\(\int x^m (c+a^2 c x^2)^{3/2} \arctan (a x)^2 \, dx\) [359]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.08 (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 = {} \[ \int x^m \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2 \, dx=\int x^m \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2 \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 1.10 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int x^m \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2 \, dx=\int x^m \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2 \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 1.46 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

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

[In]

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

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int x^m \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2 \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.56 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int x^m \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x^{m} \arctan \left (a x\right )^{2} \,d x } \]

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

integrate(x^m*(a^2*c*x^2+c)^(3/2)*arctan(a*x)^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 = 0.52 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int x^m \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2 \, dx=\int x^m\,{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^{3/2} \,d x \]

[In]

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

[Out]

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