\(\int \frac {x \arctan (a x)^{3/2}}{c+a^2 c x^2} \, dx\) [777]

Optimal result

Integrand size = 22, antiderivative size = 22 \[ \int \frac {x \arctan (a x)^{3/2}}{c+a^2 c x^2} \, dx=\frac {2 x \arctan (a x)^{5/2}}{5 a c}-\frac {2 \text {Int}\left (\arctan (a x)^{5/2},x\right )}{5 a c} \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.03 (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 = {} \[ \int \frac {x \arctan (a x)^{3/2}}{c+a^2 c x^2} \, dx=\int \frac {x \arctan (a x)^{3/2}}{c+a^2 c x^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.80 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {x \arctan (a x)^{3/2}}{c+a^2 c x^2} \, dx=\int \frac {x \arctan (a x)^{3/2}}{c+a^2 c x^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 4.20 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

integrate(x*arctan(a*x)^(3/2)/(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 = 1.42 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {x \arctan (a x)^{3/2}}{c+a^2 c x^2} \, dx=\frac {\int \frac {x \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}}{a^{2} x^{2} + 1}\, dx}{c} \]

[In]

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

[Out]

Integral(x*atan(a*x)**(3/2)/(a**2*x**2 + 1), x)/c

Maxima [F(-2)]

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

[In]

integrate(x*arctan(a*x)^(3/2)/(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 = 74.30 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.14 \[ \int \frac {x \arctan (a x)^{3/2}}{c+a^2 c x^2} \, dx=\int { \frac {x \arctan \left (a x\right )^{\frac {3}{2}}}{a^{2} c x^{2} + c} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.44 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {x \arctan (a x)^{3/2}}{c+a^2 c x^2} \, dx=\int \frac {x\,{\mathrm {atan}\left (a\,x\right )}^{3/2}}{c\,a^2\,x^2+c} \,d x \]

[In]

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

[Out]

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