Optimal. Leaf size=203 \[ \frac {2 B n \text {Li}_2\left (\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{h (b f-a g)}-\frac {\log \left (1-\frac {(c+d x) (b f-a g)}{(a+b x) (d f-c g)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{h (b f-a g)}+\frac {2 B^2 n^2 \text {Li}_3\left (\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right )}{h (b f-a g)} \]
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Rubi [A] time = 0.76, antiderivative size = 371, normalized size of antiderivative = 1.83, number of steps used = 11, number of rules used = 8, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.186, Rules used = {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 31
Rule 36
Rule 2315
Rule 2502
Rule 2503
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}{(f+g x) (a h+b h x)} \, dx &=\int \left (\frac {A^2}{h (a+b x) (f+g x)}+\frac {2 A B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{h (a+b x) (f+g x)}+\frac {B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{h (a+b x) (f+g x)}\right ) \, dx\\ &=\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*}
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Mathematica [B] time = 1.05, size = 1415, normalized size = 6.97 \[ \text {result too large to display} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.04, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {B^{2} \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )^{2} + 2 \, A B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A^{2}}{b g h x^{2} + a f h + {\left (b f + a g\right )} h x}, x\right ) \]
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 \[ \int \frac {{\left (B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{2}}{{\left (b h x + a h\right )} {\left (g x + f\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 3.17, size = 0, normalized size = 0.00 \[ \int \frac {\left (B \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )+A \right )^{2}}{\left (g x +f \right ) \left (b h x +a h \right )}\, dx \]
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 \[ A^{2} {\left (\frac {\log \left (b x + a\right )}{{\left (b f - a g\right )} h} - \frac {\log \left (g x + f\right )}{{\left (b f - a g\right )} h}\right )} + \int \frac {B^{2} \log \left ({\left (b x + a\right )}^{n}\right )^{2} + B^{2} \log \left ({\left (d x + c\right )}^{n}\right )^{2} + B^{2} \log \relax (e)^{2} + 2 \, A B \log \relax (e) + 2 \, {\left (B^{2} \log \relax (e) + A B\right )} \log \left ({\left (b x + a\right )}^{n}\right ) - 2 \, {\left (B^{2} \log \left ({\left (b x + a\right )}^{n}\right ) + B^{2} \log \relax (e) + A B\right )} \log \left ({\left (d x + c\right )}^{n}\right )}{b g h x^{2} + a f h + {\left (b f h + a g h\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (A+B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )\right )}^2}{\left (f+g\,x\right )\,\left (a\,h+b\,h\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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