\(\int \frac {\log (f x^m)}{a+b \log (c (d+e x)^n)} \, dx\) [375]

Optimal result

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

[Out]

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

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.05 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {\log \left (f x^m\right )}{a+b \log \left (c (d+e x)^n\right )} \, dx=\int \frac {\log \left (f x^m\right )}{a+b \log \left (c (d+e x)^n\right )} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.04 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

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

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 2.76 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {\log \left (f x^m\right )}{a+b \log \left (c (d+e x)^n\right )} \, dx=\int \frac {\log {\left (f x^{m} \right )}}{a + b \log {\left (c \left (d + e x\right )^{n} \right )}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {\log \left (f x^m\right )}{a+b \log \left (c (d+e x)^n\right )} \, dx=\int { \frac {\log \left (f x^{m}\right )}{b \log \left ({\left (e x + d\right )}^{n} c\right ) + a} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {\log \left (f x^m\right )}{a+b \log \left (c (d+e x)^n\right )} \, dx=\int { \frac {\log \left (f x^{m}\right )}{b \log \left ({\left (e x + d\right )}^{n} c\right ) + a} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.21 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {\log \left (f x^m\right )}{a+b \log \left (c (d+e x)^n\right )} \, dx=\int \frac {\ln \left (f\,x^m\right )}{a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )} \,d x \]

[In]

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

[Out]

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