\(\int \frac {x^m}{(c+a^2 c x^2)^{5/2} \arctan (a x)^3} \, dx\) [685]

Optimal result

Integrand size = 24, antiderivative size = 24 \[ \int \frac {x^m}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\text {Int}\left (\frac {x^m}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3},x\right ) \]

[Out]

Unintegrable(x^m/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^3,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 = {} \[ \int \frac {x^m}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\int \frac {x^m}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx \]

[In]

Int[x^m/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3),x]

[Out]

Defer[Int][x^m/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {x^m}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 1.08 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {x^m}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\int \frac {x^m}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx \]

[In]

Integrate[x^m/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3),x]

[Out]

Integrate[x^m/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3), x]

Maple [N/A] (verified)

Not integrable

Time = 15.07 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92

\[\int \frac {x^{m}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}} \arctan \left (a x \right )^{3}}d x\]

[In]

int(x^m/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^3,x)

[Out]

int(x^m/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^3,x)

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.58 \[ \int \frac {x^m}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\int { \frac {x^{m}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )^{3}} \,d x } \]

[In]

integrate(x^m/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^3,x, algorithm="fricas")

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {x^m}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\text {Timed out} \]

[In]

integrate(x**m/(a**2*c*x**2+c)**(5/2)/atan(a*x)**3,x)

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.44 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {x^m}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\int { \frac {x^{m}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )^{3}} \,d x } \]

[In]

integrate(x^m/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^3,x, algorithm="maxima")

[Out]

integrate(x^m/((a^2*c*x^2 + c)^(5/2)*arctan(a*x)^3), x)

Giac [N/A]

Not integrable

Time = 41.72 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.12 \[ \int \frac {x^m}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\int { \frac {x^{m}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )^{3}} \,d x } \]

[In]

integrate(x^m/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^3,x, algorithm="giac")

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.65 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {x^m}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\int \frac {x^m}{{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]

[In]

int(x^m/(atan(a*x)^3*(c + a^2*c*x^2)^(5/2)),x)

[Out]

int(x^m/(atan(a*x)^3*(c + a^2*c*x^2)^(5/2)), x)