3.295 \(\int (c-a^2 c x^2)^{5/2} \sin ^{-1}(a x)^3 \, dx\)

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

[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])

________________________________________________________________________________________

Rubi [A]  time = 0.547163, antiderivative size = 533, normalized size of antiderivative = 1., 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 = {4649, 4647, 4641, 4627, 4707, 30, 4677, 14, 261} \[ -\frac{65 a^3 c^2 x^4 \sqrt{c-a^2 c x^2}}{2304 \sqrt{1-a^2 x^2}}+\frac{865 a c^2 x^2 \sqrt{c-a^2 c x^2}}{2304 \sqrt{1-a^2 x^2}}-\frac{c^2 \left (1-a^2 x^2\right )^{5/2} \sqrt{c-a^2 c x^2}}{216 a}-\frac{15 a c^2 x^2 \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^2}{32 \sqrt{1-a^2 x^2}}+\frac{5}{16} c^2 x \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^3-\frac{245}{384} c^2 x \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)-\frac{1}{36} c^2 x \left (1-a^2 x^2\right )^2 \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)-\frac{65}{576} c^2 x \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)+\frac{5 c^2 \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^4}{64 a \sqrt{1-a^2 x^2}}+\frac{c^2 \left (1-a^2 x^2\right )^{5/2} \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^2}{12 a}+\frac{5 c^2 \left (1-a^2 x^2\right )^{3/2} \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^2}{32 a}+\frac{115 c^2 \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^2}{768 a \sqrt{1-a^2 x^2}}+\frac{1}{6} x \left (c-a^2 c x^2\right )^{5/2} \sin ^{-1}(a x)^3+\frac{5}{24} c x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^3 \]

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 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 4627

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 4707

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

Rule 30

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

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 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 261

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]

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*}

Mathematica [A]  time = 0.800163, size = 179, normalized size = 0.34 \[ \frac{c^2 \sqrt{c-a^2 c x^2} \left (4320 \sin ^{-1}(a x)^4+288 \left (45 \sin \left (2 \sin ^{-1}(a x)\right )+9 \sin \left (4 \sin ^{-1}(a x)\right )+\sin \left (6 \sin ^{-1}(a x)\right )\right ) \sin ^{-1}(a x)^3-12 \left (1620 \sin \left (2 \sin ^{-1}(a x)\right )+81 \sin \left (4 \sin ^{-1}(a x)\right )+4 \sin \left (6 \sin ^{-1}(a x)\right )\right ) \sin ^{-1}(a x)+72 \sin ^{-1}(a x)^2 \left (270 \cos \left (2 \sin ^{-1}(a x)\right )+27 \cos \left (4 \sin ^{-1}(a x)\right )+2 \cos \left (6 \sin ^{-1}(a x)\right )\right )-9720 \cos \left (2 \sin ^{-1}(a x)\right )-243 \cos \left (4 \sin ^{-1}(a x)\right )-8 \cos \left (6 \sin ^{-1}(a x)\right )\right )}{55296 a \sqrt{1-a^2 x^2}} \]

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]  time = 0.247, size = 875, normalized size = 1.6 \begin{align*} \text{result too large to display} \end{align*}

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)

[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)*x^6*a^6+32*x^7*a^7+48*I*(-a^2*x^2+1)^(1/2)*x^4*a^4-64*a^5*x^5-18*I*(-a^2*x^2+1)^(1/
2)*x^2*a^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)-3/4096*(-c*(a^2*x^2-1))^(1/2)*(-8*I*(-a^2*x^2+1)^(1/2)*x^4*a^4+8*a^5*x^5+8*I*(-a^2*x^2+1)^(1/2)*x^
2*a^2-12*a^3*x^3-I*(-a^2*x^2+1)^(1/2)+4*a*x)*(24*I*arcsin(a*x)^2+32*arcsin(a*x)^3-3*I-12*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)*x^2*a^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)+15/512*(-c*(a^2*x^2-1))^(1/2)*(2*I*(-
a^2*x^2+1)^(1/2)*x^2*a^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*arcsi
n(a*x))*c^2/a/(a^2*x^2-1)-3/4096*(-c*(a^2*x^2-1))^(1/2)*(8*I*(-a^2*x^2+1)^(1/2)*x^4*a^4+8*a^5*x^5-8*I*(-a^2*x^
2+1)^(1/2)*x^2*a^2-12*a^3*x^3+I*(-a^2*x^2+1)^(1/2)+4*a*x)*(-24*I*arcsin(a*x)^2+32*arcsin(a*x)^3+3*I-12*arcsin(
a*x))*c^2/a/(a^2*x^2-1)+1/13824*(-c*(a^2*x^2-1))^(1/2)*(32*I*(-a^2*x^2+1)^(1/2)*x^6*a^6+32*x^7*a^7-48*I*(-a^2*
x^2+1)^(1/2)*x^4*a^4-64*a^5*x^5+18*I*(-a^2*x^2+1)^(1/2)*x^2*a^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)

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

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]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a^{4} c^{2} x^{4} - 2 \, a^{2} c^{2} x^{2} + c^{2}\right )} \sqrt{-a^{2} c x^{2} + c} \arcsin \left (a x\right )^{3}, x\right ) \end{align*}

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

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]

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