\(\int \frac {(f+g x^n)^2 \log ^q(c (d+e x^n)^p)}{x} \, dx\) [382]

Optimal result

Integrand size = 27, antiderivative size = 27 \[ \int \frac {\left (f+g x^n\right )^2 \log ^q\left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\frac {2^{-1-q} g^2 \left (d+e x^n\right )^2 \left (c \left (d+e x^n\right )^p\right )^{-2/p} \Gamma \left (1+q,-\frac {2 \log \left (c \left (d+e x^n\right )^p\right )}{p}\right ) \log ^q\left (c \left (d+e x^n\right )^p\right ) \left (-\frac {\log \left (c \left (d+e x^n\right )^p\right )}{p}\right )^{-q}}{e^2 n}+\frac {2 f g \left (d+e x^n\right ) \left (c \left (d+e x^n\right )^p\right )^{-1/p} \Gamma \left (1+q,-\frac {\log \left (c \left (d+e x^n\right )^p\right )}{p}\right ) \log ^q\left (c \left (d+e x^n\right )^p\right ) \left (-\frac {\log \left (c \left (d+e x^n\right )^p\right )}{p}\right )^{-q}}{e n}-\frac {d g^2 \left (d+e x^n\right ) \left (c \left (d+e x^n\right )^p\right )^{-1/p} \Gamma \left (1+q,-\frac {\log \left (c \left (d+e x^n\right )^p\right )}{p}\right ) \log ^q\left (c \left (d+e x^n\right )^p\right ) \left (-\frac {\log \left (c \left (d+e x^n\right )^p\right )}{p}\right )^{-q}}{e^2 n}+f^2 \text {Int}\left (\frac {\log ^q\left (c \left (d+e x^n\right )^p\right )}{x},x\right ) \]

[Out]

2^(-1-q)*g^2*(d+e*x^n)^2*GAMMA(1+q,-2*ln(c*(d+e*x^n)^p)/p)*ln(c*(d+e*x^n)^p)^q/e^2/n/((c*(d+e*x^n)^p)^(2/p))/(
(-ln(c*(d+e*x^n)^p)/p)^q)+2*f*g*(d+e*x^n)*GAMMA(1+q,-ln(c*(d+e*x^n)^p)/p)*ln(c*(d+e*x^n)^p)^q/e/n/((c*(d+e*x^n
)^p)^(1/p))/((-ln(c*(d+e*x^n)^p)/p)^q)-d*g^2*(d+e*x^n)*GAMMA(1+q,-ln(c*(d+e*x^n)^p)/p)*ln(c*(d+e*x^n)^p)^q/e^2
/n/((c*(d+e*x^n)^p)^(1/p))/((-ln(c*(d+e*x^n)^p)/p)^q)+f^2*Unintegrable(ln(c*(d+e*x^n)^p)^q/x,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 = {} \[ \int \frac {\left (f+g x^n\right )^2 \log ^q\left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\int \frac {\left (f+g x^n\right )^2 \log ^q\left (c \left (d+e x^n\right )^p\right )}{x} \, dx \]

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (f+g x^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 = 0.19 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {\left (f+g x^n\right )^2 \log ^q\left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\int \frac {\left (f+g x^n\right )^2 \log ^q\left (c \left (d+e x^n\right )^p\right )}{x} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.48 \[ \int \frac {\left (f+g x^n\right )^2 \log ^q\left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\int { \frac {{\left (g x^{n} + f\right )}^{2} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )^{q}}{x} \,d x } \]

[In]

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

[Out]

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

Sympy [F(-2)]

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

[In]

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

[Out]

Exception raised: HeuristicGCDFailed >> no luck

Maxima [F(-2)]

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

[In]

integrate((f+g*x^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 = 2.75 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {\left (f+g x^n\right )^2 \log ^q\left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\int { \frac {{\left (g x^{n} + f\right )}^{2} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )^{q}}{x} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.51 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {\left (f+g x^n\right )^2 \log ^q\left (c \left (d+e x^n\right )^p\right )}{x} \, dx=\int \frac {{\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )}^q\,{\left (f+g\,x^n\right )}^2}{x} \,d x \]

[In]

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

[Out]

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