3.447 \(\int (c-a^2 c x^2)^{3/2} \sin ^{-1}(a x)^{3/2} \, dx\)

Optimal. Leaf size=363 \[ -\frac{3 \sqrt{\frac{\pi }{2}} c \sqrt{c-a^2 c x^2} \text{FresnelC}\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{512 a \sqrt{1-a^2 x^2}}-\frac{3 \sqrt{\pi } c \sqrt{c-a^2 c x^2} \text{FresnelC}\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{32 a \sqrt{1-a^2 x^2}}+\frac{3 c \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}}{20 a \sqrt{1-a^2 x^2}}+\frac{1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^{3/2}+\frac{3}{8} c x \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}+\frac{3 c \left (1-a^2 x^2\right )^{3/2} \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}}{32 a}-\frac{9 a c x^2 \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}}{32 \sqrt{1-a^2 x^2}}+\frac{27 c \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}}{256 a \sqrt{1-a^2 x^2}} \]

[Out]

(27*c*Sqrt[c - a^2*c*x^2]*Sqrt[ArcSin[a*x]])/(256*a*Sqrt[1 - a^2*x^2]) - (9*a*c*x^2*Sqrt[c - a^2*c*x^2]*Sqrt[A
rcSin[a*x]])/(32*Sqrt[1 - a^2*x^2]) + (3*c*(1 - a^2*x^2)^(3/2)*Sqrt[c - a^2*c*x^2]*Sqrt[ArcSin[a*x]])/(32*a) +
(3*c*x*Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^(3/2))/8 + (x*(c - a^2*c*x^2)^(3/2)*ArcSin[a*x]^(3/2))/4 + (3*c*Sqrt[c
- a^2*c*x^2]*ArcSin[a*x]^(5/2))/(20*a*Sqrt[1 - a^2*x^2]) - (3*c*Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*FresnelC[2*Sqr
t[2/Pi]*Sqrt[ArcSin[a*x]]])/(512*a*Sqrt[1 - a^2*x^2]) - (3*c*Sqrt[Pi]*Sqrt[c - a^2*c*x^2]*FresnelC[(2*Sqrt[Arc
Sin[a*x]])/Sqrt[Pi]])/(32*a*Sqrt[1 - a^2*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.433735, antiderivative size = 363, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {4649, 4647, 4641, 4629, 4723, 3312, 3304, 3352, 4677, 4661} \[ -\frac{3 \sqrt{\frac{\pi }{2}} c \sqrt{c-a^2 c x^2} \text{FresnelC}\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{512 a \sqrt{1-a^2 x^2}}-\frac{3 \sqrt{\pi } c \sqrt{c-a^2 c x^2} \text{FresnelC}\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{32 a \sqrt{1-a^2 x^2}}+\frac{3 c \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}}{20 a \sqrt{1-a^2 x^2}}+\frac{1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^{3/2}+\frac{3}{8} c x \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}+\frac{3 c \left (1-a^2 x^2\right )^{3/2} \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}}{32 a}-\frac{9 a c x^2 \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}}{32 \sqrt{1-a^2 x^2}}+\frac{27 c \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}}{256 a \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(c - a^2*c*x^2)^(3/2)*ArcSin[a*x]^(3/2),x]

[Out]

(27*c*Sqrt[c - a^2*c*x^2]*Sqrt[ArcSin[a*x]])/(256*a*Sqrt[1 - a^2*x^2]) - (9*a*c*x^2*Sqrt[c - a^2*c*x^2]*Sqrt[A
rcSin[a*x]])/(32*Sqrt[1 - a^2*x^2]) + (3*c*(1 - a^2*x^2)^(3/2)*Sqrt[c - a^2*c*x^2]*Sqrt[ArcSin[a*x]])/(32*a) +
(3*c*x*Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^(3/2))/8 + (x*(c - a^2*c*x^2)^(3/2)*ArcSin[a*x]^(3/2))/4 + (3*c*Sqrt[c
- a^2*c*x^2]*ArcSin[a*x]^(5/2))/(20*a*Sqrt[1 - a^2*x^2]) - (3*c*Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*FresnelC[2*Sqr
t[2/Pi]*Sqrt[ArcSin[a*x]]])/(512*a*Sqrt[1 - a^2*x^2]) - (3*c*Sqrt[Pi]*Sqrt[c - a^2*c*x^2]*FresnelC[(2*Sqrt[Arc
Sin[a*x]])/Sqrt[Pi]])/(32*a*Sqrt[1 - a^2*x^2])

Rule 4649

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(x*(d + e*x^2)^p*(
a + b*ArcSin[c*x])^n)/(2*p + 1), x] + (Dist[(2*d*p)/(2*p + 1), Int[(d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x])^n,
x], x] - Dist[(b*c*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/((2*p + 1)*(1 - c^2*x^2)^FracPart[p]), Int[x*(1 - c
^2*x^2)^(p - 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && Gt
Q[n, 0] && GtQ[p, 0]

