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

Optimal. Leaf size=26 \[ \text{Unintegrable}\left (\frac{\left (a+b x^4+c x^2\right )^p}{c+e x^2},x\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0109996, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\left (a+c x^2+b x^4\right )^p}{c+e x^2} \, dx \]

Verification is Not applicable to the result.

[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*} \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\\ \end{align*}

Mathematica [A]  time = 0.130838, size = 0, normalized size = 0. \[ \int \frac{\left (a+c x^2+b x^4\right )^p}{c+e x^2} \, dx \]

Verification is Not applicable to the result.

[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 [A]  time = 0.037, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( b{x}^{4}+c{x}^{2}+a \right ) ^{p}}{e{x}^{2}+c}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{4} + c x^{2} + a\right )}^{p}}{e x^{2} + c}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b x^{4} + c x^{2} + a\right )}^{p}}{e x^{2} + c}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{4} + c x^{2} + a\right )}^{p}}{e x^{2} + c}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)