\(\int \frac {(a+c x^2+b x^4)^p}{c+e x^2} \, dx\) [404]

Optimal result

Integrand size = 24, antiderivative size = 24 \[ \int \frac {\left (a+c x^2+b x^4\right )^p}{c+e x^2} \, dx=\text {Int}\left (\frac {\left (a+c x^2+b x^4\right )^p}{c+e x^2},x\right ) \]

[Out]

Unintegrable((b*x^4+c*x^2+a)^p/(e*x^2+c),x)

Rubi [N/A]

Not integrable

Time = 0.01 (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 = {} \[ \int \frac {\left (a+c x^2+b x^4\right )^p}{c+e x^2} \, dx=\int \frac {\left (a+c x^2+b x^4\right )^p}{c+e x^2} \, dx \]

[In]

Int[(a + c*x^2 + b*x^4)^p/(c + e*x^2),x]

[Out]

Defer[Int][(a + c*x^2 + b*x^4)^p/(c + e*x^2), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (a+c x^2+b x^4\right )^p}{c+e x^2} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.90 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\left (a+c x^2+b x^4\right )^p}{c+e x^2} \, dx=\int \frac {\left (a+c x^2+b x^4\right )^p}{c+e x^2} \, dx \]

[In]

Integrate[(a + c*x^2 + b*x^4)^p/(c + e*x^2),x]

[Out]

Integrate[(a + c*x^2 + b*x^4)^p/(c + e*x^2), x]

Maple [N/A]

Not integrable

Time = 0.05 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00

\[\int \frac {\left (b \,x^{4}+c \,x^{2}+a \right )^{p}}{e \,x^{2}+c}d x\]

[In]

int((b*x^4+c*x^2+a)^p/(e*x^2+c),x)

[Out]

int((b*x^4+c*x^2+a)^p/(e*x^2+c),x)

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\left (a+c x^2+b x^4\right )^p}{c+e x^2} \, dx=\int { \frac {{\left (b x^{4} + c x^{2} + a\right )}^{p}}{e x^{2} + c} \,d x } \]

[In]

integrate((b*x^4+c*x^2+a)^p/(e*x^2+c),x, algorithm="fricas")

[Out]

integral((b*x^4 + c*x^2 + a)^p/(e*x^2 + c), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+c x^2+b x^4\right )^p}{c+e x^2} \, dx=\text {Timed out} \]

[In]

integrate((b*x**4+c*x**2+a)**p/(e*x**2+c),x)

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.22 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\left (a+c x^2+b x^4\right )^p}{c+e x^2} \, dx=\int { \frac {{\left (b x^{4} + c x^{2} + a\right )}^{p}}{e x^{2} + c} \,d x } \]

[In]

integrate((b*x^4+c*x^2+a)^p/(e*x^2+c),x, algorithm="maxima")

[Out]

integrate((b*x^4 + c*x^2 + a)^p/(e*x^2 + c), x)

Giac [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\left (a+c x^2+b x^4\right )^p}{c+e x^2} \, dx=\int { \frac {{\left (b x^{4} + c x^{2} + a\right )}^{p}}{e x^{2} + c} \,d x } \]

[In]

integrate((b*x^4+c*x^2+a)^p/(e*x^2+c),x, algorithm="giac")

[Out]

integrate((b*x^4 + c*x^2 + a)^p/(e*x^2 + c), x)

Mupad [N/A]

Not integrable

Time = 7.81 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\left (a+c x^2+b x^4\right )^p}{c+e x^2} \, dx=\int \frac {{\left (b\,x^4+c\,x^2+a\right )}^p}{e\,x^2+c} \,d x \]

[In]

int((a + b*x^4 + c*x^2)^p/(c + e*x^2),x)

[Out]

int((a + b*x^4 + c*x^2)^p/(c + e*x^2), x)