Rule 4647

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(x*Sqrt[d + e*x^2]*(
a + b*ArcSin[c*x])^n)/2, x] + (Dist[Sqrt[d + e*x^2]/(2*Sqrt[1 - c^2*x^2]), Int[(a + b*ArcSin[c*x])^n/Sqrt[1 -
c^2*x^2], x], x] - Dist[(b*c*n*Sqrt[d + e*x^2])/(2*Sqrt[1 - c^2*x^2]), Int[x*(a + b*ArcSin[c*x])^(n - 1), x],
x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0]

Rule 4641

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(a + b*ArcSin[c*x])^
(n + 1)/(b*c*Sqrt[d]*(n + 1)), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0] && NeQ[n,
-1]

Rule 4629

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[(x^(m + 1)*(a + b*ArcSin[c*x])^n)/(m
+ 1), x] - Dist[(b*c*n)/(m + 1), Int[(x^(m + 1)*(a + b*ArcSin[c*x])^(n - 1))/Sqrt[1 - c^2*x^2], x], x] /; Fre
eQ[{a, b, c}, x] && IGtQ[m, 0] && GtQ[n, 0]

Rule 4723

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

Rule 3312

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 3304

Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Cos[(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 3352

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

Rule 4677

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x^2)^
(p + 1)*(a + b*ArcSin[c*x])^n)/(2*e*(p + 1)), x] + Dist[(b*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*c*(p + 1
)*(1 - c^2*x^2)^FracPart[p]), Int[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x] /; FreeQ[{a, b,
c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]

Rule 4661

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[d^p/c, Subst[Int[(
a + b*x)^n*Cos[x]^(2*p + 1), x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && I
GtQ[2*p, 0] && (IntegerQ[p] || GtQ[d, 0])

Rubi steps

\begin{align*} \int \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^{3/2} \, dx &=\frac{1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^{3/2}+\frac{1}{4} (3 c) \int \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2} \, dx-\frac{\left (3 a c \sqrt{c-a^2 c x^2}\right ) \int x \left (1-a^2 x^2\right ) \sqrt{\sin ^{-1}(a x)} \, dx}{8 \sqrt{1-a^2 x^2}}\\ &=\frac{3 c \left (1-a^2 x^2\right )^{3/2} \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}}{32 a}+\frac{3}{8} c x \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}+\frac{1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^{3/2}-\frac{\left (3 c \sqrt{c-a^2 c x^2}\right ) \int \frac{\left (1-a^2 x^2\right )^{3/2}}{\sqrt{\sin ^{-1}(a x)}} \, dx}{64 \sqrt{1-a^2 x^2}}+\frac{\left (3 c \sqrt{c-a^2 c x^2}\right ) \int \frac{\sin ^{-1}(a x)^{3/2}}{\sqrt{1-a^2 x^2}} \, dx}{8 \sqrt{1-a^2 x^2}}-\frac{\left (9 a c \sqrt{c-a^2 c x^2}\right ) \int x \sqrt{\sin ^{-1}(a x)} \, dx}{16 \sqrt{1-a^2 x^2}}\\ &=-\frac{9 a c x^2 \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}}{32 \sqrt{1-a^2 x^2}}+\frac{3 c \left (1-a^2 x^2\right )^{3/2} \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}}{32 a}+\frac{3}{8} c x \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}+\frac{1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^{3/2}+\frac{3 c \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}}{20 a \sqrt{1-a^2 x^2}}-\frac{\left (3 c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\cos ^4(x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{64 a \sqrt{1-a^2 x^2}}+\frac{\left (9 a^2 c \sqrt{c-a^2 c x^2}\right ) \int \frac{x^2}{\sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}} \, dx}{64 \sqrt{1-a^2 x^2}}\\ &=-\frac{9 a c x^2 \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}}{32 \sqrt{1-a^2 x^2}}+\frac{3 c \left (1-a^2 x^2\right )^{3/2} \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}}{32 a}+\frac{3}{8} c x \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}+\frac{1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^{3/2}+\frac{3 c \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}}{20 a \sqrt{1-a^2 x^2}}-\frac{\left (3 c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{3}{8 \sqrt{x}}+\frac{\cos (2 x)}{2 \sqrt{x}}+\frac{\cos (4 x)}{8 \sqrt{x}}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{64 a \sqrt{1-a^2 x^2}}+\frac{\left (9 c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\sin ^2(x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{64 a \sqrt{1-a^2 x^2}}\\ &=-\frac{9 c \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}}{256 a \sqrt{1-a^2 x^2}}-\frac{9 a c x^2 \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}}{32 \sqrt{1-a^2 x^2}}+\frac{3 c \left (1-a^2 x^2\right )^{3/2} \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}}{32 a}+\frac{3}{8} c x \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}+\frac{1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^{3/2}+\frac{3 c \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}}{20 a \sqrt{1-a^2 x^2}}-\frac{\left (3 c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\cos (4 x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{512 a \sqrt{1-a^2 x^2}}-\frac{\left (3 c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\cos (2 x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{128 a \sqrt{1-a^2 x^2}}+\frac{\left (9 c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{1}{2 \sqrt{x}}-\frac{\cos (2 x)}{2 \sqrt{x}}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{64 a \sqrt{1-a^2 x^2}}\\ &=\frac{27 c \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}}{256 a \sqrt{1-a^2 x^2}}-\frac{9 a c x^2 \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}}{32 \sqrt{1-a^2 x^2}}+\frac{3 c \left (1-a^2 x^2\right )^{3/2} \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}}{32 a}+\frac{3}{8} c x \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}+\frac{1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^{3/2}+\frac{3 c \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}}{20 a \sqrt{1-a^2 x^2}}-\frac{\left (3 c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \cos \left (4 x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{256 a \sqrt{1-a^2 x^2}}-\frac{\left (3 c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{64 a \sqrt{1-a^2 x^2}}-\frac{\left (9 c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\cos (2 x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{128 a \sqrt{1-a^2 x^2}}\\ &=\frac{27 c \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}}{256 a \sqrt{1-a^2 x^2}}-\frac{9 a c x^2 \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}}{32 \sqrt{1-a^2 x^2}}+\frac{3 c \left (1-a^2 x^2\right )^{3/2} \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}}{32 a}+\frac{3}{8} c x \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}+\frac{1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^{3/2}+\frac{3 c \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}}{20 a \sqrt{1-a^2 x^2}}-\frac{3 c \sqrt{\frac{\pi }{2}} \sqrt{c-a^2 c x^2} C\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{512 a \sqrt{1-a^2 x^2}}-\frac{3 c \sqrt{\pi } \sqrt{c-a^2 c x^2} C\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{128 a \sqrt{1-a^2 x^2}}-\frac{\left (9 c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{64 a \sqrt{1-a^2 x^2}}\\ &=\frac{27 c \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}}{256 a \sqrt{1-a^2 x^2}}-\frac{9 a c x^2 \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}}{32 \sqrt{1-a^2 x^2}}+\frac{3 c \left (1-a^2 x^2\right )^{3/2} \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}}{32 a}+\frac{3}{8} c x \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}+\frac{1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^{3/2}+\frac{3 c \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}}{20 a \sqrt{1-a^2 x^2}}-\frac{3 c \sqrt{\frac{\pi }{2}} \sqrt{c-a^2 c x^2} C\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{512 a \sqrt{1-a^2 x^2}}-\frac{3 c \sqrt{\pi } \sqrt{c-a^2 c x^2} C\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{32 a \sqrt{1-a^2 x^2}}\\ \end{align*}

