3.3.95 \(\int (c-a^2 c x^2)^{5/2} \text {ArcSin}(a x)^3 \, dx\) [295]

Optimal. Leaf size=533 \[ \frac {865 a c^2 x^2 \sqrt {c-a^2 c x^2}}{2304 \sqrt {1-a^2 x^2}}-\frac {65 a^3 c^2 x^4 \sqrt {c-a^2 c x^2}}{2304 \sqrt {1-a^2 x^2}}-\frac {c^2 \left (1-a^2 x^2\right )^{5/2} \sqrt {c-a^2 c x^2}}{216 a}-\frac {245}{384} c^2 x \sqrt {c-a^2 c x^2} \text {ArcSin}(a x)-\frac {65}{576} c^2 x \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2} \text {ArcSin}(a x)-\frac {1}{36} c^2 x \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2} \text {ArcSin}(a x)+\frac {115 c^2 \sqrt {c-a^2 c x^2} \text {ArcSin}(a x)^2}{768 a \sqrt {1-a^2 x^2}}-\frac {15 a c^2 x^2 \sqrt {c-a^2 c x^2} \text {ArcSin}(a x)^2}{32 \sqrt {1-a^2 x^2}}+\frac {5 c^2 \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2} \text {ArcSin}(a x)^2}{32 a}+\frac {c^2 \left (1-a^2 x^2\right )^{5/2} \sqrt {c-a^2 c x^2} \text {ArcSin}(a x)^2}{12 a}+\frac {5}{16} c^2 x \sqrt {c-a^2 c x^2} \text {ArcSin}(a x)^3+\frac {5}{24} c x \left (c-a^2 c x^2\right )^{3/2} \text {ArcSin}(a x)^3+\frac {1}{6} x \left (c-a^2 c x^2\right )^{5/2} \text {ArcSin}(a x)^3+\frac {5 c^2 \sqrt {c-a^2 c x^2} \text {ArcSin}(a x)^4}{64 a \sqrt {1-a^2 x^2}} \]

[Out]

5/24*c*x*(-a^2*c*x^2+c)^(3/2)*arcsin(a*x)^3+1/6*x*(-a^2*c*x^2+c)^(5/2)*arcsin(a*x)^3-1/216*c^2*(-a^2*x^2+1)^(5
/2)*(-a^2*c*x^2+c)^(1/2)/a-245/384*c^2*x*arcsin(a*x)*(-a^2*c*x^2+c)^(1/2)-65/576*c^2*x*(-a^2*x^2+1)*arcsin(a*x
)*(-a^2*c*x^2+c)^(1/2)-1/36*c^2*x*(-a^2*x^2+1)^2*arcsin(a*x)*(-a^2*c*x^2+c)^(1/2)+5/32*c^2*(-a^2*x^2+1)^(3/2)*
arcsin(a*x)^2*(-a^2*c*x^2+c)^(1/2)/a+1/12*c^2*(-a^2*x^2+1)^(5/2)*arcsin(a*x)^2*(-a^2*c*x^2+c)^(1/2)/a+5/16*c^2
*x*arcsin(a*x)^3*(-a^2*c*x^2+c)^(1/2)+865/2304*a*c^2*x^2*(-a^2*c*x^2+c)^(1/2)/(-a^2*x^2+1)^(1/2)-65/2304*a^3*c
^2*x^4*(-a^2*c*x^2+c)^(1/2)/(-a^2*x^2+1)^(1/2)+115/768*c^2*arcsin(a*x)^2*(-a^2*c*x^2+c)^(1/2)/a/(-a^2*x^2+1)^(
1/2)-15/32*a*c^2*x^2*arcsin(a*x)^2*(-a^2*c*x^2+c)^(1/2)/(-a^2*x^2+1)^(1/2)+5/64*c^2*arcsin(a*x)^4*(-a^2*c*x^2+
c)^(1/2)/a/(-a^2*x^2+1)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.39, antiderivative size = 533, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 9, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {4743, 4741, 4737, 4723, 4795, 30, 4767, 14, 267} \begin {gather*} -\frac {15 a c^2 x^2 \text {ArcSin}(a x)^2 \sqrt {c-a^2 c x^2}}{32 \sqrt {1-a^2 x^2}}+\frac {5}{16} c^2 x \text {ArcSin}(a x)^3 \sqrt {c-a^2 c x^2}-\frac {245}{384} c^2 x \text {ArcSin}(a x) \sqrt {c-a^2 c x^2}-\frac {1}{36} c^2 x \left (1-a^2 x^2\right )^2 \text {ArcSin}(a x) \sqrt {c-a^2 c x^2}-\frac {65}{576} c^2 x \left (1-a^2 x^2\right ) \text {ArcSin}(a x) \sqrt {c-a^2 c x^2}+\frac {5 c^2 \text {ArcSin}(a x)^4 \sqrt {c-a^2 c x^2}}{64 a \sqrt {1-a^2 x^2}}+\frac {c^2 \left (1-a^2 x^2\right )^{5/2} \text {ArcSin}(a x)^2 \sqrt {c-a^2 c x^2}}{12 a}+\frac {5 c^2 \left (1-a^2 x^2\right )^{3/2} \text {ArcSin}(a x)^2 \sqrt {c-a^2 c x^2}}{32 a}+\frac {115 c^2 \text {ArcSin}(a x)^2 \sqrt {c-a^2 c x^2}}{768 a \sqrt {1-a^2 x^2}}+\frac {1}{6} x \text {ArcSin}(a x)^3 \left (c-a^2 c x^2\right )^{5/2}+\frac {5}{24} c x \text {ArcSin}(a x)^3 \left (c-a^2 c x^2\right )^{3/2}+\frac {865 a c^2 x^2 \sqrt {c-a^2 c x^2}}{2304 \sqrt {1-a^2 x^2}}-\frac {c^2 \left (1-a^2 x^2\right )^{5/2} \sqrt {c-a^2 c x^2}}{216 a}-\frac {65 a^3 c^2 x^4 \sqrt {c-a^2 c x^2}}{2304 \sqrt {1-a^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c - a^2*c*x^2)^(5/2)*ArcSin[a*x]^3,x]

