3.5.47 \(\int (c-a^2 c x^2)^{3/2} \text {ArcSin}(a x)^{3/2} \, dx\) [447]

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

[Out]

1/4*x*(-a^2*c*x^2+c)^(3/2)*arcsin(a*x)^(3/2)+3/8*c*x*arcsin(a*x)^(3/2)*(-a^2*c*x^2+c)^(1/2)+3/20*c*arcsin(a*x)
^(5/2)*(-a^2*c*x^2+c)^(1/2)/a/(-a^2*x^2+1)^(1/2)-3/1024*c*FresnelC(2*2^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))*2^(1/
2)*Pi^(1/2)*(-a^2*c*x^2+c)^(1/2)/a/(-a^2*x^2+1)^(1/2)-3/32*c*FresnelC(2*arcsin(a*x)^(1/2)/Pi^(1/2))*Pi^(1/2)*(
-a^2*c*x^2+c)^(1/2)/a/(-a^2*x^2+1)^(1/2)+3/32*c*(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c)^(1/2)*arcsin(a*x)^(1/2)/a+27
/256*c*(-a^2*c*x^2+c)^(1/2)*arcsin(a*x)^(1/2)/a/(-a^2*x^2+1)^(1/2)-9/32*a*c*x^2*(-a^2*c*x^2+c)^(1/2)*arcsin(a*
x)^(1/2)/(-a^2*x^2+1)^(1/2)

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Rubi [A]
time = 0.30, antiderivative size = 363, normalized size of antiderivative = 1.00, 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 = {4743, 4741, 4737, 4725, 4809, 3393, 3385, 3433, 4767, 4753} \begin {gather*} -\frac {3 \sqrt {\frac {\pi }{2}} c \sqrt {c-a^2 c x^2} \text {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\text {ArcSin}(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 {\text {ArcSin}(a x)}}{\sqrt {\pi }}\right )}{32 a \sqrt {1-a^2 x^2}}+\frac {3 c \text {ArcSin}(a x)^{5/2} \sqrt {c-a^2 c x^2}}{20 a \sqrt {1-a^2 x^2}}+\frac {1}{4} x \text {ArcSin}(a x)^{3/2} \left (c-a^2 c x^2\right )^{3/2}+\frac {3}{8} c x \text {ArcSin}(a x)^{3/2} \sqrt {c-a^2 c x^2}+\frac {3 c \left (1-a^2 x^2\right )^{3/2} \sqrt {\text {ArcSin}(a x)} \sqrt {c-a^2 c x^2}}{32 a}-\frac {9 a c x^2 \sqrt {\text {ArcSin}(a x)} \sqrt {c-a^2 c x^2}}{32 \sqrt {1-a^2 x^2}}+\frac {27 c \sqrt {\text {ArcSin}(a x)} \sqrt {c-a^2 c x^2}}{256 a \sqrt {1-a^2 x^2}} \end {gather*}

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 3385

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 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 3433

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

Rule 4725

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 4737

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

Rule 4741

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[(1/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]], Int[(a + b*ArcSin[c*x])^n/S
qrt[1 - c^2*x^2], x], x] - Dist[b*c*(n/2)*Simp[Sqrt[d + e*x^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 4743

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/(2*p + 1))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p], Int[x*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcS
in[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0]

Rule 4753

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

Rule 4767

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/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^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 4809

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

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Mathematica [C] Result contains complex when optimal does not.
time = 0.30, size = 186, normalized size = 0.51 \begin {gather*} \frac {c \sqrt {c-a^2 c x^2} \left (-240 \sqrt {\pi } \sqrt {\text {ArcSin}(a x)^2} \text {FresnelC}\left (\frac {2 \sqrt {\text {ArcSin}(a x)}}{\sqrt {\pi }}\right )+\sqrt {\text {ArcSin}(a x)} \left (5 \sqrt {i \text {ArcSin}(a x)} \text {Gamma}\left (\frac {5}{2},-4 i \text {ArcSin}(a x)\right )+5 \sqrt {-i \text {ArcSin}(a x)} \text {Gamma}\left (\frac {5}{2},4 i \text {ArcSin}(a x)\right )+32 \sqrt {\text {ArcSin}(a x)^2} \left (12 \text {ArcSin}(a x)^2+15 \cos (2 \text {ArcSin}(a x))+20 \text {ArcSin}(a x) \sin (2 \text {ArcSin}(a x))\right )\right )\right )}{2560 a \sqrt {1-a^2 x^2} \sqrt {\text {ArcSin}(a x)^2}} \end {gather*}

Antiderivative was successfully verified.

[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 = 0.29, size = 0, normalized size = 0.00 \[\int \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} \arcsin \left (a x \right )^{\frac {3}{2}}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

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 >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\mathrm {asin}\left (a\,x\right )}^{3/2}\,{\left (c-a^2\,c\,x^2\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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