\(\int \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2} \, dx\) [800]

Optimal result

Integrand size = 23, antiderivative size = 23 \[ \int \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2} \, dx=-\frac {3 \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}}{4 a}+\frac {1}{2} x \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2}+\frac {3}{8} c \text {Int}\left (\frac {1}{\sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}},x\right )+\frac {1}{2} c \text {Int}\left (\frac {\arctan (a x)^{3/2}}{\sqrt {c+a^2 c x^2}},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.08 (sec) , antiderivative size = 23, 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 = {} \[ \int \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2} \, dx=\int \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2} \, dx=\int \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 13.71 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

integrate(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 = 26.60 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2} \, dx=\int \sqrt {c \left (a^{2} x^{2} + 1\right )} \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}\, dx \]

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

integrate(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 [F(-2)]

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

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [N/A]

Not integrable

Time = 0.34 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2} \, dx=\int {\mathrm {atan}\left (a\,x\right )}^{3/2}\,\sqrt {c\,a^2\,x^2+c} \,d x \]

[In]

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

[Out]

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