\(\int \frac {\arctan (a x)^{3/2}}{x^3 \sqrt {c+a^2 c x^2}} \, dx\) [819]

Optimal result

Integrand size = 26, antiderivative size = 26 \[ \int \frac {\arctan (a x)^{3/2}}{x^3 \sqrt {c+a^2 c x^2}} \, dx=-\frac {3 a \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}}{4 c x}-\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^{3/2}}{2 c x^2}+\frac {3}{8} a^2 \text {Int}\left (\frac {1}{x \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}},x\right )-\frac {1}{2} a^2 \text {Int}\left (\frac {\arctan (a x)^{3/2}}{x \sqrt {c+a^2 c x^2}},x\right ) \]

[Out]

-1/2*arctan(a*x)^(3/2)*(a^2*c*x^2+c)^(1/2)/c/x^2-3/4*a*(a^2*c*x^2+c)^(1/2)*arctan(a*x)^(1/2)/c/x-1/2*a^2*Unint
egrable(arctan(a*x)^(3/2)/x/(a^2*c*x^2+c)^(1/2),x)+3/8*a^2*Unintegrable(1/x/(a^2*c*x^2+c)^(1/2)/arctan(a*x)^(1
/2),x)

Rubi [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 26, 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 = {} \[ \int \frac {\arctan (a x)^{3/2}}{x^3 \sqrt {c+a^2 c x^2}} \, dx=\int \frac {\arctan (a x)^{3/2}}{x^3 \sqrt {c+a^2 c x^2}} \, dx \]

[In]

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

[Out]

(-3*a*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])/(4*c*x) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/(2*c*x^2) + (3*
a^2*Defer[Int][1/(x*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x])/8 - (a^2*Defer[Int][ArcTan[a*x]^(3/2)/(x*Sqrt[
c + a^2*c*x^2]), x])/2

Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^{3/2}}{2 c x^2}+\frac {1}{4} (3 a) \int \frac {\sqrt {\arctan (a x)}}{x^2 \sqrt {c+a^2 c x^2}} \, dx-\frac {1}{2} a^2 \int \frac {\arctan (a x)^{3/2}}{x \sqrt {c+a^2 c x^2}} \, dx \\ & = -\frac {3 a \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}}{4 c x}-\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^{3/2}}{2 c x^2}+\frac {1}{8} \left (3 a^2\right ) \int \frac {1}{x \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}} \, dx-\frac {1}{2} a^2 \int \frac {\arctan (a x)^{3/2}}{x \sqrt {c+a^2 c x^2}} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 7.36 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {\arctan (a x)^{3/2}}{x^3 \sqrt {c+a^2 c x^2}} \, dx=\int \frac {\arctan (a x)^{3/2}}{x^3 \sqrt {c+a^2 c x^2}} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 15.28 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85

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

[In]

int(arctan(a*x)^(3/2)/x^3/(a^2*c*x^2+c)^(1/2),x)

[Out]

int(arctan(a*x)^(3/2)/x^3/(a^2*c*x^2+c)^(1/2),x)

Fricas [F(-2)]

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

[In]

integrate(arctan(a*x)^(3/2)/x^3/(a^2*c*x^2+c)^(1/2),x, algorithm="fricas")

[Out]

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

Sympy [N/A]

Not integrable

Time = 33.40 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {\arctan (a x)^{3/2}}{x^3 \sqrt {c+a^2 c x^2}} \, dx=\int \frac {\operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}}{x^{3} \sqrt {c \left (a^{2} x^{2} + 1\right )}}\, dx \]

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

integrate(arctan(a*x)^(3/2)/x^3/(a^2*c*x^2+c)^(1/2),x, algorithm="maxima")

[Out]

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

Giac [N/A]

Not integrable

Time = 149.98 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.12 \[ \int \frac {\arctan (a x)^{3/2}}{x^3 \sqrt {c+a^2 c x^2}} \, dx=\int { \frac {\arctan \left (a x\right )^{\frac {3}{2}}}{\sqrt {a^{2} c x^{2} + c} x^{3}} \,d x } \]

[In]

integrate(arctan(a*x)^(3/2)/x^3/(a^2*c*x^2+c)^(1/2),x, algorithm="giac")

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.34 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {\arctan (a x)^{3/2}}{x^3 \sqrt {c+a^2 c x^2}} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^{3/2}}{x^3\,\sqrt {c\,a^2\,x^2+c}} \,d x \]

[In]

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

[Out]

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