Optimal. Leaf size=88 \[ \frac{(a+b x)^{m+1} (c+d x)^{-m-1} \left (e (a+b x)^n (c+d x)^{-n}\right )^{-\frac{m+1}{n}} \text{Ei}\left (\frac{(m+1) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{n}\right )}{n (b c-a d)} \]
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Rubi [A] time = 0.10988, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.025, Rules used = {2510} \[ \frac{(a+b x)^{m+1} (c+d x)^{-m-1} \left (e (a+b x)^n (c+d x)^{-n}\right )^{-\frac{m+1}{n}} \text{Ei}\left (\frac{(m+1) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{n}\right )}{n (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 2510
Rubi steps
\begin{align*} \int \frac{(a+b x)^m (c+d x)^{-2-m}}{\log \left (e (a+b x)^n (c+d x)^{-n}\right )} \, dx &=\frac{(a+b x)^{1+m} (c+d x)^{-1-m} \left (e (a+b x)^n (c+d x)^{-n}\right )^{-\frac{1+m}{n}} \text{Ei}\left (\frac{(1+m) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{n}\right )}{(b c-a d) n}\\ \end{align*}
Mathematica [F] time = 0.359671, size = 0, normalized size = 0. \[ \int \frac{(a+b x)^m (c+d x)^{-2-m}}{\log \left (e (a+b x)^n (c+d x)^{-n}\right )} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 1.237, size = 0, normalized size = 0. \begin{align*} \int{ \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{-2-m} \left ( \ln \left ({\frac{e \left ( bx+a \right ) ^{n}}{ \left ( dx+c \right ) ^{n}}} \right ) \right ) ^{-1}}\, 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*} \int \frac{{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 2}}{\log \left (\frac{{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )}\,{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{{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 2}}{\log \left (\frac{{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )}, 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 x + a\right )}^{m}{\left (d x + c\right )}^{-m - 2}}{\log \left (\frac{{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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