\(\int \frac {x \arctan (a x)^{3/2}}{\sqrt {c+a^2 c x^2}} \, dx\) [815]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.08 (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 = {} \[ \int \frac {x \arctan (a x)^{3/2}}{\sqrt {c+a^2 c x^2}} \, dx=\int \frac {x \arctan (a x)^{3/2}}{\sqrt {c+a^2 c x^2}} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 1.00 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {x \arctan (a x)^{3/2}}{\sqrt {c+a^2 c x^2}} \, dx=\int \frac {x \arctan (a x)^{3/2}}{\sqrt {c+a^2 c x^2}} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 7.18 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

integrate(x*arctan(a*x)^(3/2)/(a^2*c*x^2+c)^(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 = 36.95 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {x \arctan (a x)^{3/2}}{\sqrt {c+a^2 c x^2}} \, dx=\int \frac {x \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}}{\sqrt {c \left (a^{2} x^{2} + 1\right )}}\, dx \]

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 151.32 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.12 \[ \int \frac {x \arctan (a x)^{3/2}}{\sqrt {c+a^2 c x^2}} \, dx=\int { \frac {x \arctan \left (a x\right )^{\frac {3}{2}}}{\sqrt {a^{2} c x^{2} + c}} \,d x } \]

[In]

integrate(x*arctan(a*x)^(3/2)/(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 = 22, normalized size of antiderivative = 0.92 \[ \int \frac {x \arctan (a x)^{3/2}}{\sqrt {c+a^2 c x^2}} \, dx=\int \frac {x\,{\mathrm {atan}\left (a\,x\right )}^{3/2}}{\sqrt {c\,a^2\,x^2+c}} \,d x \]

[In]

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

[Out]

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