Optimal. Leaf size=77 \[ \frac {x^{1-m} (f x)^{m-1} \log \left (\frac {e x^m}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e m}+\frac {b n x^{1-m} (f x)^{m-1} \text {Li}_2\left (-\frac {e x^m}{d}\right )}{e m^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.19, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2339, 2337, 2391} \[ \frac {b n x^{1-m} (f x)^{m-1} \text {PolyLog}\left (2,-\frac {e x^m}{d}\right )}{e m^2}+\frac {x^{1-m} (f x)^{m-1} \log \left (\frac {e x^m}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e m} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2337
Rule 2339
Rule 2391
Rubi steps
\begin {align*} \int \frac {(f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{d+e x^m} \, dx &=\left (x^{1-m} (f x)^{-1+m}\right ) \int \frac {x^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{d+e x^m} \, dx\\ &=\frac {x^{1-m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x^m}{d}\right )}{e m}-\frac {\left (b n x^{1-m} (f x)^{-1+m}\right ) \int \frac {\log \left (1+\frac {e x^m}{d}\right )}{x} \, dx}{e m}\\ &=\frac {x^{1-m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x^m}{d}\right )}{e m}+\frac {b n x^{1-m} (f x)^{-1+m} \text {Li}_2\left (-\frac {e x^m}{d}\right )}{e m^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.15, size = 141, normalized size = 1.83 \[ \frac {x^{-m} (f x)^m \left (m \log (x) \left (a m+b m \log \left (c x^n\right )+b n \log \left (d+e x^m\right )-b n \log \left (d-d x^m\right )\right )+a m \log \left (d-d x^m\right )+b m \log \left (c x^n\right ) \log \left (d-d x^m\right )-b n \text {Li}_2\left (\frac {e x^m}{d}+1\right )-b n \log \left (-\frac {e x^m}{d}\right ) \log \left (d+e x^m\right )-b m^2 n \log ^2(x)\right )}{e f m^2} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.44, size = 77, normalized size = 1.00 \[ \frac {b f^{m - 1} m n \log \relax (x) \log \left (\frac {e x^{m} + d}{d}\right ) + b f^{m - 1} n {\rm Li}_2\left (-\frac {e x^{m} + d}{d} + 1\right ) + {\left (b m \log \relax (c) + a m\right )} f^{m - 1} \log \left (e x^{m} + d\right )}{e m^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \left (f x\right )^{m - 1}}{e x^{m} + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 1.06, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right ) \left (f x \right )^{m -1}}{e \,x^{m}+d}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ b \int \frac {f^{m} x^{m} \log \relax (c) + f^{m} x^{m} \log \left (x^{n}\right )}{e f x x^{m} + d f x}\,{d x} + \frac {a f^{m - 1} \log \left (\frac {e x^{m} + d}{e}\right )}{e m} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (f\,x\right )}^{m-1}\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{d+e\,x^m} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (f x\right )^{m - 1} \left (a + b \log {\left (c x^{n} \right )}\right )}{d + e x^{m}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________