[Out]

(865*a*c^2*x^2*Sqrt[c - a^2*c*x^2])/(2304*Sqrt[1 - a^2*x^2]) - (65*a^3*c^2*x^4*Sqrt[c - a^2*c*x^2])/(2304*Sqrt
[1 - a^2*x^2]) - (c^2*(1 - a^2*x^2)^(5/2)*Sqrt[c - a^2*c*x^2])/(216*a) - (245*c^2*x*Sqrt[c - a^2*c*x^2]*ArcSin
[a*x])/384 - (65*c^2*x*(1 - a^2*x^2)*Sqrt[c - a^2*c*x^2]*ArcSin[a*x])/576 - (c^2*x*(1 - a^2*x^2)^2*Sqrt[c - a^
2*c*x^2]*ArcSin[a*x])/36 + (115*c^2*Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^2)/(768*a*Sqrt[1 - a^2*x^2]) - (15*a*c^2*x
^2*Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^2)/(32*Sqrt[1 - a^2*x^2]) + (5*c^2*(1 - a^2*x^2)^(3/2)*Sqrt[c - a^2*c*x^2]*
ArcSin[a*x]^2)/(32*a) + (c^2*(1 - a^2*x^2)^(5/2)*Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^2)/(12*a) + (5*c^2*x*Sqrt[c -
a^2*c*x^2]*ArcSin[a*x]^3)/16 + (5*c*x*(c - a^2*c*x^2)^(3/2)*ArcSin[a*x]^3)/24 + (x*(c - a^2*c*x^2)^(5/2)*ArcS
in[a*x]^3)/6 + (5*c^2*Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^4)/(64*a*Sqrt[1 - a^2*x^2])

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 267

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 4723

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

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 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 4795

