\(\int \frac {x^3 \sqrt {\arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx\) [734]

Optimal result

Integrand size = 26, antiderivative size = 26 \[ \int \frac {x^3 \sqrt {\arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=-\frac {2 \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}}{3 a^4 c}+\frac {x^2 \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}}{3 a^2 c}+\frac {\text {Int}\left (\frac {1}{\sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}},x\right )}{3 a^3}-\frac {\text {Int}\left (\frac {x^2}{\sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}},x\right )}{6 a} \]

[Out]

-2/3*(a^2*c*x^2+c)^(1/2)*arctan(a*x)^(1/2)/a^4/c+1/3*x^2*(a^2*c*x^2+c)^(1/2)*arctan(a*x)^(1/2)/a^2/c+1/3*Unint
egrable(1/(a^2*c*x^2+c)^(1/2)/arctan(a*x)^(1/2),x)/a^3-1/6*Unintegrable(x^2/(a^2*c*x^2+c)^(1/2)/arctan(a*x)^(1
/2),x)/a

Rubi [N/A]

Not integrable

Time = 0.22 (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 = {} \[ \int \frac {x^3 \sqrt {\arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=\int \frac {x^3 \sqrt {\arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx \]

[In]

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

[Out]

(-2*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])/(3*a^4*c) + (x^2*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])/(3*a^2*c) +
Defer[Int][1/(Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x]/(3*a^3) - Defer[Int][x^2/(Sqrt[c + a^2*c*x^2]*Sqrt[A
rcTan[a*x]]), x]/(6*a)

Rubi steps \begin{align*} \text {integral}& = \frac {x^2 \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}}{3 a^2 c}-\frac {2 \int \frac {x \sqrt {\arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx}{3 a^2}-\frac {\int \frac {x^2}{\sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}} \, dx}{6 a} \\ & = -\frac {2 \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}}{3 a^4 c}+\frac {x^2 \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}}{3 a^2 c}+\frac {\int \frac {1}{\sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}} \, dx}{3 a^3}-\frac {\int \frac {x^2}{\sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}} \, dx}{6 a} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 4.89 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {x^3 \sqrt {\arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=\int \frac {x^3 \sqrt {\arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 14.58 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

integrate(x^3*arctan(a*x)^(1/2)/(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 = 7.97 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {x^3 \sqrt {\arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=\int \frac {x^{3} \sqrt {\operatorname {atan}{\left (a x \right )}}}{\sqrt {c \left (a^{2} x^{2} + 1\right )}}\, dx \]

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

integrate(x^3*arctan(a*x)^(1/2)/(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. \[ \int \frac {x^3 \sqrt {\arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x^3*arctan(a*x)^(1/2)/(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 = 24, normalized size of antiderivative = 0.92 \[ \int \frac {x^3 \sqrt {\arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=\int \frac {x^3\,\sqrt {\mathrm {atan}\left (a\,x\right )}}{\sqrt {c\,a^2\,x^2+c}} \,d x \]

[In]

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

[Out]

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