Optimal. Leaf size=69 \[ \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{d f m \left (d+e x^m\right )}-\frac {b n x^{-m} (f x)^m \log \left (d+e x^m\right )}{d e f m^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.07, antiderivative size = 69, 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 = {2373, 274, 266}
\begin {gather*} \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{d f m \left (d+e x^m\right )}-\frac {b n x^{-m} (f x)^m \log \left (d+e x^m\right )}{d e f m^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 266
Rule 274
Rule 2373
Rubi steps
\begin {align*} \int \frac {(f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^m\right )^2} \, dx &=\frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{d f m \left (d+e x^m\right )}-\frac {(b n) \int \frac {(f x)^{-1+m}}{d+e x^m} \, dx}{d m}\\ &=\frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{d f m \left (d+e x^m\right )}-\frac {\left (b n x^{-m} (f x)^m\right ) \int \frac {x^{-1+m}}{d+e x^m} \, dx}{d f m}\\ &=\frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{d f m \left (d+e x^m\right )}-\frac {b n x^{-m} (f x)^m \log \left (d+e x^m\right )}{d e f m^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.07, size = 89, normalized size = 1.29 \begin {gather*} -\frac {x^{-m} (f x)^m \left (a d m-b m n \left (d+e x^m\right ) \log (x)+b d m \log \left (c x^n\right )+b d n \log \left (d+e x^m\right )+b e n x^m \log \left (d+e x^m\right )\right )}{d e f m^2 \left (d+e x^m\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {\left (f x \right )^{-1+m} \left (a +b \ln \left (c \,x^{n}\right )\right )}{\left (d +e \,x^{m}\right )^{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.36, size = 108, normalized size = 1.57 \begin {gather*} b f^{m} n {\left (\frac {{\left (m \log \left (x\right ) + 2\right )} e^{\left (-1\right )}}{d f m^{2}} - \frac {e^{\left (-1\right )} \log \left (d e + e^{\left (m \log \left (x\right ) + 2\right )}\right )}{d f m^{2}}\right )} - \frac {b f^{m} \log \left (c x^{n}\right )}{d f m e + f m e^{\left (m \log \left (x\right ) + 2\right )}} - \frac {a f^{m}}{d f m e + f m e^{\left (m \log \left (x\right ) + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.38, size = 92, normalized size = 1.33 \begin {gather*} \frac {b f^{m - 1} m n x^{m} e \log \left (x\right ) - {\left (b d m \log \left (c\right ) + a d m\right )} f^{m - 1} - {\left (b f^{m - 1} n x^{m} e + b d f^{m - 1} n\right )} \log \left (x^{m} e + d\right )}{d m^{2} x^{m} e^{2} + d^{2} m^{2} e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (f x\right )^{m - 1} \left (a + b \log {\left (c x^{n} \right )}\right )}{\left (d + e x^{m}\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 206 vs.
\(2 (70) = 140\).
time = 7.41, size = 206, normalized size = 2.99 \begin {gather*} \frac {b f^{m} m n x x^{m} e \log \left (x\right )}{d f m^{2} x x^{m} e^{2} + d^{2} f m^{2} x e} - \frac {b f^{m} n x x^{m} e \log \left (x^{m} e + d\right )}{d f m^{2} x x^{m} e^{2} + d^{2} f m^{2} x e} - \frac {b d f^{m} n x \log \left (x^{m} e + d\right )}{d f m^{2} x x^{m} e^{2} + d^{2} f m^{2} x e} - \frac {b d f^{m} m x \log \left (c\right )}{d f m^{2} x x^{m} e^{2} + d^{2} f m^{2} x e} - \frac {a d f^{m} m x}{d f m^{2} x x^{m} e^{2} + d^{2} f m^{2} x e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (f\,x\right )}^{m-1}\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{{\left (d+e\,x^m\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________