Optimal. Leaf size=83 \[ \frac {\operatorname {Subst}\left (\text {Int}\left (\frac {1}{(a+b x) (f+g x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2},x\right ),e \left (\frac {a+b x}{c+d x}\right )^n,e (a+b x)^n (c+d x)^{-n}\right )}{h} \]
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Rubi [A] time = 0.46, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {1}{\left (a f h+b g h x^2+h (b f x+a g x)\right ) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1}{\left (a f h+b g h x^2+h (b f x+a g x)\right ) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2} \, dx &=\int \frac {1}{h (a+b x) (f+g x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2} \, dx\\ &=\frac {\int \frac {1}{(a+b x) (f+g x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2} \, dx}{h}\\ &=\frac {\int \left (\frac {b}{(b f-a g) (a+b x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}-\frac {g}{(b f-a g) (f+g x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}\right ) \, dx}{h}\\ &=\frac {b \int \frac {1}{(a+b x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2} \, dx}{(b f-a g) h}-\frac {g \int \frac {1}{(f+g x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2} \, dx}{(b f-a g) h}\\ \end {align*}
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Mathematica [A] time = 0.27, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a f h+b g h x^2+h (b f x+a g x)\right ) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.88, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{A^{2} b g h x^{2} + A^{2} a f h + {\left (A^{2} b f + A^{2} a g\right )} h x + {\left (B^{2} b g h x^{2} + B^{2} a f h + {\left (B^{2} b f + B^{2} a g\right )} h x\right )} \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )^{2} + 2 \, {\left (A B b g h x^{2} + A B a f h + {\left (A B b f + A B a g\right )} h x\right )} \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b g h x^{2} + a f h + {\left (b f x + a g x\right )} h\right )} {\left (B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 5.41, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b g h \,x^{2}+a f h +\left (a g x +b x f \right ) h \right ) \left (B \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )+A \right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ {\left (d f - c g\right )} \int \frac {1}{{\left (b c f^{2} h n - a d f^{2} h n\right )} A B + {\left (b c f^{2} h n \log \relax (e) - a d f^{2} h n \log \relax (e)\right )} B^{2} + {\left ({\left (b c g^{2} h n - a d g^{2} h n\right )} A B + {\left (b c g^{2} h n \log \relax (e) - a d g^{2} h n \log \relax (e)\right )} B^{2}\right )} x^{2} + 2 \, {\left ({\left (b c f g h n - a d f g h n\right )} A B + {\left (b c f g h n \log \relax (e) - a d f g h n \log \relax (e)\right )} B^{2}\right )} x + {\left ({\left (b c g^{2} h n - a d g^{2} h n\right )} B^{2} x^{2} + 2 \, {\left (b c f g h n - a d f g h n\right )} B^{2} x + {\left (b c f^{2} h n - a d f^{2} h n\right )} B^{2}\right )} \log \left ({\left (b x + a\right )}^{n}\right ) - {\left ({\left (b c g^{2} h n - a d g^{2} h n\right )} B^{2} x^{2} + 2 \, {\left (b c f g h n - a d f g h n\right )} B^{2} x + {\left (b c f^{2} h n - a d f^{2} h n\right )} B^{2}\right )} \log \left ({\left (d x + c\right )}^{n}\right )}\,{d x} - \frac {d x + c}{{\left (b c f h n - a d f h n\right )} A B + {\left (b c f h n \log \relax (e) - a d f h n \log \relax (e)\right )} B^{2} + {\left ({\left (b c g h n - a d g h n\right )} A B + {\left (b c g h n \log \relax (e) - a d g h n \log \relax (e)\right )} B^{2}\right )} x + {\left ({\left (b c g h n - a d g h n\right )} B^{2} x + {\left (b c f h n - a d f h n\right )} B^{2}\right )} \log \left ({\left (b x + a\right )}^{n}\right ) - {\left ({\left (b c g h n - a d g h n\right )} B^{2} x + {\left (b c f h n - a d f h n\right )} B^{2}\right )} \log \left ({\left (d x + c\right )}^{n}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (A+B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )\right )}^2\,\left (h\,\left (a\,g\,x+b\,f\,x\right )+a\,f\,h+b\,g\,h\,x^2\right )} \,d x \]
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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