\(\int (c+a^2 c x^2)^{3/2} \sqrt {\arctan (a x)} \, dx\) [728]

Optimal result

Integrand size = 23, antiderivative size = 23 \[ \int \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)} \, dx=\text {Int}\left (\left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)},x\right ) \]

[Out]

Unintegrable((a^2*c*x^2+c)^(3/2)*arctan(a*x)^(1/2),x)

Rubi [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)} \, dx=\int \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)} \, dx \]

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 1.85 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)} \, dx=\int \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 13.08 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83

\[\int \left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} \sqrt {\arctan \left (a x \right )}d x\]

[In]

int((a^2*c*x^2+c)^(3/2)*arctan(a*x)^(1/2),x)

[Out]

int((a^2*c*x^2+c)^(3/2)*arctan(a*x)^(1/2),x)

Fricas [F(-2)]

Exception generated. \[ \int \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((a^2*c*x^2+c)^(3/2)*arctan(a*x)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [N/A]

Not integrable

Time = 32.73 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)} \, dx=\int \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \sqrt {\operatorname {atan}{\left (a x \right )}}\, dx \]

[In]

integrate((a**2*c*x**2+c)**(3/2)*atan(a*x)**(1/2),x)

[Out]

Integral((c*(a**2*x**2 + 1))**(3/2)*sqrt(atan(a*x)), x)

Maxima [F(-2)]

Exception generated. \[ \int \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate((a^2*c*x^2+c)^(3/2)*arctan(a*x)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negative exponent.

Giac [F(-2)]

Exception generated. \[ \int \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((a^2*c*x^2+c)^(3/2)*arctan(a*x)^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [N/A]

Not integrable

Time = 0.35 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)} \, dx=\int \sqrt {\mathrm {atan}\left (a\,x\right )}\,{\left (c\,a^2\,x^2+c\right )}^{3/2} \,d x \]

[In]

int(atan(a*x)^(1/2)*(c + a^2*c*x^2)^(3/2),x)

[Out]

int(atan(a*x)^(1/2)*(c + a^2*c*x^2)^(3/2), x)