Optimal. Leaf size=309 \[ -\frac{2 b d m n \text{PolyLog}\left (2,\frac{e x}{d}+1\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e}+\frac{2 b^2 d m n^2 \text{PolyLog}\left (2,\frac{e x}{d}+1\right )}{e}+\frac{2 b^2 d m n^2 \text{PolyLog}\left (3,\frac{e x}{d}+1\right )}{e}+\frac{(d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac{m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac{d m \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-2 a b n x \log \left (f x^m\right )+2 a b m n x+2 b m n x (a-b n)-\frac{2 b^2 n (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e}+\frac{4 b^2 m n (d+e x) \log \left (c (d+e x)^n\right )}{e}+\frac{2 b^2 d m n \log \left (-\frac{e x}{d}\right ) \log \left (c (d+e x)^n\right )}{e}+2 b^2 n^2 x \log \left (f x^m\right )-4 b^2 m n^2 x \]
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Rubi [A] time = 0.452542, antiderivative size = 309, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 12, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.522, Rules used = {2389, 2296, 2295, 2423, 2411, 43, 2351, 2317, 2391, 2353, 2374, 6589} \[ -\frac{2 b d m n \text{PolyLog}\left (2,\frac{e x}{d}+1\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e}+\frac{2 b^2 d m n^2 \text{PolyLog}\left (2,\frac{e x}{d}+1\right )}{e}+\frac{2 b^2 d m n^2 \text{PolyLog}\left (3,\frac{e x}{d}+1\right )}{e}+\frac{(d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac{m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac{d m \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-2 a b n x \log \left (f x^m\right )+2 a b m n x+2 b m n x (a-b n)-\frac{2 b^2 n (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e}+\frac{4 b^2 m n (d+e x) \log \left (c (d+e x)^n\right )}{e}+\frac{2 b^2 d m n \log \left (-\frac{e x}{d}\right ) \log \left (c (d+e x)^n\right )}{e}+2 b^2 n^2 x \log \left (f x^m\right )-4 b^2 m n^2 x \]
Antiderivative was successfully verified.
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Rule 2389
Rule 2296
Rule 2295
Rule 2423
Rule 2411
Rule 43
Rule 2351
Rule 2317
Rule 2391
Rule 2353
Rule 2374
Rule 6589
Rubi steps
\begin{align*} \int \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx &=-2 a b n x \log \left (f x^m\right )+2 b^2 n^2 x \log \left (f x^m\right )-\frac{2 b^2 n (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e}+\frac{(d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-m \int \left (-2 a b n+2 b^2 n^2-\frac{2 b^2 n (d+e x) \log \left (c (d+e x)^n\right )}{e x}+\frac{(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e x}\right ) \, dx\\ &=2 b m n (a-b n) x-2 a b n x \log \left (f x^m\right )+2 b^2 n^2 x \log \left (f x^m\right )-\frac{2 b^2 n (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e}+\frac{(d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac{m \int \frac{(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x} \, dx}{e}+\frac{\left (2 b^2 m n\right ) \int \frac{(d+e x) \log \left (c (d+e x)^n\right )}{x} \, dx}{e}\\ &=2 b m n (a-b n) x-2 a b n x \log \left (f x^m\right )+2 b^2 n^2 x \log \left (f x^m\right )-\frac{2 b^2 n (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e}+\frac{(d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac{m \operatorname{Subst}\left (\int \frac{x \left (a+b \log \left (c x^n\right )\right )^2}{-\frac{d}{e}+\frac{x}{e}} \, dx,x,d+e x\right )}{e^2}+\frac{\left (2 b^2 m n\right ) \operatorname{Subst}\left (\int \frac{x \log \left (c x^n\right )}{-\frac{d}{e}+\frac{x}{e}} \, dx,x,d+e x\right )}{e^2}\\ &=2 b m n (a-b n) x-2 a b n x \log \left (f x^m\right )+2 b^2 n^2 x \log \left (f x^m\right )-\frac{2 b^2 n (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e}+\frac{(d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac{m \operatorname{Subst}\left (\int \left (e \left (a+b \log \left (c x^n\right )\right )^2-\frac{d e \left (a+b \log \left (c x^n\right )\right )^2}{d-x}\right ) \, dx,x,d+e x\right )}{e^2}+\frac{\left (2 b^2 m n\right ) \operatorname{Subst}\left (\int \left (e \log \left (c x^n\right )-\frac{d e \log \left (c x^n\right )}{d-x}\right ) \, dx,x,d+e x\right )}{e^2}\\ &=2 b m n (a-b n) x-2 a b n x \log \left (f x^m\right )+2 b^2 n^2 x \log \left (f x^m\right )-\frac{2 b^2 n (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e}+\frac{(d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac{m \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{e}+\frac{(d m) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{d-x} \, dx,x,d+e x\right )}{e}+\frac{\left (2 b^2 m n\right ) \operatorname{Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e}-\frac{\left (2 b^2 d m n\right ) \operatorname{Subst}\left (\int \frac{\log \left (c x^n\right )}{d-x} \, dx,x,d+e x\right )}{e}\\ &=-2 b^2 m n^2 x+2 b m n (a-b n) x-2 a b n x \log \left (f x^m\right )+2 b^2 n^2 x \log \left (f x^m\right )+\frac{2 b^2 m n (d+e x) \log \left (c (d+e x)^n\right )}{e}+\frac{2 b^2 d m n \log \left (-\frac{e x}{d}\right ) \log \left (c (d+e x)^n\right )}{e}-\frac{2 b^2 n (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e}-\frac{m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac{d m \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac{(d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac{(2 b m n) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{e}+\frac{(2 b d m n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1-\frac{x}{d}\right )}{x} \, dx,x,d+e x\right )}{e}-\frac{\left (2 b^2 d m n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{d}\right )}{x} \, dx,x,d+e x\right )}{e}\\ &=2 a b m n x-2 b^2 m n^2 x+2 b m n (a-b n) x-2 a b n x \log \left (f x^m\right )+2 b^2 n^2 x \log \left (f x^m\right )+\frac{2 b^2 m n (d+e x) \log \left (c (d+e x)^n\right )}{e}+\frac{2 b^2 d m n \log \left (-\frac{e x}{d}\right ) \log \left (c (d+e x)^n\right )}{e}-\frac{2 b^2 n (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e}-\frac{m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac{d m \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac{(d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac{2 b^2 d m n^2 \text{Li}_2\left (1+\frac{e x}{d}\right )}{e}-\frac{2 b d m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (1+\frac{e x}{d}\right )}{e}+\frac{\left (2 b^2 m n\right ) \operatorname{Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e}+\frac{\left (2 b^2 d m n^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{x}{d}\right )}{x} \, dx,x,d+e x\right )}{e}\\ &=2 a b m n x-4 b^2 m n^2 x+2 b m n (a-b n) x-2 a b n x \log \left (f x^m\right )+2 b^2 n^2 x \log \left (f x^m\right )+\frac{4 b^2 m n (d+e x) \log \left (c (d+e x)^n\right )}{e}+\frac{2 b^2 d m n \log \left (-\frac{e x}{d}\right ) \log \left (c (d+e x)^n\right )}{e}-\frac{2 b^2 n (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e}-\frac{m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac{d m \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac{(d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac{2 b^2 d m n^2 \text{Li}_2\left (1+\frac{e x}{d}\right )}{e}-\frac{2 b d m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (1+\frac{e x}{d}\right )}{e}+\frac{2 b^2 d m n^2 \text{Li}_3\left (1+\frac{e x}{d}\right )}{e}\\ \end{align*}
