Optimal. Leaf size=163 \[ \frac {(a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{i^2 (c+d x) (b c-a d)}-\frac {2 A B n (a+b x)}{i^2 (c+d x) (b c-a d)}-\frac {2 B^2 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{i^2 (c+d x) (b c-a d)}+\frac {2 B^2 n^2 (a+b x)}{i^2 (c+d x) (b c-a d)} \]
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Rubi [C] time = 0.76, antiderivative size = 514, normalized size of antiderivative = 3.15, number of steps used = 24, number of rules used = 11, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.314, Rules used = {2525, 12, 2528, 2524, 2418, 2390, 2301, 2394, 2393, 2391, 44} \[ \frac {2 b B^2 n^2 \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{d i^2 (b c-a d)}+\frac {2 b B^2 n^2 \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{d i^2 (b c-a d)}+\frac {2 b B n \log (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{d i^2 (b c-a d)}+\frac {2 B n \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{d i^2 (c+d x)}-\frac {2 b B n \log (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{d i^2 (b c-a d)}-\frac {\left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{d i^2 (c+d x)}-\frac {b B^2 n^2 \log ^2(a+b x)}{d i^2 (b c-a d)}-\frac {b B^2 n^2 \log ^2(c+d x)}{d i^2 (b c-a d)}-\frac {2 b B^2 n^2 \log (a+b x)}{d i^2 (b c-a d)}+\frac {2 b B^2 n^2 \log (c+d x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{d i^2 (b c-a d)}+\frac {2 b B^2 n^2 \log (c+d x)}{d i^2 (b c-a d)}+\frac {2 b B^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{d i^2 (b c-a d)}-\frac {2 B^2 n^2}{d i^2 (c+d x)} \]
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}{(197 c+197 d x)^2} \, dx &=-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{38809 d (c+d x)}+\frac {(2 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 )}{197 (a+b x) (c+d x)^2} \, dx}{197 d}\\ &=-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{38809 d (c+d x)}+\frac {(2 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) (c+d x)^2} \, dx}{38809 d}\\ &=-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{38809 d (c+d x)}+\frac {(2 B (b c-a d) n) \int \left (\frac {b^2 \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)}-\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) (c+d x)^2}-\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 (c+d x)}\right ) \, dx}{38809 d}\\ &=-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{38809 d (c+d x)}-\frac {(2 B n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c+d x)^2} \, dx}{38809}-\frac {(2 b B n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{38809 (b c-a d)}+\frac {\left (2 b^2 B n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{38809 d (b c-a d)}\\ &=\frac {2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{38809 d (c+d x)}+\frac {2 b B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{38809 d (b c-a d)}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{38809 d (c+d x)}-\frac {2 b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{38809 d (b c-a d)}-\frac {\left (2 B^2 n^2\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^2} \, dx}{38809 d}-\frac {\left (2 b B^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}{38809 d (b c-a d)}+\frac {\left (2 b B^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}{38809 d (b c-a d)}\\ &=\frac {2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{38809 d (c+d x)}+\frac {2 b B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{38809 d (b c-a d)}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{38809 d (c+d x)}-\frac {2 b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{38809 d (b c-a d)}-\frac {\left (2 b B^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}{38809 d (b c-a d)}+\frac {\left (2 b B^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}{38809 d (b c-a d)}-\frac {\left (2 B^2 (b c-a d) n^2\right ) \int \frac {1}{(a+b x) (c+d x)^2} \, dx}{38809 d}\\ &=\frac {2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{38809 d (c+d x)}+\frac {2 b B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{38809 d (b c-a