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

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

[Out]

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

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Rubi [A]  time = 0.0059268, 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 = {} \[ \int \left (a+b \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )\right )^n \, dx \]

Verification is Not applicable to the result.

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

Mathematica [A]  time = 0.406539, size = 0, normalized size = 0. \[ \int \left (a+b \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )\right )^n \, dx \]

Verification is Not applicable to the result.

[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]

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Maple [A]  time = 0.659, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\ln \left ( c \left ( d+{\frac{e}{gx+f}} \right ) ^{p} \right ) \right ) ^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[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)