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

Optimal. Leaf size=27 \[ \text {Int}\left (\frac {\left (a+c x^2+b x^4\right )^p}{\left (c+e x^2\right )^2},x\right ) \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\left (a+c x^2+b x^4\right )^p}{\left (c+e x^2\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

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

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Mathematica [A]
time = 0.35, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+c x^2+b x^4\right )^p}{\left (c+e x^2\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[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]

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Maple [A]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\left (b \,x^{4}+c \,x^{2}+a \right )^{p}}{\left (e \,x^{2}+c \right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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/(x^2*e + c)^2, x)

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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/(x^4*e^2 + 2*c*x^2*e + c^2), x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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/(x^2*e + c)^2, x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {{\left (b\,x^4+c\,x^2+a\right )}^p}{{\left (e\,x^2+c\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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