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

Optimal. Leaf size=65 \[ \frac{\left (a^2 c x^2+c\right )^{3/2} \sqrt{\tan ^{-1}(a x)}}{3 a^2 c}-\frac{\text{Unintegrable}\left (\frac{\sqrt{a^2 c x^2+c}}{\sqrt{\tan ^{-1}(a x)}},x\right )}{6 a} \]

[Out]

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

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Rubi [A]  time = 0.103673, 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 = {} \[ \int x \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

((c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]])/(3*a^2*c) - Defer[Int][Sqrt[c + a^2*c*x^2]/Sqrt[ArcTan[a*x]], x]/(6*
a)

Rubi steps

\begin{align*} \int x \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)} \, dx &=\frac{\left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}}{3 a^2 c}-\frac{\int \frac{\sqrt{c+a^2 c x^2}}{\sqrt{\tan ^{-1}(a x)}} \, dx}{6 a}\\ \end{align*}

Mathematica [A]  time = 6.86393, size = 0, normalized size = 0. \[ \int x \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.974, size = 0, normalized size = 0. \begin{align*} \int x\sqrt{{a}^{2}c{x}^{2}+c}\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)^(1/2)*arctan(a*x)^(1/2),x)

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x*sqrt(c*(a**2*x**2 + 1))*sqrt(atan(a*x)), x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a^{2} c x^{2} + c} 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)^(1/2)*arctan(a*x)^(1/2),x, algorithm="giac")

[Out]

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