {\left ({\left (119 \, b^{5} c - 77 \, a b^{4} d + 44 \, a^{2} b^{3} e - 20 \, a^{3} b^{2} f\right )} x^{17} + 2 \, {\left (119 \, a b^{4} c - 77 \, a^{2} b^{3} d + 44 \, a^{3} b^{2} e - 20 \, a^{4} b f\right )} x^{14} + {\left (119 \, a^{2} b^{3} c - 77 \, a^{3} b^{2} d + 44 \, a^{4} b e - 20 \, a^{5} f\right )} x^{11}\right )} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b^{2} x^{2} + a b x \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} + a^{2} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}}\right ) - 440 \, {\left ({\left (119 \, b^{5} c - 77 \, a b^{4} d + 44 \, a^{2} b^{3} e - 20 \, a^{3} b^{2} f\right )} x^{17} + 2 \, {\left (119 \, a b^{4} c - 77 \, a^{2} b^{3} d + 44 \, a^{3} b^{2} e - 20 \, a^{4} b f\right )} x^{14} + {\left (119 \, a^{2} b^{3} c - 77 \, a^{3} b^{2} d + 44 \, a^{4} b e - 20 \, a^{5} f\right )} x^{11}\right )} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b x - a \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}}\right )}{11880 \, {\left (a^{6} b^{2} x^{17} + 2 \, a^{7} b x^{14} + a^{8} x^{11}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 440, normalized size = 1.16 \[ \frac {\sqrt {3} {\left (119 \, \left (-a b^{2}\right )^{\frac {1}{3}} b^{3} c - 77 \, \left (-a b^{2}\right )^{\frac {1}{3}} a b^{2} d - 20 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{3} f + 44 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{2} b e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{7}} - \frac {{\left (119 \, b^{4} c - 77 \, a b^{3} d - 20 \, a^{3} b f + 44 \, a^{2} b^{2} e\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a^{7}} + \frac {{\left (119 \, \left (-a b^{2}\right )^{\frac {1}{3}} b^{3} c - 77 \, \left (-a b^{2}\right )^{\frac {1}{3}} a b^{2} d - 20 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{3} f + 44 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{2} b e\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a^{7}} + \frac {29 \, b^{5} c x^{4} - 23 \, a b^{4} d x^{4} - 11 \, a^{3} b^{2} f x^{4} + 17 \, a^{2} b^{3} x^{4} e + 32 \, a b^{4} c x - 26 \, a^{2} b^{3} d x - 14 \, a^{4} b f x + 20 \, a^{3} b^{2} x e}{18 \, {\left (b x^{3} + a\right )}^{2} a^{6}} + \frac {2200 \, b^{3} c x^{9} - 1320 \, a b^{2} d x^{9} - 220 \, a^{3} f x^{9} + 660 \, a^{2} b x^{9} e - 528 \, a b^{2} c x^{6} + 264 \, a^{2} b d x^{6} - 88 \, a^{3} x^{6} e + 165 \, a^{2} b c x^{3} - 55 \, a^{3} d x^{3} - 40 \, a^{3} c}{440 \, a^{6} x^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 651, normalized size = 1.71 \[ -\frac {11 b^{2} f \,x^{4}}{18 \left (b \,x^{3}+a \right )^{2} a^{3}}+\frac {17 b^{3} e \,x^{4}}{18 \left (b \,x^{3}+a \right )^{2} a^{4}}-\frac {23 b^{4} d \,x^{4}}{18 \left (b \,x^{3}+a \right )^{2} a^{5}}+\frac {29 b^{5} c \,x^{4}}{18 \left (b \,x^{3}+a \right )^{2} a^{6}}-\frac {7 b f x}{9 \left (b \,x^{3}+a \right )^{2} a^{2}}+\frac {10 b^{2} e x}{9 \left (b \,x^{3}+a \right )^{2} a^{3}}-\frac {13 b^{3} d x}{9 \left (b \,x^{3}+a \right )^{2} a^{4}}+\frac {16 b^{4} c x}{9 \left (b \,x^{3}+a \right )^{2} a^{5}}-\frac {20 \sqrt {3}\, 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 {2}{3}} a^{3}}-\frac {20 f \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{3}}+\frac {10 f \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 {2}{3}} a^{3}}+\frac {44 \sqrt {3}\, b 