d)}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{38809 d (c+d x)}-\frac {2 b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{38809 d (b c-a d)}+\frac {\left (2 b B^2 n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{38809 (b c-a d)}-\frac {\left (2 b B^2 n^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{38809 (b c-a d)}-\frac {\left (2 b^2 B^2 n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{38809 d (b c-a d)}+\frac {\left (2 b^2 B^2 n^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{38809 d (b c-a d)}-\frac {\left (2 B^2 (b c-a d) n^2\right ) \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}{38809 d}\\ &=-\frac {2 B^2 n^2}{38809 d (c+d x)}-\frac {2 b B^2 n^2 \log (a+b x)}{38809 d (b c-a d)}+\frac {2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{38809 d (c+d x)}+\frac {2 b B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{38809 d (b c-a d)}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{38809 d (c+d x)}+\frac {2 b B^2 n^2 \log (c+d x)}{38809 d (b c-a d)}+\frac {2 b B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{38809 d (b c-a d)}-\frac {2 b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{38809 d (b c-a d)}+\frac {2 b B^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{38809 d (b c-a d)}-\frac {\left (2 b B^2 n^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{38809 (b c-a d)}-\frac {\left (2 b B^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{38809 d (b c-a d)}-\frac {\left (2 b B^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{38809 d (b c-a d)}-\frac {\left (2 b^2 B^2 n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{38809 d (b c-a d)}\\ &=-\frac {2 B^2 n^2}{38809 d (c+d x)}-\frac {2 b B^2 n^2 \log (a+b x)}{38809 d (b c-a d)}-\frac {b B^2 n^2 \log ^2(a+b x)}{38809 d (b c-a d)}+\frac {2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{38809 d (c+d x)}+\frac {2 b B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{38809 d (b c-a d)}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{38809 d (c+d x)}+\frac {2 b B^2 n^2 \log (c+d x)}{38809 d (b c-a d)}+\frac {2 b B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{38809 d (b c-a d)}-\frac {2 b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{38809 d (b c-a d)}-\frac {b B^2 n^2 \log ^2(c+d x)}{38809 d (b c-a d)}+\frac {2 b B^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{38809 d (b c-a d)}-\frac {\left (2 b B^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 )}{38809 d (b c-a d)}-\frac {\left (2 b B^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 )}{38809 d (b c-a d)}\\ &=-\frac {2 B^2 n^2}{38809 d (c+d x)}-\frac {2 b B^2 n^2 \log (a+b x)}{38809 d (b c-a d)}-\frac {b B^2 n^2 \log ^2(a+b x)}{38809 d (b c-a d)}+\frac {2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{38809 d (c+d x)}+\frac {2 b B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{38809 d (b c-a d)}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{38809 d (c+d x)}+\frac {2 b B^2 n^2 \log (c+d x)}{38809 d (b c-a d)}+\frac {2 b B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{38809 d (b c-a d)}-\frac {2 b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{38809 d (b c-a d)}-\frac {b B^2 n^2 \log ^2(c+d x)}{38809 d (b c-a d)}+\frac {2 b B^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{38809 d (b c-a d)}+\frac {2 b B^2 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{38809 d (b c-a d)}+\frac {2 b B^2 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{38809 d (b c-a d)}\\ \end {align*}
