\(\int \frac {x^3 \arctan (a x)^{5/2}}{(c+a^2 c x^2)^2} \, dx\) [865]

Optimal result

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

[Out]

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

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 3.59 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {x^3 \arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^2} \, dx=\int \frac {x^3 \arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 7.58 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92

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

[In]

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

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int \frac {x^3 \arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^2} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [N/A]

Not integrable

Time = 15.15 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.33 \[ \int \frac {x^3 \arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {\int \frac {x^{3} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}}{a^{4} x^{4} + 2 a^{2} x^{2} + 1}\, dx}{c^{2}} \]

[In]

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

[Out]

Integral(x**3*atan(a*x)**(5/2)/(a**4*x**4 + 2*a**2*x**2 + 1), x)/c**2

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^3 \arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^2} \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 118.13 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.12 \[ \int \frac {x^3 \arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^2} \, dx=\int { \frac {x^{3} \arctan \left (a x\right )^{\frac {5}{2}}}{{\left (a^{2} c x^{2} + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.48 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {x^3 \arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^2} \, dx=\int \frac {x^3\,{\mathrm {atan}\left (a\,x\right )}^{5/2}}{{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]

[In]

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

[Out]

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