Optimal. Leaf size=266 \[ -\frac {b^2 (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 g^3 i (a+b x)^2 (b c-a d)^3}+\frac {d^2 \log \left (\frac {a+b x}{c+d x}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g^3 i (b c-a d)^3}+\frac {2 b d (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g^3 i (a+b x) (b c-a d)^3}-\frac {B d^2 n \log ^2\left (\frac {a+b x}{c+d x}\right )}{2 g^3 i (b c-a d)^3}-\frac {B n (c+d x)^2 \left (b-\frac {4 d (a+b x)}{c+d x}\right )^2}{4 g^3 i (a+b x)^2 (b c-a d)^3} \]
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Rubi [C] time = 0.83, antiderivative size = 557, normalized size of antiderivative = 2.09, number of steps used = 26, number of rules used = 11, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.256, Rules used = {2528, 2525, 12, 44, 2524, 2418, 2390, 2301, 2394, 2393, 2391} \[ \frac {B d^2 n \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{g^3 i (b c-a d)^3}+\frac {B d^2 n \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{g^3 i (b c-a d)^3}+\frac {d^2 \log (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g^3 i (b c-a d)^3}-\frac {d^2 \log (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g^3 i (b c-a d)^3}+\frac {d \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g^3 i (a+b x) (b c-a d)^2}-\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{2 g^3 i (a+b x)^2 (b c-a d)}-\frac {B d^2 n \log ^2(a+b x)}{2 g^3 i (b c-a d)^3}-\frac {B d^2 n \log ^2(c+d x)}{2 g^3 i (b c-a d)^3}+\frac {3 B d^2 n \log (a+b x)}{2 g^3 i (b c-a d)^3}-\frac {3 B d^2 n \log (c+d x)}{2 g^3 i (b c-a d)^3}+\frac {B d^2 n \log (c+d x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{g^3 i (b c-a d)^3}+\frac {B d^2 n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{g^3 i (b c-a d)^3}+\frac {3 B d n}{2 g^3 i (a+b x) (b c-a d)^2}-\frac {B n}{4 g^3 i (a+b x)^2 (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 12
Rule 44
Rule 2301
Rule 2390
Rule 2391
Rule 2393
Rule 2394
Rule 2418
Rule 2524
Rule 2525
Rule 2528
Rubi steps
\begin {align*} \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(141 c+141 d x) (a g+b g x)^3} \, dx &=\int \left (\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{141 (b c-a d) g^3 (a+b x)^3}-\frac {b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{141 (b c-a d)^2 g^3 (a+b x)^2}+\frac {b d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{141 (b c-a d)^3 g^3 (a+b x)}-\frac {d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{141 (b c-a d)^3 g^3 (c+d x)}\right ) \, dx\\ &=\frac {\left (b d^2\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{141 (b c-a d)^3 g^3}-\frac {d^3 \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{141 (b c-a d)^3 g^3}-\frac {(b d) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^2} \, dx}{141 (b c-a d)^2 g^3}+\frac {b \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^3} \, dx}{141 (b c-a d) g^3}\\ &=-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{282 (b c-a d) g^3 (a+b x)^2}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{141 (b c-a d)^2 g^3 (a+b x)}+\frac {d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{141 (b c-a d)^3 g^3}-\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{141 (b c-a d)^3 g^3}-\frac {\left (B d^2 n\right ) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{141 (b c-a d)^3 g^3}+\frac {\left (B d^2 n\right ) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{141 (b c-a d)^3 g^3}-\frac {(B d n) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{141 (b c-a d)^2 g^3}+\frac {(B n) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{282 (b c-a d) g^3}\\ &=-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{282 (b c-a d) g^3 (a+b