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

Optimal. Leaf size=25 \[ \text{Unintegrable}\left (\frac{\left (a^2 c x^2+c\right )^{3/2}}{\sqrt{\tan ^{-1}(a x)}},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0363264, 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 = {} \[ \int \frac{\left (c+a^2 c x^2\right )^{3/2}}{\sqrt{\tan ^{-1}(a x)}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.36435, size = 0, normalized size = 0. \[ \int \frac{\left (c+a^2 c x^2\right )^{3/2}}{\sqrt{\tan ^{-1}(a x)}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.668, size = 0, normalized size = 0. \begin{align*} \int{ \left ({a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}}{\frac{1}{\sqrt{\arctan \left ( ax \right ) }}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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)/sqrt(arctan(a*x)), x)