\(\int x^2 (c+a^2 c x^2)^2 \arctan (a x)^{5/2} \, dx\) [844]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.04 (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 = {} \[ \int x^2 \left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2} \, dx=\int x^2 \left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 2.80 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int x^2 \left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2} \, dx=\int x^2 \left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 4.28 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 80.17 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.12 \[ \int x^2 \left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2} \, dx=c^{2} \left (\int x^{2} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}\, dx + \int 2 a^{2} x^{4} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}\, dx + \int a^{4} x^{6} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}\, dx\right ) \]

[In]

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

[Out]

c**2*(Integral(x**2*atan(a*x)**(5/2), x) + Integral(2*a**2*x**4*atan(a*x)**(5/2), x) + Integral(a**4*x**6*atan
(a*x)**(5/2), x))

Maxima [F(-2)]

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

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 112.39 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.12 \[ \int x^2 \left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2} \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{2} x^{2} \arctan \left (a x\right )^{\frac {5}{2}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.47 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int x^2 \left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2} \, dx=\int x^2\,{\mathrm {atan}\left (a\,x\right )}^{5/2}\,{\left (c\,a^2\,x^2+c\right )}^2 \,d x \]

[In]

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

[Out]

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