\(\int \frac {(f+g x^3)^2}{\log ^2(c (d+e x^2)^p)} \, dx\) [306]

Optimal result

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

[Out]

Unintegrable((g*x^3+f)^2/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 = {} \[ \int \frac {\left (f+g x^3\right )^2}{\log ^2\left (c \left (d+e x^2\right )^p\right )} \, dx=\int \frac {\left (f+g x^3\right )^2}{\log ^2\left (c \left (d+e x^2\right )^p\right )} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.42 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\left (f+g x^3\right )^2}{\log ^2\left (c \left (d+e x^2\right )^p\right )} \, dx=\int \frac {\left (f+g x^3\right )^2}{\log ^2\left (c \left (d+e x^2\right )^p\right )} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.46 \[ \int \frac {\left (f+g x^3\right )^2}{\log ^2\left (c \left (d+e x^2\right )^p\right )} \, dx=\int { \frac {{\left (g x^{3} + f\right )}^{2}}{\log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{2}} \,d x } \]

[In]

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

[Out]

integral((g^2*x^6 + 2*f*g*x^3 + f^2)/log((e*x^2 + d)^p*c)^2, x)

Sympy [N/A]

Not integrable

Time = 18.46 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {\left (f+g x^3\right )^2}{\log ^2\left (c \left (d+e x^2\right )^p\right )} \, dx=\int \frac {\left (f + g x^{3}\right )^{2}}{\log {\left (c \left (d + e x^{2}\right )^{p} \right )}^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 152, normalized size of antiderivative = 6.33 \[ \int \frac {\left (f+g x^3\right )^2}{\log ^2\left (c \left (d+e x^2\right )^p\right )} \, dx=\int { \frac {{\left (g x^{3} + f\right )}^{2}}{\log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{2}} \,d x } \]

[In]

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

[Out]

-1/2*(e*g^2*x^8 + d*g^2*x^6 + 2*e*f*g*x^5 + 2*d*f*g*x^3 + e*f^2*x^2 + d*f^2)/(e*p*x*log((e*x^2 + d)^p) + e*p*x
*log(c)) + integrate(1/2*(7*e*g^2*x^8 + 5*d*g^2*x^6 + 8*e*f*g*x^5 + 4*d*f*g*x^3 + e*f^2*x^2 - d*f^2)/(e*p*x^2*
log((e*x^2 + d)^p) + e*p*x^2*log(c)), x)

Giac [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\left (f+g x^3\right )^2}{\log ^2\left (c \left (d+e x^2\right )^p\right )} \, dx=\int { \frac {{\left (g x^{3} + f\right )}^{2}}{\log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.46 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\left (f+g x^3\right )^2}{\log ^2\left (c \left (d+e x^2\right )^p\right )} \, dx=\int \frac {{\left (g\,x^3+f\right )}^2}{{\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}^2} \,d x \]

[In]

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

[Out]

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