\(\int \frac {\arctan (a x)^{5/2}}{x^3 (c+a^2 c x^2)} \, dx\) [862]

Optimal result

Integrand size = 24, antiderivative size = 24 \[ \int \frac {\arctan (a x)^{5/2}}{x^3 \left (c+a^2 c x^2\right )} \, dx=\frac {2 i a^2 \arctan (a x)^{7/2}}{7 c}+\frac {\text {Int}\left (\frac {\arctan (a x)^{5/2}}{x^3},x\right )}{c}-\frac {i a^2 \text {Int}\left (\frac {\arctan (a x)^{5/2}}{x (i+a x)},x\right )}{c} \]

[Out]

2/7*I*a^2*arctan(a*x)^(7/2)/c+Unintegrable(arctan(a*x)^(5/2)/x^3,x)/c-I*a^2*Unintegrable(arctan(a*x)^(5/2)/x/(
I+a*x),x)/c

Rubi [N/A]

Not integrable

Time = 0.14 (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 = {} \[ \int \frac {\arctan (a x)^{5/2}}{x^3 \left (c+a^2 c x^2\right )} \, dx=\int \frac {\arctan (a x)^{5/2}}{x^3 \left (c+a^2 c x^2\right )} \, dx \]

[In]

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

[Out]

(((2*I)/7)*a^2*ArcTan[a*x]^(7/2))/c + Defer[Int][ArcTan[a*x]^(5/2)/x^3, x]/c - (I*a^2*Defer[Int][ArcTan[a*x]^(
5/2)/(x*(I + a*x)), x])/c

Rubi steps \begin{align*} \text {integral}& = -\left (a^2 \int \frac {\arctan (a x)^{5/2}}{x \left (c+a^2 c x^2\right )} \, dx\right )+\frac {\int \frac {\arctan (a x)^{5/2}}{x^3} \, dx}{c} \\ & = \frac {2 i a^2 \arctan (a x)^{7/2}}{7 c}+\frac {\int \frac {\arctan (a x)^{5/2}}{x^3} \, dx}{c}-\frac {\left (i a^2\right ) \int \frac {\arctan (a x)^{5/2}}{x (i+a x)} \, dx}{c} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 1.83 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\arctan (a x)^{5/2}}{x^3 \left (c+a^2 c x^2\right )} \, dx=\int \frac {\arctan (a x)^{5/2}}{x^3 \left (c+a^2 c x^2\right )} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 1.66 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 5.41 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {\arctan (a x)^{5/2}}{x^3 \left (c+a^2 c x^2\right )} \, dx=\frac {\int \frac {\operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}}{a^{2} x^{5} + x^{3}}\, dx}{c} \]

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 107.95 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.12 \[ \int \frac {\arctan (a x)^{5/2}}{x^3 \left (c+a^2 c x^2\right )} \, dx=\int { \frac {\arctan \left (a x\right )^{\frac {5}{2}}}{{\left (a^{2} c x^{2} + c\right )} x^{3}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.50 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {\arctan (a x)^{5/2}}{x^3 \left (c+a^2 c x^2\right )} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^{5/2}}{x^3\,\left (c\,a^2\,x^2+c\right )} \,d x \]

[In]

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

[Out]

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