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 = 0.11, antiderivative size = 88, normalized size of antiderivative = 1.00, 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*}
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Mathematica [F] time = 0.40, size = 0, normalized size = 0.00 \[ \int \frac {(a+b x)^m (c+d x)^{-2-m}}{\log \left (e (a+b x)^n (c+d x)^{-n}\right )} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.85, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 2}}{\log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )}, x\right ) \]
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 \[ \int \frac {{\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 2}}{\log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.87, size = 0, normalized size = 0.00 \[ \int \frac {\left (b x +a \right )^{m} \left (d x +c \right )^{-m -2}}{\ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )}\, dx \]
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 \[ \int \frac {{\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 2}}{\log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,x\right )}^m}{\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )\,{\left (c+d\,x\right )}^{m+2}} \,d x \]
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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