\(\int x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^{5/2} \, dx\) [879]

Optimal result

Integrand size = 26, antiderivative size = 26 \[ \int x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^{5/2} \, dx=\text {Int}\left (x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^{5/2},x\right ) \]

[Out]

Unintegrable(x^2*arctan(a*x)^(5/2)*(a^2*c*x^2+c)^(1/2),x)

Rubi [N/A]

Not integrable

Time = 0.07 (sec) , antiderivative size = 26, 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 = {} \[ \int x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^{5/2} \, dx=\int x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^{5/2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 3.88 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^{5/2} \, dx=\int x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^{5/2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 5.57 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85

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

[In]

int(x^2*arctan(a*x)^(5/2)*(a^2*c*x^2+c)^(1/2),x)

[Out]

int(x^2*arctan(a*x)^(5/2)*(a^2*c*x^2+c)^(1/2),x)

Fricas [F(-2)]

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

[In]

integrate(x^2*arctan(a*x)^(5/2)*(a^2*c*x^2+c)^(1/2),x, algorithm="fricas")

[Out]

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

Sympy [F(-1)]

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

[In]

integrate(x**2*atan(a*x)**(5/2)*(a**2*c*x**2+c)**(1/2),x)

[Out]

Timed out

Maxima [F(-2)]

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

[In]

integrate(x^2*arctan(a*x)^(5/2)*(a^2*c*x^2+c)^(1/2),x, algorithm="maxima")

[Out]

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

Giac [N/A]

Not integrable

Time = 123.40 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.12 \[ \int x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^{5/2} \, dx=\int { \sqrt {a^{2} c x^{2} + c} x^{2} \arctan \left (a x\right )^{\frac {5}{2}} \,d x } \]

[In]

integrate(x^2*arctan(a*x)^(5/2)*(a^2*c*x^2+c)^(1/2),x, algorithm="giac")

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.34 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^{5/2} \, dx=\int x^2\,{\mathrm {atan}\left (a\,x\right )}^{5/2}\,\sqrt {c\,a^2\,x^2+c} \,d x \]

[In]

int(x^2*atan(a*x)^(5/2)*(c + a^2*c*x^2)^(1/2),x)

[Out]

int(x^2*atan(a*x)^(5/2)*(c + a^2*c*x^2)^(1/2), x)