\(\int \frac {(a+c x^2+b x^4)^p}{(c+e x^2)^2} \, dx\) [405]

Optimal result

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

[Out]

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

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.86 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\left (a+c x^2+b x^4\right )^p}{\left (c+e x^2\right )^2} \, dx=\int \frac {\left (a+c x^2+b x^4\right )^p}{\left (c+e x^2\right )^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.06 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.54 \[ \int \frac {\left (a+c x^2+b x^4\right )^p}{\left (c+e x^2\right )^2} \, dx=\int { \frac {{\left (b x^{4} + c x^{2} + a\right )}^{p}}{{\left (e x^{2} + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.21 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\left (a+c x^2+b x^4\right )^p}{\left (c+e x^2\right )^2} \, dx=\int { \frac {{\left (b x^{4} + c x^{2} + a\right )}^{p}}{{\left (e x^{2} + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\left (a+c x^2+b x^4\right )^p}{\left (c+e x^2\right )^2} \, dx=\int { \frac {{\left (b x^{4} + c x^{2} + a\right )}^{p}}{{\left (e x^{2} + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 8.06 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\left (a+c x^2+b x^4\right )^p}{\left (c+e x^2\right )^2} \, dx=\int \frac {{\left (b\,x^4+c\,x^2+a\right )}^p}{{\left (e\,x^2+c\right )}^2} \,d x \]

[In]

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

[Out]

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