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

Optimal. Leaf size=60 \[ \frac{c^2 \left (a^2 x^2+1\right )^3 \sqrt{\tan ^{-1}(a x)}}{6 a^2}-\frac{\text{Unintegrable}\left (\frac{\left (a^2 c x^2+c\right )^2}{\sqrt{\tan ^{-1}(a x)}},x\right )}{12 a} \]

[Out]

(c^2*(1 + a^2*x^2)^3*Sqrt[ArcTan[a*x]])/(6*a^2) - Unintegrable[(c + a^2*c*x^2)^2/Sqrt[ArcTan[a*x]], x]/(12*a)

________________________________________________________________________________________

Rubi [A]  time = 0.0720387, 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 = {} \[ \int x \left (c+a^2 c x^2\right )^2 \sqrt{\tan ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

(c^2*(1 + a^2*x^2)^3*Sqrt[ArcTan[a*x]])/(6*a^2) - Defer[Int][(c + a^2*c*x^2)^2/Sqrt[ArcTan[a*x]], x]/(12*a)

Rubi steps

\begin{align*} \int x \left (c+a^2 c x^2\right )^2 \sqrt{\tan ^{-1}(a x)} \, dx &=\frac{c^2 \left (1+a^2 x^2\right )^3 \sqrt{\tan ^{-1}(a x)}}{6 a^2}-\frac{\int \frac{\left (c+a^2 c x^2\right )^2}{\sqrt{\tan ^{-1}(a x)}} \, dx}{12 a}\\ \end{align*}

Mathematica [A]  time = 1.77285, size = 0, normalized size = 0. \[ \int x \left (c+a^2 c x^2\right )^2 \sqrt{\tan ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.527, size = 0, normalized size = 0. \begin{align*} \int x \left ({a}^{2}c{x}^{2}+c \right ) ^{2}\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)^2*arctan(a*x)^(1/2),x)

[Out]

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

________________________________________________________________________________________

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)^2*arctan(a*x)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

________________________________________________________________________________________

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)^2*arctan(a*x)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

________________________________________________________________________________________

Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} c^{2} \left (\int x \sqrt{\operatorname{atan}{\left (a x \right )}}\, dx + \int 2 a^{2} x^{3} \sqrt{\operatorname{atan}{\left (a x \right )}}\, dx + \int a^{4} x^{5} \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)**2*atan(a*x)**(1/2),x)

[Out]

c**2*(Integral(x*sqrt(atan(a*x)), x) + Integral(2*a**2*x**3*sqrt(atan(a*x)), x) + Integral(a**4*x**5*sqrt(atan
(a*x)), x))

________________________________________________________________________________________

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

[Out]

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