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

Rubi steps

\begin {align*} \int \left (c-a^2 c x^2\right )^{5/2} \sin ^{-1}(a x)^3 \, dx &=\frac {1}{6} x \left (c-a^2 c x^2\right )^{5/2} \sin ^{-1}(a x)^3+\frac {1}{6} (5 c) \int \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^3 \, dx-\frac {\left (a c^2 \sqrt {c-a^2 c x^2}\right ) \int x \left (1-a^2 x^2\right )^2 \sin ^{-1}(a x)^2 \, dx}{2 \sqrt {1-a^2 x^2}}\\ &=\frac {c^2 \left (1-a^2 x^2\right )^{5/2} \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^2}{12 a}+\frac {5}{24} c x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^3+\frac {1}{6} x \left (c-a^2 c x^2\right )^{5/2} \sin ^{-1}(a x)^3+\frac {1}{8} \left (5 c^2\right ) \int \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^3 \, dx-\frac {\left (c^2 \sqrt {c-a^2 c x^2}\right ) \int \left (1-a^2 x^2\right )^{5/2} \sin ^{-1}(a x) \, dx}{6 \sqrt {1-a^2 x^2}}-\frac {\left (5 a c^2 \sqrt {c-a^2 c x^2}\right ) \int x \left (1-a^2 x^2\right ) \sin ^{-1}(a x)^2 \, dx}{8 \sqrt {1-a^2 x^2}}\\ &=-\frac {1}{36} c^2 x \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)+\frac {5 c^2 \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^2}{32 a}+\frac {c^2 \left (1-a^2 x^2\right )^{5/2} \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^2}{12 a}+\frac {5}{16} c^2 x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^3+\frac {5}{24} c x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^3+\frac {1}{6} x \left (c-a^2 c x^2\right )^{5/2} \sin ^{-1}(a x)^3-\frac {\left (5 c^2 \sqrt {c-a^2 c x^2}\right ) \int \left (1-a^2 x^2\right )^{3/2} \sin ^{-1}(a x) \, dx}{36 \sqrt {1-a^2 x^2}}-\frac {\left (5 c^2 \sqrt {c-a^2 c x^2}\right ) \int \left (1-a^2 x^2\right )^{3/2} \sin ^{-1}(a x) \, dx}{16 \sqrt {1-a^2 x^2}}+\frac {\left (5 c^2 \sqrt {c-a^2 c x^2}\right ) \int \frac {\sin ^{-1}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx}{16 \sqrt {1-a^2 x^2}}+\frac {\left (a c^2 \sqrt {c-a^2 c x^2}\right ) \int x \left (1-a^2 x^2\right )^2 \, dx}{36 \sqrt {1-a^2 x^2}}-\frac {\left (15 a c^2 \sqrt {c-a^2 c x^2}\right ) \int x \sin ^{-1}(a x)^2 \, dx}{16 \sqrt {1-a^2 x^2}}\\ &=-\frac {c^2 \left (1-a^2 x^2\right )^{5/2} \sqrt {c-a^2 c x^2}}{216 a}-\frac {65}{576} c^2 x \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)-\frac {1}{36} c^2 x \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)-\frac {15 a c^2 x^2 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^2}{32 \sqrt {1-a^2 x^2}}+\frac {5 c^2 \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^2}{32 a}+\frac {c^2 \left (1-a^2 x^2\right )^{5/2} \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^2}{12 a}+\frac {5}{16} c^2 x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^3+\frac {5}{24} c x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^3+\frac {1}{6} x \left (c-a^2 c x^2\right )^{5/2} \sin ^{-1}(a x)^3+\frac {5 c^2 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^4}{64 a \sqrt {1-a^2 x^2}}-\frac {\left (5 c^2 \sqrt {c-a^2 c x^2}\right ) \int \sqrt {1-a^2 x^2} \sin ^{-1}(a x) \, dx}{48 \sqrt {1-a^2 x^2}}-\frac {\left (15 c^2 \sqrt {c-a^2 c x^2}\right ) \int \sqrt {1-a^2 x^2} \sin ^{-1}(a x) \, dx}{64 \sqrt {1-a^2 x^2}}+\frac {\left (5 a c^2 \sqrt {c-a^2 c x^2}\right ) \int x \left (1-a^2 x^2\right ) \, dx}{144 \sqrt {1-a^2 x^2}}+\frac {\left (5 a c^2 \sqrt {c-a^2 c x^2}\right ) \int x \left (1-a^2 x^2\right ) \, dx}{64 \sqrt {1-a^2 x^2}}+\frac {\left (15 a^2 c^2 \sqrt {c-a^2 c x^2}\right ) \int \frac {x^2 \sin ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{16 \sqrt {1-a^2 x^2}}\\ &=-\frac {c^2 \left (1-a^2 x^2\right )^{5/2} \sqrt {c-a^2 