Optimal. Leaf size=254 \[ \frac {b^2 \log \left (\frac {a+b x}{c+d x}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g i^3 (b c-a d)^3}+\frac {d^2 (a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 g i^3 (c+d x)^2 (b c-a d)^3}-\frac {2 b d (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g i^3 (c+d x) (b c-a d)^3}-\frac {b^2 B n \log ^2\left (\frac {a+b x}{c+d x}\right )}{2 g i^3 (b c-a d)^3}-\frac {B n \left (4 b-\frac {d (a+b x)}{c+d x}\right )^2}{4 g i^3 (b c-a d)^3} \]
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Rubi [C] time = 0.87, antiderivative size = 557, normalized size of antiderivative = 2.19, 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, 2524, 2418, 2390, 2301, 2394, 2393, 2391, 2525, 12, 44} \[ \frac {b^2 B n \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{g i^3 (b c-a d)^3}+\frac {b^2 B n \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{g i^3 (b c-a d)^3}+\frac {b^2 \log (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g i^3 (b c-a d)^3}-\frac {b^2 \log (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g i^3 (b c-a d)^3}+\frac {b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g i^3 (c+d x) (b c-a d)^2}+\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{2 g i^3 (c+d x)^2 (b c-a d)}-\frac {b^2 B n \log ^2(a+b x)}{2 g i^3 (b c-a d)^3}-\frac {b^2 B n \log ^2(c+d x)}{2 g i^3 (b c-a d)^3}-\frac {3 b^2 B n \log (a+b x)}{2 g i^3 (b c-a d)^3}+\frac {3 b^2 B n \log (c+d x)}{2 g i^3 (b c-a d)^3}+\frac {b^2 B n \log (c+d x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{g i^3 (b c-a d)^3}+\frac {b^2 B n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{g i^3 (b c-a d)^3}-\frac {3 b B n}{2 g i^3 (c+d x) (b c-a d)^2}-\frac {B n}{4 g i^3 (c+d x)^2 (b c-a d)} \]
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 )}{(155 c+155 d x)^3 (a g+b g x)} \, dx &=\int \left (\frac {b^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3723875 (b c-a d)^3 g (a+b x)}-\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3723875 (b c-a d) g (c+d x)^3}-\frac {b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3723875 (b c-a d)^2 g (c+d x)^2}-\frac {b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3723875 (b c-a d)^3 g (c+d x)}\right ) \, dx\\ &=\frac {b^3 \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{3723875 (b c-a d)^3 g}-\frac {\left (b^2 d\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{3723875 (b c-a d)^3 g}-\frac {(b d) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c+d x)^2} \, dx}{3723875 (b c-a d)^2 g}-\frac {d \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c+d x)^3} \, dx}{3723875 (b c-a d) g}\\ &=\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{7447750 (b c-a d) g (c+d x)^2}+\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3723875 (b c-a d)^2 g (c+d x)}+\frac {b^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3723875 (b c-a d)^3 g}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3723875 (b c-a d)^3 g}-\frac {\left (b^2 B 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}{3723875 (b c-a d)^3 g}+\frac {\left (b^2 B 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}{3723875 (b c-a d)^3 g}-\frac {(b B n) \int \frac {b c-a d}{(a+b x) (c+d x)^2} \, dx}{3723875 (b c-a d)^2 g}-\frac {(B n) \int \frac {b c-a d}{(a+b x) (c+d x)^3} \, dx}{7447750 (b c-a d) g}\\ &=\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{7447750 (b c-a d) g (c+d x)^2}+\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3723875 (b c-a d)^2 g (c+d x)}+\frac {b^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3723875 (b c-a d)^3 g}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3723875 (b c-a d)^3 g}-\frac {(B n) \int \frac {1}{(a+b x) (c+d x)^3} \, dx}{7447750 g}-\frac {\left (b^2 B n\right ) \int \left (\frac {b \log (a+b x)}{a+b x}-\frac {d \log (a+b x)}{c+d x}\right ) \, dx}{3723875 (b c-a d)^3 g}+\frac {\left (b^2 B n\right ) \int \left (\frac {b \log (c+d x)}{a+b x}-\frac {d \log (c+d x)}{c+d x}\right ) \, dx}{3723875 (b c-a d)^3 g}-\frac {(b B n) \int \frac {1}{(a+b x) (c+d x)^2} \, dx}{3723875 (b c-a d) g}\\ &=\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{7447750 (b c-a d) g (c+d x)^2}+\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3723875 (b c-a d)^2 g (c+d x)}+\frac {b^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3723875 (b c-a d)^3 g}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3723875 (b c-a d)^3 g}-\frac {(B n) \int \left (\frac {b^3}{(b c-a d)^3 (a+b x)}-\frac {d}{(b c-a d) (c+d x)^3}-\frac {b d}{(b c-a d)^2 (c+d x)^2}-\frac {b^2 d}{(b c-a d)^3 (c+d x)}\right ) \, dx}{7447750 g}-\frac {\left (b^3 B n\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{3723875 (b c-a d)^3 g}+\frac {\left (b^3 B n\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{3723875 (b c-a d)^3 g}+\frac {\left (b^2 B d n\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{3723875 (b c-a d)^3 g}-\frac {\left (b^2 B d n\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{3723875 (b c-a d)^3 g}-\frac {(b B n) \int \left (\frac {b^2}{(b c-a d)^2 (a+b x)}-\frac {d}{(b c-a d) (c+d x)^2}-\frac {b d}{(b c-a d)^2 (c+d x)}\right ) \, dx}{3723875 (b c-a d) g}\\ &=-\frac {B n}{14895500 (b c-a d) g (c+d x)^2}-\frac {3 b B n}{7447750 (b c-a d)^2 g (c+d x)}-\frac {3 b^2 B n \log (a+b x)}{7447750 (b c-a d)^3 g}+\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{7447750 (b c-a d) g (c+d x)^2}+\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3723875 (b c-a d)^2 g (c+d x)}+\frac {b^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3723875 (b c-a d)^3 g}+\frac {3 b^2 B n \log (c+d x)}{7447750 (b c-a d)^3 g}+\frac {b^2 B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3723875 (b c-a d)^3 g}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3723875 (b c-a d)^3 g}+\frac {b^2 B n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{3723875 (b c-a d)^3 g}-\frac {\left (b^2 B n\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{3723875 (b c-a d)^3 g}-\frac {\left (b^2 B n\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{3723875 (b c-a d)^3 g}-\frac {\left (b^3 B n\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{3723875 (b c-a d)^3 g}-\frac {\left (b^2 B d n\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{3723875 (b c-a d)^3 g}\\ &=-\frac {B n}{14895500 (b c-a d) g (c+d x)^2}-\frac {3 b B n}{7447750 (b c-a d)^2 g (c+d x)}-\frac {3 b^2 B n \log (a+b x)}{7447750 (b c-a d)^3 g}-\frac {b^2 B n \log ^2(a+b x)}{7447750 (b c-a d)^3 g}+\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{7447750 (b c-a d) g (c+d x)^2}+\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3723875 (b c-a d)^2 g (c+d x)}+\frac {b^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3723875 (b c-a d)^3 g}+\frac {3 b^2 B n \log (c+d x)}{7447750 (b c-a d)^3 g}+\frac {b^2 B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3723875 (b c-a d)^3 g}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3723875 (b c-a d)^3 g}-\frac {b^2 B n \log ^2(c+d x)}{7447750 (b c-a d)^3 g}+\frac {b^2 B n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{3723875 (b c-a d)^3 g}-\frac {\left (b^2 B 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 )}{3723875 (b c-a d)^3 g}-\frac {\left (b^2 B 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 )}{3723875 (b c-a d)^3 g}\\ &=-\frac {B n}{14895500 (b c-a d) g (c+d x)^2}-\frac {3 b B n}{7447750 (b c-a d)^2 g (c+d x)}-\frac {3 b^2 B n \log (a+b x)}{7447750 (b c-a d)^3 g}-\frac {b^2 B n \log ^2(a+b x)}{7447750 (b c-a d)^3 g}+\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{7447750 (b c-a d) g (c+d x)^2}+\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3723875 (b c-a d)^2 g (c+d x)}+\frac {b^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3723875 (b c-a d)^3 g}+\frac {3 b^2 B n \log (c+d x)}{7447750 (b c-a d)^3 g}+\frac {b^2 B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3723875 (b c-a d)^3 g}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3723875 (b c-a d)^3 g}-\frac {b^2 B n \log ^2(c+d x)}{7447750 (b c-a d)^3 g}+\frac {b^2 B n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{3723875 (b c-a d)^3 g}+\frac {b^2 B n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{3723875 (b c-a d)^3 g}+\frac {b^2 B n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{3723875 (b c-a d)^3 g}\\ \end {align*}
