\(\int \frac {\log (c (d+e x^n)^p)}{(f+g x)^3} \, dx\) [218]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{(f+g x)^3} \, dx=\text {Int}\left (\frac {\log \left (c \left (d+e x^n\right )^p\right )}{(f+g x)^3},x\right ) \]

[Out]

Unintegrable(ln(c*(d+e*x^n)^p)/(g*x+f)^3,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 = {} \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{(f+g x)^3} \, dx=\int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{(f+g x)^3} \, dx \]

[In]

Int[Log[c*(d + e*x^n)^p]/(f + g*x)^3,x]

[Out]

Defer[Int][Log[c*(d + e*x^n)^p]/(f + g*x)^3, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{(f+g x)^3} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.36 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{(f+g x)^3} \, dx=\int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{(f+g x)^3} \, dx \]

[In]

Integrate[Log[c*(d + e*x^n)^p]/(f + g*x)^3,x]

[Out]

Integrate[Log[c*(d + e*x^n)^p]/(f + g*x)^3, x]

Maple [N/A]

Not integrable

Time = 0.22 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

\[\int \frac {\ln \left (c \left (d +e \,x^{n}\right )^{p}\right )}{\left (g x +f \right )^{3}}d x\]

[In]

int(ln(c*(d+e*x^n)^p)/(g*x+f)^3,x)

[Out]

int(ln(c*(d+e*x^n)^p)/(g*x+f)^3,x)

Fricas [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.20 \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{(f+g x)^3} \, dx=\int { \frac {\log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{{\left (g x + f\right )}^{3}} \,d x } \]

[In]

integrate(log(c*(d+e*x^n)^p)/(g*x+f)^3,x, algorithm="fricas")

[Out]

integral(log((e*x^n + d)^p*c)/(g^3*x^3 + 3*f*g^2*x^2 + 3*f^2*g*x + f^3), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{(f+g x)^3} \, dx=\text {Timed out} \]

[In]

integrate(ln(c*(d+e*x**n)**p)/(g*x+f)**3,x)

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.36 (sec) , antiderivative size = 174, normalized size of antiderivative = 8.70 \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{(f+g x)^3} \, dx=\int { \frac {\log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{{\left (g x + f\right )}^{3}} \,d x } \]

[In]

integrate(log(c*(d+e*x^n)^p)/(g*x+f)^3,x, algorithm="maxima")

[Out]

-d*n*p*integrate(1/2/(d*g^3*x^3 + 2*d*f*g^2*x^2 + d*f^2*g*x + (e*g^3*x^3 + 2*e*f*g^2*x^2 + e*f^2*g*x)*x^n), x)
+ 1/2*(f*g*n*p*x + f^2*n*p - f^2*log((e*x^n + d)^p) - f^2*log(c) + (g^2*n*p*x^2 + 2*f*g*n*p*x + f^2*n*p)*log(
x))/(f^2*g^3*x^2 + 2*f^3*g^2*x + f^4*g) - 1/2*n*p*log(g*x + f)/(f^2*g)

Giac [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{(f+g x)^3} \, dx=\int { \frac {\log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{{\left (g x + f\right )}^{3}} \,d x } \]

[In]

integrate(log(c*(d+e*x^n)^p)/(g*x+f)^3,x, algorithm="giac")

[Out]

integrate(log((e*x^n + d)^p*c)/(g*x + f)^3, x)

Mupad [N/A]

Not integrable

Time = 1.50 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{(f+g x)^3} \, dx=\int \frac {\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )}{{\left (f+g\,x\right )}^3} \,d x \]

[In]

int(log(c*(d + e*x^n)^p)/(f + g*x)^3,x)

[Out]

int(log(c*(d + e*x^n)^p)/(f + g*x)^3, x)