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 = 0.38, 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 = {}
\begin {gather*} \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 \end {gather*}
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 = 1.25, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 0.24, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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