Optimal. Leaf size=135 \[ \frac {2 B n (b c-a d) \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b d}+\frac {(a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{b}+\frac {2 B^2 n^2 (b c-a d) \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{b d} \]
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Rubi [B] time = 0.62, antiderivative size = 275, normalized size of antiderivative = 2.04, number of steps used = 20, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {2523, 12, 2528, 2524, 2418, 2390, 2301, 2394, 2393, 2391} \[ \frac {2 a B^2 n^2 \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{b}+\frac {2 B^2 c n^2 \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{d}+\frac {2 a B n \log (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b}-\frac {2 B c n \log (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{d}+x \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2+\frac {2 B^2 c n^2 \log (c+d x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{d}+\frac {2 a B^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b}-\frac {a B^2 n^2 \log ^2(a+b x)}{b}-\frac {B^2 c n^2 \log ^2(c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2301
Rule 2390
Rule 2391
Rule 2393
Rule 2394
Rule 2418
Rule 2523
Rule 2524
Rule 2528
Rubi steps
\begin {align*} \int \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx &=x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2-(2 B n) \int \frac {(b c-a d) x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) (c+d x)} \, dx\\ &=x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2-(2 B (b c-a d) n) \int \frac {x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) (c+d x)} \, dx\\ &=x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2-(2 B (b c-a d) n) \int \left (-\frac {a \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)}+\frac {c \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)}\right ) \, dx\\ &=x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2+(2 a B n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx-(2 B c n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx\\ &=\frac {2 a B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b}+x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2-\frac {2 B c n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{d}-\frac {\left (2 a 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}{b}+\frac {\left (2 B^2 c 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}{d}\\ &=\frac {2 a B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b}+x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2-\frac {2 B c n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{d}-\frac {\left (2 a 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}{b}+\frac {\left (2 B^2 c 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}{d}\\ &=\frac {2 a B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b}+x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2-\frac {2 B c n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{d}-\left (2 a B^2 n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx-\left (2 B^2 c n^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx+\frac {\left (2 b B^2 c n^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{d}+\frac {\left (2 a B^2 d n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b}\\ &=\frac {2 a B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b}+x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2+\frac {2 B^2 c n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{d}-\frac {2 B c n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{d}+\frac {2 a B^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b}-\left (2 a B^2 n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx-\frac {\left (2 a B^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b}-\left (2 B^2 c n^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx-\frac {\left (2 B^2 c n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{d}\\ &=-\frac {a B^2 n^2 \log ^2(a+b x)}{b}+\frac {2 a B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b}+x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2+\frac {2 B^2 c n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{d}-\frac {2 B c n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{d}-\frac {B^2 c n^2 \log ^2(c+d x)}{d}+\frac {2 a B^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b}-\frac {\left (2 a 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 )}{b}-\frac {\left (2 B^2 c 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 )}{d}\\ &=-\frac {a B^2 n^2 \log ^2(a+b x)}{b}+\frac {2 a B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b}+x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2+\frac {2 B^2 c n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{d}-\frac {2 B c n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{d}-\frac {B^2 c n^2 \log ^2(c+d x)}{d}+\frac {2 a B^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b}+\frac {2 a B^2 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b}+\frac {2 B^2 c n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{d}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 226, normalized size = 1.67 \[ \frac {B n \left (2 a d \log (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )-2 b c \log (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )-a B d n \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )\right )+b B c n \left (2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac {d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )\right )}{b d}+x \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2 \]
Antiderivative was successfully verified.
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fricas [F] time = 1.10, size = 0, normalized size = 0.00 \[ {\rm integral}\left (B^{2} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )^{2} + 2 \, A B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A^{2}, x\right ) \]
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 = 0.15, size = 0, normalized size = 0.00 \[ \int \left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ 2 \, A B n {\left (\frac {a \log \left (b x + a\right )}{b} - \frac {c \log \left (d x + c\right )}{d}\right )} + 2 \, A B x \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A^{2} x + B^{2} {\left (\frac {2 \, b c n^{2} \log \left (b x + a\right ) \log \left (d x + c\right ) - b c n^{2} \log \left (d x + c\right )^{2} + b d x \log \left ({\left (b x + a\right )}^{n}\right )^{2} + b d x \log \left ({\left (d x + c\right )}^{n}\right )^{2} + 2 \, {\left (a d n \log \left (b x + a\right ) - b c n \log \left (d x + c\right ) + b d x \log \relax (e)\right )} \log \left ({\left (b x + a\right )}^{n}\right ) - 2 \, {\left (a d n \log \left (b x + a\right ) - b c n \log \left (d x + c\right ) + b d x \log \left ({\left (b x + a\right )}^{n}\right ) + b d x \log \relax (e)\right )} \log \left ({\left (d x + c\right )}^{n}\right )}{b d} - \int -\frac {b^{2} d x^{2} \log \relax (e)^{2} + a b c \log \relax (e)^{2} - {\left ({\left (2 \, n \log \relax (e) - \log \relax (e)^{2}\right )} b^{2} c - {\left (2 \, n \log \relax (e) + \log \relax (e)^{2}\right )} a b d\right )} x - 2 \, {\left (b^{2} c n^{2} x + 2 \, a b c n^{2} - a^{2} d n^{2}\right )} \log \left (b x + a\right )}{b^{2} d x^{2} + a b c + {\left (b^{2} c + a b d\right )} x}\,{d x}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}^2 \,d x \]
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 \[ \int \left (A + B \log {\left (e \left (\frac {a + b x}{c + d x}\right )^{n} \right )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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