x)^2}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{141 (b c-a d)^2 g^3 (a+b x)}+\frac {d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{141 (b c-a d)^3 g^3}-\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{141 (b c-a d)^3 g^3}+\frac {(B n) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{282 g^3}-\frac {\left (B d^2 n\right ) \int \left (\frac {b \log (a+b x)}{a+b x}-\frac {d \log (a+b x)}{c+d x}\right ) \, dx}{141 (b c-a d)^3 g^3}+\frac {\left (B d^2 n\right ) \int \left (\frac {b \log (c+d x)}{a+b x}-\frac {d \log (c+d x)}{c+d x}\right ) \, dx}{141 (b c-a d)^3 g^3}-\frac {(B d n) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{141 (b c-a d) g^3}\\ &=-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{282 (b c-a d) g^3 (a+b x)^2}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{141 (b c-a d)^2 g^3 (a+b x)}+\frac {d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{141 (b c-a d)^3 g^3}-\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{141 (b c-a d)^3 g^3}+\frac {(B n) \int \left (\frac {b}{(b c-a d) (a+b x)^3}-\frac {b d}{(b c-a d)^2 (a+b x)^2}+\frac {b d^2}{(b c-a d)^3 (a+b x)}-\frac {d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{282 g^3}-\frac {\left (b B d^2 n\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{141 (b c-a d)^3 g^3}+\frac {\left (b B d^2 n\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{141 (b c-a d)^3 g^3}+\frac {\left (B d^3 n\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{141 (b c-a d)^3 g^3}-\frac {\left (B d^3 n\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{141 (b c-a d)^3 g^3}-\frac {(B d n) \int \left (\frac {b}{(b c-a d) (a+b x)^2}-\frac {b d}{(b c-a d)^2 (a+b x)}+\frac {d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{141 (b c-a d) g^3}\\ &=-\frac {B n}{564 (b c-a d) g^3 (a+b x)^2}+\frac {B d n}{94 (b c-a d)^2 g^3 (a+b x)}+\frac {B d^2 n \log (a+b x)}{94 (b c-a d)^3 g^3}-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{282 (b c-a d) g^3 (a+b x)^2}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{141 (b c-a d)^2 g^3 (a+b x)}+\frac {d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{141 (b c-a d)^3 g^3}-\frac {B d^2 n \log (c+d x)}{94 (b c-a d)^3 g^3}+\frac {B d^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{141 (b c-a d)^3 g^3}-\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{141 (b c-a d)^3 g^3}+\frac {B d^2 n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{141 (b c-a d)^3 g^3}-\frac {\left (B d^2 n\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{141 (b c-a d)^3 g^3}-\frac {\left (B d^2 n\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{141 (b c-a d)^3 g^3}-\frac {\left (b B d^2 n\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{141 (b c-a d)^3 g^3}-\frac {\left (B d^3 n\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{141 (b c-a d)^3 g^3}\\ &=-\frac {B n}{564 (b c-a d) g^3 (a+b x)^2}+\frac {B d n}{94 (b c-a d)^2 g^3 (a+b x)}+\frac {B d^2 n \log (a+b x)}{94 (b c-a d)^3 g^3}-\frac {B d^2 n \log ^2(a+b x)}{282 (b c-a d)^3 g^3}-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{282 (b c-a d) g^3 (a+b x)^2}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{141 (b c-a d)^2 g^3 (a+b x)}+\frac {d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{141 (b c-a d)^3 g^3}-\frac {B d^2 n \log (c+d x)}{94 (b c-a d)^3 g^3}+\frac {B d^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{141 (b c-a d)^3 g^3}-\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{141 (b c-a d)^3 g^3}-\frac {B d^2 n \log ^2(c+d x)}{282 (b c-a d)^3 g^3}+\frac {B d^2 n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{141 (b c-a d)^3 g^3}-\frac {\left (B d^2 