Optimal. Leaf size=141 \[ \frac{e^{\frac{2 d}{e m}} \left (f x^m\right )^{2/m} \left (a+b \log \left (c x^n\right )\right ) \text{Ei}\left (-\frac{2 \left (d+e \log \left (f x^m\right )\right )}{e m}\right )}{e m x^2}-\frac{b n e^{\frac{2 d}{e m}} \left (f x^m\right )^{2/m} \left (d+e \log \left (f x^m\right )\right ) \text{Ei}\left (-\frac{2 \left (d+e \log \left (f x^m\right )\right )}{e m}\right )}{e^2 m^2 x^2}-\frac{b n}{2 e m x^2} \]
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Rubi [A] time = 0.16907, antiderivative size = 141, normalized size of antiderivative = 1., 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} \[ \frac{e^{\frac{2 d}{e m}} \left (f x^m\right )^{2/m} \left (a+b \log \left (c x^n\right )\right ) \text{Ei}\left (-\frac{2 \left (d+e \log \left (f x^m\right )\right )}{e m}\right )}{e m x^2}-\frac{b n e^{\frac{2 d}{e m}} \left (f x^m\right )^{2/m} \left (d+e \log \left (f x^m\right )\right ) \text{Ei}\left (-\frac{2 \left (d+e \log \left (f x^m\right )\right )}{e m}\right )}{e^2 m^2 x^2}-\frac{b n}{2 e m x^2} \]
Antiderivative was successfully verified.
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Rule 2310
Rule 2178
Rule 2366
Rule 12
Rule 15
Rule 6482
Rubi steps
\begin{align*} \int \frac{a+b \log \left (c x^n\right )}{x^3 \left (d+e \log \left (f x^m\right )\right )} \, dx &=\frac{e^{\frac{2 d}{e m}} \left (f x^m\right )^{2/m} \text{Ei}\left (-\frac{2 \left (d+e \log \left (f x^m\right )\right )}{e m}\right ) \left (a+b \log \left (c x^n\right )\right )}{e m x^2}-(b n) \int \frac{e^{\frac{2 d}{e m}} \left (f x^m\right )^{2/m} \text{Ei}\left (-\frac{2 \left (d+e \log \left (f x^m\right )\right )}{e m}\right )}{e m x^3} \, dx\\ &=\frac{e^{\frac{2 d}{e m}} \left (f x^m\right )^{2/m} \text{Ei}\left (-\frac{2 \left (d+e \log \left (f x^m\right )\right )}{e m}\right ) \left (a+b \log \left (c x^n\right )\right )}{e m x^2}-\frac{\left (b e^{\frac{2 d}{e m}} n\right ) \int \frac{\left (f x^m\right )^{2/m} \text{Ei}\left (-\frac{2 \left (d+e \log \left (f x^m\right )\right )}{e m}\right )}{x^3} \, dx}{e m}\\ &=\frac{e^{\frac{2 d}{e m}} \left (f x^m\right )^{2/m} \text{Ei}\left (-\frac{2 \left (d+e \log \left (f x^m\right )\right )}{e m}\right ) \left (a+b \log \left (c x^n\right )\right )}{e m x^2}-\frac{\left (b e^{\frac{2 d}{e m}} n \left (f x^m\right )^{2/m}\right ) \int \frac{\text{Ei}\left (-\frac{2 \left (d+e \log \left (f x^m\right )\right )}{e m}\right )}{x} \, dx}{e m x^2}\\ &=\frac{e^{\frac{2 d}{e m}} \left (f x^m\right )^{2/m} \text{Ei}\left (-\frac{2 \left (d+e \log \left (f x^m\right )\right )}{e m}\right ) \left (a+b \log \left (c x^n\right )\right )}{e m x^2}-\frac{\left (b e^{\frac{2 d}{e m}} n \left (f x^m\right )^{2/m}\right ) \operatorname{Subst}\left (\int \text{Ei}\left (-\frac{2 (d+e x)}{e m}\right ) \, dx,x,\log \left (f x^m\right )\right )}{e m^2 x^2}\\ &=\frac{e^{\frac{2 d}{e m}} \left (f x^m\right )^{2/m} \text{Ei}\left (-\frac{2 \left (d+e \log \left (f x^m\right )\right )}{e m}\right ) \left (a+b \log \left (c x^n\right )\right )}{e m x^2}+\frac{\left (b e^{\frac{2 d}{e m}} n \left (f x^m\right )^{2/m}\right ) \operatorname{Subst}\left (\int \text{Ei}(x) \, dx,x,-\frac{2 d}{e m}-\frac{2 \log \left (f x^m\right )}{m}\right )}{2 e m x^2}\\ &=-\frac{b n}{2 e m x^2}-\frac{b e^{\frac{2 d}{e m}} n \left (f x^m\right )^{2/m} \text{Ei}\left (-\frac{2 d}{e m}-\frac{2 \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^2}+\frac{e^{\frac{2 d}{e m}} \left (f x^m\right )^{2/m} \text{Ei}\left (-\frac{2 \left (d+e \log \left (f x^m\right )\right )}{e m}\right ) \left (a+b \log \left (c x^n\right )\right )}{e m x^2}\\ \end{align*}
Mathematica [A] time = 0.130263, size = 94, normalized size = 0.67 \[ \frac{2 e^{\frac{2 d}{e m}} \left (f x^m\right )^{2/m} \text{Ei}\left (-\frac{2 \left (d+e \log \left (f x^m\right )\right )}{e m}\right ) \left (a e m+b e m \log \left (c x^n\right )-b d n-b e n \log \left (f x^m\right )\right )-b e m n}{2 e^2 m^2 x^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.352, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\ln \left ( c{x}^{n} \right ) }{{x}^{3} \left ( d+e\ln \left ( f{x}^{m} \right ) \right ) }}\, 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{b \log \left (c x^{n}\right ) + a}{{\left (e \log \left (f x^{m}\right ) + d\right )} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.808807, size = 225, normalized size = 1.6 \begin{align*} -\frac{b e m n - 2 \,{\left (b e m x^{2} \log \left (c\right ) - b e n x^{2} \log \left (f\right ) +{\left (a e m - b d n\right )} x^{2}\right )} e^{\left (\frac{2 \,{\left (e \log \left (f\right ) + d\right )}}{e m}\right )} \logintegral \left (\frac{e^{\left (-\frac{2 \,{\left (e \log \left (f\right ) + d\right )}}{e m}\right )}}{x^{2}}\right )}{2 \, e^{2} m^{2} x^{2}} \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{b \log \left (c x^{n}\right ) + a}{{\left (e \log \left (f x^{m}\right ) + d\right )} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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