3.376 \(\int \frac{\log (f x^m)}{(a+b \log (c (d+e x)^n))^2} \, dx\)

Optimal. Leaf size=25 \[ \text{Unintegrable}\left (\frac{\log \left (f x^m\right )}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2},x\right ) \]

[Out]

Unintegrable[Log[f*x^m]/(a + b*Log[c*(d + e*x)^n])^2, x]

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Rubi [A]  time = 0.0108033, 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 = {} \[ \int \frac{\log \left (f x^m\right )}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Log[f*x^m]/(a + b*Log[c*(d + e*x)^n])^2,x]

[Out]

Defer[Int][Log[f*x^m]/(a + b*Log[c*(d + e*x)^n])^2, x]

Rubi steps

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

Mathematica [A]  time = 0.462445, size = 0, normalized size = 0. \[ \int \frac{\log \left (f x^m\right )}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Log[f*x^m]/(a + b*Log[c*(d + e*x)^n])^2,x]

[Out]

Integrate[Log[f*x^m]/(a + b*Log[c*(d + e*x)^n])^2, x]

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Maple [A]  time = 10.89, size = 0, normalized size = 0. \begin{align*} \int{\frac{\ln \left ( f{x}^{m} \right ) }{ \left ( a+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(f*x^m)/(a+b*ln(c*(e*x+d)^n))^2,x)

[Out]

int(ln(f*x^m)/(a+b*ln(c*(e*x+d)^n))^2,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(e*x*log(f) + d*log(f) + (e*x + d)*log(x^m))/(b^2*e*n*log((e*x + d)^n) + b^2*e*n*log(c) + a*b*e*n) + integrat
e((e*(m + log(f))*x + e*x*log(x^m) + d*m)/(b^2*e*n*x*log((e*x + d)^n) + (b^2*e*n*log(c) + a*b*e*n)*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(log(f*x^m)/(b^2*log((e*x + d)^n*c)^2 + 2*a*b*log((e*x + d)^n*c) + a^2), 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(f*x**m)/(a+b*ln(c*(e*x+d)**n))**2,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(log(f*x^m)/(b*log((e*x + d)^n*c) + a)^2, x)