Optimal. Leaf size=141 \[ \frac {b n x^2}{2 e m}-\frac {b e^{-\frac {2 d}{e m}} n x^2 \left (f x^m\right )^{-2/m} \text {Ei}\left (\frac {2 \left (d+e \log \left (f x^m\right )\right )}{e m}\right ) \left (d+e \log \left (f x^m\right )\right )}{e^2 m^2}+\frac {e^{-\frac {2 d}{e m}} x^2 \left (f x^m\right )^{-2/m} \text {Ei}\left (\frac {2 \left (d+e \log \left (f x^m\right )\right )}{e m}\right ) \left (a+b \log \left (c x^n\right )\right )}{e m} \]
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Rubi [A]
time = 0.10, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2347, 2209,
2413, 12, 15, 6617} \begin {gather*} \frac {x^2 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}-\frac {b n x^2 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}+\frac {b n x^2}{2 e m} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 15
Rule 2209
Rule 2347
Rule 2413
Rule 6617
Rubi steps
\begin {align*} \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{d+e \log \left (f x^m\right )} \, dx &=\frac {e^{-\frac {2 d}{e m}} x^2 \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}-(b n) \int \frac {e^{-\frac {2 d}{e m}} x \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} \, dx\\ &=\frac {e^{-\frac {2 d}{e m}} x^2 \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}-\frac {\left (b e^{-\frac {2 d}{e m}} n\right ) \int x \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 ) \, dx}{e m}\\ &=\frac {e^{-\frac {2 d}{e m}} x^2 \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}-\frac {\left (b e^{-\frac {2 d}{e m}} n x^2 \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}\\ &=\frac {e^{-\frac {2 d}{e m}} x^2 \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}-\frac {\left (b e^{-\frac {2 d}{e m}} n x^2 \left (f x^m\right )^{-2/m}\right ) \text {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}\\ &=\frac {e^{-\frac {2 d}{e m}} x^2 \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}-\frac {\left (b e^{-\frac {2 d}{e m}} n x^2 \left (f x^m\right )^{-2/m}\right ) \text {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}\\ &=\frac {b n x^2}{2 e m}-\frac {b e^{-\frac {2 d}{e m}} n x^2 \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}+\frac {e^{-\frac {2 d}{e m}} x^2 \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}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 93, normalized size = 0.66 \begin {gather*} \frac {x^2 \left (b e m n+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 d n-b e n \log \left (f x^m\right )+b e m \log \left (c x^n\right )\right )\right )}{2 e^2 m^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.40, size = 2350, normalized size = 16.67
method | result | size |
risch | \(\text {Expression too large to display}\) | \(2350\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 95, normalized size = 0.67 \begin {gather*} \frac {{\left (b m n x^{2} e^{\left (\frac {2 \, {\left (e \log \left (f\right ) + d\right )} e^{\left (-1\right )}}{m} + 1\right )} + 2 \, {\left (b m e \log \left (c\right ) - b n e \log \left (f\right ) - b d n + a m e\right )} \operatorname {log\_integral}\left (x^{2} e^{\left (\frac {2 \, {\left (e \log \left (f\right ) + d\right )} e^{\left (-1\right )}}{m}\right )}\right )\right )} e^{\left (-\frac {2 \, {\left (e \log \left (f\right ) + d\right )} e^{\left (-1\right )}}{m} - 2\right )}}{2 \, m^{2}} \end {gather*}
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 \begin {gather*} \int \frac {x \left (a + b \log {\left (c x^{n} \right )}\right )}{d + e \log {\left (f x^{m} \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.76, size = 206, normalized size = 1.46 \begin {gather*} \frac {b n x^{2} e^{\left (-1\right )}}{2 \, m} - \frac {b d n {\rm Ei}\left (\frac {2 \, d e^{\left (-1\right )}}{m} + \frac {2 \, \log \left (f\right )}{m} + 2 \, \log \left (x\right )\right ) e^{\left (-\frac {2 \, d e^{\left (-1\right )}}{m} - 2\right )}}{f^{\frac {2}{m}} m^{2}} + \frac {b {\rm Ei}\left (\frac {2 \, d e^{\left (-1\right )}}{m} + \frac {2 \, \log \left (f\right )}{m} + 2 \, \log \left (x\right )\right ) e^{\left (-\frac {2 \, d e^{\left (-1\right )}}{m} - 1\right )} \log \left (c\right )}{f^{\frac {2}{m}} m} - \frac {b n {\rm Ei}\left (\frac {2 \, d e^{\left (-1\right )}}{m} + \frac {2 \, \log \left (f\right )}{m} + 2 \, \log \left (x\right )\right ) e^{\left (-\frac {2 \, d e^{\left (-1\right )}}{m} - 1\right )} \log \left (f\right )}{f^{\frac {2}{m}} m^{2}} + \frac {a {\rm Ei}\left (\frac {2 \, d e^{\left (-1\right )}}{m} + \frac {2 \, \log \left (f\right )}{m} + 2 \, \log \left (x\right )\right ) e^{\left (-\frac {2 \, d e^{\left (-1\right )}}{m} - 1\right )}}{f^{\frac {2}{m}} m} \end {gather*}
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 \begin {gather*} \int \frac {x\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{d+e\,\ln \left (f\,x^m\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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