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Mathematica [C] time = 0.42, size = 434, normalized size = 1.71 \[ \frac {4 b^2 (c+d x)^2 \log (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )-4 b^2 (c+d x)^2 \log (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )+2 (b c-a d)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )+4 b (c+d x) (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )-2 b^2 B n (c+d x)^2 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )\right )+2 b^2 B n (c+d x)^2 \left (2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac {d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )-B n \left (2 b^2 (c+d x)^2 \log (a+b x)+2 b (c+d x) (b c-a d)+(b c-a d)^2-2 b^2 (c+d x)^2 \log (c+d x)\right )-4 b B n (c+d x) (b (c+d x) \log (a+b x)-a d-b (c+d x) \log (c+d x)+b c)}{4 g i^3 (c+d x)^2 (b c-a d)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.92, size = 487, normalized size = 1.92 \[ \frac {6 \, A b^{2} c^{2} - 8 \, A a b c d + 2 \, A a^{2} d^{2} + 2 \, {\left (B b^{2} d^{2} n x^{2} + 2 \, B b^{2} c d n x + B b^{2} c^{2} n\right )} \log \left (\frac {b x + a}{d x + c}\right )^{2} - {\left (7 \, B b^{2} c^{2} - 8 \, B a b c d + B a^{2} d^{2}\right )} n + 2 \, {\left (2 \, A b^{2} c d - 2 \, A a b d^{2} - 3 \, {\left (B b^{2} c d - B a b d^{2}\right )} n\right )} x + 2 \, {\left (3 \, B b^{2} c^{2} - 4 \, B a b c d + B a^{2} d^{2} + 2 \, {\left (B b^{2} c d - B a b d^{2}\right )} x + 2 \, {\left (B b^{2} d^{2} x^{2} + 2 \, B b^{2} c d x + B b^{2} c^{2}\right )} \log \left (\frac {b x + a}{d x + c}\right )\right )} \log \relax (e) + 2 \, {\left (2 \, A b^{2} c^{2} - {\left (3 \, B b^{2} d^{2} n - 2 \, A b^{2} d^{2}\right )} x^{2} - {\left (4 \, B a b c d - B a^{2} d^{2}\right )} n + 2 \, {\left (2 \, A b^{2} c d - {\left (2 \, B b^{2} c d + B a b d^{2}\right )} n\right )} x\right )} \log \left (\frac {b x + a}{d x + c}\right )}{4 \, {\left ({\left (b^{3} c^{3} d^{2} - 3 \, a b^{2} c^{2} d^{3} + 3 \, a^{2} b c d^{4} - a^{3} d^{5}\right )} g i^{3} x^{2} + 2 \, {\left (b^{3} c^{4} d - 3 \, a b^{2} c^{3} d^{2} + 3 \, a^{2} b c^{2} d^{3} - a^{3} c d^{4}\right )} g i^{3} x + {\left (b^{3} c^{5} - 3 \, a b^{2} c^{4} d + 3 \, a^{2} b c^{3} d^{2} - a^{3} c^{2} d^{3}\right )} g i^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 4.01, size = 370, normalized size = 1.46 \[ \frac {1}{4} \, {\left (\frac {2 \, B b^{2} i n \log \left (\frac {b x + a}{d x + c}\right )^{2}}{b^{2} c^{2} g - 2 \, a b c d g + a^{2} d^{2} g} - 2 \, {\left (\frac {4 \, {\left (b x + a\right )} B b d i n}{{\left (b^{2} c^{2} g - 2 \, a b c d g + a^{2} d^{2} g\right )} {\left (d x + c\right )}} - \frac {{\left (b x + a\right )}^{2} B d^{2} i n}{{\left (b^{2} c^{2} g - 2 \, a b c d g + a^{2} d^{2} g\right )} {\left (d x + c\right )}^{2}}\right )} \log \left (\frac {b x + a}{d x + c}\right ) - \frac {4 \, {\left (A b^{2} + B b^{2}\right )} \log \left (\frac {b x + a}{d x + c}\right )}{b^{2} c^{2} g i - 2 \, a b c d g i + a^{2} d^{2} g i} - \frac {{\left (B d^{2} i n - 2 \, A d^{2} i - 2 \, B d^{2} i\right )} {\left (b x + a\right )}^{2}}{{\left (b^{2} c^{2} g - 2 \, a b c d g + a^{2} d^{2} g\right )} {\left (d x + c\right )}^{2}} + \frac {8 \, {\left (B b d i n - A b d i - B b d i\right )} {\left (b x + a\right )}}{{\left (b^{2} c^{2} g - 2 \, a b c d g + a^{2} d^{2} g\right )} {\left (d x + c\right )}}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )} \]
