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

Optimal result

Integrand size = 23, antiderivative size = 23 \[ \int \frac {\log \left (f x^m\right )}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\text {Int}\left (\frac {\log \left (f x^m\right )}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\log \left (f x^m\right )}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\int \frac {\log \left (f x^m\right )}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx \]

[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*} \text {integral}& = \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 [N/A]

Not integrable

Time = 0.35 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {\log \left (f x^m\right )}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\int \frac {\log \left (f x^m\right )}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx \]

[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]

Maple [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00

\[\int \frac {\ln \left (f \,x^{m}\right )}{{\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}^{2}}d x\]

[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)

Fricas [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.96 \[ \int \frac {\log \left (f x^m\right )}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\int { \frac {\log \left (f x^{m}\right )}{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2}} \,d x } \]

[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)

Sympy [N/A]

Not integrable

Time = 31.82 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {\log \left (f x^m\right )}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\int \frac {\log {\left (f x^{m} \right )}}{\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{2}}\, dx \]

[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)

Maxima [N/A]

Not integrable

Time = 0.38 (sec) , antiderivative size = 108, normalized size of antiderivative = 4.70 \[ \int \frac {\log \left (f x^m\right )}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\int { \frac {\log \left (f x^{m}\right )}{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2}} \,d x } \]

[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)

Giac [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {\log \left (f x^m\right )}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\int { \frac {\log \left (f x^{m}\right )}{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2}} \,d x } \]

[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)

Mupad [N/A]

Not integrable

Time = 1.23 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {\log \left (f x^m\right )}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\int \frac {\ln \left (f\,x^m\right )}{{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2} \,d x \]

[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)