Optimal. Leaf size=269 \[ \frac {2 b c-a d}{a^3 x}-\frac {c}{4 a^2 x^4}+\frac {x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^3 b \left (a+b x^3\right )}+\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (2 a^3 f+a^2 b e-4 a b^2 d+7 b^3 c\right )}{18 a^{10/3} b^{5/3}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (2 a^3 f+a^2 b e-4 a b^2 d+7 b^3 c\right )}{9 a^{10/3} b^{5/3}}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (2 a^3 f+a^2 b e-4 a b^2 d+7 b^3 c\right )}{3 \sqrt {3} a^{10/3} b^{5/3}} \]
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Rubi [A] time = 0.29, antiderivative size = 269, normalized size of antiderivative = 1.00, 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 = {1829, 1488, 292, 31, 634, 617, 204, 628} \[ \frac {x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^3 b \left (a+b x^3\right )}+\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^2 b e+2 a^3 f-4 a b^2 d+7 b^3 c\right )}{18 a^{10/3} b^{5/3}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^2 b e+2 a^3 f-4 a b^2 d+7 b^3 c\right )}{9 a^{10/3} b^{5/3}}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (a^2 b e+2 a^3 f-4 a b^2 d+7 b^3 c\right )}{3 \sqrt {3} a^{10/3} b^{5/3}}+\frac {2 b c-a d}{a^3 x}-\frac {c}{4 a^2 x^4} \]
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 292
Rule 617
Rule 628
Rule 634
Rule 1488
Rule 1829
Rubi steps
\begin {align*} \int \frac {c+d x^3+e x^6+f x^9}{x^5 \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 a^3 b \left (a+b x^3\right )}-\frac {\int \frac {-3 b^3 c+3 b^3 \left (\frac {b c}{a}-d\right ) x^3-b^2 \left (\frac {b^3 c}{a^2}-\frac {b^2 d}{a}+b e+2 a f\right ) x^6}{x^5 \left (a+b x^3\right )} \, dx}{3 a b^3}\\ &=\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^3 b \left (a+b x^3\right )}-\frac {\int \left (-\frac {3 b^3 c}{a x^5}-\frac {3 b^3 (-2 b c+a d)}{a^2 x^2}-\frac {b^2 \left (7 b^3 c-4 a b^2 d+a^2 b e+2 a^3 f\right ) x}{a^2 \left (a+b x^3\right )}\right ) \, dx}{3 a b^3}\\ &=-\frac {c}{4 a^2 x^4}+\frac {2 b c-a d}{a^3 x}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^3 b \left (a+b x^3\right )}+\frac {\left (7 b^3 c-4 a b^2 d+a^2 b e+2 a^3 f\right ) \int \frac {x}{a+b x^3} \, dx}{3 a^3 b}\\ &=-\frac {c}{4 a^2 x^4}+\frac {2 b c-a d}{a^3 x}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^3 b \left (a+b x^3\right )}-\frac {\left (7 b^3 c-4 a b^2 d+a^2 b e+2 a^3 f\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a^{10/3} b^{4/3}}+\frac {\left (7 b^3 c-4 a b^2 d+a^2 b e+2 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 a^{10/3} b^{4/3}}\\ &=-\frac {c}{4 a^2 x^4}+\frac {2 b c-a d}{a^3 x}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^3 b \left (a+b x^3\right )}-\frac {\left (7 b^3 c-4 a b^2 d+a^2 b e+2 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{10/3} b^{5/3}}+\frac {\left (7 b^3 c-4 a b^2 d+a^2 b e+2 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 a^{10/3} b^{5/3}}+\frac {\left (7 b^3 c-4 a b^2 d+a^2 b e+2 a^3 f\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 a^3 b^{4/3}}\\ &=-\frac {c}{4 a^2 x^4}+\frac {2 b c-a d}{a^3 x}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^3 b \left (a+b x^3\right )}-\frac {\left (7 b^3 c-4 a b^2 d+a^2 b e+2 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{10/3} b^{5/3}}+\frac {\left (7 b^3 c-4 a b^2 d+a^2 b e+2 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{10/3} b^{5/3}}+\frac {\left (7 b^3 c-4 a b^2 d+a^2 b e+2 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 a^{10/3} b^{5/3}}\\ &=-\frac {c}{4 a^2 x^4}+\frac {2 b c-a d}{a^3 x}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^3 b \left (a+b x^3\right )}-\frac {\left (7 b^3 c-4 a b^2 d+a^2 b e+2 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} a^{10/3} b^{5/3}}-\frac {\left (7 b^3 c-4 a b^2 d+a^2 b e+2 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{10/3} b^{5/3}}+\frac {\left (7 b^3 c-4 a b^2 d+a^2 b e+2 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{10/3} b^{5/3}}\\ \end {align*}