n\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{141 (b c-a d)^3 g^3}-\frac {\left (B d^2 n\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{141 (b c-a d)^3 g^3}\\ &=-\frac {B n}{564 (b c-a d) g^3 (a+b x)^2}+\frac {B d n}{94 (b c-a d)^2 g^3 (a+b x)}+\frac {B d^2 n \log (a+b x)}{94 (b c-a d)^3 g^3}-\frac {B d^2 n \log ^2(a+b x)}{282 (b c-a d)^3 g^3}-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{282 (b c-a d) g^3 (a+b x)^2}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{141 (b c-a d)^2 g^3 (a+b x)}+\frac {d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{141 (b c-a d)^3 g^3}-\frac {B d^2 n \log (c+d x)}{94 (b c-a d)^3 g^3}+\frac {B d^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{141 (b c-a d)^3 g^3}-\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{141 (b c-a d)^3 g^3}-\frac {B d^2 n \log ^2(c+d x)}{282 (b c-a d)^3 g^3}+\frac {B d^2 n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{141 (b c-a d)^3 g^3}+\frac {B d^2 n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{141 (b c-a d)^3 g^3}+\frac {B d^2 n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{141 (b c-a d)^3 g^3}\\ \end {align*}
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Mathematica [C] time = 0.41, size = 434, normalized size = 1.63 \[ \frac {4 d^2 (a+b x)^2 \log (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )-4 d^2 (a+b x)^2 \log (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )-2 (b c-a d)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )+4 d (a+b x) (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )-2 B d^2 n (a+b x)^2 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )\right )+2 B d^2 n (a+b x)^2 \left (2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac {d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )-B n \left (2 d^2 (a+b x)^2 \log (c+d x)+2 d (a+b x) (a d-b c)+(b c-a d)^2-2 d^2 (a+b x)^2 \log (a+b x)\right )+4 B d n (a+b x) (-d (a+b x) \log (c+d x)+d (a+b x) \log (a+b x)-a d+b c)}{4 g^3 i (a+b x)^2 (b c-a d)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 483, normalized size = 1.82 \[ -\frac {2 \, A b^{2} c^{2} - 8 \, A a b c d + 6 \, A a^{2} d^{2} - 2 \, {\left (B b^{2} d^{2} n x^{2} + 2 \, B a b d^{2} n x + B a^{2} d^{2} n\right )} \log \left (\frac {b x + a}{d x + c}\right )^{2} + {\left (B b^{2} c^{2} - 8 \, B a b c d + 7 \, B a^{2} d^{2}\right )} n - 2 \, {\left (2 \, A b^{2} c d - 2 \, A a b d^{2} + 3 \, {\left (B b^{2} c d - B a b d^{2}\right )} n\right )} x + 2 \, {\left (B b^{2} c^{2} - 4 \, B a b c d + 3 \, B a^{2} d^{2} - 2 \, {\left (B b^{2} c d - B a b d^{2}\right )} x - 2 \, {\left (B b^{2} d^{2} x^{2} + 2 \, B a b d^{2} x + B a^{2} d^{2}\right )} \log \left (\frac {b x + a}{d x + c}\right )\right )} \log \relax (e) - 2 \, {\left (2 \, A a^{2} d^{2} + {\left (3 \, B b^{2} d^{2} n + 2 \, A b^{2} d^{2}\right )} x^{2} - {\left (B b^{2} c^{2} - 4 \, B a b c d\right )} n + 2 \, {\left (2 \, A a b d^{2} + {\left (B b^{2} c d + 2 \, B a b d^{2}\right )} n\right )} x\right )} \log \left (\frac {b x + a}{d x + c}\right )}{4 \, {\left ({\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} g^{3} i x^{2} + 2 \, {\left (a b^{4} c^{3} - 3 \, a^{2} b^{3} c^{2} d + 3 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}\right )} g^{3} i x + {\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} g^{3} i\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [F] time = 0.47, size = 0, normalized size = 0.00 \[ \int \frac {B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A}{\left (b g x +a g \right )^{3} \left (d i x +c i \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.73, size = 888, normalized size = 3.34 \[ \frac {1}{2} \, B {\left (\frac {2 \, b