c x^2}}{216 a}-\frac {245}{384} c^2 x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)-\frac {65}{576} c^2 x \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)-\frac {1}{36} c^2 x \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)-\frac {15 a c^2 x^2 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^2}{32 \sqrt {1-a^2 x^2}}+\frac {5 c^2 \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^2}{32 a}+\frac {c^2 \left (1-a^2 x^2\right )^{5/2} \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^2}{12 a}+\frac {5}{16} c^2 x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^3+\frac {5}{24} c x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^3+\frac {1}{6} x \left (c-a^2 c x^2\right )^{5/2} \sin ^{-1}(a x)^3+\frac {5 c^2 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^4}{64 a \sqrt {1-a^2 x^2}}-\frac {\left (5 c^2 \sqrt {c-a^2 c x^2}\right ) \int \frac {\sin ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{96 \sqrt {1-a^2 x^2}}-\frac {\left (15 c^2 \sqrt {c-a^2 c x^2}\right ) \int \frac {\sin ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{128 \sqrt {1-a^2 x^2}}+\frac {\left (15 c^2 \sqrt {c-a^2 c x^2}\right ) \int \frac {\sin ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{32 \sqrt {1-a^2 x^2}}+\frac {\left (5 a c^2 \sqrt {c-a^2 c x^2}\right ) \int \left (x-a^2 x^3\right ) \, dx}{144 \sqrt {1-a^2 x^2}}+\frac {\left (5 a c^2 \sqrt {c-a^2 c x^2}\right ) \int x \, dx}{96 \sqrt {1-a^2 x^2}}+\frac {\left (5 a c^2 \sqrt {c-a^2 c x^2}\right ) \int \left (x-a^2 x^3\right ) \, dx}{64 \sqrt {1-a^2 x^2}}+\frac {\left (15 a c^2 \sqrt {c-a^2 c x^2}\right ) \int x \, dx}{128 \sqrt {1-a^2 x^2}}+\frac {\left (15 a c^2 \sqrt {c-a^2 c x^2}\right ) \int x \, dx}{32 \sqrt {1-a^2 x^2}}\\ &=\frac {865 a c^2 x^2 \sqrt {c-a^2 c x^2}}{2304 \sqrt {1-a^2 x^2}}-\frac {65 a^3 c^2 x^4 \sqrt {c-a^2 c x^2}}{2304 \sqrt {1-a^2 x^2}}-\frac {c^2 \left (1-a^2 x^2\right )^{5/2} \sqrt {c-a^2 c x^2}}{216 a}-\frac {245}{384} c^2 x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)-\frac {65}{576} c^2 x \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)-\frac {1}{36} c^2 x \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)+\frac {115 c^2 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^2}{768 a \sqrt {1-a^2 x^2}}-\frac {15 a c^2 x^2 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^2}{32 \sqrt {1-a^2 x^2}}+\frac {5 c^2 \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^2}{32 a}+\frac {c^2 \left (1-a^2 x^2\right )^{5/2} \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^2}{12 a}+\frac {5}{16} c^2 x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^3+\frac {5}{24} c x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^3+\frac {1}{6} x \left (c-a^2 c x^2\right )^{5/2} \sin ^{-1}(a x)^3+\frac {5 c^2 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^4}{64 a \sqrt {1-a^2 x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.56, size = 179, normalized size = 0.34 \begin {gather*} \frac {c^2 \sqrt {c-a^2 c x^2} \left (4320 \text {ArcSin}(a x)^4-9720 \cos (2 \text {ArcSin}(a x))-243 \cos (4 \text {ArcSin}(a x))-8 \cos (6 \text {ArcSin}(a x))+72 \text {ArcSin}(a x)^2 (270 \cos (2 \text {ArcSin}(a x))+27 \cos (4 \text {ArcSin}(a x))+2 \cos (6 \text {ArcSin}(a x)))+288 \text {ArcSin}(a x)^3 (45 \sin (2 \text {ArcSin}(a x))+9 \sin (4 \text {ArcSin}(a x))+\sin (6 \text {ArcSin}(a x)))-12 \text {ArcSin}(a x) (1620 \sin (2 \text {ArcSin}(a x))+81 \sin (4 \text {ArcSin}(a x))+4 \sin (6 \text {ArcSin}(a x)))\right )}{55296 a \sqrt {1-a^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c - a^2*c*x^2)^(5/2)*ArcSin[a*x]^3,x]

