\(\int \frac {\arctan (a x)^{5/2}}{x (c+a^2 c x^2)^3} \, dx\) [877]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.05 (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 = {} \[ \int \frac {\arctan (a x)^{5/2}}{x \left (c+a^2 c x^2\right )^3} \, dx=\int \frac {\arctan (a x)^{5/2}}{x \left (c+a^2 c x^2\right )^3} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 2.04 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\arctan (a x)^{5/2}}{x \left (c+a^2 c x^2\right )^3} \, dx=\int \frac {\arctan (a x)^{5/2}}{x \left (c+a^2 c x^2\right )^3} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 3.77 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 10.94 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.54 \[ \int \frac {\arctan (a x)^{5/2}}{x \left (c+a^2 c x^2\right )^3} \, dx=\frac {\int \frac {\operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}}{a^{6} x^{7} + 3 a^{4} x^{5} + 3 a^{2} x^{3} + x}\, dx}{c^{3}} \]

[In]

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

[Out]

Integral(atan(a*x)**(5/2)/(a**6*x**7 + 3*a**4*x**5 + 3*a**2*x**3 + x), x)/c**3

Maxima [F(-2)]

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

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 125.50 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.12 \[ \int \frac {\arctan (a x)^{5/2}}{x \left (c+a^2 c x^2\right )^3} \, dx=\int { \frac {\arctan \left (a x\right )^{\frac {5}{2}}}{{\left (a^{2} c x^{2} + c\right )}^{3} x} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.53 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {\arctan (a x)^{5/2}}{x \left (c+a^2 c x^2\right )^3} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^{5/2}}{x\,{\left (c\,a^2\,x^2+c\right )}^3} \,d x \]

[In]

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

[Out]

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