\(\int \frac {x^m \sqrt {\arctan (a x)}}{(c+a^2 c x^2)^3} \, dx\) [713]

Optimal result

Integrand size = 24, antiderivative size = 24 \[ \int \frac {x^m \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=\text {Int}\left (\frac {x^m \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^3},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.05 (sec) , antiderivative size = 24, 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 = {} \[ \int \frac {x^m \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=\int \frac {x^m \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 1.58 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {x^m \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=\int \frac {x^m \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 9.55 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 50, normalized size of antiderivative = 2.08 \[ \int \frac {x^m \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=\int { \frac {x^{m} \sqrt {\arctan \left (a x\right )}}{{\left (a^{2} c x^{2} + c\right )}^{3}} \,d x } \]

[In]

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

[Out]

integral(x^m*sqrt(arctan(a*x))/(a^6*c^3*x^6 + 3*a^4*c^3*x^4 + 3*a^2*c^3*x^2 + c^3), x)

Sympy [N/A]

Not integrable

Time = 114.91 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.71 \[ \int \frac {x^m \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {\int \frac {x^{m} \sqrt {\operatorname {atan}{\left (a x \right )}}}{a^{6} x^{6} + 3 a^{4} x^{4} + 3 a^{2} x^{2} + 1}\, dx}{c^{3}} \]

[In]

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

[Out]

Integral(x**m*sqrt(atan(a*x))/(a**6*x**6 + 3*a**4*x**4 + 3*a**2*x**2 + 1), x)/c**3

Maxima [F(-2)]

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

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 52.26 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.12 \[ \int \frac {x^m \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=\int { \frac {x^{m} \sqrt {\arctan \left (a x\right )}}{{\left (a^{2} c x^{2} + c\right )}^{3}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.54 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {x^m \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=\int \frac {x^m\,\sqrt {\mathrm {atan}\left (a\,x\right )}}{{\left (c\,a^2\,x^2+c\right )}^3} \,d x \]

[In]

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

[Out]

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