\(\int \frac {\log (c (d+e x^n)^p)}{(f+g x)^2} \, dx\) [217]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{(f+g x)^2} \, dx=\text {Int}\left (\frac {\log \left (c \left (d+e x^n\right )^p\right )}{(f+g x)^2},x\right ) \]

[Out]

Unintegrable(ln(c*(d+e*x^n)^p)/(g*x+f)^2,x)

Rubi [N/A]

Not integrable

Time = 0.01 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{(f+g x)^2} \, dx=\int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{(f+g x)^2} \, dx \]

[In]

Int[Log[c*(d + e*x^n)^p]/(f + g*x)^2,x]

[Out]

Defer[Int][Log[c*(d + e*x^n)^p]/(f + g*x)^2, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{(f+g x)^2} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.36 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{(f+g x)^2} \, dx=\int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{(f+g x)^2} \, dx \]

[In]

Integrate[Log[c*(d + e*x^n)^p]/(f + g*x)^2,x]

[Out]

Integrate[Log[c*(d + e*x^n)^p]/(f + g*x)^2, x]

Maple [N/A]

Not integrable

Time = 0.21 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

\[\int \frac {\ln \left (c \left (d +e \,x^{n}\right )^{p}\right )}{\left (g x +f \right )^{2}}d x\]

[In]

int(ln(c*(d+e*x^n)^p)/(g*x+f)^2,x)

[Out]

int(ln(c*(d+e*x^n)^p)/(g*x+f)^2,x)

Fricas [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.65 \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{(f+g x)^2} \, dx=\int { \frac {\log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{{\left (g x + f\right )}^{2}} \,d x } \]

[In]

integrate(log(c*(d+e*x^n)^p)/(g*x+f)^2,x, algorithm="fricas")

[Out]

integral(log((e*x^n + d)^p*c)/(g^2*x^2 + 2*f*g*x + f^2), x)

Sympy [N/A]

Not integrable

Time = 43.22 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{(f+g x)^2} \, dx=\int \frac {\log {\left (c \left (d + e x^{n}\right )^{p} \right )}}{\left (f + g x\right )^{2}}\, dx \]

[In]

integrate(ln(c*(d+e*x**n)**p)/(g*x+f)**2,x)

[Out]

Integral(log(c*(d + e*x**n)**p)/(f + g*x)**2, x)

Maxima [N/A]

Not integrable

Time = 0.35 (sec) , antiderivative size = 105, normalized size of antiderivative = 5.25 \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{(f+g x)^2} \, dx=\int { \frac {\log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{{\left (g x + f\right )}^{2}} \,d x } \]

[In]

integrate(log(c*(d+e*x^n)^p)/(g*x+f)^2,x, algorithm="maxima")

[Out]

-d*n*p*integrate(1/(d*g^2*x^2 + d*f*g*x + (e*g^2*x^2 + e*f*g*x)*x^n), x) - n*p*log(g*x + f)/(f*g) - (f*log((e*
x^n + d)^p) + f*log(c) - (g*n*p*x + f*n*p)*log(x))/(f*g^2*x + f^2*g)

Giac [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{(f+g x)^2} \, dx=\int { \frac {\log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{{\left (g x + f\right )}^{2}} \,d x } \]

[In]

integrate(log(c*(d+e*x^n)^p)/(g*x+f)^2,x, algorithm="giac")

[Out]

integrate(log((e*x^n + d)^p*c)/(g*x + f)^2, x)

Mupad [N/A]

Not integrable

Time = 1.41 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{(f+g x)^2} \, dx=\int \frac {\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )}{{\left (f+g\,x\right )}^2} \,d x \]

[In]

int(log(c*(d + e*x^n)^p)/(f + g*x)^2,x)

[Out]

int(log(c*(d + e*x^n)^p)/(f + g*x)^2, x)