\(\int \frac {(f+g x^{-2 n})^2 \log ^q(c (d+e x^n)^p)}{x} \, dx\) [384]

Optimal result

Integrand size = 29, antiderivative size = 29 \[ \int \frac {\left (f+g x^{-2 n}\right )^2 \log ^q\left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\text {Int}\left (\frac {\left (f+g x^{-2 n}\right )^2 \log ^q\left (c \left (d+e x^n\right )^p\right )}{x},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.07 (sec) , antiderivative size = 29, 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 = {} \[ \int \frac {\left (f+g x^{-2 n}\right )^2 \log ^q\left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\int \frac {\left (f+g x^{-2 n}\right )^2 \log ^q\left (c \left (d+e x^n\right )^p\right )}{x} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {\left (f+g x^{-2 n}\right )^2 \log ^q\left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\int \frac {\left (f+g x^{-2 n}\right )^2 \log ^q\left (c \left (d+e x^n\right )^p\right )}{x} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.69 \[ \int \frac {\left (f+g x^{-2 n}\right )^2 \log ^q\left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\int { \frac {{\left (f + \frac {g}{x^{2 \, n}}\right )}^{2} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )^{q}}{x} \,d x } \]

[In]

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

[Out]

integral((f^2*x^(4*n) + 2*f*g*x^(2*n) + g^2)*log((e*x^n + d)^p*c)^q/(x*x^(4*n)), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (f+g x^{-2 n}\right )^2 \log ^q\left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (f+g x^{-2 n}\right )^2 \log ^q\left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate((f+g/(x^(2*n)))^2*log(c*(d+e*x^n)^p)^q/x,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 = 2.49 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.14 \[ \int \frac {\left (f+g x^{-2 n}\right )^2 \log ^q\left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\int { \frac {{\left (f + \frac {g}{x^{2 \, n}}\right )}^{2} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )^{q}}{x} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.51 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.14 \[ \int \frac {\left (f+g x^{-2 n}\right )^2 \log ^q\left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\int \frac {{\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )}^q\,{\left (f+\frac {g}{x^{2\,n}}\right )}^2}{x} \,d x \]

[In]

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

[Out]

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