Optimal. Leaf size=288 \[ -\frac {b B n (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 g^3 (a+b x)^2 (b c-a d)^2}+\frac {2 B d n (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g^3 (a+b x) (b c-a d)^2}-\frac {b (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 g^3 (a+b x)^2 (b c-a d)^2}+\frac {d (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{g^3 (a+b x) (b c-a d)^2}-\frac {b B^2 n^2 (c+d x)^2}{4 g^3 (a+b x)^2 (b c-a d)^2}+\frac {2 B^2 d n^2 (c+d x)}{g^3 (a+b x) (b c-a d)^2} \]
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Rubi [C] time = 0.92, antiderivative size = 626, normalized size of antiderivative = 2.17, number of steps used = 28, number of rules used = 11, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.314, Rules used = {2525, 12, 2528, 44, 2524, 2418, 2390, 2301, 2394, 2393, 2391} \[ \frac {B^2 d^2 n^2 \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{b g^3 (b c-a d)^2}+\frac {B^2 d^2 n^2 \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{b g^3 (b c-a d)^2}+\frac {B d^2 n \log (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b g^3 (b c-a d)^2}-\frac {B d^2 n \log (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b g^3 (b c-a d)^2}+\frac {B d n \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b g^3 (a+b x) (b c-a d)}-\frac {\left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 b g^3 (a+b x)^2}-\frac {B n \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 b g^3 (a+b x)^2}-\frac {B^2 d^2 n^2 \log ^2(a+b x)}{2 b g^3 (b c-a d)^2}-\frac {B^2 d^2 n^2 \log ^2(c+d x)}{2 b g^3 (b c-a d)^2}+\frac {3 B^2 d^2 n^2 \log (a+b x)}{2 b g^3 (b c-a d)^2}-\frac {3 B^2 d^2 n^2 \log (c+d x)}{2 b g^3 (b c-a d)^2}+\frac {B^2 d^2 n^2 \log (c+d x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{b g^3 (b c-a d)^2}+\frac {B^2 d^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b g^3 (b c-a d)^2}+\frac {3 B^2 d n^2}{2 b g^3 (a+b x) (b c-a d)}-\frac {B^2 n^2}{4 b g^3 (a+b x)^2} \]
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 {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^3} \, dx &=-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 b g^3 (a+b x)^2}+\frac {(B n) \int \frac {(b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{g^2 (a+b x)^3 (c+d x)} \, dx}{b g}\\ &=-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 b g^3 (a+b x)^2}+\frac {(B (b c-a d) n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^3 (c+d x)} \, dx}{b g^3}\\ &=-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 b g^3 (a+b x)^2}+\frac {(B (b c-a d) n) \int \left (\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) (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 )}{(b c-a d)^2 (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 )}{(b c-a d)^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 )}{(b c-a d)^3 (c+d x)}\right ) \, dx}{b g^3}\\ &=-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 b g^3 (a+b x)^2}+\frac {(B n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^3} \, dx}{g^3}+\frac {\left (B d^2 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{(b c-a d)^2 g^3}-\frac {\left (B d^3 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{b (b c-a d)^2 g^3}-\frac {(B d n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^2} \, dx}{(b c-a d) g^3}\\ &=-\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b g^3 (a+b x)^2}+\frac {B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b (b c-a d) g^3 (a+b