[Out]

(c^2*Sqrt[c - a^2*c*x^2]*(4320*ArcSin[a*x]^4 - 9720*Cos[2*ArcSin[a*x]] - 243*Cos[4*ArcSin[a*x]] - 8*Cos[6*ArcS
in[a*x]] + 72*ArcSin[a*x]^2*(270*Cos[2*ArcSin[a*x]] + 27*Cos[4*ArcSin[a*x]] + 2*Cos[6*ArcSin[a*x]]) + 288*ArcS
in[a*x]^3*(45*Sin[2*ArcSin[a*x]] + 9*Sin[4*ArcSin[a*x]] + Sin[6*ArcSin[a*x]]) - 12*ArcSin[a*x]*(1620*Sin[2*Arc
Sin[a*x]] + 81*Sin[4*ArcSin[a*x]] + 4*Sin[6*ArcSin[a*x]])))/(55296*a*Sqrt[1 - a^2*x^2])

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Maple [C] Result contains complex when optimal does not.
time = 0.25, size = 699, normalized size = 1.31

method	result	size
default	\(-\frac {5 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \arcsin \left (a x \right )^{4} c^{2}}{64 a \left (a^{2} x^{2}-1\right )}+\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (-32 i \sqrt {-a^{2} x^{2}+1}\, a^{6} x^{6}+32 a^{7} x^{7}+48 i \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}-64 a^{5} x^{5}-18 i \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+38 a^{3} x^{3}+i \sqrt {-a^{2} x^{2}+1}-6 a x \right ) \left (18 i \arcsin \left (a x \right )^{2}+36 \arcsin \left (a x \right )^{3}-i-6 \arcsin \left (a x \right )\right ) c^{2}}{13824 a \left (a^{2} x^{2}-1\right )}+\frac {15 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (2 i \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+2 a^{3} x^{3}-i \sqrt {-a^{2} x^{2}+1}-2 a x \right ) \left (-6 i \arcsin \left (a x \right )^{2}+4 \arcsin \left (a x \right )^{3}+3 i-6 \arcsin \left (a x \right )\right ) c^{2}}{512 a \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (i a^{2} x^{2}-a x \sqrt {-a^{2} x^{2}+1}-i\right ) \left (2088 i \arcsin \left (a x \right )^{2}+2304 \arcsin \left (a x \right )^{3}-251 i-924 \arcsin \left (a x \right )\right ) \cos \left (5 \arcsin \left (a x \right )\right ) c^{2}}{110592 a \left (a^{2} x^{2}-1\right )}+\frac {5 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (i \sqrt {-a^{2} x^{2}+1}\, a x +a^{2} x^{2}-1\right ) \left (360 i \arcsin \left (a x \right )^{2}+576 \arcsin \left (a x \right )^{3}-47 i-204 \arcsin \left (a x \right )\right ) \sin \left (5 \arcsin \left (a x \right )\right ) c^{2}}{110592 a \left (a^{2} x^{2}-1\right )}-\frac {3 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (i a^{2} x^{2}-a x \sqrt {-a^{2} x^{2}+1}-i\right ) \left (264 i \arcsin \left (a x \right )^{2}+128 \arcsin \left (a x \right )^{3}-123 i-228 \arcsin \left (a x \right )\right ) \cos \left (3 \arcsin \left (a x \right )\right ) c^{2}}{4096 a \left (a^{2} x^{2}-1\right )}+\frac {9 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (i \sqrt {-a^{2} x^{2}+1}\, a x +a^{2} x^{2}-1\right ) \left (72 i \arcsin \left (a x \right )^{2}+64 \arcsin \left (a x \right )^{3}-39 i-84 \arcsin \left (a x \right )\right ) \sin \left (3 \arcsin \left (a x \right )\right ) c^{2}}{4096 a \left (a^{2} x^{2}-1\right )}\)	\(699\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-a^2*c*x^2+c)^(5/2)*arcsin(a*x)^3,x,method=_RETURNVERBOSE)

