Optimal. Leaf size=273 \[ \frac{2 b f \text{PolyLog}\left (2,-\frac{f h-e i}{i (e+f x)}\right ) (a+b \log (c (e+f x)))}{d (f h-e i)^2}+\frac{2 b^2 f \text{PolyLog}\left (2,-\frac{i (e+f x)}{f h-e i}\right )}{d (f h-e i)^2}+\frac{2 b^2 f \text{PolyLog}\left (3,-\frac{f h-e i}{i (e+f x)}\right )}{d (f h-e i)^2}+\frac{2 b f \log \left (\frac{f (h+i x)}{f h-e i}\right ) (a+b \log (c (e+f x)))}{d (f h-e i)^2}-\frac{i (e+f x) (a+b \log (c (e+f x)))^2}{d (h+i x) (f h-e i)^2}-\frac{f \log \left (\frac{f h-e i}{i (e+f x)}+1\right ) (a+b \log (c (e+f x)))^2}{d (f h-e i)^2} \]
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Rubi [A] time = 0.637081, antiderivative size = 300, normalized size of antiderivative = 1.1, number of steps used = 12, number of rules used = 11, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.344, Rules used = {2411, 12, 2347, 2344, 2302, 30, 2317, 2374, 6589, 2318, 2391} \[ -\frac{2 b f \text{PolyLog}\left (2,-\frac{i (e+f x)}{f h-e i}\right ) (a+b \log (c (e+f x)))}{d (f h-e i)^2}+\frac{2 b^2 f \text{PolyLog}\left (2,-\frac{i (e+f x)}{f h-e i}\right )}{d (f h-e i)^2}+\frac{2 b^2 f \text{PolyLog}\left (3,-\frac{i (e+f x)}{f h-e i}\right )}{d (f h-e i)^2}+\frac{f (a+b \log (c (e+f x)))^3}{3 b d (f h-e i)^2}-\frac{f \log \left (\frac{f (h+i x)}{f h-e i}\right ) (a+b \log (c (e+f x)))^2}{d (f h-e i)^2}-\frac{i (e+f x) (a+b \log (c (e+f x)))^2}{d (h+i x) (f h-e i)^2}+\frac{2 b f \log \left (\frac{f (h+i x)}{f h-e i}\right ) (a+b \log (c (e+f x)))}{d (f h-e i)^2} \]
Antiderivative was successfully verified.
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Rule 2411
Rule 12
Rule 2347
Rule 2344
Rule 2302
Rule 30
Rule 2317
Rule 2374
Rule 6589
Rule 2318
Rule 2391
Rubi steps
\begin{align*} \int \frac{(a+b \log (c (e+f x)))^2}{(h+189 x)^2 (d e+d f x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a+b \log (c x))^2}{d x \left (\frac{-189 e+f h}{f}+\frac{189 x}{f}\right )^2} \, dx,x,e+f x\right )}{f}\\ &=\frac{\operatorname{Subst}\left (\int \frac{(a+b \log (c x))^2}{x \left (\frac{-189 e+f h}{f}+\frac{189 x}{f}\right )^2} \, dx,x,e+f x\right )}{d f}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{(a+b \log (c x))^2}{x \left (\frac{-189 e+f h}{f}+\frac{189 x}{f}\right )} \, dx,x,e+f x\right )}{d (189 e-f h)}+\frac{189 \operatorname{Subst}\left (\int \frac{(a+b \log (c x))^2}{\left (\frac{-189 e+f h}{f}+\frac{189 x}{f}\right )^2} \, dx,x,e+f x\right )}{d f (189 e-f h)}\\ &=-\frac{189 (e+f x) (a+b \log (c (e+f x)))^2}{d (189 e-f h)^2 (h+189 x)}-\frac{189 \operatorname{Subst}\left (\int \frac{(a+b \log (c x))^2}{\frac{-189 e+f h}{f}+\frac{189 x}{f}} \, dx,x,e+f x\right )}{d (189 e-f h)^2}+\frac{(378 b) \operatorname{Subst}\left (\int \frac{a+b \log (c x)}{\frac{-189 e+f h}{f}+\frac{189 x}{f}} \, dx,x,e+f x\right )}{d (189 e-f h)^2}+\frac{f \operatorname{Subst}\left (\int \frac{(a+b \log (c x))^2}{x} \, dx,x,e+f x\right )}{d (189 