\(\int \frac {x (c+a^2 c x^2)^2}{\arctan (a x)^{5/2}} \, dx\) [1047]

Optimal result

Integrand size = 22, antiderivative size = 22 \[ \int \frac {x \left (c+a^2 c x^2\right )^2}{\arctan (a x)^{5/2}} \, dx=\text {Int}\left (\frac {x \left (c+a^2 c x^2\right )^2}{\arctan (a x)^{5/2}},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 22, 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 = {} \[ \int \frac {x \left (c+a^2 c x^2\right )^2}{\arctan (a x)^{5/2}} \, dx=\int \frac {x \left (c+a^2 c x^2\right )^2}{\arctan (a x)^{5/2}} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 1.22 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {x \left (c+a^2 c x^2\right )^2}{\arctan (a x)^{5/2}} \, dx=\int \frac {x \left (c+a^2 c x^2\right )^2}{\arctan (a x)^{5/2}} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 1.18 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

integrate(x*(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 = 5.12 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.23 \[ \int \frac {x \left (c+a^2 c x^2\right )^2}{\arctan (a x)^{5/2}} \, dx=c^{2} \left (\int \frac {x}{\operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}}\, dx + \int \frac {2 a^{2} x^{3}}{\operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}}\, dx + \int \frac {a^{4} x^{5}}{\operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}}\, dx\right ) \]

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

integrate(x*(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 [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [N/A]

Not integrable

Time = 0.55 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {x \left (c+a^2 c x^2\right )^2}{\arctan (a x)^{5/2}} \, dx=\int \frac {x\,{\left (c\,a^2\,x^2+c\right )}^2}{{\mathrm {atan}\left (a\,x\right )}^{5/2}} \,d x \]

[In]

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

[Out]

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