\(\int (c+a^2 c x^2)^2 \sqrt {\arctan (a x)} \, dx\) [692]

Optimal result

Integrand size = 21, antiderivative size = 21 \[ \int \left (c+a^2 c x^2\right )^2 \sqrt {\arctan (a x)} \, dx=\text {Int}\left (\left (c+a^2 c x^2\right )^2 \sqrt {\arctan (a x)},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 21, 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 = {} \[ \int \left (c+a^2 c x^2\right )^2 \sqrt {\arctan (a x)} \, dx=\int \left (c+a^2 c x^2\right )^2 \sqrt {\arctan (a x)} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 1.08 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \left (c+a^2 c x^2\right )^2 \sqrt {\arctan (a x)} \, dx=\int \left (c+a^2 c x^2\right )^2 \sqrt {\arctan (a x)} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 3.51 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

integrate((a^2*c*x^2+c)^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 = 2.34 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.29 \[ \int \left (c+a^2 c x^2\right )^2 \sqrt {\arctan (a x)} \, dx=c^{2} \left (\int 2 a^{2} x^{2} \sqrt {\operatorname {atan}{\left (a x \right )}}\, dx + \int a^{4} x^{4} \sqrt {\operatorname {atan}{\left (a x \right )}}\, dx + \int \sqrt {\operatorname {atan}{\left (a x \right )}}\, dx\right ) \]

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 52.87 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.14 \[ \int \left (c+a^2 c x^2\right )^2 \sqrt {\arctan (a x)} \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{2} \sqrt {\arctan \left (a x\right )} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.45 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \left (c+a^2 c x^2\right )^2 \sqrt {\arctan (a x)} \, dx=\int \sqrt {\mathrm {atan}\left (a\,x\right )}\,{\left (c\,a^2\,x^2+c\right )}^2 \,d x \]

[In]

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

[Out]

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