\(\int x \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)} \, dx\) [723]

Optimal result

Integrand size = 24, antiderivative size = 24 \[ \int x \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)} \, dx=\frac {\left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}{3 a^2 c}-\frac {\text {Int}\left (\frac {\sqrt {c+a^2 c x^2}}{\sqrt {\arctan (a x)}},x\right )}{6 a} \]

[Out]

1/3*(a^2*c*x^2+c)^(3/2)*arctan(a*x)^(1/2)/a^2/c-1/6*Unintegrable((a^2*c*x^2+c)^(1/2)/arctan(a*x)^(1/2),x)/a

Rubi [N/A]

Not integrable

Time = 0.07 (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 = {} \[ \int x \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)} \, dx=\int x \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)} \, dx \]

[In]

Int[x*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]],x]

[Out]

((c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]])/(3*a^2*c) - Defer[Int][Sqrt[c + a^2*c*x^2]/Sqrt[ArcTan[a*x]], x]/(6*
a)

Rubi steps \begin{align*} \text {integral}& = \frac {\left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}{3 a^2 c}-\frac {\int \frac {\sqrt {c+a^2 c x^2}}{\sqrt {\arctan (a x)}} \, dx}{6 a} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 3.67 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int x \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)} \, dx=\int x \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)} \, dx \]

[In]

Integrate[x*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]],x]

[Out]

Integrate[x*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]], x]

Maple [N/A] (verified)

Not integrable

Time = 9.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

integrate(x*(a^2*c*x^2+c)^(1/2)*arctan(a*x)^(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 = 2.87 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int x \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)} \, dx=\int x \sqrt {c \left (a^{2} x^{2} + 1\right )} \sqrt {\operatorname {atan}{\left (a x \right )}}\, dx \]

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

integrate(x*(a^2*c*x^2+c)^(1/2)*arctan(a*x)^(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 = 0.36 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int x \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)} \, dx=\int x\,\sqrt {\mathrm {atan}\left (a\,x\right )}\,\sqrt {c\,a^2\,x^2+c} \,d x \]

[In]

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

[Out]

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