3.966 \(\int \frac{x^4}{(c+a^2 c x^2)^{5/2} \sqrt{\tan ^{-1}(a x)}} \, dx\)

Optimal. Leaf size=28 \[ \text{Unintegrable}\left (\frac{x^4}{\left (a^2 c x^2+c\right )^{5/2} \sqrt{\tan ^{-1}(a x)}},x\right ) \]

[Out]

Unintegrable[x^4/((c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]), x]

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Rubi [A]  time = 0.120415, 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 = {} \[ \int \frac{x^4}{\left (c+a^2 c x^2\right )^{5/2} \sqrt{\tan ^{-1}(a x)}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[x^4/((c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]),x]

[Out]

Defer[Int][x^4/((c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]), x]

Rubi steps

\begin{align*} \int \frac{x^4}{\left (c+a^2 c x^2\right )^{5/2} \sqrt{\tan ^{-1}(a x)}} \, dx &=\int \frac{x^4}{\left (c+a^2 c x^2\right )^{5/2} \sqrt{\tan ^{-1}(a x)}} \, dx\\ \end{align*}

Mathematica [A]  time = 3.64429, size = 0, normalized size = 0. \[ \int \frac{x^4}{\left (c+a^2 c x^2\right )^{5/2} \sqrt{\tan ^{-1}(a x)}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[x^4/((c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]),x]

[Out]

Integrate[x^4/((c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]), x]

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Maple [A]  time = 2.194, size = 0, normalized size = 0. \begin{align*} \int{{x}^{4} \left ({a}^{2}c{x}^{2}+c \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt{\arctan \left ( ax \right ) }}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^(1/2),x)

[Out]

int(x^4/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^(1/2),x)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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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(x**4/(a**2*c*x**2+c)**(5/2)/atan(a*x)**(1/2),x)

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{{\left (a^{2} c x^{2} + c\right )}^{\frac{5}{2}} \sqrt{\arctan \left (a x\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^(1/2),x, algorithm="giac")

[Out]

integrate(x^4/((a^2*c*x^2 + c)^(5/2)*sqrt(arctan(a*x))), x)