3.218 \(\int \frac{\log (c (d+e x^n)^p)}{(f+g x)^3} \, dx\)

Optimal. Leaf size=22 \[ \text{Unintegrable}\left (\frac{\log \left (c \left (d+e x^n\right )^p\right )}{(f+g x)^3},x\right ) \]

[Out]

Unintegrable[Log[c*(d + e*x^n)^p]/(f + g*x)^3, x]

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Rubi [A]  time = 0.0122042, 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 = {} \[ \int \frac{\log \left (c \left (d+e x^n\right )^p\right )}{(f+g x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Log[c*(d + e*x^n)^p]/(f + g*x)^3,x]

[Out]

Defer[Int][Log[c*(d + e*x^n)^p]/(f + g*x)^3, x]

Rubi steps

\begin{align*} \int \frac{\log \left (c \left (d+e x^n\right )^p\right )}{(f+g x)^3} \, dx &=\int \frac{\log \left (c \left (d+e x^n\right )^p\right )}{(f+g x)^3} \, dx\\ \end{align*}

Mathematica [A]  time = 0.204131, size = 0, normalized size = 0. \[ \int \frac{\log \left (c \left (d+e x^n\right )^p\right )}{(f+g x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Log[c*(d + e*x^n)^p]/(f + g*x)^3,x]

[Out]

Integrate[Log[c*(d + e*x^n)^p]/(f + g*x)^3, x]

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Maple [A]  time = 2.204, size = 0, normalized size = 0. \begin{align*} \int{\frac{\ln \left ( c \left ( d+e{x}^{n} \right ) ^{p} \right ) }{ \left ( gx+f \right ) ^{3}}}\, 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)^3,x)

[Out]

int(ln(c*(d+e*x^n)^p)/(g*x+f)^3,x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -d n p \int \frac{1}{2 \,{\left (d g^{3} x^{3} + 2 \, d f g^{2} x^{2} + d f^{2} g x +{\left (e g^{3} x^{3} + 2 \, e f g^{2} x^{2} + e f^{2} g x\right )} x^{n}\right )}}\,{d x} + \frac{f g n p x + f^{2} n p - f^{2} \log \left ({\left (e x^{n} + d\right )}^{p}\right ) - f^{2} \log \left (c\right ) +{\left (g^{2} n p x^{2} + 2 \, f g n p x + f^{2} n p\right )} \log \left (x\right )}{2 \,{\left (f^{2} g^{3} x^{2} + 2 \, f^{3} g^{2} x + f^{4} g\right )}} - \frac{n p \log \left (g x + f\right )}{2 \, 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)^3,x, algorithm="maxima")

[Out]

-d*n*p*integrate(1/2/(d*g^3*x^3 + 2*d*f*g^2*x^2 + d*f^2*g*x + (e*g^3*x^3 + 2*e*f*g^2*x^2 + e*f^2*g*x)*x^n), x)
+ 1/2*(f*g*n*p*x + f^2*n*p - f^2*log((e*x^n + d)^p) - f^2*log(c) + (g^2*n*p*x^2 + 2*f*g*n*p*x + f^2*n*p)*log(
x))/(f^2*g^3*x^2 + 2*f^3*g^2*x + f^4*g) - 1/2*n*p*log(g*x + f)/(f^2*g)

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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^{3} x^{3} + 3 \, f g^{2} x^{2} + 3 \, f^{2} g x + f^{3}}, 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)^3,x, algorithm="fricas")

[Out]

integral(log((e*x^n + d)^p*c)/(g^3*x^3 + 3*f*g^2*x^2 + 3*f^2*g*x + f^3), 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(ln(c*(d+e*x**n)**p)/(g*x+f)**3,x)

[Out]

Timed out

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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 )}^{3}}\,{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)^3,x, algorithm="giac")

[Out]

integrate(log((e*x^n + d)^p*c)/(g*x + f)^3, x)