Integrand size = 27, antiderivative size = 150 \[ \int \frac {(f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^m\right )^3} \, dx=\frac {b n x^{1-m} (f x)^{-1+m}}{2 d e m^2 \left (d+e x^m\right )}+\frac {b n x^{1-m} (f x)^{-1+m} \log (x)}{2 d^2 e m}-\frac {x^{1-m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{2 e m \left (d+e x^m\right )^2}-\frac {b n x^{1-m} (f x)^{-1+m} \log \left (d+e x^m\right )}{2 d^2 e m^2} \]
[Out]
Time = 0.16 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {2377, 2376, 272, 46} \[ \int \frac {(f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^m\right )^3} \, dx=-\frac {x^{1-m} (f x)^{m-1} \left (a+b \log \left (c x^n\right )\right )}{2 e m \left (d+e x^m\right )^2}-\frac {b n x^{1-m} (f x)^{m-1} \log \left (d+e x^m\right )}{2 d^2 e m^2}+\frac {b n x^{1-m} \log (x) (f x)^{m-1}}{2 d^2 e m}+\frac {b n x^{1-m} (f x)^{m-1}}{2 d e m^2 \left (d+e x^m\right )} \]
[In]
[Out]
Rule 46
Rule 272
Rule 2376
Rule 2377
Rubi steps \begin{align*} \text {integral}& = \left (x^{1-m} (f x)^{-1+m}\right ) \int \frac {x^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^m\right )^3} \, dx \\ & = -\frac {x^{1-m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{2 e m \left (d+e x^m\right )^2}+\frac {\left (b n x^{1-m} (f x)^{-1+m}\right ) \int \frac {1}{x \left (d+e x^m\right )^2} \, dx}{2 e m} \\ & = -\frac {x^{1-m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{2 e m \left (d+e x^m\right )^2}+\frac {\left (b n x^{1-m} (f x)^{-1+m}\right ) \text {Subst}\left (\int \frac {1}{x (d+e x)^2} \, dx,x,x^m\right )}{2 e m^2} \\ & = -\frac {x^{1-m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{2 e m \left (d+e x^m\right )^2}+\frac {\left (b n x^{1-m} (f x)^{-1+m}\right ) \text {Subst}\left (\int \left (\frac {1}{d^2 x}-\frac {e}{d (d+e x)^2}-\frac {e}{d^2 (d+e x)}\right ) \, dx,x,x^m\right )}{2 e m^2} \\ & = \frac {b n x^{1-m} (f x)^{-1+m}}{2 d e m^2 \left (d+e x^m\right )}+\frac {b n x^{1-m} (f x)^{-1+m} \log (x)}{2 d^2 e m}-\frac {x^{1-m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{2 e m \left (d+e x^m\right )^2}-\frac {b n x^{1-m} (f x)^{-1+m} \log \left (d+e x^m\right )}{2 d^2 e m^2} \\ \end{align*}
Time = 0.28 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.91 \[ \int \frac {(f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^m\right )^3} \, dx=\frac {x^{-m} (f x)^m \left (-a d^2 m+b d^2 n+b d e n x^m+b m n \left (d+e x^m\right )^2 \log (x)-b d^2 m \log \left (c x^n\right )-b d^2 n \log \left (d+e x^m\right )-2 b d e n x^m \log \left (d+e x^m\right )-b e^2 n x^{2 m} \log \left (d+e x^m\right )\right )}{2 d^2 e f m^2 \left (d+e x^m\right )^2} \]
[In]
[Out]
\[\int \frac {\left (f x \right )^{m -1} \left (a +b \ln \left (c \,x^{n}\right )\right )}{\left (d +e \,x^{m}\right )^{3}}d x\]
[In]
[Out]
none
Time = 0.31 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.11 \[ \int \frac {(f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^m\right )^3} \, dx=\frac {b e^{2} f^{m - 1} m n x^{2 \, m} \log \left (x\right ) + {\left (2 \, b d e m n \log \left (x\right ) + b d e n\right )} f^{m - 1} x^{m} - {\left (b d^{2} m \log \left (c\right ) + a d^{2} m - b d^{2} n\right )} f^{m - 1} - {\left (b e^{2} f^{m - 1} n x^{2 \, m} + 2 \, b d e f^{m - 1} n x^{m} + b d^{2} f^{m - 1} n\right )} \log \left (e x^{m} + d\right )}{2 \, {\left (d^{2} e^{3} m^{2} x^{2 \, m} + 2 \, d^{3} e^{2} m^{2} x^{m} + d^{4} e m^{2}\right )}} \]
[In]
[Out]
\[ \int \frac {(f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^m\right )^3} \, dx=\int \frac {\left (f x\right )^{m - 1} \left (a + b \log {\left (c x^{n} \right )}\right )}{\left (d + e x^{m}\right )^{3}}\, dx \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.01 \[ \int \frac {(f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^m\right )^3} \, dx=\frac {1}{2} \, b f^{m} n {\left (\frac {1}{{\left (d e^{2} f m x^{m} + d^{2} e f m\right )} m} + \frac {\log \left (x\right )}{d^{2} e f m} - \frac {\log \left (e x^{m} + d\right )}{d^{2} e f m^{2}}\right )} - \frac {b f^{m} \log \left (c x^{n}\right )}{2 \, {\left (e^{3} f m x^{2 \, m} + 2 \, d e^{2} f m x^{m} + d^{2} e f m\right )}} - \frac {a f^{m}}{2 \, {\left (e^{3} f m x^{2 \, m} + 2 \, d e^{2} f m x^{m} + d^{2} e f m\right )}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 633 vs. \(2 (142) = 284\).
