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

Optimal. Leaf size=56 \[ \frac{c \left (a^2 x^2+1\right )^2 \sqrt{\tan ^{-1}(a x)}}{4 a^2}-\frac{\text{Unintegrable}\left (\frac{a^2 c x^2+c}{\sqrt{\tan ^{-1}(a x)}},x\right )}{8 a} \]

[Out]

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

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Rubi [A]  time = 0.0461141, 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 = {} \[ \int x \left (c+a^2 c x^2\right ) \sqrt{\tan ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 2.28078, size = 0, normalized size = 0. \[ \int x \left (c+a^2 c x^2\right ) \sqrt{\tan ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.377, size = 0, normalized size = 0. \begin{align*} \int x \left ({a}^{2}c{x}^{2}+c \right ) \sqrt{\arctan \left ( ax \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x*(a^2*c*x^2+c)*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*(a^2*c*x^2+c)*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*(a^2*c*x^2+c)*arctan(a*x)^(1/2),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 x \sqrt{\operatorname{atan}{\left (a x \right )}}\, dx + \int a^{2} x^{3} \sqrt{\operatorname{atan}{\left (a x \right )}}\, dx\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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