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Mathematica [A] time = 0.27, size = 255, normalized size = 0.95 \[ \frac {-\frac {9 a^{4/3} c}{x^4}-\frac {12 \sqrt [3]{a} x^2 \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{b \left (a+b x^3\right )}-\frac {4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (2 a^3 f+a^2 b e-4 a b^2 d+7 b^3 c\right )}{b^{5/3}}-\frac {4 \sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right ) \left (2 a^3 f+a^2 b e-4 a b^2 d+7 b^3 c\right )}{b^{5/3}}+\frac {2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (2 a^3 f+a^2 b e-4 a b^2 d+7 b^3 c\right )}{b^{5/3}}-\frac {36 \sqrt [3]{a} (a d-2 b c)}{x}}{36 a^{10/3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 902, normalized size = 3.35 \[ \left [-\frac {9 \, a^{3} b^{3} c - 12 \, {\left (7 \, a b^{5} c - 4 \, a^{2} b^{4} d + a^{3} b^{3} e - a^{4} b^{2} f\right )} x^{6} - 9 \, {\left (7 \, a^{2} b^{4} c - 4 \, a^{3} b^{3} d\right )} x^{3} - 6 \, \sqrt {\frac {1}{3}} {\left ({\left (7 \, a b^{5} c - 4 \, a^{2} b^{4} d + a^{3} b^{3} e + 2 \, a^{4} b^{2} f\right )} x^{7} + {\left (7 \, a^{2} b^{4} c - 4 \, a^{3} b^{3} d + a^{4} b^{2} e + 2 \, a^{5} b f\right )} x^{4}\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 ) - 2 \, {\left ({\left (7 \, b^{4} c - 4 \, a b^{3} d + a^{2} b^{2} e + 2 \, a^{3} b f\right )} x^{7} + {\left (7 \, a b^{3} c - 4 \, a^{2} b^{2} d + a^{3} b e + 2 \, a^{4} f\right )} x^{4}\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 ) + 4 \, {\left ({\left (7 \, b^{4} c - 4 \, a b^{3} d + a^{2} b^{2} e + 2 \, a^{3} b f\right )} x^{7} + {\left (7 \, a b^{3} c - 4 \, a^{2} b^{2} d + a^{3} b e + 2 \, a^{4} f\right )} x^{4}\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b x - \left (-a b^{2}\right )^{\frac {1}{3}}\right )}{36 \, {\left (a^{4} b^{4} x^{7} + a^{5} b^{3} x^{4}\right )}}, -\frac {9 \, a^{3} b^{3} c - 12 \, {\left (7 \, a b^{5} c - 4 \, a^{2} b^{4} d + a^{3} b^{3} e - a^{4} b^{2} f\right )} x^{6} - 9 \, {\left (7 \, a^{2} b^{4} c - 4 \, a^{3} b^{3} d\right )} x^{3} - 12 \, \sqrt {\frac {1}{3}} {\left ({\left (7 \, a b^{5} c - 4 \, a^{2} b^{4} d + a^{3} b^{3} e + 2 \, a^{4} b^{2} f\right )} x^{7} + {\left (7 \, a^{2} b^{4} c - 4 \, a^{3} b^{3} d + a^{4} b^{2} e + 2 \, a^{5} b f\right )} x^{4}\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 ) - 2 \, {\left ({\left (7 \, b^{4} c - 4 \, a b^{3} d + a^{2} b^{2} e + 2 \, a^{3} b f\right )} x^{7} + {\left (7 \, a b^{3} c - 4 \, a^{2} b^{2} d + a^{3} b e + 2 \, a^{4} f\right )} x^{4}\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 ) + 4 \, {\left ({\left (7 \, b^{4} c - 4 \, a b^{3} d + a^{2} b^{2} e + 2 \, a^{3} b f\right )} x^{7} + {\left (7 \, a b^{3} c - 4 \, a^{2} b^{2} d + a^{3} b e + 2 \, a^{4} f\right )} x^{4}\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b x - \left (-a b^{2}\right )^{\frac {1}{3}}\right )}{36 \, {\left (a^{4} b^{4} x^{7} + a^{5} b^{3} x^{4}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 310, normalized size = 1.15 \[ \frac {\sqrt {3} {\left (7 \, b^{3} c - 4 \, a b^{2} d + 2 \, a^{3} f + a^{2} b e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{3} b} - \frac {{\left (7 \, b^{3} c - 4 \, a b^{2} d + 2 \, a^{3} f + a^{2} b e\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{3} b} - \frac {{\left (7 \, b^{3} c \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 4 \, a b^{2} d \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 2 \, a^{3} f \left (-\frac {a}{b}\right )^{\frac {1}{3}} + a^{2} b \left (-\frac {a}{b}\right )^{\frac {1}{3}} e\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a^{4} b} + \frac {b^{3} c x^{2} - a b^{2} d x^{2} - a^{3} f x^{2} + a^{2} b x^{2} e}{3 \, {\left (b x^{3} + a\right )} a^{3} b} + \frac {8 \, b c x^{3} - 4 \, a d x^{3} - a c}{4 \, a^{3} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 