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

Optimal. Leaf size=23 \[ \text {Int}\left (\frac {\log \left (c \left (d+e x^n\right )^p\right )}{(f+g x)^2},x\right ) \]

[Out]

Unintegrable(ln(c*(d+e*x^n)^p)/(g*x+f)^2,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 = {} \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{(f+g x)^2} \, dx \]

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*}

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Mathematica [A]  time = 0.19, size = 0, normalized size = 0.00 \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{(f+g x)^2} \, dx \]

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]

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fricas [A]  time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{g^{2} x^{2} + 2 \, f g x + f^{2}}, x\right ) \]

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)

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

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)

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maple [A]  time = 1.37, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (c \left (e \,x^{n}+d \right )^{p}\right )}{\left (g x +f \right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )}{{\left (f+g\,x\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log {\left (c \left (d + e x^{n}\right )^{p} \right )}}{\left (f + g x\right )^{2}}\, dx \]

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]

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

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