Mathematica [A] time = 0.473595, size = 549, normalized size = 1.78 \[ -2 b e m n \left (\frac{x (\log (x)-1)}{e}-\frac{d \left (\frac{\text{PolyLog}\left (2,-\frac{e x}{d}\right )}{e}+\frac{\log (x) \log \left (\frac{d+e x}{d}\right )}{e}\right )}{e}\right ) \left (a+b \left (\log \left (c (d+e x)^n\right )-n \log (d+e x)\right )\right )+b^2 m n^2 \left (-2 e \left (\frac{d \text{PolyLog}\left (2,-\frac{e x}{d}\right )-e x \log (d+e x)-d \log (d+e x)+\log (x) \left (e x \log (d+e x)+d \log \left (\frac{e x}{d}+1\right )-e x\right )+2 e x}{e^2}-\frac{d \left (\text{PolyLog}\left (3,\frac{d+e x}{d}\right )-\log (d+e x) \text{PolyLog}\left (2,\frac{d+e x}{d}\right )+\frac{1}{2} \left (\log (x)-\log \left (-\frac{e x}{d}\right )\right ) \log ^2(d+e x)\right )}{e^2}\right )+2 e \left (-\frac{d \log ^2(d+e x)}{2 e^2}+\frac{d \log (d+e x)}{e^2}+\frac{x \log (d+e x)}{e}-\frac{x}{e}\right )-x \log ^2(d+e x)+x \log (x) \log ^2(d+e x)\right )-2 b n x \left (\log \left (f x^m\right )+m (-\log (x))-m\right ) \left (a+b \left (\log \left (c (d+e x)^n\right )-n \log (d+e x)\right )\right )+2 b n x \log (d+e x) \left (\log \left (f x^m\right )-m\right ) \left (a+b \left (\log \left (c (d+e x)^n\right )-n \log (d+e x)\right )\right )+\frac{2 b d n \log (d+e x) \left (\log \left (f x^m\right )+m (-\log (x))-m\right ) \left (a+b \left (\log \left (c (d+e x)^n\right )-n \log (d+e x)\right )\right )}{e}-x \left (m-\log \left (f x^m\right )\right ) \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )^2+b^2 n^2 \left (x \log ^2(d+e x)-2 e \left (-\frac{d \log ^2(d+e x)}{2 e^2}+\frac{d \log (d+e x)}{e^2}+\frac{x \log (d+e x)}{e}-\frac{x}{e}\right )\right ) \left (\log \left (f x^m\right )-m \log (x)\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 1.871, size = 0, normalized size = 0. \begin{align*} \int \ln \left ( f{x}^{m} \right ) \left ( a+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -{\left (b^{2}{\left (m - \log \left (f\right )\right )} x - b^{2} x \log \left (x^{m}\right )\right )} \log \left ({\left (e x + d\right )}^{n}\right )^{2} + \int \frac{b^{2} d \log \left (c\right )^{2} \log \left (f\right ) + 2 \, a b d \log \left (c\right ) \log \left (f\right ) + a^{2} d \log \left (f\right ) +{\left (b^{2} e \log \left (c\right )^{2} \log \left (f\right ) + 2 \, a b e \log \left (c\right ) \log \left (f\right ) + a^{2} e \log \left (f\right )\right )} x + 2 \,{\left (b^{2} d \log \left (c\right ) \log \left (f\right ) + a b d \log \left (f\right ) +{\left (a b e \log \left (f\right ) +{\left (e \log \left (c\right ) \log \left (f\right ) +{\left (m n - n \log \left (f\right )\right )} e\right )} b^{2}\right )} x +{\left (b^{2} d \log \left (c\right ) + a b d -{\left ({\left (e n - e \log \left (c\right )\right )} b^{2} - a b e\right )} x\right )} \log \left (x^{m}\right )\right )} \log \left ({\left (e x + d\right )}^{n}\right ) +{\left (b^{2} d \log \left (c\right )^{2} + 2 \, a b d \log \left (c\right ) + a^{2} d +{\left (b^{2} e \log \left (c\right )^{2} + 2 \, a b e \log \left (c\right ) + a^{2} e\right )} x\right )} \log \left (x^{m}\right )}{e x + d}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (b^{2} \log \left ({\left (e x + d\right )}^{n} c\right )^{2} \log \left (f x^{m}\right ) + 2 \, a b \log \left ({\left (e x + d\right )}^{n} c\right ) \log \left (f x^{m}\right ) + a^{2} \log \left (f x^{m}\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} \log \left (f x^{m}\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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