3.1026 \(\int \frac{x^2}{(c+a^2 c x^2)^{3/2} \tan ^{-1}(a x)^{3/2}} \, dx\)

Optimal. Leaf size=126 \[ 2 a \text{Unintegrable}\left (\frac{x^3}{\left (a^2 c x^2+c\right )^{3/2} \sqrt{\tan ^{-1}(a x)}},x\right )+\frac{4 \sqrt{2 \pi } \sqrt{a^2 x^2+1} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{a^3 c \sqrt{a^2 c x^2+c}}-\frac{2 x^2}{a c \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.431362, 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 = {} \[ \int \frac{x^2}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

(-2*x^2)/(a*c*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]) + (4*Sqrt[2*Pi]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[2/Pi]*Sqr
t[ArcTan[a*x]]])/(a^3*c*Sqrt[c + a^2*c*x^2]) + 2*a*Defer[Int][x^3/((c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x
]

Rubi steps

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

Mathematica [A]  time = 2.2166, size = 0, normalized size = 0. \[ \int \frac{x^2}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 2.893, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2} \left ({a}^{2}c{x}^{2}+c \right ) ^{-{\frac{3}{2}}} \left ( \arctan \left ( ax \right ) \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: UnboundLocalError

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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