x)}+\frac {B d^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b (b c-a d)^2 g^3}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 b g^3 (a+b x)^2}-\frac {B d^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b (b c-a d)^2 g^3}+\frac {\left (B^2 n^2\right ) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{2 b g^3}-\frac {\left (B^2 d^2 n^2\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}{b (b c-a d)^2 g^3}+\frac {\left (B^2 d^2 n^2\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}{b (b c-a d)^2 g^3}-\frac {\left (B^2 d n^2\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b (b c-a d) g^3}\\ &=-\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b g^3 (a+b x)^2}+\frac {B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b (b c-a d) g^3 (a+b x)}+\frac {B d^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b (b c-a d)^2 g^3}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 b g^3 (a+b x)^2}-\frac {B d^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b (b c-a d)^2 g^3}-\frac {\left (B^2 d n^2\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{b g^3}-\frac {\left (B^2 d^2 n^2\right ) \int \left (\frac {b \log (a+b x)}{a+b x}-\frac {d \log (a+b x)}{c+d x}\right ) \, dx}{b (b c-a d)^2 g^3}+\frac {\left (B^2 d^2 n^2\right ) \int \left (\frac {b \log (c+d x)}{a+b x}-\frac {d \log (c+d x)}{c+d x}\right ) \, dx}{b (b c-a d)^2 g^3}+\frac {\left (B^2 (b c-a d) n^2\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{2 b g^3}\\ &=-\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b g^3 (a+b x)^2}+\frac {B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b (b c-a d) g^3 (a+b x)}+\frac {B d^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b (b c-a d)^2 g^3}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 b g^3 (a+b x)^2}-\frac {B d^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b (b c-a d)^2 g^3}-\frac {\left (B^2 d n^2\right ) \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}{b g^3}-\frac {\left (B^2 d^2 n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{(b c-a d)^2 g^3}+\frac {\left (B^2 d^2 n^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{(b c-a d)^2 g^3}+\frac {\left (B^2 d^3 n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b (b c-a d)^2 g^3}-\frac {\left (B^2 d^3 n^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{b (b c-a d)^2 g^3}+\frac {\left (B^2 (b c-a d) n^2\right ) \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}{2 b g^3}\\ &=-\frac {B^2 n^2}{4 b g^3 (a+b x)^2}+\frac {3 B^2 d n^2}{2 b (b c-a d) g^3 (a+b x)}+\frac {3 B^2 d^2 n^2 \log (a+b x)}{2 b (b c-a d)^2 g^3}-\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b g^3 (a+b x)^2}+\frac {B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b (b c-a d) g^3 (a+b x)}+\frac {B d^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b (b c-a d)^2 g^3}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 b g^3 (a+b x)^2}-\frac {3 B^2 d^2 n^2 \log (c+d x)}{2 b (b c-a d)^2 g^3}+\frac {B^2 d^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b (b c-a d)^2 g^3}-\frac {B d^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b (b c-a d)^2 g^3}+\frac {B^2 d^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b (b c-a d)^2 g^3}-\frac {\left (B^2 d^2 n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{(b c-a d)^2 g^3}-\frac {\left (B^2 d^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b (b c-a d)^2 g^3}-\frac {\left (B^2 