\(\int \frac {(c+a^2 c x^2)^3 \arctan (a x)^{5/2}}{x^2} \, dx\) [854]

Optimal result

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

[Out]

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

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 1.50 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^{5/2}}{x^2} \, dx=\int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^{5/2}}{x^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 2.87 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

integrate((a^2*c*x^2+c)^3*arctan(a*x)^(5/2)/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 = 61.46 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.75 \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^{5/2}}{x^2} \, dx=c^{3} \left (\int 3 a^{2} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}\, dx + \int \frac {\operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}}{x^{2}}\, dx + \int 3 a^{4} x^{2} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}\, dx + \int a^{6} x^{4} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}\, dx\right ) \]

[In]

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

[Out]

c**3*(Integral(3*a**2*atan(a*x)**(5/2), x) + Integral(atan(a*x)**(5/2)/x**2, x) + Integral(3*a**4*x**2*atan(a*
x)**(5/2), x) + Integral(a**6*x**4*atan(a*x)**(5/2), x))

Maxima [F(-2)]

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

[In]

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

[Out]

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

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [N/A]

Not integrable

Time = 0.49 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^{5/2}}{x^2} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^{5/2}\,{\left (c\,a^2\,x^2+c\right )}^3}{x^2} \,d x \]

[In]

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

[Out]

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