e-f h)^2}\\ &=\frac{2 b f \log \left (-\frac{f (h+189 x)}{189 e-f h}\right ) (a+b \log (c (e+f x)))}{d (189 e-f h)^2}-\frac{189 (e+f x) (a+b \log (c (e+f x)))^2}{d (189 e-f h)^2 (h+189 x)}-\frac{f \log \left (-\frac{f (h+189 x)}{189 e-f h}\right ) (a+b \log (c (e+f x)))^2}{d (189 e-f h)^2}+\frac{f \operatorname{Subst}\left (\int x^2 \, dx,x,a+b \log (c (e+f x))\right )}{b d (189 e-f h)^2}+\frac{(2 b f) \operatorname{Subst}\left (\int \frac{(a+b \log (c x)) \log \left (1+\frac{189 x}{-189 e+f h}\right )}{x} \, dx,x,e+f x\right )}{d (189 e-f h)^2}-\frac{\left (2 b^2 f\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{189 x}{-189 e+f h}\right )}{x} \, dx,x,e+f x\right )}{d (189 e-f h)^2}\\ &=\frac{2 b f \log \left (-\frac{f (h+189 x)}{189 e-f h}\right ) (a+b \log (c (e+f x)))}{d (189 e-f h)^2}-\frac{189 (e+f x) (a+b \log (c (e+f x)))^2}{d (189 e-f h)^2 (h+189 x)}-\frac{f \log \left (-\frac{f (h+189 x)}{189 e-f h}\right ) (a+b \log (c (e+f x)))^2}{d (189 e-f h)^2}+\frac{f (a+b \log (c (e+f x)))^3}{3 b d (189 e-f h)^2}+\frac{2 b^2 f \text{Li}_2\left (\frac{189 (e+f x)}{189 e-f h}\right )}{d (189 e-f h)^2}-\frac{2 b f (a+b \log (c (e+f x))) \text{Li}_2\left (\frac{189 (e+f x)}{189 e-f h}\right )}{d (189 e-f h)^2}+\frac{\left (2 b^2 f\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{189 x}{-189 e+f h}\right )}{x} \, dx,x,e+f x\right )}{d (189 e-f h)^2}\\ &=\frac{2 b f \log \left (-\frac{f (h+189 x)}{189 e-f h}\right ) (a+b \log (c (e+f x)))}{d (189 e-f h)^2}-\frac{189 (e+f x) (a+b \log (c (e+f x)))^2}{d (189 e-f h)^2 (h+189 x)}-\frac{f \log \left (-\frac{f (h+189 x)}{189 e-f h}\right ) (a+b \log (c (e+f x)))^2}{d (189 e-f h)^2}+\frac{f (a+b \log (c (e+f x)))^3}{3 b d (189 e-f h)^2}+\frac{2 b^2 f \text{Li}_2\left (\frac{189 (e+f x)}{189 e-f h}\right )}{d (189 e-f h)^2}-\frac{2 b f (a+b \log (c (e+f x))) \text{Li}_2\left (\frac{189 (e+f x)}{189 e-f h}\right )}{d (189 e-f h)^2}+\frac{2 b^2 f \text{Li}_3\left (\frac{189 (e+f x)}{189 e-f h}\right )}{d (189 e-f h)^2}\\ \end{align*}
Mathematica [A] time = 0.51819, size = 360, normalized size = 1.32 \[ \frac{3 a b \left (-2 f (h+i x) \left (\text{PolyLog}\left (2,\frac{i (e+f x)}{e i-f h}\right )+\log (c (e+f x)) \log \left (\frac{f (h+i x)}{f h-e i}\right )\right )+f (h+i x) \log ^2(c (e+f x))+2 (f h-e i) \log (c (e+f x))-2 f (h+i x) \log (e+f x)+2 f (h+i x) \log (h+i x)\right )+b^2 \left (-6 f (h+i x) (\log (c (e+f x))-1) \text{PolyLog}\left (2,\frac{i (e+f x)}{e i-f h}\right )+6 f (h+i x) \text{PolyLog}\left (3,\frac{i (e+f x)}{e i-f h}\right )+\log (c (e+f x)) \left (f (h+i x) \log ^2(c (e+f x))-3 \log (c (e+f x)) \left (f (h+i x) \log \left (\frac{f (h+i x)}{f h-e i}\right )+i (e+f x)\right )+6 f (h+i x) \log \left (\frac{f (h+i x)}{f h-e i}\right )\right )\right )+3 a^2 f (h+i x) \log (e+f x)+3 a^2 (f h-e i)-3 a^2 f (h+i x) \log (h+i x)}{3 d (h+i x) (f h-e i)^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 