\(\int \frac {1}{(f+g x^3) \log ^2(c (d+e x^2)^p)} \, dx\) [308]

Optimal result

Integrand size = 24, antiderivative size = 24 \[ \int \frac {1}{\left (f+g x^3\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )} \, dx=\text {Int}\left (\frac {1}{\left (f+g x^3\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )},x\right ) \]

[Out]

Unintegrable(1/(g*x^3+f)/ln(c*(e*x^2+d)^p)^2,x)

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 24, 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 = {} \[ \int \frac {1}{\left (f+g x^3\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )} \, dx=\int \frac {1}{\left (f+g x^3\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )} \, dx \]

[In]

Int[1/((f + g*x^3)*Log[c*(d + e*x^2)^p]^2),x]

[Out]

Defer[Int][1/((f + g*x^3)*Log[c*(d + e*x^2)^p]^2), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\left (f+g x^3\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 9.47 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {1}{\left (f+g x^3\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )} \, dx=\int \frac {1}{\left (f+g x^3\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )} \, dx \]

[In]

Integrate[1/((f + g*x^3)*Log[c*(d + e*x^2)^p]^2),x]

[Out]

Integrate[1/((f + g*x^3)*Log[c*(d + e*x^2)^p]^2), x]

Maple [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00

\[\int \frac {1}{\left (g \,x^{3}+f \right ) {\ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}^{2}}d x\]

[In]

int(1/(g*x^3+f)/ln(c*(e*x^2+d)^p)^2,x)

[Out]

int(1/(g*x^3+f)/ln(c*(e*x^2+d)^p)^2,x)

Fricas [N/A]

Not integrable

Time = 0.34 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {1}{\left (f+g x^3\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )} \, dx=\int { \frac {1}{{\left (g x^{3} + f\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{2}} \,d x } \]

[In]

integrate(1/(g*x^3+f)/log(c*(e*x^2+d)^p)^2,x, algorithm="fricas")

[Out]

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

Sympy [N/A]

Not integrable

Time = 139.76 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {1}{\left (f+g x^3\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )} \, dx=\int \frac {1}{\left (f + g x^{3}\right ) \log {\left (c \left (d + e x^{2}\right )^{p} \right )}^{2}}\, dx \]

[In]

integrate(1/(g*x**3+f)/ln(c*(e*x**2+d)**p)**2,x)

[Out]

Integral(1/((f + g*x**3)*log(c*(d + e*x**2)**p)**2), x)

Maxima [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 159, normalized size of antiderivative = 6.62 \[ \int \frac {1}{\left (f+g x^3\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )} \, dx=\int { \frac {1}{{\left (g x^{3} + f\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{2}} \,d x } \]

[In]

integrate(1/(g*x^3+f)/log(c*(e*x^2+d)^p)^2,x, algorithm="maxima")

[Out]

-1/2*(e*x^2 + d)/(e*g*p*x^4*log(c) + e*f*p*x*log(c) + (e*g*p*x^4 + e*f*p*x)*log((e*x^2 + d)^p)) - integrate(1/
2*(2*e*g*x^5 + 4*d*g*x^3 - e*f*x^2 + d*f)/(e*g^2*p*x^8*log(c) + 2*e*f*g*p*x^5*log(c) + e*f^2*p*x^2*log(c) + (e
*g^2*p*x^8 + 2*e*f*g*p*x^5 + e*f^2*p*x^2)*log((e*x^2 + d)^p)), x)

Giac [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {1}{\left (f+g x^3\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )} \, dx=\int { \frac {1}{{\left (g x^{3} + f\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{2}} \,d x } \]

[In]

integrate(1/(g*x^3+f)/log(c*(e*x^2+d)^p)^2,x, algorithm="giac")

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.50 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {1}{\left (f+g x^3\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )} \, dx=\int \frac {1}{{\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}^2\,\left (g\,x^3+f\right )} \,d x \]

[In]

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

[Out]

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