Optimal. Leaf size=80 \[ -\frac {(f x)^{-n} \log \left (c \left (d+e x^n\right )^p\right )}{f n}+\frac {e p x^n \log (x) (f x)^{-n}}{d f}-\frac {e p x^n (f x)^{-n} \log \left (d+e x^n\right )}{d f n} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2455, 19, 266, 36, 29, 31} \[ -\frac {(f x)^{-n} \log \left (c \left (d+e x^n\right )^p\right )}{f n}+\frac {e p x^n \log (x) (f x)^{-n}}{d f}-\frac {e p x^n (f x)^{-n} \log \left (d+e x^n\right )}{d f n} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 19
Rule 29
Rule 31
Rule 36
Rule 266
Rule 2455
Rubi steps
\begin {align*} \int (f x)^{-1-n} \log \left (c \left (d+e x^n\right )^p\right ) \, dx &=-\frac {(f x)^{-n} \log \left (c \left (d+e x^n\right )^p\right )}{f n}+\frac {(e p) \int \frac {x^{-1+n} (f x)^{-n}}{d+e x^n} \, dx}{f}\\ &=-\frac {(f x)^{-n} \log \left (c \left (d+e x^n\right )^p\right )}{f n}+\frac {\left (e p x^n (f x)^{-n}\right ) \int \frac {1}{x \left (d+e x^n\right )} \, dx}{f}\\ &=-\frac {(f x)^{-n} \log \left (c \left (d+e x^n\right )^p\right )}{f n}+\frac {\left (e p x^n (f x)^{-n}\right ) \operatorname {Subst}\left (\int \frac {1}{x (d+e x)} \, dx,x,x^n\right )}{f n}\\ &=-\frac {(f x)^{-n} \log \left (c \left (d+e x^n\right )^p\right )}{f n}+\frac {\left (e p x^n (f x)^{-n}\right ) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^n\right )}{d f n}-\frac {\left (e^2 p x^n (f x)^{-n}\right ) \operatorname {Subst}\left (\int \frac {1}{d+e x} \, dx,x,x^n\right )}{d f n}\\ &=\frac {e p x^n (f x)^{-n} \log (x)}{d f}-\frac {e p x^n (f x)^{-n} \log \left (d+e x^n\right )}{d f n}-\frac {(f x)^{-n} \log \left (c \left (d+e x^n\right )^p\right )}{f n}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 57, normalized size = 0.71 \[ -\frac {(f x)^{-n} \left (d \log \left (c \left (d+e x^n\right )^p\right )+e p x^n \log \left (d+e x^n\right )-e n p x^n \log (x)\right )}{d f n} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.50, size = 75, normalized size = 0.94 \[ \frac {e f^{-n - 1} n p x^{n} \log \relax (x) - d f^{-n - 1} \log \relax (c) - {\left (e f^{-n - 1} p x^{n} + d f^{-n - 1} p\right )} \log \left (e x^{n} + d\right )}{d n x^{n}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (f x\right )^{-n - 1} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 1.53, size = 0, normalized size = 0.00 \[ \int \left (f x \right )^{-n -1} \ln \left (c \left (e \,x^{n}+d \right )^{p}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.73, size = 71, normalized size = 0.89 \[ \frac {e p {\left (\frac {\log \relax (x)}{d f^{n}} - \frac {\log \left (\frac {e x^{n} + d}{e}\right )}{d f^{n} n}\right )}}{f} - \frac {\log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{\left (f x\right )^{n} f n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )}{{\left (f\,x\right )}^{n+1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________