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

Optimal. Leaf size=79 \[ \frac{1}{8} c \text{Unintegrable}\left (\frac{1}{\sqrt{\tan ^{-1}(a x)}},x\right )+\frac{2}{3} c \text{Unintegrable}\left (\tan ^{-1}(a x)^{3/2},x\right )+\frac{1}{3} c x \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^{3/2}-\frac{c \left (a^2 x^2+1\right ) \sqrt{\tan ^{-1}(a x)}}{4 a} \]

[Out]

-(c*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]])/(4*a) + (c*x*(1 + a^2*x^2)*ArcTan[a*x]^(3/2))/3 + (c*Unintegrable[1/Sqrt[
ArcTan[a*x]], x])/8 + (2*c*Unintegrable[ArcTan[a*x]^(3/2), x])/3

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Rubi [A]  time = 0.0235658, 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 = {} \[ \int \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^{3/2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

-(c*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]])/(4*a) + (c*x*(1 + a^2*x^2)*ArcTan[a*x]^(3/2))/3 + (c*Defer[Int][1/Sqrt[Ar
cTan[a*x]], x])/8 + (2*c*Defer[Int][ArcTan[a*x]^(3/2), x])/3

Rubi steps

\begin{align*} \int \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^{3/2} \, dx &=-\frac{c \left (1+a^2 x^2\right ) \sqrt{\tan ^{-1}(a x)}}{4 a}+\frac{1}{3} c x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^{3/2}+\frac{1}{8} c \int \frac{1}{\sqrt{\tan ^{-1}(a x)}} \, dx+\frac{1}{3} (2 c) \int \tan ^{-1}(a x)^{3/2} \, dx\\ \end{align*}

Mathematica [A]  time = 3.92188, size = 0, normalized size = 0. \[ \int \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^{3/2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.317, size = 0, normalized size = 0. \begin{align*} \int \left ({a}^{2}c{x}^{2}+c \right ) \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)

[Out]

int((a^2*c*x^2+c)*arctan(a*x)^(3/2),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)*arctan(a*x)^(3/2),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)*arctan(a*x)^(3/2),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 a^{2} x^{2} \operatorname{atan}^{\frac{3}{2}}{\left (a x \right )}\, dx + \int \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)

[Out]

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

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a^{2} c x^{2} + c\right )} \arctan \left (a x\right )^{\frac{3}{2}}\,{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, algorithm="giac")

[Out]

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