\(\int \frac {x^2 \sqrt {\arctan (a x)}}{(c+a^2 c x^2)^{5/2}} \, dx\) [752]

Optimal result

Integrand size = 26, antiderivative size = 163 \[ \int \frac {x^2 \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\frac {x^3 \sqrt {\arctan (a x)}}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {1+a^2 x^2} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{4 a^3 c^2 \sqrt {c+a^2 c x^2}}+\frac {\sqrt {\frac {\pi }{6}} \sqrt {1+a^2 x^2} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arctan (a x)}\right )}{12 a^3 c^2 \sqrt {c+a^2 c x^2}} \]

[Out]

1/72*FresnelS(6^(1/2)/Pi^(1/2)*arctan(a*x)^(1/2))*6^(1/2)*Pi^(1/2)*(a^2*x^2+1)^(1/2)/a^3/c^2/(a^2*c*x^2+c)^(1/
2)-1/8*FresnelS(2^(1/2)/Pi^(1/2)*arctan(a*x)^(1/2))*2^(1/2)*Pi^(1/2)*(a^2*x^2+1)^(1/2)/a^3/c^2/(a^2*c*x^2+c)^(
1/2)+1/3*x^3*arctan(a*x)^(1/2)/c/(a^2*c*x^2+c)^(3/2)

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {5064, 5091, 5090, 3393, 3386, 3432} \[ \int \frac {x^2 \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\frac {x^3 \sqrt {\arctan (a x)}}{3 c \left (a^2 c x^2+c\right )^{3/2}}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {a^2 x^2+1} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{4 a^3 c^2 \sqrt {a^2 c x^2+c}}+\frac {\sqrt {\frac {\pi }{6}} \sqrt {a^2 x^2+1} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arctan (a x)}\right )}{12 a^3 c^2 \sqrt {a^2 c x^2+c}} \]

[In]

Int[(x^2*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^(5/2),x]

[Out]

(x^3*Sqrt[ArcTan[a*x]])/(3*c*(c + a^2*c*x^2)^(3/2)) - (Sqrt[Pi/2]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[2/Pi]*Sqrt[A
rcTan[a*x]]])/(4*a^3*c^2*Sqrt[c + a^2*c*x^2]) + (Sqrt[Pi/6]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcTan[
a*x]]])/(12*a^3*c^2*Sqrt[c + a^2*c*x^2])

Rule 3386

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Sin[f*(x^2/d)], x], x,
Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]

Rule 3393

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

Rule 3432

Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[d, 2]))*FresnelS[Sqrt[2/Pi]*Rt[d, 2
]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]

Rule 5064

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> Simp
[(f*x)^(m + 1)*(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^p/(d*f*(m + 1))), x] - Dist[b*c*(p/(f*(m + 1))), Int[(
f*x)^(m + 1)*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[e,
c^2*d] && EqQ[m + 2*q + 3, 0] && GtQ[p, 0] && NeQ[m, -1]

Rule 5090

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Dist[d^q/c^(m
+ 1), Subst[Int[(a + b*x)^p*(Sin[x]^m/Cos[x]^(m + 2*(q + 1))), x], x, ArcTan[c*x]], x] /; FreeQ[{a, b, c, d,
e, p}, x] && EqQ[e, c^2*d] && IGtQ[m, 0] && ILtQ[m + 2*q + 1, 0] && (IntegerQ[q] || GtQ[d, 0])

Rule 5091

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Dist[d^(q + 1
/2)*(Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]), Int[x^m*(1 + c^2*x^2)^q*(a + b*ArcTan[c*x])^p, x], x] /; FreeQ[{a, b,
c, d, e, p}, x] && EqQ[e, c^2*d] && IGtQ[m, 0] && ILtQ[m + 2*q + 1, 0] &&  !(IntegerQ[q] || GtQ[d, 0])

