\(\int \frac {1}{a+b \log (c (d+\frac {e}{f+g x})^p)} \, dx\) [640]

Optimal result

Integrand size = 22, antiderivative size = 22 \[ \int \frac {1}{a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )} \, dx=\text {Int}\left (\frac {1}{a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )},x\right ) \]

[Out]

Unintegrable(1/(a+b*ln(c*(d+e/(g*x+f))^p)),x)

Rubi [N/A]

Not integrable

Time = 0.01 (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 = {} \[ \int \frac {1}{a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )} \, dx=\int \frac {1}{a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )} \, dx \]

[In]

Int[(a + b*Log[c*(d + e/(f + g*x))^p])^(-1),x]

[Out]

Defer[Int][(a + b*Log[c*(d + e/(f + g*x))^p])^(-1), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )} \, dx=\int \frac {1}{a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )} \, dx \]

[In]

Integrate[(a + b*Log[c*(d + e/(f + g*x))^p])^(-1),x]

[Out]

Integrate[(a + b*Log[c*(d + e/(f + g*x))^p])^(-1), x]

Maple [N/A]

Not integrable

Time = 0.08 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00

\[\int \frac {1}{a +b \ln \left (c \left (d +\frac {e}{g x +f}\right )^{p}\right )}d x\]

[In]

int(1/(a+b*ln(c*(d+e/(g*x+f))^p)),x)

[Out]

int(1/(a+b*ln(c*(d+e/(g*x+f))^p)),x)

Fricas [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.36 \[ \int \frac {1}{a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )} \, dx=\int { \frac {1}{b \log \left (c {\left (d + \frac {e}{g x + f}\right )}^{p}\right ) + a} \,d x } \]

[In]

integrate(1/(a+b*log(c*(d+e/(g*x+f))^p)),x, algorithm="fricas")

[Out]

integral(1/(b*log(c*((d*g*x + d*f + e)/(g*x + f))^p) + a), x)

Sympy [N/A]

Not integrable

Time = 1.15 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86 \[ \int \frac {1}{a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )} \, dx=\int \frac {1}{a + b \log {\left (c \left (d + \frac {e}{f + g x}\right )^{p} \right )}}\, dx \]

[In]

integrate(1/(a+b*ln(c*(d+e/(g*x+f))**p)),x)

[Out]

Integral(1/(a + b*log(c*(d + e/(f + g*x))**p)), x)

Maxima [N/A]

Not integrable

Time = 0.35 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )} \, dx=\int { \frac {1}{b \log \left (c {\left (d + \frac {e}{g x + f}\right )}^{p}\right ) + a} \,d x } \]

[In]

integrate(1/(a+b*log(c*(d+e/(g*x+f))^p)),x, algorithm="maxima")

[Out]

integrate(1/(b*log(c*(d + e/(g*x + f))^p) + a), x)

Giac [N/A]

Not integrable

Time = 1.04 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )} \, dx=\int { \frac {1}{b \log \left (c {\left (d + \frac {e}{g x + f}\right )}^{p}\right ) + a} \,d x } \]

[In]

integrate(1/(a+b*log(c*(d+e/(g*x+f))^p)),x, algorithm="giac")

[Out]

integrate(1/(b*log(c*(d + e/(g*x + f))^p) + a), x)

Mupad [N/A]

Not integrable

Time = 1.48 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )} \, dx=\int \frac {1}{a+b\,\ln \left (c\,{\left (d+\frac {e}{f+g\,x}\right )}^p\right )} \,d x \]

[In]

int(1/(a + b*log(c*(d + e/(f + g*x))^p)),x)

[Out]

int(1/(a + b*log(c*(d + e/(f + g*x))^p)), x)