d x - b c + 3 \, a d}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} g^{3} i x^{2} + 2 \, {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} g^{3} i x + {\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )} g^{3} i} + \frac {2 \, d^{2} \log \left (b x + a\right )}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} g^{3} i} - \frac {2 \, d^{2} \log \left (d x + c\right )}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} g^{3} i}\right )} \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right ) - \frac {{\left (b^{2} c^{2} - 8 \, a b c d + 7 \, a^{2} d^{2} + 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \log \left (b x + a\right )^{2} + 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \log \left (d x + c\right )^{2} - 6 \, {\left (b^{2} c d - a b d^{2}\right )} x - 6 \, {\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \log \left (b x + a\right ) + 2 \, {\left (3 \, b^{2} d^{2} x^{2} + 6 \, a b d^{2} x + 3 \, a^{2} d^{2} - 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )\right )} B n}{4 \, {\left (a^{2} b^{3} c^{3} g^{3} i - 3 \, a^{3} b^{2} c^{2} d g^{3} i + 3 \, a^{4} b c d^{2} g^{3} i - a^{5} d^{3} g^{3} i + {\left (b^{5} c^{3} g^{3} i - 3 \, a b^{4} c^{2} d g^{3} i + 3 \, a^{2} b^{3} c d^{2} g^{3} i - a^{3} b^{2} d^{3} g^{3} i\right )} x^{2} + 2 \, {\left (a b^{4} c^{3} g^{3} i - 3 \, a^{2} b^{3} c^{2} d g^{3} i + 3 \, a^{3} b^{2} c d^{2} g^{3} i - a^{4} b d^{3} g^{3} i\right )} x\right )}} + \frac {1}{2} \, A {\left (\frac {2 \, b d x - b c + 3 \, a d}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} g^{3} i x^{2} + 2 \, {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} g^{3} i x + {\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )} g^{3} i} + \frac {2 \, d^{2} \log \left (b x + a\right )}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} g^{3} i} - \frac {2 \, d^{2} \log \left (d x + c\right )}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} g^{3} i}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.34, size = 573, normalized size = 2.15 \[ \frac {B\,d^2\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,\left (\frac {g^3\,i\,n\,\left (a\,d-b\,c\right )\,\left (2\,a\,d-b\,c\right )}{2\,d^2}+\frac {a\,g^3\,i\,n\,\left (a\,d-b\,c\right )}{2\,d}+\frac {b\,g^3\,i\,n\,x\,\left (a\,d-b\,c\right )}{d}\right )}{g^3\,i\,n\,\left (a\,d-b\,c\right )\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )\,\left (i\,a^2\,g^3+2\,i\,a\,b\,g^3\,x+i\,b^2\,g^3\,x^2\right )}-\frac {B\,d^2\,{\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}^2}{2\,g^3\,i\,n\,\left (a\,d-b\,c\right )\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}-\frac {\frac {6\,A\,a\,d-2\,A\,b\,c+7\,B\,a\,d\,n-B\,b\,c\,n}{2\,\left (a\,d-b\,c\right )}+\frac {d\,x\,\left (2\,A\,b+3\,B\,b\,n\right )}{a\,d-b\,c}}{x^2\,\left (2\,b^3\,c\,g^3\,i-2\,a\,b^2\,d\,g^3\,i\right )+x\,\left (4\,a\,b^2\,c\,g^3\,i-4\,a^2\,b\,d\,g^3\,i\right )-2\,a^3\,d\,g^3\,i+2\,a^2\,b\,c\,g^3\,i}+\frac {d^2\,\mathrm {atan}\left (\frac {d^2\,\left (A+\frac {3\,B\,n}{2}\right )\,\left (2\,i\,a^3\,d^3\,g^3-2\,i\,a^2\,b\,c\,d^2\,g^3-2\,i\,a\,b^2\,c^2\,d\,g^3+2\,i\,b^3\,c^3\,g^3\right )\,1{}\mathrm {i}}{g^3\,i\,\left (2\,A\,d^2+3\,B\,d^2\,n\right )\,{\left (a\,d-b\,c\right )}^3}+\frac {b\,d^3\,x\,\left (A+\frac {3\,B\,n}{2}\right )\,\left (i\,a^2\,d^2\,g^3-2\,i\,a\,b\,c\,d\,g^3+i\,b^2\,c^2\,g^3\right )\,4{}\mathrm {i}}{g^3\,i\,\left (2\,A\,d^2+3\,B\,d^2\,n\right )\,{\left (a\,d-b\,c\right )}^3}\right )\,\left (A+\frac {3\,B\,n}{2}\right )\,2{}\mathrm {i}}{g^3\,i\,{\left (a\,d-b\,c\right )}^3} \]
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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