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

Optimal. Leaf size=24 \[ \text{Unintegrable}\left (\frac{\left (a^2 c x^2+c\right ) \tan ^{-1}(a x)^{3/2}}{x},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0320078, 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 = {} \[ \int \frac{\left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^{3/2}}{x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 1.71671, size = 0, normalized size = 0. \[ \int \frac{\left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^{3/2}}{x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.413, size = 0, normalized size = 0. \begin{align*} \int{\frac{{a}^{2}c{x}^{2}+c}{x} \left ( \arctan \left ( ax \right ) \right ) ^{{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((a^2*c*x^2+c)*arctan(a*x)^(3/2)/x,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)*arctan(a*x)^(3/2)/x,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)*arctan(a*x)^(3/2)/x,x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} c \left (\int \frac{\operatorname{atan}^{\frac{3}{2}}{\left (a x \right )}}{x}\, dx + \int a^{2} x \operatorname{atan}^{\frac{3}{2}}{\left (a x \right )}\, dx\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a**2*c*x**2+c)*atan(a*x)**(3/2)/x,x)

[Out]

c*(Integral(atan(a*x)**(3/2)/x, x) + Integral(a**2*x*atan(a*x)**(3/2), x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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