Optimal. Leaf size=131 \[ \frac {2 B n \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{b}-\frac {\log \left (1-\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{b}+\frac {2 B^2 n^2 \text {Li}_3\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b} \]
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Rubi [A] time = 0.50, antiderivative size = 227, normalized size of antiderivative = 1.73, number of steps used = 10, number of rules used = 8, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {6742, 2488, 2411, 2343, 2333, 2315, 2506, 6610} \[ \frac {2 A B n \text {PolyLog}\left (2,\frac {b c-a d}{d (a+b x)}+1\right )}{b}+\frac {2 B^2 n \text {PolyLog}\left (2,\frac {b c-a d}{d (a+b x)}+1\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b}+\frac {2 B^2 n^2 \text {PolyLog}\left (3,\frac {b c-a d}{d (a+b x)}+1\right )}{b}+\frac {A^2 \log (a+b x)}{b}-\frac {2 A B \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b}-\frac {B^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 2315
Rule 2333
Rule 2343
Rule 2411
Rule 2488
Rule 2506
Rule 6610
Rule 6742
Rubi steps
\begin {align*} \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{a+b x} \, dx &=\int \left (\frac {A^2}{a+b x}+\frac {2 A B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{a+b x}+\frac {B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{a+b x}\right ) \, dx\\ &=\frac {A^2 \log (a+b x)}{b}+(2 A B) \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{a+b x} \, dx+B^2 \int \frac {\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{a+b x} \, dx\\ &=\frac {A^2 \log (a+b x)}{b}-\frac {2 A B \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b}-\frac {B^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b}+\frac {(2 A B (b c-a d) n) \int \frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{b}+\frac {\left (2 B^2 (b c-a d) n\right ) \int \frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x) (c+d x)} \, dx}{b}\\ &=\frac {A^2 \log (a+b x)}{b}-\frac {2 A B \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b}-\frac {B^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b}+\frac {2 B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b}+\frac {(2 A B (b c-a d) n) \operatorname {Subst}\left (\int \frac {\log \left (-\frac {b c-a d}{d x}\right )}{x \left (\frac {b c-a d}{b}+\frac {d x}{b}\right )} \, dx,x,a+b x\right )}{b^2}-\frac {\left (2 B^2 (b c-a d) n^2\right ) \int \frac {\text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{b}\\ &=\frac {A^2 \log (a+b x)}{b}-\frac {2 A B \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b}-\frac {B^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b}+\frac {2 B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b}+\frac {2 B^2 n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b}-\frac {(2 A B (b c-a d) n) \operatorname {Subst}\left (\int \frac {\log \left (-\frac {(b c-a d) x}{d}\right )}{\left (\frac {b c-a d}{b}+\frac {d}{b x}\right ) x} \, dx,x,\frac {1}{a+b x}\right )}{b^2}\\ &=\frac {A^2 \log (a+b x)}{b}-\frac {2 A B \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b}-\frac {B^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b}+\frac {2 B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b}+\frac {2 B^2 n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b}-\frac {(2 A B (b c-a d) n) \operatorname {Subst}\left (\int \frac {\log \left (-\frac {(b c-a d) x}{d}\right )}{\frac {d}{b}+\frac {(b c-a d) x}{b}} \, dx,x,\frac {1}{a+b x}\right )}{b^2}\\ &=\frac {A^2 \log (a+b x)}{b}-\frac {2 A B \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b}-\frac {B^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b}+\frac {2 A B n \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b}+\frac {2 B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b}+\frac {2 B^2 n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b}\\ \end {align*}
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Mathematica [B] time = 0.19, size = 269, normalized size = 2.05 \[ \frac {A^2 \log (a+b x)-2 A B \log \left (\frac {a d-b c}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )+2 A B n \text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )-A B n \log ^2\left (\frac {a d-b c}{d (a+b x)}\right )-2 A B n \log \left (\frac {a d-b c}{d (a+b x)}\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )+2 B^2 n \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )-B^2 \log \left (\frac {a d-b c}{d (a+b x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )+2 B^2 n^2 \text {Li}_3\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.78, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {B^{2} \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )^{2} + 2 \, A B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A^{2}}{b x + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{2}}{b x + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 2.57, size = 0, normalized size = 0.00 \[ \int \frac {\left (B \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )+A \right )^{2}}{b x +a}\, dx \]
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 \[ \frac {B^{2} \log \left (b x + a\right ) \log \left ({\left (d x + c\right )}^{n}\right )^{2}}{b} + \frac {A^{2} \log \left (b x + a\right )}{b} - \int -\frac {B^{2} b c \log \relax (e)^{2} + 2 \, A B b c \log \relax (e) + {\left (B^{2} b d x + B^{2} b c\right )} \log \left ({\left (b x + a\right )}^{n}\right )^{2} + {\left (B^{2} b d \log \relax (e)^{2} + 2 \, A B b d \log \relax (e)\right )} x + 2 \, {\left (B^{2} b c \log \relax (e) + A B b c + {\left (B^{2} b d \log \relax (e) + A B b d\right )} x\right )} \log \left ({\left (b x + a\right )}^{n}\right ) - 2 \, {\left (B^{2} b c \log \relax (e) + A B b c + {\left (B^{2} b d \log \relax (e) + A B b d\right )} x + {\left (B^{2} b d n x + B^{2} a d n\right )} \log \left (b x + a\right ) + {\left (B^{2} b d x + B^{2} b c\right )} \log \left ({\left (b x + a\right )}^{n}\right )\right )} \log \left ({\left (d x + c\right )}^{n}\right )}{b^{2} d x^{2} + a b c + {\left (b^{2} c + a b d\right )} x}\,{d x} \]
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 \[ \int \frac {{\left (A+B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )\right )}^2}{a+b\,x} \,d x \]
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 \[ \int \frac {\left (A + B \log {\left (e \left (a + b x\right )^{n} \left (c + d x\right )^{- n} \right )}\right )^{2}}{a + b x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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