Time = 0.36 (sec) , antiderivative size = 633, normalized size of antiderivative = 4.22 \[ \int \frac {(f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^m\right )^3} \, dx=\frac {b e^{2} f^{m} m n x^{2} x^{2 \, m} \log \left (x\right )}{2 \, {\left (d^{2} e^{3} f m^{2} x^{2} x^{2 \, m} + 2 \, d^{3} e^{2} f m^{2} x^{2} x^{m} + d^{4} e f m^{2} x^{2}\right )}} + \frac {b d e f^{m} m n x^{2} x^{m} \log \left (x\right )}{d^{2} e^{3} f m^{2} x^{2} x^{2 \, m} + 2 \, d^{3} e^{2} f m^{2} x^{2} x^{m} + d^{4} e f m^{2} x^{2}} - \frac {b e^{2} f^{m} n x^{2} x^{2 \, m} \log \left (e x^{m} + d\right )}{2 \, {\left (d^{2} e^{3} f m^{2} x^{2} x^{2 \, m} + 2 \, d^{3} e^{2} f m^{2} x^{2} x^{m} + d^{4} e f m^{2} x^{2}\right )}} - \frac {b d e f^{m} n x^{2} x^{m} \log \left (e x^{m} + d\right )}{d^{2} e^{3} f m^{2} x^{2} x^{2 \, m} + 2 \, d^{3} e^{2} f m^{2} x^{2} x^{m} + d^{4} e f m^{2} x^{2}} + \frac {b d e f^{m} n x^{2} x^{m}}{2 \, {\left (d^{2} e^{3} f m^{2} x^{2} x^{2 \, m} + 2 \, d^{3} e^{2} f m^{2} x^{2} x^{m} + d^{4} e f m^{2} x^{2}\right )}} - \frac {b d^{2} f^{m} n x^{2} \log \left (e x^{m} + d\right )}{2 \, {\left (d^{2} e^{3} f m^{2} x^{2} x^{2 \, m} + 2 \, d^{3} e^{2} f m^{2} x^{2} x^{m} + d^{4} e f m^{2} x^{2}\right )}} - \frac {b d^{2} f^{m} m x^{2} \log \left (c\right )}{2 \, {\left (d^{2} e^{3} f m^{2} x^{2} x^{2 \, m} + 2 \, d^{3} e^{2} f m^{2} x^{2} x^{m} + d^{4} e f m^{2} x^{2}\right )}} - \frac {a d^{2} f^{m} m x^{2}}{2 \, {\left (d^{2} e^{3} f m^{2} x^{2} x^{2 \, m} + 2 \, d^{3} e^{2} f m^{2} x^{2} x^{m} + d^{4} e f m^{2} x^{2}\right )}} + \frac {b d^{2} f^{m} n x^{2}}{2 \, {\left (d^{2} e^{3} f m^{2} x^{2} x^{2 \, m} + 2 \, d^{3} e^{2} f m^{2} x^{2} x^{m} + d^{4} e f m^{2} x^{2}\right )}} \]
[In]
[Out]
Timed out. \[ \int \frac {(f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^m\right )^3} \, dx=\int \frac {{\left (f\,x\right )}^{m-1}\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{{\left (d+e\,x^m\right )}^3} \,d x \]
[In]
[Out]