486, normalized size = 1.81 \[ \frac {e \,x^{2}}{3 \left (b \,x^{3}+a \right ) a}-\frac {b d \,x^{2}}{3 \left (b \,x^{3}+a \right ) a^{2}}+\frac {b^{2} c \,x^{2}}{3 \left (b \,x^{3}+a \right ) a^{3}}-\frac {f \,x^{2}}{3 \left (b \,x^{3}+a \right ) b}+\frac {\sqrt {3}\, 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}} a b}-\frac {e \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} a b}+\frac {e \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}} a b}-\frac {4 \sqrt {3}\, 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}} a^{2}}+\frac {4 d \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{2}}-\frac {2 d \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}} a^{2}}+\frac {7 \sqrt {3}\, b 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}} a^{3}}-\frac {7 b c \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{3}}+\frac {7 b c \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}} a^{3}}+\frac {2 \sqrt {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^{2}}-\frac {2 f \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{2}}+\frac {f \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}}-\frac {d}{a^{2} x}+\frac {2 b c}{a^{3} x}-\frac {c}{4 a^{2} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.93, size = 267, normalized size = 0.99 \[ \frac {4 \, {\left (7 \, b^{3} c - 4 \, a b^{2} d + a^{2} b e - a^{3} f\right )} x^{6} - 3 \, a^{2} b c + 3 \, {\left (7 \, a b^{2} c - 4 \, a^{2} b d\right )} x^{3}}{12 \, {\left (a^{3} b^{2} x^{7} + a^{4} b x^{4}\right )}} + \frac {\sqrt {3} {\left (7 \, b^{3} c - 4 \, a b^{2} d + a^{2} b e + 2 \, a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a^{3} b^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {{\left (7 \, b^{3} c - 4 \, a b^{2} d + a^{2} b e + 2 \, a^{3} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, a^{3} b^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {{\left (7 \, b^{3} c - 4 \, a b^{2} d + a^{2} b e + 2 \, a^{3} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, a^{3} b^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.18, size = 247, normalized size = 0.92 \[ -\frac {\frac {c}{4\,a}+\frac {x^3\,\left (4\,a\,d-7\,b\,c\right )}{4\,a^2}-\frac {x^6\,\left (-f\,a^3+e\,a^2\,b-4\,d\,a\,b^2+7\,c\,b^3\right )}{3\,a^3\,b}}{b\,x^7+a\,x^4}-\frac {\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (2\,f\,a^3+e\,a^2\,b-4\,d\,a\,b^2+7\,c\,b^3\right )}{9\,a^{10/3}\,b^{5/3}}+\frac {\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (2\,f\,a^3+e\,a^2\,b-4\,d\,a\,b^2+7\,c\,b^3\right )}{9\,a^{10/3}\,b^{5/3}}-\frac {\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (2\,f\,a^3+e\,a^2\,b-4\,d\,a\,b^2+7\,c\,b^3\right )}{9\,a^{10/3}\,b^{5/3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 177.03, size = 473, normalized size = 1.76 \[ \operatorname {RootSum} {\left (729 t^{3} a^{10} b^{5} + 8 a^{9} f^{3} + 12 a^{8} b e f^{2} - 48 a^{7} b^{2} d f^{2} + 6 a^{7} b^{2} e^{2} f + 84 a^{6} b^{3} c f^{2} - 48 a^{6} b^{3} d e f + a^{6} b^{3} e^{3} + 84 a^{5} b^{4} c e f + 96 a^{5} b^{4} d^{2} f - 12 a^{5} b^{4} d e^{2} - 336 a^{4} b^{5} c d f + 21 a^{4} b^{5} c e^{2} + 48 a^{4} b^{5} d^{2} e + 294 a^{3} b^{6} c^{2} f - 168 a^{3} b^{6} c d e - 64 a^{3} b^{6} d^{3} + 147 a^{2} b^{7} c^{2} e + 336 a^{2} b^{7} c d^{2} - 588 a b^{8} c^{2} d + 343 b^{9} c^{3}, \left (t \mapsto t \log {\left (\frac {81 t^{2} a^{7} b^{3}}{4 a^{6} f^{2} + 4 a^{5} b e f - 16 a^{4} b^{2} d f + a^{4} b^{2} e^{2} + 28 a^{3} b^{3} c f - 8 a^{3} b^{3} d e + 14 a^{2} b^{4} c e + 16 a^{2} b^{4} d^{2} - 56 a b^{5} c d + 49 b^{6} c^{2}} + x \right )} \right )\right )} + \frac {- 3 a^{2} b c + x^{6} \left (- 4 a^{3} f + 4 a^{2} b e - 16 a b^{2} d + 28 b^{3} c\right ) + x^{3} \left (- 12 a^{2} b d + 21 a b^{2} c\right )}{12 a^{4} b x^{4} + 12 a^{3} b^{2} x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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