\(\int \frac {(c+a^2 c x^2)^2 \arctan (a x)^{3/2}}{x^2} \, dx\) [767]

Optimal result

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

[Out]

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

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 1.18 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)^{3/2}}{x^2} \, dx=\int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)^{3/2}}{x^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 5.16 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

integrate((a^2*c*x^2+c)^2*arctan(a*x)^(3/2)/x^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 = 7.60 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.00 \[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)^{3/2}}{x^2} \, dx=c^{2} \left (\int 2 a^{2} \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}\, dx + \int \frac {\operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}}{x^{2}}\, dx + \int a^{4} x^{2} \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}\, dx\right ) \]

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 253.70 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.12 \[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)^{3/2}}{x^2} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{2} \arctan \left (a x\right )^{\frac {3}{2}}}{x^{2}} \,d x } \]

[In]

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

[In]

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

[Out]

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