\(\int \frac {x^m \arctan (a x)^{3/2}}{c+a^2 c x^2} \, dx\) [774]

Optimal result

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

[Out]

Unintegrable(x^m*arctan(a*x)^(3/2)/(a^2*c*x^2+c),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 = {} \[ \int \frac {x^m \arctan (a x)^{3/2}}{c+a^2 c x^2} \, dx=\int \frac {x^m \arctan (a x)^{3/2}}{c+a^2 c x^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.65 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {x^m \arctan (a x)^{3/2}}{c+a^2 c x^2} \, dx=\int \frac {x^m \arctan (a x)^{3/2}}{c+a^2 c x^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 5.83 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {x^m \arctan (a x)^{3/2}}{c+a^2 c x^2} \, dx=\int { \frac {x^{m} \arctan \left (a x\right )^{\frac {3}{2}}}{a^{2} c x^{2} + c} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 71.78 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {x^m \arctan (a x)^{3/2}}{c+a^2 c x^2} \, dx=\frac {\int \frac {x^{m} \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}}{a^{2} x^{2} + 1}\, dx}{c} \]

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 52.02 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.12 \[ \int \frac {x^m \arctan (a x)^{3/2}}{c+a^2 c x^2} \, dx=\int { \frac {x^{m} \arctan \left (a x\right )^{\frac {3}{2}}}{a^{2} c x^{2} + c} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.53 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {x^m \arctan (a x)^{3/2}}{c+a^2 c x^2} \, dx=\int \frac {x^m\,{\mathrm {atan}\left (a\,x\right )}^{3/2}}{c\,a^2\,x^2+c} \,d x \]

[In]

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

[Out]

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