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

Optimal. Leaf size=22 \[ \text{Unintegrable}\left (\frac{\log \left (c \left (d+e x^n\right )^p\right )}{(f+g x)^2},x\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0122881, 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 = {} \[ \int \frac{\log \left (c \left (d+e x^n\right )^p\right )}{(f+g x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin{align*} \int \frac{\log \left (c \left (d+e x^n\right )^p\right )}{(f+g x)^2} \, dx &=\int \frac{\log \left (c \left (d+e x^n\right )^p\right )}{(f+g x)^2} \, dx\\ \end{align*}

Mathematica [A]  time = 0.181424, size = 0, normalized size = 0. \[ \int \frac{\log \left (c \left (d+e x^n\right )^p\right )}{(f+g x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 2.131, size = 0, normalized size = 0. \begin{align*} \int{\frac{\ln \left ( c \left ( d+e{x}^{n} \right ) ^{p} \right ) }{ \left ( gx+f \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -d n p \int \frac{1}{d g^{2} x^{2} + d f g x +{\left (e g^{2} x^{2} + e f g x\right )} x^{n}}\,{d x} - \frac{n p \log \left (g x + f\right )}{f g} - \frac{f \log \left ({\left (e x^{n} + d\right )}^{p}\right ) + f \log \left (c\right ) -{\left (g n p x + f n p\right )} \log \left (x\right )}{f g^{2} x + f^{2} g} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-d*n*p*integrate(1/(d*g^2*x^2 + d*f*g*x + (e*g^2*x^2 + e*f*g*x)*x^n), x) - n*p*log(g*x + f)/(f*g) - (f*log((e*
x^n + d)^p) + f*log(c) - (g*n*p*x + f*n*p)*log(x))/(f*g^2*x + f^2*g)

________________________________________________________________________________________

Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{g^{2} x^{2} + 2 \, f g x + f^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{{\left (g x + f\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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