2.26, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c \left ( fx+e \right ) \right ) \right ) ^{2}}{ \left ( dfx+de \right ) \left ( ix+h \right ) ^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.43, size = 840, normalized size = 3.08 \begin{align*} a^{2}{\left (\frac{f \log \left (f x + e\right )}{d f^{2} h^{2} - 2 \, d e f h i + d e^{2} i^{2}} - \frac{f \log \left (i x + h\right )}{d f^{2} h^{2} - 2 \, d e f h i + d e^{2} i^{2}} + \frac{1}{d f h^{2} - d e h i +{\left (d f h i - d e i^{2}\right )} x}\right )} - \frac{{\left (\log \left (f x + e\right )^{2} \log \left (\frac{f i x + e i}{f h - e i} + 1\right ) + 2 \,{\rm Li}_2\left (-\frac{f i x + e i}{f h - e i}\right ) \log \left (f x + e\right ) - 2 \,{\rm Li}_{3}(-\frac{f i x + e i}{f h - e i})\right )} b^{2} f}{{\left (f^{2} h^{2} - 2 \, e f h i + e^{2} i^{2}\right )} d} + \frac{3 \,{\left (f h - e i\right )} b^{2} \log \left (c\right )^{2} +{\left (b^{2} f i x + b^{2} f h\right )} \log \left (f x + e\right )^{3} + 6 \,{\left (f h - e i\right )} a b \log \left (c\right ) + 3 \,{\left (a b f h +{\left (f h \log \left (c\right ) - e i\right )} b^{2} +{\left (a b f i +{\left (f i \log \left (c\right ) - f i\right )} b^{2}\right )} x\right )} \log \left (f x + e\right )^{2} + 3 \,{\left (2 \,{\left (f h \log \left (c\right ) - e i\right )} a b +{\left (f h \log \left (c\right )^{2} - 2 \, e i \log \left (c\right )\right )} b^{2} +{\left (2 \,{\left (f i \log \left (c\right ) - f i\right )} a b +{\left (f i \log \left (c\right )^{2} - 2 \, f i \log \left (c\right )\right )} b^{2}\right )} x\right )} \log \left (f x + e\right )}{3 \,{\left ({\left (f^{2} h^{2} i - 2 \, e f h i^{2} + e^{2} i^{3}\right )} d x +{\left (f^{2} h^{3} - 2 \, e f h^{2} i + e^{2} h i^{2}\right )} d\right )}} - \frac{2 \,{\left ({\left (f \log \left (c\right ) - f\right )} b^{2} + a b f\right )}{\left (\log \left (f x + e\right ) \log \left (\frac{f i x + e i}{f h - e i} + 1\right ) +{\rm Li}_2\left (-\frac{f i x + e i}{f h - e i}\right )\right )}}{{\left (f^{2} h^{2} - 2 \, e f h i + e^{2} i^{2}\right )} d} - \frac{{\left (2 \,{\left (f \log \left (c\right ) - f\right )} a b +{\left (f \log \left (c\right )^{2} - 2 \, f \log \left (c\right )\right )} b^{2}\right )} \log \left (i x + h\right )}{{\left (f^{2} h^{2} - 2 \, e f h i + e^{2} i^{2}\right )} d} \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 (\frac{b^{2} \log \left (c f x + c e\right )^{2} + 2 \, a b \log \left (c f x + c e\right ) + a^{2}}{d f i^{2} x^{3} + d e h^{2} +{\left (2 \, d f h i + d e i^{2}\right )} x^{2} +{\left (d f h^{2} + 2 \, d e h i\right )} x}, 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 \frac{{\left (b \log \left ({\left (f x + e\right )} c\right ) + a\right )}^{2}}{{\left (d f x + d e\right )}{\left (i x + h\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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