\(\int \frac {\log ^3(c (d+e x^2)^p)}{f+g x^2} \, dx\) [278]

Optimal result

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

[Out]

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

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 3.98 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\log ^3\left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx=\int \frac {\log ^3\left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

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

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 17.20 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {\log ^3\left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx=\int \frac {\log {\left (c \left (d + e x^{2}\right )^{p} \right )}^{3}}{f + g x^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 2.03 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\log ^3\left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx=\int { \frac {\log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{3}}{g x^{2} + f} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.37 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\log ^3\left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx=\int { \frac {\log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{3}}{g x^{2} + f} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.36 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\log ^3\left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx=\int \frac {{\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}^3}{g\,x^2+f} \,d x \]

[In]

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

[Out]

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