\(\int \frac {\log ^q(c (d+e x^n)^p)}{x (f+g x^n)} \, dx\) [386]

Optimal result

Integrand size = 27, antiderivative size = 27 \[ \int \frac {\log ^q\left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^n\right )} \, dx=\text {Int}\left (\frac {\log ^q\left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^n\right )},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.05 (sec) , antiderivative size = 27, 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 = {} \[ \int \frac {\log ^q\left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^n\right )} \, dx=\int \frac {\log ^q\left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^n\right )} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.82 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {\log ^q\left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^n\right )} \, dx=\int \frac {\log ^q\left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^n\right )} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.41 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.34 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {\log ^q\left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^n\right )} \, dx=\int { \frac {\log \left ({\left (e x^{n} + d\right )}^{p} c\right )^{q}}{{\left (g x^{n} + f\right )} x} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 9.21 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int \frac {\log ^q\left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^n\right )} \, dx=\int \frac {\log {\left (c \left (d + e x^{n}\right )^{p} \right )}^{q}}{x \left (f + g x^{n}\right )}\, dx \]

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {\log ^q\left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^n\right )} \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: In function CAR, the value of the first argument is  0which is not
of the expected type LIST

Giac [N/A]

Not integrable

Time = 2.47 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {\log ^q\left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^n\right )} \, dx=\int { \frac {\log \left ({\left (e x^{n} + d\right )}^{p} c\right )^{q}}{{\left (g x^{n} + f\right )} x} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.53 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {\log ^q\left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^n\right )} \, dx=\int \frac {{\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )}^q}{x\,\left (f+g\,x^n\right )} \,d x \]

[In]

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

[Out]

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