Optimal. Leaf size=66 \[ \frac {(d x)^{m+1} e^{-\frac {a (m+1)}{b n}} \left (c x^n\right )^{-\frac {m+1}{n}} \text {Ei}\left (\frac {(m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b d n} \]
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Rubi [A] time = 0.07, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2310, 2178} \[ \frac {(d x)^{m+1} e^{-\frac {a (m+1)}{b n}} \left (c x^n\right )^{-\frac {m+1}{n}} \text {Ei}\left (\frac {(m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b d n} \]
Antiderivative was successfully verified.
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Rule 2178
Rule 2310
Rubi steps
\begin {align*} \int \frac {(d x)^m}{a+b \log \left (c x^n\right )} \, dx &=\frac {\left ((d x)^{1+m} \left (c x^n\right )^{-\frac {1+m}{n}}\right ) \operatorname {Subst}\left (\int \frac {e^{\frac {(1+m) x}{n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )}{d n}\\ &=\frac {e^{-\frac {a (1+m)}{b n}} (d x)^{1+m} \left (c x^n\right )^{-\frac {1+m}{n}} \text {Ei}\left (\frac {(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b d n}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 67, normalized size = 1.02 \[ \frac {x^{-m} (d x)^m \exp \left (-\frac {(m+1) \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )}{b n}\right ) \text {Ei}\left (\frac {(m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 68, normalized size = 1.03 \[ \frac {{\rm Ei}\left (\frac {{\left (b m + b\right )} n \log \relax (x) + a m + {\left (b m + b\right )} \log \relax (c) + a}{b n}\right ) e^{\left (\frac {b m n \log \relax (d) - a m - {\left (b m + b\right )} \log \relax (c) - a}{b n}\right )}}{b n} \]
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 \[ \int \frac {\left (d x\right )^{m}}{b \log \left (c x^{n}\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.52, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x \right )^{m}}{b \ln \left (c \,x^{n}\right )+a}\, dx \]
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 \[ \int \frac {\left (d x\right )^{m}}{b \log \left (c x^{n}\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (d\,x\right )}^m}{a+b\,\ln \left (c\,x^n\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x\right )^{m}}{a + b \log {\left (c x^{n} \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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