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 \[ \int \frac {B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A}{\left (b g x +a g \right ) \left (d i x +c i \right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.98, size = 888, normalized size = 3.50 \[ \frac {1}{2} \, B {\left (\frac {2 \, b d x + 3 \, b c - a d}{{\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} g i^{3} x^{2} + 2 \, {\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} g i^{3} x + {\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2}\right )} g i^{3}} + \frac {2 \, b^{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 i^{3}} - \frac {2 \, b^{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 i^{3}}\right )} \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right ) - \frac {{\left (7 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \log \left (b x + a\right )^{2} + 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{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 \, b^{2} c d x + b^{2} c^{2}\right )} \log \left (b x + a\right ) - 2 \, {\left (3 \, b^{2} d^{2} x^{2} + 6 \, b^{2} c d x + 3 \, b^{2} c^{2} + 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )\right )} B n}{4 \, {\left (b^{3} c^{5} g i^{3} - 3 \, a b^{2} c^{4} d g i^{3} + 3 \, a^{2} b c^{3} d^{2} g i^{3} - a^{3} c^{2} d^{3} g i^{3} + {\left (b^{3} c^{3} d^{2} g i^{3} - 3 \, a b^{2} c^{2} d^{3} g i^{3} + 3 \, a^{2} b c d^{4} g i^{3} - a^{3} d^{5} g i^{3}\right )} x^{2} + 2 \, {\left (b^{3} c^{4} d g i^{3} - 3 \, a b^{2} c^{3} d^{2} g i^{3} + 3 \, a^{2} b c^{2} d^{3} g i^{3} - a^{3} c d^{4} g i^{3}\right )} x\right )}} + \frac {1}{2} \, A {\left (\frac {2 \, b d x + 3 \, b c - a d}{{\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} g i^{3} x^{2} + 2 \, {\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} g i^{3} x + {\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2}\right )} g i^{3}} + \frac {2 \, b^{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 i^{3}} - \frac {2 \, b^{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 i^{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.65, size = 573, normalized size = 2.26 \[ \frac {B\,b^2\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,\left (\frac {c\,g\,i^3\,n\,\left (a\,d-b\,c\right )}{2\,b}-\frac {g\,i^3\,n\,\left (a\,d-b\,c\right )\,\left (a\,d-2\,b\,c\right )}{2\,b^2}+\frac {d\,g\,i^3\,n\,x\,\left (a\,d-b\,c\right )}{b}\right )}{g\,i^3\,n\,\left (a\,d-b\,c\right )\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )\,\left (g\,c^2\,i^3+2\,g\,c\,d\,i^3\,x+g\,d^2\,i^3\,x^2\right )}-\frac {B\,b^2\,{\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}^2}{2\,g\,i^3\,n\,\left (a\,d-b\,c\right )\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}-\frac {\frac {2\,A\,a\,d-6\,A\,b\,c-B\,a\,d\,n+7\,B\,b\,c\,n}{2\,\left (a\,d-b\,c\right )}-\frac {b\,x\,\left (2\,A\,d-3\,B\,d\,n\right )}{a\,d-b\,c}}{x^2\,\left (2\,a\,d^3\,g\,i^3-2\,b\,c\,d^2\,g\,i^3\right )+x\,\left (4\,a\,c\,d^2\,g\,i^3-4\,b\,c^2\,d\,g\,i^3\right )-2\,b\,c^3\,g\,i^3+2\,a\,c^2\,d\,g\,i^3}+\frac {b^2\,\mathrm {atan}\left (\frac {b^2\,\left (A-\frac {3\,B\,n}{2}\right )\,\left (2\,g\,a^3\,d^3\,i^3-2\,g\,a^2\,b\,c\,d^2\,i^3-2\,g\,a\,b^2\,c^2\,d\,i^3+2\,g\,b^3\,c^3\,i^3\right )\,1{}\mathrm {i}}{g\,i^3\,\left (2\,A\,b^2-3\,B\,b^2\,n\right )\,{\left (a\,d-b\,c\right )}^3}+\frac {b^3\,d\,x\,\left (A-\frac {3\,B\,n}{2}\right )\,\left (g\,a^2\,d^2\,i^3-2\,g\,a\,b\,c\,d\,i^3+g\,b^2\,c^2\,i^3\right )\,4{}\mathrm {i}}{g\,i^3\,\left (2\,A\,b^2-3\,B\,b^2\,n\right )\,{\left (a\,d-b\,c\right )}^3}\right )\,\left (A-\frac {3\,B\,n}{2}\right )\,2{}\mathrm {i}}{g\,i^3\,{\left (a\,d-b\,c\right )}^3} \]
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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