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

Optimal. Leaf size=23 \[ \text {Int}\left (\frac {\log \left (c \left (d+e x^n\right )^p\right )}{(f+g x)^3},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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*}

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Mathematica [A]  time = 0.22, size = 0, normalized size = 0.00 \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{(f+g x)^3} \, dx \]

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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fricas [A]  time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{g^{3} x^{3} + 3 \, f g^{2} x^{2} + 3 \, f^{2} g x + f^{3}}, x\right ) \]

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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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{{\left (g x + f\right )}^{3}}\,{d x} \]

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)

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maple [A]  time = 1.54, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (c \left (e \,x^{n}+d \right )^{p}\right )}{\left (g x +f \right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ -d n p \int \frac {1}{2 \, {\left (d g^{3} x^{3} + 2 \, d f g^{2} x^{2} + d f^{2} g x + {\left (e g^{3} x^{3} + 2 \, e f g^{2} x^{2} + e f^{2} g x\right )} x^{n}\right )}}\,{d x} + \frac {f g n p x + f^{2} n p - f^{2} \log \left ({\left (e x^{n} + d\right )}^{p}\right ) - f^{2} \log \relax (c) + {\left (g^{2} n p x^{2} + 2 \, f g n p x + f^{2} n p\right )} \log \relax (x)}{2 \, {\left (f^{2} g^{3} x^{2} + 2 \, f^{3} g^{2} x + f^{4} g\right )}} - \frac {n p \log \left (g x + f\right )}{2 \, f^{2} g} \]

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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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )}{{\left (f+g\,x\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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