Optimal. Leaf size=131 \[ -\frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2 \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{b}+\frac {2 B n \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b}+\frac {2 B^2 n^2 \text {Li}_3\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b} \]
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Rubi [A]
time = 0.12, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {2573, 2549,
2379, 2421, 6724} \begin {gather*} \frac {2 B n \text {PolyLog}\left (2,\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 {2 B^2 n^2 \text {PolyLog}\left (3,\frac {b (c+d x)}{d (a+b x)}\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 2379
Rule 2421
Rule 2549
Rule 2573
Rule 6724
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) \text {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) \text {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) \text {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] Leaf count is larger than twice the leaf count of optimal. \(443\) vs. \(2(131)=262\).
time = 0.13, size = 443, normalized size = 3.38 \begin {gather*} \frac {B^2 n^2 \log ^3(a+b x)+3 B n \log ^2(a+b x) \left (A+B \left (-n \log (a+b x)+n \log (c+d x)+\log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )\right )+3 \log (a+b x) \left (A+B \left (-n \log (a+b x)+n \log (c+d x)+\log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )\right )^2-6 A B n \left (\log \left (\frac {d (a+b x)}{-b c+a d}\right ) \log (c+d x)+\text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )-6 B^2 n \left (-n \log (a+b x)+n \log (c+d x)+\log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \left (\log \left (\frac {d (a+b x)}{-b c+a d}\right ) \log (c+d x)+\text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )-6 B^2 n^2 \left (\frac {1}{2} \log ^2(a+b x) \left (\log (c+d x)-\log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-\log (a+b x) \text {Li}_2\left (\frac {d (a+b x)}{-b c+a d}\right )+\text {Li}_3\left (\frac {d (a+b x)}{-b c+a d}\right )\right )+3 B^2 n^2 \left (\log \left (\frac {d (a+b x)}{-b c+a d}\right ) \log ^2(c+d x)+2 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )-2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )\right )}{3 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.08, size = 0, normalized size = 0.00 \[\int \frac {\left (A +B \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )\right )^{2}}{b x +a}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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 \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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