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

Optimal. Leaf size=24 \[ \text{Unintegrable}\left (\frac{1}{a+b \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )},x\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0059505, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{1}{a+b \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )} \, dx \]

Verification is Not applicable to the result.

[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*} \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\\ \end{align*}

Mathematica [A]  time = 0.538266, size = 0, normalized size = 0. \[ \int \frac{1}{a+b \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )} \, dx \]

Verification is Not applicable to the result.

[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 [A]  time = 0.123, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\ln \left ( c \left ( d+{\frac{e}{gx+f}} \right ) ^{p} \right ) \right ) ^{-1}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{b \log \left (c{\left (d + \frac{e}{g x + f}\right )}^{p}\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{b \log \left (c \left (\frac{d g x + d f + e}{g x + f}\right )^{p}\right ) + a}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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 [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{b \log \left (c{\left (d + \frac{e}{g x + f}\right )}^{p}\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)