d^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{b (b c-a d)^2 g^3}-\frac {\left (B^2 d^3 n^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b (b c-a d)^2 g^3}\\ &=-\frac {B^2 n^2}{4 b g^3 (a+b x)^2}+\frac {3 B^2 d n^2}{2 b (b c-a d) g^3 (a+b x)}+\frac {3 B^2 d^2 n^2 \log (a+b x)}{2 b (b c-a d)^2 g^3}-\frac {B^2 d^2 n^2 \log ^2(a+b x)}{2 b (b c-a d)^2 g^3}-\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b g^3 (a+b x)^2}+\frac {B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b (b c-a d) g^3 (a+b x)}+\frac {B d^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b (b c-a d)^2 g^3}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 b g^3 (a+b x)^2}-\frac {3 B^2 d^2 n^2 \log (c+d x)}{2 b (b c-a d)^2 g^3}+\frac {B^2 d^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b (b c-a d)^2 g^3}-\frac {B d^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b (b c-a d)^2 g^3}-\frac {B^2 d^2 n^2 \log ^2(c+d x)}{2 b (b c-a d)^2 g^3}+\frac {B^2 d^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b (b c-a d)^2 g^3}-\frac {\left (B^2 d^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b (b c-a d)^2 g^3}-\frac {\left (B^2 d^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{b (b c-a d)^2 g^3}\\ &=-\frac {B^2 n^2}{4 b g^3 (a+b x)^2}+\frac {3 B^2 d n^2}{2 b (b c-a d) g^3 (a+b x)}+\frac {3 B^2 d^2 n^2 \log (a+b x)}{2 b (b c-a d)^2 g^3}-\frac {B^2 d^2 n^2 \log ^2(a+b x)}{2 b (b c-a d)^2 g^3}-\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b g^3 (a+b x)^2}+\frac {B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b (b c-a d) g^3 (a+b x)}+\frac {B d^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b (b c-a d)^2 g^3}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 b g^3 (a+b x)^2}-\frac {3 B^2 d^2 n^2 \log (c+d x)}{2 b (b c-a d)^2 g^3}+\frac {B^2 d^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b (b c-a d)^2 g^3}-\frac {B d^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b (b c-a d)^2 g^3}-\frac {B^2 d^2 n^2 \log ^2(c+d x)}{2 b (b c-a d)^2 g^3}+\frac {B^2 d^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b (b c-a d)^2 g^3}+\frac {B^2 d^2 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b (b c-a d)^2 g^3}+\frac {B^2 d^2 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b (b c-a d)^2 g^3}\\ \end {align*}
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Mathematica [C] time = 0.49, size = 463, normalized size = 1.61 \[ -\frac {\frac {B n \left (-4 d^2 (a+b x)^2 \log (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )+4 d^2 (a+b x)^2 \log (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )+2 (b c-a d)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )+4 d (a+b x) (a d-b c) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )+2 B d^2 n (a+b x)^2 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )\right )-2 B d^2 n (a+b x)^2 \left (2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac {d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )+B n \left (2 d^2 (a+b x)^2 \log (c+d x)+2 d (a+b x) (a d-b c)+(b c-a d)^2-2 d^2 (a+b x)^2 \log (a+b x)\right )-4 B d n (a+b x) (-d (a+b x) \log (c+d x)+d (a+b x) \log (a+b x)-a d+b c)\right )}{(b c-a d)^2}+2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 b g^3 (a+b x)^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.86, size = 651, normalized size = 2.26 \[ -\frac {2 \, A^{2} b^{2} c^{2} - 4 \, A^{2} a b c d + 2 \, A^{2} a^{2} d^{2} + {\left (B^{2} b^{2} c^{2} - 8 \, B^{2} a b c d + 7 \, B^{2} a^{2} d^{2}\right )} n^{2} + 2 \, {\left (B^{2} b^{2} c^{2} - 2 \, B^{2} a b c d + B^{2} a^{2} d^{2}\right )} \log \relax (e)^{2} - 2 \, {\left (B^{2} b^{2} d^{2} n^{2} x^{2} + 2 \, B^{2} a b d^{2} n^{2} x - {\left (B^{2} b^{2} c^{2} - 2 \, B^{2} a b c d\right )} n^{2}\right )} \log \left (\frac {b x + a}{d x + c}\right )^{2} + 2 \, {\left (A B b^{2} c^{2} - 4 \, A B a b c d + 3 \, A B a^{2} d^{2}\right )} n - 2 \, {\left (3 \, {\left (B^{2} b^{2} c d - B^{2} a b d^{2}\right )} n^{2} + 2 \, {\left (A B b^{2} c d - A B a b d^{2}\right )} n\right )} x + 2 \, {\left (2 \, A B b^{2} c^{2} - 4 \, A B a b c d + 2 \, A B a^{2} d^{2} - 2 \, {\left (B^{2} b^{2} c d - B^{2} a b d^{2}\right )} n x + {\left (B^{2} b^{2} c^{2} - 4 \, B^{2} a b c d + 3 \, B^{2} a^{2} d^{2}\right )} n - 2 \, {\left (B^{2} b^{2} d^{2} n x^{2} + 2 \, B^{2} a b d^{2} n x - {\left (B^{2} b^{2} c^{2} - 2 \, B^{2} a b c d\right )} n\right )} \log \left (\frac {b x + a}{d x + c}\right )\right )} \log \relax (e) + 2 \, {\left ({\left (B^{2} b^{2} c^{2} - 4 \, B^{2} a b c d\right )} n^{2} - {\left (3 \, B^{2} b^{2} d^{2} n^{2} + 2 \, A B b^{2} d^{2} n\right )} x^{2} + 2 \, {\left (A B b^{2} c^{2} - 2 \, A B a b c d\right )} n - 2 \, {\left (2 \, A B a b d^{2} n + {\left (B^{2} b^{2} c d + 2 \, B^{2} a b d^{2}\right )} n^{2}\right )} x\right )} \log \left (\frac {b x + a}{d x + c}\right )}{4 \, {\left ({\left (b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )} g^{3} x^{2} + 2 \, {\left (a b^{4} c^{2} - 2 \, a^{2} b^{3} c d + a^{3} b^{2} d^{2}\right )} g^{3} x + {\left (a^{2} b^{3} c^{2} - 2 \, a^{3} b^{2} c d + a^{4} b d^{2}\right )} g^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 10.43, size = 458, normalized size = 1.59 \[ -\frac {1}{4} \, {\left (\frac {2 \, {\left (B^{2} b n^{2} - \frac {2 \, {\left (b x + a\right )} B^{2} d n^{2}}{d x + c}\right )} \log \left (\frac {b x + a}{d x + c}\right )^{2}}{\frac {{\left (b x + a\right )}^{2} b c g^{3}}{{\left (d x + c\right )}^{2}} - \frac {{\left (b x + a\right )}^{2} a d g^{3}}{{\left (d x + c\right )}^{2}}} + \frac {2 \, {\left (B^{2} b n^{2} - \frac {4 \, {\left (b x + a\right )} B^{2} d n^{2}}{d x + c} + 2 \, A B b n + 2 \, B^{2} b n - \frac {4 \, {\left (b x + a\right )} A B d n}{d x + c} - \frac {4 \, {\left (b x + a\right )} B^{2} d n}{d x + c}\right )} \log \left (\frac {b x + a}{d x + c}\right )}{\frac {{\left (b x + a\right )}^{2} b c g^{3}}{{\left (d x + c\right )}^{2}} - \frac {{\left (b x + a\right )}^{2} a d g^{3}}{{\left (d x + c\right )}^{2}}} + \frac {B^{2} b n^{2} - \frac {8 \, {\left (b x + a\right )} B^{2} d n^{2}}{d x + c} + 2 \, A B b n + 2 \, B^{2} b n - \frac {8 \, {\left (b x + a\right )} A B d n}{d x + c} - \frac {8 \, {\left (b x + a\right )} B^{2} d n}{d x + c} + 2 \, A^{2} b + 4 \, A B b + 2 \, B^{2} b - \frac {4 \, {\left (b x + a\right )} A^{2} d}{d x + c} - \frac {8 \, {\left (b x + a\right )} A B d}{d x + c} - \frac {4 \, {\left (b x + a\right )} B^{2} d}{d x + c}}{\frac {{\left (b x + a\right )}^{2} b c g^{3}}{{\left (d x + c\right )}^{2}} - \frac {{\left (b x + a\right )}^{2} a d g^{3}}{{\left (d x + c\right )}^{2}}}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.28, size = 0, normalized size = 0.00 \[ \int \frac {\left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )^{2}}{\left (b g x +a g \right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.74, size = 861, normalized size = 2.99 \[ \frac {1}{2} \, A B n {\left (\frac {2 \, b d x - b c + 3 \, a d}{{\left (b^{4} c - a b^{3} d\right )} g^{3} x^{2} + 2 \, {\left (a b^{3} c - a^{2} b^{2} d\right )} g^{3} x + {\left (a^{2} b^{2} c - a^{3} b d\right )} g^{3}} + \frac {2 \, d^{2} \log \left (b x + a\right )}{{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} g^{3}} - \frac {2 \, d^{2} \log \left (d x + c\right )}{{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} g^{3}}\right )} + \frac {1}{4} \, {\left (2 \, n {\left (\frac {2 \, b d x - b c + 3 \, a d}{{\left (b^{4} c - a b^{3} d\right )} g^{3} x^{2} + 2 \, {\left (a b^{3} c - a^{2} b^{2} d\right )} g^{3} x + {\left (a^{2} b^{2} c - a^{3} b d\right )} g^{3}} + \frac {2 \, d^{2} \log \left (b x + a\right )}{{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} g^{3}} - \frac {2 \, d^{2} \log \left (d x + c\right )}{{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} g^{3}}\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 )} n^{2}}{a^{2} b^{3} c^{2} g^{3} - 2 \, a^{3} b^{2} c d g^{3} + a^{4} b d^{2} g^{3} + {\left (b^{5} c^{2} g^{3} - 2 \, a b^{4} c d g^{3} + a^{2} b^{3} d^{2} g^{3}\right )} x^{2} + 2 \, {\left (a b^{4} c^{2} g^{3} - 2 \, a^{2} b^{3} c d g^{3} + a^{3} b^{2} d^{2} g^{3}\right )} x}\right )} B^{2} - \frac {B^{2} \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right )^{2}}{2 \, {\left (b^{3} g^{3} x^{2} + 2 \, a b^{2} g^{3} x + a^{2} b g^{3}\right )}} - \frac {A B \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right )}{b^{3} g^{3} x^{2} + 2 \, a b^{2} g^{3} x + a^{2} b g^{3}} - \frac {A^{2}}{2 \, {\left (b^{3} g^{3} x^{2} + 2 \, a b^{2} g^{3} x + a^{2} b g^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.17, size = 506, normalized size = 1.76 \[ -{\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}^2\,\left (\frac {B^2}{2\,b\,\left (a^2\,g^3+2\,a\,b\,g^3\,x+b^2\,g^3\,x^2\right )}-\frac {B^2\,d^2}{2\,b\,g^3\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}\right )-\frac {\frac {2\,A^2\,a\,d-2\,A^2\,b\,c+7\,B^2\,a\,d\,n^2-B^2\,b\,c\,n^2+6\,A\,B\,a\,d\,n-2\,A\,B\,b\,c\,n}{2\,\left (a\,d-b\,c\right )}+\frac {d\,x\,\left (3\,b\,B^2\,n^2+2\,A\,b\,B\,n\right )}{a\,d-b\,c}}{2\,a^2\,b\,g^3+4\,a\,b^2\,g^3\,x+2\,b^3\,g^3\,x^2}-\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,\left (\frac {A\,B}{a^2\,b\,g^3+2\,a\,b^2\,g^3\,x+b^3\,g^3\,x^2}+\frac {B^2\,d^2\,\left (\frac {b\,g^3\,n\,\left (a\,d-b\,c\right )\,\left (2\,a\,d-b\,c\right )}{2\,d^2}+\frac {b^2\,g^3\,n\,x\,\left (a\,d-b\,c\right )}{d}+\frac {a\,b\,g^3\,n\,\left (a\,d-b\,c\right )}{2\,d}\right )}{b\,g^3\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )\,\left (a^2\,b\,g^3+2\,a\,b^2\,g^3\,x+b^3\,g^3\,x^2\right )}\right )-\frac {B\,d^2\,n\,\mathrm {atan}\left (\frac {\left (2\,b\,d\,x-\frac {2\,b^3\,c^2\,g^3-2\,a^2\,b\,d^2\,g^3}{2\,b\,g^3\,\left (a\,d-b\,c\right )}\right )\,1{}\mathrm {i}}{a\,d-b\,c}\right )\,\left (2\,A+3\,B\,n\right )\,1{}\mathrm {i}}{b\,g^3\,{\left (a\,d-b\,c\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {A^{2}}{a^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}\, dx + \int \frac {B^{2} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}^{2}}{a^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}\, dx + \int \frac {2 A B \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{a^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}\, dx}{g^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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