Optimal. Leaf size=133 \[ \frac {e^{\frac {d}{e m}} \left (f x^m\right )^{\frac {1}{m}} \left (a+b \log \left (c x^n\right )\right ) \text {Ei}\left (-\frac {d+e \log \left (f x^m\right )}{e m}\right )}{e m x}-\frac {b n e^{\frac {d}{e m}} \left (f x^m\right )^{\frac {1}{m}} \left (d+e \log \left (f x^m\right )\right ) \text {Ei}\left (-\frac {d+e \log \left (f x^m\right )}{e m}\right )}{e^2 m^2 x}-\frac {b n}{e m x} \]
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Rubi [A] time = 0.17, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2310, 2178, 2366, 12, 15, 6482} \[ \frac {e^{\frac {d}{e m}} \left (f x^m\right )^{\frac {1}{m}} \left (a+b \log \left (c x^n\right )\right ) \text {Ei}\left (-\frac {d+e \log \left (f x^m\right )}{e m}\right )}{e m x}-\frac {b n e^{\frac {d}{e m}} \left (f x^m\right )^{\frac {1}{m}} \left (d+e \log \left (f x^m\right )\right ) \text {Ei}\left (-\frac {d+e \log \left (f x^m\right )}{e m}\right )}{e^2 m^2 x}-\frac {b n}{e m x} \]
Antiderivative was successfully verified.
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Rule 12
Rule 15
Rule 2178
Rule 2310
Rule 2366
Rule 6482
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c x^n\right )}{x^2 \left (d+e \log \left (f x^m\right )\right )} \, dx &=\frac {e^{\frac {d}{e m}} \left (f x^m\right )^{\frac {1}{m}} \text {Ei}\left (-\frac {d+e \log \left (f x^m\right )}{e m}\right ) \left (a+b \log \left (c x^n\right )\right )}{e m x}-(b n) \int \frac {e^{\frac {d}{e m}} \left (f x^m\right )^{\frac {1}{m}} \text {Ei}\left (-\frac {d+e \log \left (f x^m\right )}{e m}\right )}{e m x^2} \, dx\\ &=\frac {e^{\frac {d}{e m}} \left (f x^m\right )^{\frac {1}{m}} \text {Ei}\left (-\frac {d+e \log \left (f x^m\right )}{e m}\right ) \left (a+b \log \left (c x^n\right )\right )}{e m x}-\frac {\left (b e^{\frac {d}{e m}} n\right ) \int \frac {\left (f x^m\right )^{\frac {1}{m}} \text {Ei}\left (-\frac {d+e \log \left (f x^m\right )}{e m}\right )}{x^2} \, dx}{e m}\\ &=\frac {e^{\frac {d}{e m}} \left (f x^m\right )^{\frac {1}{m}} \text {Ei}\left (-\frac {d+e \log \left (f x^m\right )}{e m}\right ) \left (a+b \log \left (c x^n\right )\right )}{e m x}-\frac {\left (b e^{\frac {d}{e m}} n \left (f x^m\right )^{\frac {1}{m}}\right ) \int \frac {\text {Ei}\left (-\frac {d+e \log \left (f x^m\right )}{e m}\right )}{x} \, dx}{e m x}\\ &=\frac {e^{\frac {d}{e m}} \left (f x^m\right )^{\frac {1}{m}} \text {Ei}\left (-\frac {d+e \log \left (f x^m\right )}{e m}\right ) \left (a+b \log \left (c x^n\right )\right )}{e m x}-\frac {\left (b e^{\frac {d}{e m}} n \left (f x^m\right )^{\frac {1}{m}}\right ) \operatorname {Subst}\left (\int \text {Ei}\left (-\frac {d+e x}{e m}\right ) \, dx,x,\log \left (f x^m\right )\right )}{e m^2 x}\\ &=\frac {e^{\frac {d}{e m}} \left (f x^m\right )^{\frac {1}{m}} \text {Ei}\left (-\frac {d+e \log \left (f x^m\right )}{e m}\right ) \left (a+b \log \left (c x^n\right )\right )}{e m x}+\frac {\left (b e^{\frac {d}{e m}} n \left (f x^m\right )^{\frac {1}{m}}\right ) \operatorname {Subst}\left (\int \text {Ei}(x) \, dx,x,-\frac {d}{e m}-\frac {\log \left (f x^m\right )}{m}\right )}{e m x}\\ &=-\frac {b n}{e m x}-\frac {b e^{\frac {d}{e m}} n \left (f x^m\right )^{\frac {1}{m}} \text {Ei}\left (-\frac {d}{e m}-\frac {\log \left (f x^m\right )}{m}\right ) \left (\frac {d}{e m}+\frac {\log \left (f x^m\right )}{m}\right )}{e m x}+\frac {e^{\frac {d}{e m}} \left (f x^m\right )^{\frac {1}{m}} \text {Ei}\left (-\frac {d+e \log \left (f x^m\right )}{e m}\right ) \left (a+b \log \left (c x^n\right )\right )}{e m x}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 87, normalized size = 0.65 \[ \frac {e^{\frac {d}{e m}} \left (f x^m\right )^{\frac {1}{m}} \text {Ei}\left (-\frac {d+e \log \left (f x^m\right )}{e m}\right ) \left (a e m+b e m \log \left (c x^n\right )-b d n-b e n \log \left (f x^m\right )\right )-b e m n}{e^2 m^2 x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 81, normalized size = 0.61 \[ -\frac {b e m n - {\left (b e m x \log \relax (c) - b e n x \log \relax (f) + {\left (a e m - b d n\right )} x\right )} e^{\left (\frac {e \log \relax (f) + d}{e m}\right )} \operatorname {log\_integral}\left (\frac {e^{\left (-\frac {e \log \relax (f) + d}{e m}\right )}}{x}\right )}{e^{2} m^{2} x} \]
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 \[ \int \frac {b \log \left (c x^{n}\right ) + a}{{\left (e \log \left (f x^{m}\right ) + d\right )} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.27, size = 0, normalized size = 0.00 \[ \int \frac {b \ln \left (c \,x^{n}\right )+a}{\left (e \ln \left (f \,x^{m}\right )+d \right ) x^{2}}\, dx \]
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 \[ \int \frac {b \log \left (c x^{n}\right ) + a}{{\left (e \log \left (f x^{m}\right ) + d\right )} x^{2}}\,{d x} \]
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 \[ \int \frac {a+b\,\ln \left (c\,x^n\right )}{x^2\,\left (d+e\,\ln \left (f\,x^m\right )\right )} \,d x \]
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 \[ \int \frac {a + b \log {\left (c x^{n} \right )}}{x^{2} \left (d + e \log {\left (f x^{m} \right )}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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