\(\int \frac {x^m}{(c+a^2 c x^2) \arctan (a x)^{3/2}} \, dx\) [984]

Optimal result

Integrand size = 24, antiderivative size = 24 \[ \int \frac {x^m}{\left (c+a^2 c x^2\right ) \arctan (a x)^{3/2}} \, dx=-\frac {2 x^m}{a c \sqrt {\arctan (a x)}}+\frac {2 m \text {Int}\left (\frac {x^{-1+m}}{\sqrt {\arctan (a x)}},x\right )}{a c} \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.06 (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 = {} \[ \int \frac {x^m}{\left (c+a^2 c x^2\right ) \arctan (a x)^{3/2}} \, dx=\int \frac {x^m}{\left (c+a^2 c x^2\right ) \arctan (a x)^{3/2}} \, dx \]

[In]

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

[Out]

(-2*x^m)/(a*c*Sqrt[ArcTan[a*x]]) + (2*m*Defer[Int][x^(-1 + m)/Sqrt[ArcTan[a*x]], x])/(a*c)

Rubi steps \begin{align*} \text {integral}& = -\frac {2 x^m}{a c \sqrt {\arctan (a x)}}+\frac {(2 m) \int \frac {x^{-1+m}}{\sqrt {\arctan (a x)}} \, dx}{a c} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.61 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {x^m}{\left (c+a^2 c x^2\right ) \arctan (a x)^{3/2}} \, dx=\int \frac {x^m}{\left (c+a^2 c x^2\right ) \arctan (a x)^{3/2}} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 4.12 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {x^m}{\left (c+a^2 c x^2\right ) \arctan (a x)^{3/2}} \, dx=\int { \frac {x^{m}}{{\left (a^{2} c x^{2} + c\right )} \arctan \left (a x\right )^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 56.98 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.21 \[ \int \frac {x^m}{\left (c+a^2 c x^2\right ) \arctan (a x)^{3/2}} \, dx=\frac {\int \frac {x^{m}}{a^{2} x^{2} \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )} + \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}}\, dx}{c} \]

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 63.84 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.12 \[ \int \frac {x^m}{\left (c+a^2 c x^2\right ) \arctan (a x)^{3/2}} \, dx=\int { \frac {x^{m}}{{\left (a^{2} c x^{2} + c\right )} \arctan \left (a x\right )^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.63 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {x^m}{\left (c+a^2 c x^2\right ) \arctan (a x)^{3/2}} \, dx=\int \frac {x^m}{{\mathrm {atan}\left (a\,x\right )}^{3/2}\,\left (c\,a^2\,x^2+c\right )} \,d x \]

[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)