Optimal. Leaf size=298 \[ \frac {x^2 \left (3 a^2 f-2 a b e+b^2 d\right )}{2 b^4}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-11 a^3 f+8 a^2 b e-5 a b^2 d+2 b^3 c\right )}{9 \sqrt [3]{a} b^{14/3}}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (-11 a^3 f+8 a^2 b e-5 a b^2 d+2 b^3 c\right )}{3 \sqrt {3} \sqrt [3]{a} b^{14/3}}-\frac {x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^4 \left (a+b x^3\right )}+\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-11 a^3 f+8 a^2 b e-5 a b^2 d+2 b^3 c\right )}{18 \sqrt [3]{a} b^{14/3}}+\frac {x^5 (b e-2 a f)}{5 b^3}+\frac {f x^8}{8 b^2} \]
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Rubi [A] time = 0.46, antiderivative size = 298, normalized size of antiderivative = 1.00, number of steps used = 11, 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} \[ -\frac {x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 b^4 \left (a+b x^3\right )}+\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (8 a^2 b e-11 a^3 f-5 a b^2 d+2 b^3 c\right )}{18 \sqrt [3]{a} b^{14/3}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (8 a^2 b e-11 a^3 f-5 a b^2 d+2 b^3 c\right )}{9 \sqrt [3]{a} b^{14/3}}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (8 a^2 b e-11 a^3 f-5 a b^2 d+2 b^3 c\right )}{3 \sqrt {3} \sqrt [3]{a} b^{14/3}}+\frac {x^2 \left (3 a^2 f-2 a b e+b^2 d\right )}{2 b^4}+\frac {x^5 (b e-2 a f)}{5 b^3}+\frac {f x^8}{8 b^2} \]
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^4 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^2} \, dx &=-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 b^4 \left (a+b x^3\right )}-\frac {\int \frac {-2 a b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x-3 a b^2 \left (b^2 d-a b e+a^2 f\right ) x^4-3 a b^3 (b e-a f) x^7-3 a b^4 f x^{10}}{a+b x^3} \, dx}{3 a b^5}\\ &=-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 b^4 \left (a+b x^3\right )}-\frac {\int \frac {x \left (-2 a b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )-3 a b^2 \left (b^2 d-a b e+a^2 f\right ) x^3-3 a b^3 (b e-a f) x^6-3 a b^4 f x^9\right )}{a+b x^3} \, dx}{3 a b^5}\\ &=\frac {f x^8}{8 b^2}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 b^4 \left (a+b x^3\right )}-\frac {\int \frac {x \left (-16 a b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )-24 a b^3 \left (b^2 d-a b e+a^2 f\right ) x^3-24 a b^4 (b e-2 a f) x^6\right )}{a+b x^3} \, dx}{24 a b^6}\\ &=\frac {f x^8}{8 b^2}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 b^4 \left (a+b x^3\right )}-\frac {\int \left (-24 a b^2 \left (b^2 d-2 a b e+3 a^2 f\right ) x-24 a b^3 (b e-2 a f) x^4+\frac {8 \left (-2 a b^5 c+5 a^2 b^4 d-8 a^3 b^3 e+11 a^4 b^2 f\right ) x}{a+b x^3}\right ) \, dx}{24 a b^6}\\ &=\frac {\left (b^2 d-2 a b e+3 a^2 f\right ) x^2}{2 b^4}+\frac {(b e-2 a f) x^5}{5 b^3}+\frac {f x^8}{8 b^2}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 b^4 \left (a+b x^3\right )}+\frac {\left (2 b^3 c-5 a b^2 d+8 a^2 b e-11 a^3 f\right ) \int \frac {x}{a+b x^3} \, dx}{3 b^4}\\ &=\frac {\left (b^2 d-2 a b e+3 a^2 f\right ) x^2}{2 b^4}+\frac {(b e-2 a f) x^5}{5 b^3}+\frac {f x^8}{8 b^2}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 b^4 \left (a+b x^3\right )}-\frac {\left (2 b^3 c-5 a b^2 d+8 a^2 b e-11 a^3 f\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 \sqrt [3]{a} b^{13/3}}+\frac {\left (2 b^3 c-5 a b^2 d+8 a^2 b e-11 a^3 f\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 \sqrt [3]{a} b^{13/3}}\\ &=\frac {\left (b^2 d-2 a b e+3 a^2 f\right ) x^2}{2 b^4}+\frac {(b e-2 a f) x^5}{5 b^3}+\frac {f x^8}{8 b^2}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 b^4 \left (a+b x^3\right )}-\frac {\left (2 b^3 c-5 a b^2 d+8 a^2 b e-11 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 \sqrt [3]{a} b^{14/3}}+\frac {\left (2 b^3 c-5 a b^2 d+8 a^2 b e-11 a^3 f\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 \sqrt [3]{a} b^{14/3}}+\frac {\left (2 b^3 c-5 a b^2 d+8 a^2 b e-11 a^3 f\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 b^{13/3}}\\ &=\frac {\left (b^2 d-2 a b e+3 a^2 f\right ) x^2}{2 b^4}+\frac {(b e-2 a f) x^5}{5 b^3}+\frac {f x^8}{8 b^2}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 b^4 \left (a+b x^3\right )}-\frac {\left (2 b^3 c-5 a b^2 d+8 a^2 b e-11 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 \sqrt [3]{a} b^{14/3}}+\frac {\left (2 b^3 c-5 a b^2 d+8 a^2 b e-11 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 \sqrt [3]{a} b^{14/3}}+\frac {\left (2 b^3 c-5 a b^2 d+8 a^2 b e-11 a^3 f\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 \sqrt [3]{a} b^{14/3}}\\ &=\frac {\left (b^2 d-2 a b e+3 a^2 f\right ) x^2}{2 b^4}+\frac {(b e-2 a f) x^5}{5 b^3}+\frac {f x^8}{8 b^2}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 b^4 \left (a+b x^3\right )}-\frac {\left (2 b^3 c-5 a b^2 d+8 a^2 b e-11 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} \sqrt [3]{a} b^{14/3}}-\frac {\left (2 b^3 c-5 a b^2 d+8 a^2 b e-11 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 \sqrt [3]{a} b^{14/3}}+\frac {\left (2 b^3 c-5 a b^2 d+8 a^2 b e-11 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 \sqrt [3]{a} b^{14/3}}\\ \end {align*}
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Mathematica [A] time = 0.30, size = 282, normalized size = 0.95 \[ \frac {180 b^{2/3} x^2 \left (3 a^2 f-2 a b e+b^2 d\right )+\frac {40 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (11 a^3 f-8 a^2 b e+5 a b^2 d-2 b^3 c\right )}{\sqrt [3]{a}}+\frac {40 \sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right ) \left (11 a^3 f-8 a^2 b e+5 a b^2 d-2 b^3 c\right )}{\sqrt [3]{a}}-\frac {120 b^{2/3} x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{a+b x^3}+\frac {20 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-11 a^3 f+8 a^2 b e-5 a b^2 d+2 b^3 c\right )}{\sqrt [3]{a}}+72 b^{5/3} x^5 (b e-2 a f)+45 b^{8/3} f x^8}{360 b^{14/3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 920, normalized size = 3.09 \[ \left [\frac {45 \, a b^{5} f x^{11} + 9 \, {\left (8 \, a b^{5} e - 11 \, a^{2} b^{4} f\right )} x^{8} + 36 \, {\left (5 \, a b^{5} d - 8 \, a^{2} b^{4} e + 11 \, a^{3} b^{3} f\right )} x^{5} - 60 \, {\left (2 \, a b^{5} c - 5 \, a^{2} b^{4} d + 8 \, a^{3} b^{3} e - 11 \, a^{4} b^{2} f\right )} x^{2} - 60 \, \sqrt {\frac {1}{3}} {\left (2 \, a^{2} b^{4} c - 5 \, a^{3} b^{3} d + 8 \, a^{4} b^{2} e - 11 \, a^{5} b f + {\left (2 \, a b^{5} c - 5 \, a^{2} b^{4} d + 8 \, a^{3} b^{3} e - 11 \, a^{4} b^{2} f\right )} x^{3}\right )} \sqrt {-\frac {\left (a b^{2}\right )^{\frac {1}{3}}}{a}} \log \left (\frac {2 \, b^{2} x^{3} - a b - 3 \, \sqrt {\frac {1}{3}} {\left (a b x + 2 \, \left (a b^{2}\right )^{\frac {2}{3}} x^{2} - \left (a b^{2}\right )^{\frac {1}{3}} a\right )} \sqrt {-\frac {\left (a b^{2}\right )^{\frac {1}{3}}}{a}} - 3 \, \left (a b^{2}\right )^{\frac {2}{3}} x}{b x^{3} + a}\right ) + 20 \, {\left (2 \, a b^{3} c - 5 \, a^{2} b^{2} d + 8 \, a^{3} b e - 11 \, a^{4} f + {\left (2 \, b^{4} c - 5 \, a b^{3} d + 8 \, a^{2} b^{2} e - 11 \, a^{3} b f\right )} x^{3}\right )} \left (a b^{2}\right )^{\frac {2}{3}} \log \left (b^{2} x^{2} - \left (a b^{2}\right )^{\frac {1}{3}} b x + \left (a b^{2}\right )^{\frac {2}{3}}\right ) - 40 \, {\left (2 \, a b^{3} c - 5 \, a^{2} b^{2} d + 8 \, a^{3} b e - 11 \, a^{4} f + {\left (2 \, b^{4} c - 5 \, a b^{3} d + 8 \, a^{2} b^{2} e - 11 \, a^{3} b f\right )} x^{3}\right )} \left (a b^{2}\right )^{\frac {2}{3}} \log \left (b x + \left (a b^{2}\right )^{\frac {1}{3}}\right )}{360 \, {\left (a b^{7} x^{3} + a^{2} b^{6}\right )}}, \frac {45 \, a b^{5} f x^{11} + 9 \, {\left (8 \, a b^{5} e - 11 \, a^{2} b^{4} f\right )} x^{8} + 36 \, {\left (5 \, a b^{5} d - 8 \, a^{2} b^{4} e + 11 \, a^{3} b^{3} f\right )} x^{5} - 60 \, {\left (2 \, a b^{5} c - 5 \, a^{2} b^{4} d + 8 \, a^{3} b^{3} e - 11 \, a^{4} b^{2} f\right )} x^{2} - 120 \, \sqrt {\frac {1}{3}} {\left (2 \, a^{2} b^{4} c - 5 \, a^{3} b^{3} d + 8 \, a^{4} b^{2} e - 11 \, a^{5} b f + {\left (2 \, a b^{5} c - 5 \, a^{2} b^{4} d + 8 \, a^{3} b^{3} e - 11 \, a^{4} b^{2} f\right )} x^{3}\right )} \sqrt {\frac {\left (a b^{2}\right )^{\frac {1}{3}}}{a}} \arctan \left (-\frac {\sqrt {\frac {1}{3}} {\left (2 \, b x - \left (a b^{2}\right )^{\frac {1}{3}}\right )} \sqrt {\frac {\left (a b^{2}\right )^{\frac {1}{3}}}{a}}}{b}\right ) + 20 \, {\left (2 \, a b^{3} c - 5 \, a^{2} b^{2} d + 8 \, a^{3} b e - 11 \, a^{4} f + {\left (2 \, b^{4} c - 5 \, a b^{3} d + 8 \, a^{2} b^{2} e - 11 \, a^{3} b f\right )} x^{3}\right )} \left (a b^{2}\right )^{\frac {2}{3}} \log \left (b^{2} x^{2} - \left (a b^{2}\right )^{\frac {1}{3}} b x + \left (a b^{2}\right )^{\frac {2}{3}}\right ) - 40 \, {\left (2 \, a b^{3} c - 5 \, a^{2} b^{2} d + 8 \, a^{3} b e - 11 \, a^{4} f + {\left (2 \, b^{4} c - 5 \, a b^{3} d + 8 \, a^{2} b^{2} e - 11 \, a^{3} b f\right )} x^{3}\right )} \left (a b^{2}\right )^{\frac {2}{3}} \log \left (b x + \left (a b^{2}\right )^{\frac {1}{3}}\right )}{360 \, {\left (a b^{7} x^{3} + a^{2} b^{6}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 344, normalized size = 1.15 \[ \frac {\sqrt {3} {\left (2 \, b^{3} c - 5 \, a b^{2} d - 11 \, a^{3} f + 8 \, a^{2} b e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, \left (-a b^{2}\right )^{\frac {1}{3}} b^{4}} - \frac {{\left (2 \, b^{3} c - 5 \, a b^{2} d - 11 \, a^{3} f + 8 \, a^{2} b e\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, \left (-a b^{2}\right )^{\frac {1}{3}} b^{4}} - \frac {{\left (2 \, b^{3} c \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 5 \, a b^{2} d \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 11 \, a^{3} f \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 8 \, a^{2} b \left (-\frac {a}{b}\right )^{\frac {1}{3}} e\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a b^{4}} - \frac {b^{3} c x^{2} - a b^{2} d x^{2} - a^{3} f x^{2} + a^{2} b x^{2} e}{3 \, {\left (b x^{3} + a\right )} b^{4}} + \frac {5 \, b^{14} f x^{8} - 16 \, a b^{13} f x^{5} + 8 \, b^{14} x^{5} e + 20 \, b^{14} d x^{2} + 60 \, a^{2} b^{12} f x^{2} - 40 \, a b^{13} x^{2} e}{40 \, b^{16}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 529, normalized size = 1.78 \[ \frac {f \,x^{8}}{8 b^{2}}-\frac {2 a f \,x^{5}}{5 b^{3}}+\frac {e \,x^{5}}{5 b^{2}}+\frac {a^{3} f \,x^{2}}{3 \left (b \,x^{3}+a \right ) b^{4}}-\frac {a^{2} e \,x^{2}}{3 \left (b \,x^{3}+a \right ) b^{3}}+\frac {a d \,x^{2}}{3 \left (b \,x^{3}+a \right ) b^{2}}-\frac {c \,x^{2}}{3 \left (b \,x^{3}+a \right ) b}+\frac {3 a^{2} f \,x^{2}}{2 b^{4}}-\frac {a e \,x^{2}}{b^{3}}+\frac {d \,x^{2}}{2 b^{2}}-\frac {11 \sqrt {3}\, a^{3} f \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{5}}+\frac {11 a^{3} f \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{5}}-\frac {11 a^{3} f \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{5}}+\frac {8 \sqrt {3}\, a^{2} e \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{4}}-\frac {8 a^{2} e \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{4}}+\frac {4 a^{2} e \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{4}}-\frac {5 \sqrt {3}\, a d \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{3}}+\frac {5 a d \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{3}}-\frac {5 a d \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{3}}+\frac {2 \sqrt {3}\, c \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{2}}-\frac {2 c \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{2}}+\frac {c \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.08, size = 277, normalized size = 0.93 \[ -\frac {{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{2}}{3 \, {\left (b^{5} x^{3} + a b^{4}\right )}} + \frac {\sqrt {3} {\left (2 \, b^{3} c - 5 \, a b^{2} d + 8 \, a^{2} b e - 11 \, a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, b^{5} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {5 \, b^{2} f x^{8} + 8 \, {\left (b^{2} e - 2 \, a b f\right )} x^{5} + 20 \, {\left (b^{2} d - 2 \, a b e + 3 \, a^{2} f\right )} x^{2}}{40 \, b^{4}} + \frac {{\left (2 \, b^{3} c - 5 \, a b^{2} d + 8 \, a^{2} b e - 11 \, a^{3} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, b^{5} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {{\left (2 \, b^{3} c - 5 \, a b^{2} d + 8 \, a^{2} b e - 11 \, a^{3} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, b^{5} \left (\frac {a}{b}\right )^{\frac {1}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.22, size = 287, normalized size = 0.96 \[ x^5\,\left (\frac {e}{5\,b^2}-\frac {2\,a\,f}{5\,b^3}\right )-x^2\,\left (\frac {a^2\,f}{2\,b^4}-\frac {d}{2\,b^2}+\frac {a\,\left (\frac {e}{b^2}-\frac {2\,a\,f}{b^3}\right )}{b}\right )+\frac {f\,x^8}{8\,b^2}-\frac {x^2\,\left (-\frac {f\,a^3}{3}+\frac {e\,a^2\,b}{3}-\frac {d\,a\,b^2}{3}+\frac {c\,b^3}{3}\right )}{b^5\,x^3+a\,b^4}-\frac {\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (-11\,f\,a^3+8\,e\,a^2\,b-5\,d\,a\,b^2+2\,c\,b^3\right )}{9\,a^{1/3}\,b^{14/3}}+\frac {\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-11\,f\,a^3+8\,e\,a^2\,b-5\,d\,a\,b^2+2\,c\,b^3\right )}{9\,a^{1/3}\,b^{14/3}}-\frac {\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-11\,f\,a^3+8\,e\,a^2\,b-5\,d\,a\,b^2+2\,c\,b^3\right )}{9\,a^{1/3}\,b^{14/3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 51.29, size = 490, normalized size = 1.64 \[ x^{5} \left (- \frac {2 a f}{5 b^{3}} + \frac {e}{5 b^{2}}\right ) + x^{2} \left (\frac {3 a^{2} f}{2 b^{4}} - \frac {a e}{b^{3}} + \frac {d}{2 b^{2}}\right ) + \frac {x^{2} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{3 a b^{4} + 3 b^{5} x^{3}} + \operatorname {RootSum} {\left (729 t^{3} a b^{14} - 1331 a^{9} f^{3} + 2904 a^{8} b e f^{2} - 1815 a^{7} b^{2} d f^{2} - 2112 a^{7} b^{2} e^{2} f + 726 a^{6} b^{3} c f^{2} + 2640 a^{6} b^{3} d e f + 512 a^{6} b^{3} e^{3} - 1056 a^{5} b^{4} c e f - 825 a^{5} b^{4} d^{2} f - 960 a^{5} b^{4} d e^{2} + 660 a^{4} b^{5} c d f + 384 a^{4} b^{5} c e^{2} + 600 a^{4} b^{5} d^{2} e - 132 a^{3} b^{6} c^{2} f - 480 a^{3} b^{6} c d e - 125 a^{3} b^{6} d^{3} + 96 a^{2} b^{7} c^{2} e + 150 a^{2} b^{7} c d^{2} - 60 a b^{8} c^{2} d + 8 b^{9} c^{3}, \left (t \mapsto t \log {\left (\frac {81 t^{2} a b^{9}}{121 a^{6} f^{2} - 176 a^{5} b e f + 110 a^{4} b^{2} d f + 64 a^{4} b^{2} e^{2} - 44 a^{3} b^{3} c f - 80 a^{3} b^{3} d e + 32 a^{2} b^{4} c e + 25 a^{2} b^{4} d^{2} - 20 a b^{5} c d + 4 b^{6} c^{2}} + x \right )} \right )\right )} + \frac {f x^{8}}{8 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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