3.4.76 \(\int \frac {\log (f x^m)}{(a+b \log (c (d+e x)^n))^2} \, dx\) [376]

Optimal. Leaf size=26 \[ \text {Int}\left (\frac {\log \left (f x^m\right )}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2},x\right ) \]

[Out]

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

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*}

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Mathematica [A]
time = 0.32, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\log \left (f x^m\right )}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx \end {gather*}

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 = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\ln \left (f \,x^{m}\right )}{\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )^{2}}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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]

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

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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((x*e + d)^n*c)^2 + 2*a*b*log((x*e + d)^n*c) + a^2), x)

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\log {\left (f x^{m} \right )}}{\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{2}}\, dx \end {gather*}

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]

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

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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((x*e + d)^n*c) + a)^2, x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {\ln \left (f\,x^m\right )}{{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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