[Out]

-5/64*(-c*(a^2*x^2-1))^(1/2)*(-a^2*x^2+1)^(1/2)/a/(a^2*x^2-1)*arcsin(a*x)^4*c^2+1/13824*(-c*(a^2*x^2-1))^(1/2)
*(-32*I*(-a^2*x^2+1)^(1/2)*a^6*x^6+32*a^7*x^7+48*I*(-a^2*x^2+1)^(1/2)*a^4*x^4-64*a^5*x^5-18*I*(-a^2*x^2+1)^(1/
2)*a^2*x^2+38*a^3*x^3+I*(-a^2*x^2+1)^(1/2)-6*a*x)*(18*I*arcsin(a*x)^2+36*arcsin(a*x)^3-I-6*arcsin(a*x))*c^2/a/
(a^2*x^2-1)+15/512*(-c*(a^2*x^2-1))^(1/2)*(2*I*(-a^2*x^2+1)^(1/2)*a^2*x^2+2*a^3*x^3-I*(-a^2*x^2+1)^(1/2)-2*a*x
)*(-6*I*arcsin(a*x)^2+4*arcsin(a*x)^3+3*I-6*arcsin(a*x))*c^2/a/(a^2*x^2-1)-1/110592*(-c*(a^2*x^2-1))^(1/2)*(I*
a^2*x^2-a*x*(-a^2*x^2+1)^(1/2)-I)*(2088*I*arcsin(a*x)^2+2304*arcsin(a*x)^3-251*I-924*arcsin(a*x))*cos(5*arcsin
(a*x))*c^2/a/(a^2*x^2-1)+5/110592*(-c*(a^2*x^2-1))^(1/2)*(I*(-a^2*x^2+1)^(1/2)*a*x+a^2*x^2-1)*(360*I*arcsin(a*
x)^2+576*arcsin(a*x)^3-47*I-204*arcsin(a*x))*sin(5*arcsin(a*x))*c^2/a/(a^2*x^2-1)-3/4096*(-c*(a^2*x^2-1))^(1/2
)*(I*a^2*x^2-a*x*(-a^2*x^2+1)^(1/2)-I)*(264*I*arcsin(a*x)^2+128*arcsin(a*x)^3-123*I-228*arcsin(a*x))*cos(3*arc
sin(a*x))*c^2/a/(a^2*x^2-1)+9/4096*(-c*(a^2*x^2-1))^(1/2)*(I*(-a^2*x^2+1)^(1/2)*a*x+a^2*x^2-1)*(72*I*arcsin(a*
x)^2+64*arcsin(a*x)^3-39*I-84*arcsin(a*x))*sin(3*arcsin(a*x))*c^2/a/(a^2*x^2-1)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a^2*c*x^2+c)^(5/2)*arcsin(a*x)^3,x, algorithm="maxima")

[Out]

integrate((-a^2*c*x^2 + c)^(5/2)*arcsin(a*x)^3, x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a^2*c*x^2+c)^(5/2)*arcsin(a*x)^3,x, algorithm="fricas")

[Out]

integral((a^4*c^2*x^4 - 2*a^2*c^2*x^2 + c^2)*sqrt(-a^2*c*x^2 + c)*arcsin(a*x)^3, x)

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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)**(5/2)*asin(a*x)**3,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)^(5/2)*arcsin(a*x)^3,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\,{\left (c-a^2\,c\,x^2\right )}^{5/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(asin(a*x)^3*(c - a^2*c*x^2)^(5/2),x)

[Out]

int(asin(a*x)^3*(c - a^2*c*x^2)^(5/2), x)

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