Optimal. Leaf size=141 \[ \frac{x^3 e^{-\frac{3 d}{e m}} \left (f x^m\right )^{-3/m} \left (a+b \log \left (c x^n\right )\right ) \text{Ei}\left (\frac{3 \left (d+e \log \left (f x^m\right )\right )}{e m}\right )}{e m}-\frac{b n x^3 e^{-\frac{3 d}{e m}} \left (f x^m\right )^{-3/m} \left (d+e \log \left (f x^m\right )\right ) \text{Ei}\left (\frac{3 \left (d+e \log \left (f x^m\right )\right )}{e m}\right )}{e^2 m^2}+\frac{b n x^3}{3 e m} \]
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Rubi [A] time = 0.179974, 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} \[ \frac{x^3 e^{-\frac{3 d}{e m}} \left (f x^m\right )^{-3/m} \left (a+b \log \left (c x^n\right )\right ) \text{Ei}\left (\frac{3 \left (d+e \log \left (f x^m\right )\right )}{e m}\right )}{e m}-\frac{b n x^3 e^{-\frac{3 d}{e m}} \left (f x^m\right )^{-3/m} \left (d+e \log \left (f x^m\right )\right ) \text{Ei}\left (\frac{3 \left (d+e \log \left (f x^m\right )\right )}{e m}\right )}{e^2 m^2}+\frac{b n x^3}{3 e m} \]
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{x^2 \left (a+b \log \left (c x^n\right )\right )}{d+e \log \left (f x^m\right )} \, dx &=\frac{e^{-\frac{3 d}{e m}} x^3 \left (f x^m\right )^{-3/m} \text{Ei}\left (\frac{3 \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}-(b n) \int \frac{e^{-\frac{3 d}{e m}} x^2 \left (f x^m\right )^{-3/m} \text{Ei}\left (\frac{3 \left (d+e \log \left (f x^m\right )\right )}{e m}\right )}{e m} \, dx\\ &=\frac{e^{-\frac{3 d}{e m}} x^3 \left (f x^m\right )^{-3/m} \text{Ei}\left (\frac{3 \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}-\frac{\left (b e^{-\frac{3 d}{e m}} n\right ) \int x^2 \left (f x^m\right )^{-3/m} \text{Ei}\left (\frac{3 \left (d+e \log \left (f x^m\right )\right )}{e m}\right ) \, dx}{e m}\\ &=\frac{e^{-\frac{3 d}{e m}} x^3 \left (f x^m\right )^{-3/m} \text{Ei}\left (\frac{3 \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}-\frac{\left (b e^{-\frac{3 d}{e m}} n x^3 \left (f x^m\right )^{-3/m}\right ) \int \frac{\text{Ei}\left (\frac{3 \left (d+e \log \left (f x^m\right )\right )}{e m}\right )}{x} \, dx}{e m}\\ &=\frac{e^{-\frac{3 d}{e m}} x^3 \left (f x^m\right )^{-3/m} \text{Ei}\left (\frac{3 \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}-\frac{\left (b e^{-\frac{3 d}{e m}} n x^3 \left (f x^m\right )^{-3/m}\right ) \operatorname{Subst}\left (\int \text{Ei}\left (\frac{3 (d+e x)}{e m}\right ) \, dx,x,\log \left (f x^m\right )\right )}{e m^2}\\ &=\frac{e^{-\frac{3 d}{e m}} x^3 \left (f x^m\right )^{-3/m} \text{Ei}\left (\frac{3 \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}-\frac{\left (b e^{-\frac{3 d}{e m}} n x^3 \left (f x^m\right )^{-3/m}\right ) \operatorname{Subst}\left (\int \text{Ei}(x) \, dx,x,\frac{3 d}{e m}+\frac{3 \log \left (f x^m\right )}{m}\right )}{3 e m}\\ &=\frac{b n x^3}{3 e m}-\frac{b e^{-\frac{3 d}{e m}} n x^3 \left (f x^m\right )^{-3/m} \text{Ei}\left (\frac{3 d}{e m}+\frac{3 \log \left (f x^m\right )}{m}\right ) \left (\frac{d}{e m}+\frac{\log \left (f x^m\right )}{m}\right )}{e m}+\frac{e^{-\frac{3 d}{e m}} x^3 \left (f x^m\right )^{-3/m} \text{Ei}\left (\frac{3 \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}\\ \end{align*}
Mathematica [A] time = 0.16419, size = 93, normalized size = 0.66 \[ \frac{x^3 \left (3 e^{-\frac{3 d}{e m}} \left (f x^m\right )^{-3/m} \text{Ei}\left (\frac{3 \left (d+e \log \left (f x^m\right )\right )}{e m}\right ) \left (a e m+b e m \log \left (c x^n\right )-b d n-b e n \log \left (f x^m\right )\right )+b e m n\right )}{3 e^2 m^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.338, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{2} \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }{d+e\ln \left ( f{x}^{m} \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{{\left (b \log \left (c x^{n}\right ) + a\right )} x^{2}}{e \log \left (f x^{m}\right ) + d}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.924454, size = 242, normalized size = 1.72 \begin{align*} \frac{{\left (b e m n x^{3} e^{\left (\frac{3 \,{\left (e \log \left (f\right ) + d\right )}}{e m}\right )} + 3 \,{\left (b e m \log \left (c\right ) - b e n \log \left (f\right ) + a e m - b d n\right )} \logintegral \left (x^{3} e^{\left (\frac{3 \,{\left (e \log \left (f\right ) + d\right )}}{e m}\right )}\right )\right )} e^{\left (-\frac{3 \,{\left (e \log \left (f\right ) + d\right )}}{e m}\right )}}{3 \, e^{2} m^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} \left (a + b \log{\left (c x^{n} \right )}\right )}{d + e \log{\left (f x^{m} \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.40131, size = 262, normalized size = 1.86 \begin{align*} -\frac{b d n{\rm Ei}\left (\frac{3 \, d e^{\left (-1\right )}}{m} + \frac{3 \, \log \left (f\right )}{m} + 3 \, \log \left (x\right )\right ) e^{\left (-\frac{3 \, d e^{\left (-1\right )}}{m} - 2\right )}}{f^{\frac{3}{m}} m^{2}} + \frac{b{\rm Ei}\left (\frac{3 \, d e^{\left (-1\right )}}{m} + \frac{3 \, \log \left (f\right )}{m} + 3 \, \log \left (x\right )\right ) e^{\left (-\frac{3 \, d e^{\left (-1\right )}}{m} - 1\right )} \log \left (c\right )}{f^{\frac{3}{m}} m} - \frac{b n{\rm Ei}\left (\frac{3 \, d e^{\left (-1\right )}}{m} + \frac{3 \, \log \left (f\right )}{m} + 3 \, \log \left (x\right )\right ) e^{\left (-\frac{3 \, d e^{\left (-1\right )}}{m} - 1\right )} \log \left (f\right )}{f^{\frac{3}{m}} m^{2}} + \frac{a{\rm Ei}\left (\frac{3 \, d e^{\left (-1\right )}}{m} + \frac{3 \, \log \left (f\right )}{m} + 3 \, \log \left (x\right )\right ) e^{\left (-\frac{3 \, d e^{\left (-1\right )}}{m} - 1\right )}}{f^{\frac{3}{m}} m} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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