\(\int \frac {(f x)^m}{\log (c (d+e x^2)^p)} \, dx\) [161]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int \frac {(f x)^m}{\log \left (c \left (d+e x^2\right )^p\right )} \, dx=\text {Int}\left (\frac {(f x)^m}{\log \left (c \left (d+e x^2\right )^p\right )},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.01 (sec) , antiderivative size = 20, 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 = {} \[ \int \frac {(f x)^m}{\log \left (c \left (d+e x^2\right )^p\right )} \, dx=\int \frac {(f x)^m}{\log \left (c \left (d+e x^2\right )^p\right )} \, dx \]

[In]

Int[(f*x)^m/Log[c*(d + e*x^2)^p],x]

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {(f x)^m}{\log \left (c \left (d+e x^2\right )^p\right )} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.39 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {(f x)^m}{\log \left (c \left (d+e x^2\right )^p\right )} \, dx=\int \frac {(f x)^m}{\log \left (c \left (d+e x^2\right )^p\right )} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {(f x)^m}{\log \left (c \left (d+e x^2\right )^p\right )} \, dx=\int { \frac {\left (f x\right )^{m}}{\log \left ({\left (e x^{2} + d\right )}^{p} c\right )} \,d x } \]

[In]

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

[Out]

integral((f*x)^m/log((e*x^2 + d)^p*c), x)

Sympy [N/A]

Not integrable

Time = 12.99 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85 \[ \int \frac {(f x)^m}{\log \left (c \left (d+e x^2\right )^p\right )} \, dx=\int \frac {\left (f x\right )^{m}}{\log {\left (c \left (d + e x^{2}\right )^{p} \right )}}\, dx \]

[In]

integrate((f*x)**m/ln(c*(e*x**2+d)**p),x)

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {(f x)^m}{\log \left (c \left (d+e x^2\right )^p\right )} \, dx=\int { \frac {\left (f x\right )^{m}}{\log \left ({\left (e x^{2} + d\right )}^{p} c\right )} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {(f x)^m}{\log \left (c \left (d+e x^2\right )^p\right )} \, dx=\int { \frac {\left (f x\right )^{m}}{\log \left ({\left (e x^{2} + d\right )}^{p} c\right )} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.19 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {(f x)^m}{\log \left (c \left (d+e x^2\right )^p\right )} \, dx=\int \frac {{\left (f\,x\right )}^m}{\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )} \,d x \]

[In]

int((f*x)^m/log(c*(d + e*x^2)^p),x)

[Out]

int((f*x)^m/log(c*(d + e*x^2)^p), x)