Mathematica [C]  time = 0.425425, size = 186, normalized size = 0.51 \[ \frac{c \sqrt{c-a^2 c x^2} \left (-240 \sqrt{\pi } \sqrt{\sin ^{-1}(a x)^2} \text{FresnelC}\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )+\sqrt{\sin ^{-1}(a x)} \left (5 \sqrt{i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{5}{2},-4 i \sin ^{-1}(a x)\right )+5 \sqrt{-i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{5}{2},4 i \sin ^{-1}(a x)\right )+32 \sqrt{\sin ^{-1}(a x)^2} \left (12 \sin ^{-1}(a x)^2+20 \sin \left (2 \sin ^{-1}(a x)\right ) \sin ^{-1}(a x)+15 \cos \left (2 \sin ^{-1}(a x)\right )\right )\right )\right )}{2560 a \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)^2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(c - a^2*c*x^2)^(3/2)*ArcSin[a*x]^(3/2),x]

[Out]

(c*Sqrt[c - a^2*c*x^2]*(-240*Sqrt[Pi]*Sqrt[ArcSin[a*x]^2]*FresnelC[(2*Sqrt[ArcSin[a*x]])/Sqrt[Pi]] + Sqrt[ArcS
in[a*x]]*(5*Sqrt[I*ArcSin[a*x]]*Gamma[5/2, (-4*I)*ArcSin[a*x]] + 5*Sqrt[(-I)*ArcSin[a*x]]*Gamma[5/2, (4*I)*Arc
Sin[a*x]] + 32*Sqrt[ArcSin[a*x]^2]*(12*ArcSin[a*x]^2 + 15*Cos[2*ArcSin[a*x]] + 20*ArcSin[a*x]*Sin[2*ArcSin[a*x
]]))))/(2560*a*Sqrt[1 - a^2*x^2]*Sqrt[ArcSin[a*x]^2])

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Maple [F]  time = 0.179, size = 0, normalized size = 0. \begin{align*} \int \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}} \left ( \arcsin \left ( ax \right ) \right ) ^{{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-a^2*c*x^2+c)^(3/2)*arcsin(a*x)^(3/2),x)

[Out]

int((-a^2*c*x^2+c)^(3/2)*arcsin(a*x)^(3/2),x)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a^2*c*x^2+c)^(3/2)*arcsin(a*x)^(3/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a**2*c*x**2+c)**(3/2)*asin(a*x)**(3/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-a^{2} c x^{2} + c\right )}^{\frac{3}{2}} \arcsin \left (a x\right )^{\frac{3}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((-a^2*c*x^2 + c)^(3/2)*arcsin(a*x)^(3/2), x)