Optimal. Leaf size=203 \[ \frac{2 B n \text{PolyLog}\left (2,\frac{(c+d x) (b f-a g)}{(a+b x) (d f-c g)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{h (b f-a g)}+\frac{2 B^2 n^2 \text{PolyLog}\left (3,\frac{(c+d x) (b f-a g)}{(a+b x) (d f-c g)}\right )}{h (b f-a g)}-\frac{\log \left (1-\frac{(c+d x) (b f-a g)}{(a+b x) (d f-c g)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{h (b f-a g)} \]
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Rubi [A] time = 0.818899, antiderivative size = 371, normalized size of antiderivative = 1.83, number of steps used = 13, number of rules used = 10, integrand size = 51, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.196, Rules used = {6688, 12, 6742, 36, 31, 2503, 2502, 2315, 2506, 6610} \[ \frac{2 A B n \text{PolyLog}\left (2,\frac{(f+g x) (b c-a d)}{(a+b x) (d f-c g)}+1\right )}{h (b f-a g)}+\frac{2 B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text{PolyLog}\left (2,\frac{(f+g x) (b c-a d)}{(a+b x) (d f-c g)}+1\right )}{h (b f-a g)}+\frac{2 B^2 n^2 \text{PolyLog}\left (3,\frac{(f+g x) (b c-a d)}{(a+b x) (d f-c g)}+1\right )}{h (b f-a g)}+\frac{A^2 \log (a+b x)}{h (b f-a g)}-\frac{A^2 \log (f+g x)}{h (b f-a g)}-\frac{2 A B \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac{(f+g x) (b c-a d)}{(a+b x) (d f-c g)}\right )}{h (b f-a g)}-\frac{B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac{(f+g x) (b c-a d)}{(a+b x) (d f-c g)}\right )}{h (b f-a g)} \]
Antiderivative was successfully verified.
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Rule 6688
Rule 12
Rule 6742
Rule 36
Rule 31
Rule 2503
Rule 2502
Rule 2315
Rule 2506
Rule 6610
Rubi steps
\begin{align*} \int \frac{\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{a f h+b g h x^2+h (b f x+a g x)} \, dx &=\int \frac{\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{h (a+b x) (f+g x)} \, dx\\ &=\frac{\int \frac{\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{(a+b x) (f+g x)} \, dx}{h}\\ &=\frac{\int \left (\frac{A^2}{(a+b x) (f+g x)}+\frac{2 A B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x) (f+g x)}+\frac{B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x) (f+g x)}\right ) \, dx}{h}\\ &=\frac{A^2 \int \frac{1}{(a+b x) (f+g x)} \, dx}{h}+\frac{(2 A B) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x) (f+g x)} \, dx}{h}+\frac{B^2 \int \frac{\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x) (f+g x)} \, dx}{h}\\ &=-\frac{2 A B \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}-\frac{B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac{\left (A^2 b\right ) \int \frac{1}{a+b x} \, dx}{(b f-a g) h}-\frac{\left (A^2 g\right ) \int \frac{1}{f+g x} \, dx}{(b f-a g) h}+\frac{(2 A B (b c-a d) n) \int \frac{\log \left (-\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{(b f-a g) h}+\frac{\left (2 B^2 (b c-a d) n\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{(b f-a g) h}\\ &=\frac{A^2 \log (a+b x)}{(b f-a g) h}-\frac{A^2 \log (f+g x)}{(b f-a g) h}-\frac{2 A B \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}-\frac{B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\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) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}-\frac{(2 A B (b c-a d) n) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{(b c-a d) x}{d f-c g}\right )}{1+\frac{(b c-a d) x}{d f-c g}} \, dx,x,\frac{f+g x}{a+b x}\right )}{(b f-a g) (d f-c g) h}-\frac{\left (2 B^2 (b c-a d) n^2\right ) \int \frac{\text{Li}_2\left (1+\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{(b f-a g) h}\\ &=\frac{A^2 \log (a+b x)}{(b f-a g) h}-\frac{A^2 \log (f+g x)}{(b f-a g) h}-\frac{2 A B \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}-\frac{B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac{2 A B n \text{Li}_2\left (1+\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\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) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac{2 B^2 n^2 \text{Li}_3\left (1+\frac{(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}\\ \end{align*}
Mathematica [B] time = 0.835137, size = 1415, normalized size = 6.97 \[ \text{result too large to display} \]
Antiderivative was successfully verified.
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Maple [F] time = 4.911, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{afh+bgh{x}^{2}+h \left ( agx+bxf \right ) } \left ( A+B\ln \left ({\frac{e \left ( bx+a \right ) ^{n}}{ \left ( dx+c \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*} A^{2}{\left (\frac{\log \left (b x + a\right )}{{\left (b f - a g\right )} h} - \frac{\log \left (g x + f\right )}{{\left (b f - a g\right )} h}\right )} + \int \frac{B^{2} \log \left ({\left (b x + a\right )}^{n}\right )^{2} + B^{2} \log \left ({\left (d x + c\right )}^{n}\right )^{2} + B^{2} \log \left (e\right )^{2} + 2 \, A B \log \left (e\right ) + 2 \,{\left (B^{2} \log \left (e\right ) + A B\right )} \log \left ({\left (b x + a\right )}^{n}\right ) - 2 \,{\left (B^{2} \log \left ({\left (b x + a\right )}^{n}\right ) + B^{2} \log \left (e\right ) + A B\right )} \log \left ({\left (d x + c\right )}^{n}\right )}{b g h x^{2} + a f h +{\left (b f h + a g h\right )} x}\,{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 (\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 g h x^{2} + a f h +{\left (b f + a g\right )} h 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 (\frac{{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{2}}{b g h x^{2} + a f h +{\left (b f x + a g x\right )} h}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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