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Mathematica [C] time = 0.45, size = 331, normalized size = 2.03 \[ \frac {\frac {B n \left (2 (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )+2 b (c+d x) \log (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )-2 b (c+d x) \log (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )-b B n (c+d x) \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )\right )+b B n (c+d x) \left (2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac {d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )-2 B n (b (c+d x) \log (a+b x)-a d-b (c+d x) \log (c+d x)+b c)\right )}{b c-a d}-\left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{d i^2 (c+d x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 263, normalized size = 1.61 \[ -\frac {A^{2} b c - A^{2} a d + 2 \, {\left (B^{2} b c - B^{2} a d\right )} n^{2} + {\left (B^{2} b c - B^{2} a d\right )} \log \relax (e)^{2} - {\left (B^{2} b d n^{2} x + B^{2} a d n^{2}\right )} \log \left (\frac {b x + a}{d x + c}\right )^{2} - 2 \, {\left (A B b c - A B a d\right )} n + 2 \, {\left (A B b c - A B a d - {\left (B^{2} b c - B^{2} a d\right )} n - {\left (B^{2} b d n x + B^{2} a d n\right )} \log \left (\frac {b x + a}{d x + c}\right )\right )} \log \relax (e) + 2 \, {\left (B^{2} a d n^{2} - A B a d n + {\left (B^{2} b d n^{2} - A B b d n\right )} x\right )} \log \left (\frac {b x + a}{d x + c}\right )}{{\left (b c d^{2} - a d^{3}\right )} i^{2} x + {\left (b c^{2} d - a c d^{2}\right )} i^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 4.78, size = 156, normalized size = 0.96 \[ -{\left (\frac {{\left (b x + a\right )} B^{2} n^{2} \log \left (\frac {b x + a}{d x + c}\right )^{2}}{d x + c} - \frac {2 \, {\left (B^{2} n^{2} - A B n - B^{2} n\right )} {\left (b x + a\right )} \log \left (\frac {b x + a}{d x + c}\right )}{d x + c} + \frac {{\left (2 \, B^{2} n^{2} - 2 \, A B n - 2 \, B^{2} n + A^{2} + 2 \, A B + B^{2}\right )} {\left (b x + a\right )}}{d x + c}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.32, size = 0, normalized size = 0.00 \[ \int \frac {\left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )^{2}}{\left (d i x +c i \right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.92, size = 428, normalized size = 2.63 \[ 2 \, A B n {\left (\frac {1}{d^{2} i^{2} x + c d i^{2}} + \frac {b \log \left (b x + a\right )}{{\left (b c d - a d^{2}\right )} i^{2}} - \frac {b \log \left (d x + c\right )}{{\left (b c d - a d^{2}\right )} i^{2}}\right )} + {\left (2 \, n {\left (\frac {1}{d^{2} i^{2} x + c d i^{2}} + \frac {b \log \left (b x + a\right )}{{\left (b c d - a d^{2}\right )} i^{2}} - \frac {b \log \left (d x + c\right )}{{\left (b c d - a d^{2}\right )} i^{2}}\right )} \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right ) - \frac {{\left ({\left (b d x + b c\right )} \log \left (b x + a\right )^{2} + {\left (b d x + b c\right )} \log \left (d x + c\right )^{2} + 2 \, b c - 2 \, a d + 2 \, {\left (b d x + b c\right )} \log \left (b x + a\right ) - 2 \, {\left (b d x + b c + {\left (b d x + b c\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )\right )} n^{2}}{b c^{2} d i^{2} - a c d^{2} i^{2} + {\left (b c d^{2} i^{2} - a d^{3} i^{2}\right )} x}\right )} B^{2} - \frac {B^{2} \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right )^{2}}{d^{2} i^{2} x + c d i^{2}} - \frac {2 \, A B \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right )}{d^{2} i^{2} x + c d i^{2}} - \frac {A^{2}}{d^{2} i^{2} x + c d i^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.27, size = 237, normalized size = 1.45 \[ \ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,\left (\frac {2\,B^2\,n}{x\,d^2\,i^2+c\,d\,i^2}-\frac {2\,A\,B}{x\,d^2\,i^2+c\,d\,i^2}\right )-{\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}^2\,\left (\frac {B^2}{d\,\left (c\,i^2+d\,i^2\,x\right )}+\frac {B^2\,b}{d\,i^2\,\left (a\,d-b\,c\right )}\right )-\frac {A^2-2\,A\,B\,n+2\,B^2\,n^2}{x\,d^2\,i^2+c\,d\,i^2}+\frac {B\,b\,n\,\mathrm {atan}\left (\frac {\left (2\,b\,d\,x+\frac {a\,d^2\,i^2+b\,c\,d\,i^2}{d\,i^2}\right )\,1{}\mathrm {i}}{a\,d-b\,c}\right )\,\left (A-B\,n\right )\,4{}\mathrm {i}}{d\,i^2\,\left (a\,d-b\,c\right )} \]
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 \[ \frac {\int \frac {A^{2}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx + \int \frac {B^{2} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}^{2}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx + \int \frac {2 A B \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx}{i^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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