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 {2}{3}} a^{4}}+\frac {44 b e \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{4}}-\frac {22 b e \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 {2}{3}} a^{4}}-\frac {77 \sqrt {3}\, b^{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 {2}{3}} a^{5}}-\frac {77 b^{2} d \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{5}}+\frac {77 b^{2} d \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 {2}{3}} a^{5}}+\frac {119 \sqrt {3}\, b^{3} 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 {2}{3}} a^{6}}+\frac {119 b^{3} c \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{6}}-\frac {119 b^{3} c \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 {2}{3}} a^{6}}-\frac {f}{2 a^{3} x^{2}}+\frac {3 b e}{2 a^{4} x^{2}}-\frac {3 b^{2} d}{a^{5} x^{2}}+\frac {5 b^{3} c}{a^{6} x^{2}}-\frac {e}{5 a^{3} x^{5}}+\frac {3 b d}{5 a^{4} x^{5}}-\frac {6 b^{2} c}{5 a^{5} x^{5}}-\frac {d}{8 a^{3} x^{8}}+\frac {3 b c}{8 a^{4} x^{8}}-\frac {c}{11 a^{3} x^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.29, size = 376, normalized size = 0.99 \[ \frac {220 \, {\left (119 \, b^{5} c - 77 \, a b^{4} d + 44 \, a^{2} b^{3} e - 20 \, a^{3} b^{2} f\right )} x^{15} + 352 \, {\left (119 \, a b^{4} c - 77 \, a^{2} b^{3} d + 44 \, a^{3} b^{2} e - 20 \, a^{4} b f\right )} x^{12} + 99 \, {\left (119 \, a^{2} b^{3} c - 77 \, a^{3} b^{2} d + 44 \, a^{4} b e - 20 \, a^{5} f\right )} x^{9} - 18 \, {\left (119 \, a^{3} b^{2} c - 77 \, a^{4} b d + 44 \, a^{5} e\right )} x^{6} - 360 \, a^{5} c + 45 \, {\left (17 \, a^{4} b c - 11 \, a^{5} d\right )} x^{3}}{3960 \, {\left (a^{6} b^{2} x^{17} + 2 \, a^{7} b x^{14} + a^{8} x^{11}\right )}} + \frac {\sqrt {3} {\left (119 \, b^{3} c - 77 \, a b^{2} d + 44 \, a^{2} b e - 20 \, a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{6} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (119 \, b^{3} c - 77 \, a b^{2} d + 44 \, a^{2} b e - 20 \, a^{3} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a^{6} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (119 \, b^{3} c - 77 \, a b^{2} d + 44 \, a^{2} b e - 20 \, a^{3} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, a^{6} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.18, size = 359, normalized size = 0.94 \[ \frac {b^{2/3}\,\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (-20\,f\,a^3+44\,e\,a^2\,b-77\,d\,a\,b^2+119\,c\,b^3\right )}{27\,a^{20/3}}-\frac {\frac {c}{11\,a}-\frac {x^9\,\left (-20\,f\,a^3+44\,e\,a^2\,b-77\,d\,a\,b^2+119\,c\,b^3\right )}{40\,a^4}+\frac {x^3\,\left (11\,a\,d-17\,b\,c\right )}{88\,a^2}+\frac {x^6\,\left (44\,e\,a^2-77\,d\,a\,b+119\,c\,b^2\right )}{220\,a^3}-\frac {4\,b\,x^{12}\,\left (-20\,f\,a^3+44\,e\,a^2\,b-77\,d\,a\,b^2+119\,c\,b^3\right )}{45\,a^5}-\frac {b^2\,x^{15}\,\left (-20\,f\,a^3+44\,e\,a^2\,b-77\,d\,a\,b^2+119\,c\,b^3\right )}{18\,a^6}}{a^2\,x^{11}+2\,a\,b\,x^{14}+b^2\,x^{17}}+\frac {b^{2/3}\,\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-20\,f\,a^3+44\,e\,a^2\,b-77\,d\,a\,b^2+119\,c\,b^3\right )}{27\,a^{20/3}}-\frac {b^{2/3}\,\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-20\,f\,a^3+44\,e\,a^2\,b-77\,d\,a\,b^2+119\,c\,b^3\right )}{27\,a^{20/3}} \]
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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