Optimal. Leaf size=108 \[ \frac {b \text {Int}\left (\frac {\log \left (h (f+g x)^m\right )}{(a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )},x\right )}{b c-a d}-\frac {d \text {Int}\left (\frac {\log \left (h (f+g x)^m\right )}{(c+d x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )},x\right )}{b c-a d} \]
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Rubi [A] time = 0.50, 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 = {} \[ \int \frac {\log \left (h (f+g x)^m\right )}{(a+b x) (c+d x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\log \left (h (f+g x)^m\right )}{(a+b x) (c+d x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx &=\int \left (\frac {b \log \left (h (f+g x)^m\right )}{(b c-a d) (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}-\frac {d \log \left (h (f+g x)^m\right )}{(b c-a d) (c+d x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}\right ) \, dx\\ &=\frac {b \int \frac {\log \left (h (f+g x)^m\right )}{(a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx}{b c-a d}-\frac {d \int \frac {\log \left (h (f+g x)^m\right )}{(c+d x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx}{b c-a d}\\ \end {align*}
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Mathematica [A] time = 2.01, size = 0, normalized size = 0.00 \[ \int \frac {\log \left (h (f+g x)^m\right )}{(a+b x) (c+d x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left ({\left (g x + f\right )}^{m} h\right )}{{\left (b d x^{2} + a c + {\left (b c + a d\right )} x\right )} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )}, x\right ) \]
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 \[ \int \frac {\log \left ({\left (g x + f\right )}^{m} h\right )}{{\left (b x + a\right )} {\left (d x + c\right )} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 52.40, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (h \left (g x +f \right )^{m}\right )}{\left (b x +a \right ) \left (d x +c \right ) \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )}\, dx \]
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 \[ \int \frac {\log \left ({\left (g x + f\right )}^{m} h\right )}{{\left (b x + a\right )} {\left (d x + c\right )} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )}\,{d x} \]
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 \[ \int \frac {\ln \left (h\,{\left (f+g\,x\right )}^m\right )}{\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,\left (a+b\,x\right )\,\left (c+d\,x\right )} \,d x \]
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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