3.726 \(\int x^2 (c+a^2 c x^2)^{3/2} \sqrt{\tan ^{-1}(a x)} \, dx\)

Optimal. Leaf size=28 \[ \text{Unintegrable}\left (x^2 \left (a^2 c x^2+c\right )^{3/2} \sqrt{\tan ^{-1}(a x)},x\right ) \]

[Out]

Unintegrable[x^2*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]], x]

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Rubi [A]  time = 0.123861, 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 = {} \[ \int x^2 \left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

Int[x^2*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]],x]

[Out]

Defer[Int][x^2*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]], x]

Rubi steps

\begin{align*} \int x^2 \left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)} \, dx &=\int x^2 \left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)} \, dx\\ \end{align*}

Mathematica [A]  time = 3.96448, size = 0, normalized size = 0. \[ \int x^2 \left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[x^2*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]],x]

[Out]

Integrate[x^2*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]], x]

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Maple [A]  time = 1.306, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ({a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}}\sqrt{\arctan \left ( ax \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a^2*c*x^2+c)^(3/2)*arctan(a*x)^(1/2),x)

[Out]

int(x^2*(a^2*c*x^2+c)^(3/2)*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^2*(a^2*c*x^2+c)^(3/2)*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^2*(a^2*c*x^2+c)^(3/2)*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**2*(a**2*c*x**2+c)**(3/2)*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{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}} x^{2} \sqrt{\arctan \left (a x\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a^2*c*x^2+c)^(3/2)*arctan(a*x)^(1/2),x, algorithm="giac")

[Out]

integrate((a^2*c*x^2 + c)^(3/2)*x^2*sqrt(arctan(a*x)), x)