\(\int (a+b \log (c (d+\frac {e}{f+g x})^p))^n \, dx\) [635]

Optimal result

Integrand size = 22, antiderivative size = 22 \[ \int \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^n \, dx=\text {Int}\left (\left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^n,x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 22, 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 = {} \[ \int \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^n \, dx=\int \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^n \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^n \, dx=\int \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^n \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.10 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.35 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.36 \[ \int \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^n \, dx=\int { {\left (b \log \left (c {\left (d + \frac {e}{g x + f}\right )}^{p}\right ) + a\right )}^{n} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 18.08 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86 \[ \int \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^n \, dx=\int \left (a + b \log {\left (c \left (d + \frac {e}{f + g x}\right )^{p} \right )}\right )^{n}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 1.33 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^n \, dx=\int { {\left (b \log \left (c {\left (d + \frac {e}{g x + f}\right )}^{p}\right ) + a\right )}^{n} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.43 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^n \, dx=\int { {\left (b \log \left (c {\left (d + \frac {e}{g x + f}\right )}^{p}\right ) + a\right )}^{n} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.07 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^n \, dx=\int {\left (a+b\,\ln \left (c\,{\left (d+\frac {e}{f+g\,x}\right )}^p\right )\right )}^n \,d x \]

[In]

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

[Out]

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