\(\int \frac {f+g x^2}{\log ^2(c (d+e x^2)^p)} \, dx\) [285]

Optimal result

Integrand size = 22, antiderivative size = 22 \[ \int \frac {f+g x^2}{\log ^2\left (c \left (d+e x^2\right )^p\right )} \, dx=\text {Int}\left (\frac {f+g x^2}{\log ^2\left (c \left (d+e x^2\right )^p\right )},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.01 (sec) , antiderivative size = 22, 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 = {} \[ \int \frac {f+g x^2}{\log ^2\left (c \left (d+e x^2\right )^p\right )} \, dx=\int \frac {f+g x^2}{\log ^2\left (c \left (d+e x^2\right )^p\right )} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {f+g x^2}{\log ^2\left (c \left (d+e x^2\right )^p\right )} \, dx=\int \frac {f+g x^2}{\log ^2\left (c \left (d+e x^2\right )^p\right )} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {f+g x^2}{\log ^2\left (c \left (d+e x^2\right )^p\right )} \, dx=\int { \frac {g x^{2} + f}{\log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 7.32 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {f+g x^2}{\log ^2\left (c \left (d+e x^2\right )^p\right )} \, dx=\int \frac {f + g x^{2}}{\log {\left (c \left (d + e x^{2}\right )^{p} \right )}^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 101, normalized size of antiderivative = 4.59 \[ \int \frac {f+g x^2}{\log ^2\left (c \left (d+e x^2\right )^p\right )} \, dx=\int { \frac {g x^{2} + f}{\log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{2}} \,d x } \]

[In]

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

[Out]

-1/2*(e*g*x^4 + (e*f + d*g)*x^2 + d*f)/(e*p*x*log((e*x^2 + d)^p) + e*p*x*log(c)) + integrate(1/2*(3*e*g*x^4 +
(e*f + d*g)*x^2 - d*f)/(e*p*x^2*log((e*x^2 + d)^p) + e*p*x^2*log(c)), x)

Giac [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {f+g x^2}{\log ^2\left (c \left (d+e x^2\right )^p\right )} \, dx=\int { \frac {g x^{2} + f}{\log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.42 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {f+g x^2}{\log ^2\left (c \left (d+e x^2\right )^p\right )} \, dx=\int \frac {g\,x^2+f}{{\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}^2} \,d x \]

[In]

int((f + g*x^2)/log(c*(d + e*x^2)^p)^2,x)

[Out]

int((f + g*x^2)/log(c*(d + e*x^2)^p)^2, x)