Rubi steps \begin{align*} \text {integral}& = \frac {x^3 \sqrt {\arctan (a x)}}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {1}{6} a \int \frac {x^3}{\left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}} \, dx \\ & = \frac {x^3 \sqrt {\arctan (a x)}}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {\left (a \sqrt {1+a^2 x^2}\right ) \int \frac {x^3}{\left (1+a^2 x^2\right )^{5/2} \sqrt {\arctan (a x)}} \, dx}{6 c^2 \sqrt {c+a^2 c x^2}} \\ & = \frac {x^3 \sqrt {\arctan (a x)}}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {\sqrt {1+a^2 x^2} \text {Subst}\left (\int \frac {\sin ^3(x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{6 a^3 c^2 \sqrt {c+a^2 c x^2}} \\ & = \frac {x^3 \sqrt {\arctan (a x)}}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {\sqrt {1+a^2 x^2} \text {Subst}\left (\int \left (\frac {3 \sin (x)}{4 \sqrt {x}}-\frac {\sin (3 x)}{4 \sqrt {x}}\right ) \, dx,x,\arctan (a x)\right )}{6 a^3 c^2 \sqrt {c+a^2 c x^2}} \\ & = \frac {x^3 \sqrt {\arctan (a x)}}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {\sqrt {1+a^2 x^2} \text {Subst}\left (\int \frac {\sin (3 x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{24 a^3 c^2 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {1+a^2 x^2} \text {Subst}\left (\int \frac {\sin (x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{8 a^3 c^2 \sqrt {c+a^2 c x^2}} \\ & = \frac {x^3 \sqrt {\arctan (a x)}}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {\sqrt {1+a^2 x^2} \text {Subst}\left (\int \sin \left (3 x^2\right ) \, dx,x,\sqrt {\arctan (a x)}\right )}{12 a^3 c^2 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {1+a^2 x^2} \text {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {\arctan (a x)}\right )}{4 a^3 c^2 \sqrt {c+a^2 c x^2}} \\ & = \frac {x^3 \sqrt {\arctan (a x)}}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {1+a^2 x^2} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{4 a^3 c^2 \sqrt {c+a^2 c x^2}}+\frac {\sqrt {\frac {\pi }{6}} \sqrt {1+a^2 x^2} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arctan (a x)}\right )}{12 a^3 c^2 \sqrt {c+a^2 c x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.39 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.82 \[ \int \frac {x^2 \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\frac {24 a^3 x^3 \sqrt {\arctan (a x)}-9 \sqrt {2 \pi } \left (1+a^2 x^2\right )^{3/2} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )+\sqrt {6 \pi } \left (1+a^2 x^2\right )^{3/2} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arctan (a x)}\right )}{72 a^3 c^2 \left (1+a^2 x^2\right ) \sqrt {c+a^2 c x^2}} \]

[In]

Integrate[(x^2*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^(5/2),x]

[Out]

(24*a^3*x^3*Sqrt[ArcTan[a*x]] - 9*Sqrt[2*Pi]*(1 + a^2*x^2)^(3/2)*FresnelS[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]] + Sqrt
[6*Pi]*(1 + a^2*x^2)^(3/2)*FresnelS[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]])/(72*a^3*c^2*(1 + a^2*x^2)*Sqrt[c + a^2*c*x^
2])

Maple [F]

\[\int \frac {x^{2} \sqrt {\arctan \left (a x \right )}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}d x\]

[In]

int(x^2*arctan(a*x)^(1/2)/(a^2*c*x^2+c)^(5/2),x)

[Out]

int(x^2*arctan(a*x)^(1/2)/(a^2*c*x^2+c)^(5/2),x)

Fricas [F(-2)]

Exception generated. \[ \int \frac {x^2 \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x^2*arctan(a*x)^(1/2)/(a^2*c*x^2+c)^(5/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {x^2 \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\int \frac {x^{2} \sqrt {\operatorname {atan}{\left (a x \right )}}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \]

[In]

integrate(x**2*atan(a*x)**(1/2)/(a**2*c*x**2+c)**(5/2),x)

[Out]

Integral(x**2*sqrt(atan(a*x))/(c*(a**2*x**2 + 1))**(5/2), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^2 \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate(x^2*arctan(a*x)^(1/2)/(a^2*c*x^2+c)^(5/2),x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {x^2 \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {x^{2} \sqrt {\arctan \left (a x\right )}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate(x^2*arctan(a*x)^(1/2)/(a^2*c*x^2+c)^(5/2),x, algorithm="giac")

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2 \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\int \frac {x^2\,\sqrt {\mathrm {atan}\left (a\,x\right )}}{{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]

[In]

int((x^2*atan(a*x)^(1/2))/(c + a^2*c*x^2)^(5/2),x)

[Out]

int((x^2*atan(a*x)^(1/2))/(c + a^2*c*x^2)^(5/2), x)