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

Optimal. Leaf size=431 \[ \frac {15 \sqrt {\frac {\pi }{2}} c \sqrt {c-a^2 c x^2} S\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{4096 a \sqrt {1-a^2 x^2}}+\frac {15 \sqrt {\pi } c \sqrt {c-a^2 c x^2} S\left (\frac {2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {\pi }}\right )}{128 a \sqrt {1-a^2 x^2}}+\frac {3 c \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{7/2}}{28 a \sqrt {1-a^2 x^2}}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^{5/2}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}+\frac {5 c \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{32 a}-\frac {15 a c x^2 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{32 \sqrt {1-a^2 x^2}}+\frac {45 c \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{256 a \sqrt {1-a^2 x^2}}-\frac {225}{512} c x \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}-\frac {15}{256} c x \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)} \]

[Out]

1/4*x*(-a^2*c*x^2+c)^(3/2)*arcsin(a*x)^(5/2)+5/32*c*(-a^2*x^2+1)^(3/2)*arcsin(a*x)^(3/2)*(-a^2*c*x^2+c)^(1/2)/
a+3/8*c*x*arcsin(a*x)^(5/2)*(-a^2*c*x^2+c)^(1/2)+45/256*c*arcsin(a*x)^(3/2)*(-a^2*c*x^2+c)^(1/2)/a/(-a^2*x^2+1
)^(1/2)-15/32*a*c*x^2*arcsin(a*x)^(3/2)*(-a^2*c*x^2+c)^(1/2)/(-a^2*x^2+1)^(1/2)+3/28*c*arcsin(a*x)^(7/2)*(-a^2
*c*x^2+c)^(1/2)/a/(-a^2*x^2+1)^(1/2)+15/8192*c*FresnelS(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)+15/128*c*FresnelS(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)-225/512*c*x*(-a^2*c*x^2+c)^(1/2)*arcsin(a*x)^(1/2)-15/256*c*x*(-a^2*x^2+1)*(-a
^2*c*x^2+c)^(1/2)*arcsin(a*x)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.58, antiderivative size = 431, normalized size of antiderivative = 1.00, number of steps used = 27, number of rules used = 12, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4649, 4647, 4641, 4629, 4707, 4635, 4406, 12, 3305, 3351, 4677, 4723} \[ \frac {15 \sqrt {\frac {\pi }{2}} c \sqrt {c-a^2 c x^2} S\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{4096 a \sqrt {1-a^2 x^2}}+\frac {15 \sqrt {\pi } c \sqrt {c-a^2 c x^2} S\left (\frac {2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {\pi }}\right )}{128 a \sqrt {1-a^2 x^2}}+\frac {3 c \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{7/2}}{28 a \sqrt {1-a^2 x^2}}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^{5/2}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}+\frac {5 c \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{32 a}-\frac {15 a c x^2 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{32 \sqrt {1-a^2 x^2}}+\frac {45 c \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{256 a \sqrt {1-a^2 x^2}}-\frac {225}{512} c x \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}-\frac {15}{256} c x \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-225*c*x*Sqrt[c - a^2*c*x^2]*Sqrt[ArcSin[a*x]])/512 - (15*c*x*(1 - a^2*x^2)*Sqrt[c - a^2*c*x^2]*Sqrt[ArcSin[a
*x]])/256 + (45*c*Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^(3/2))/(256*a*Sqrt[1 - a^2*x^2]) - (15*a*c*x^2*Sqrt[c - a^2*
c*x^2]*ArcSin[a*x]^(3/2))/(32*Sqrt[1 - a^2*x^2]) + (5*c*(1 - a^2*x^2)^(3/2)*Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^(3
/2))/(32*a) + (3*c*x*Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^(5/2))/8 + (x*(c - a^2*c*x^2)^(3/2)*ArcSin[a*x]^(5/2))/4
+ (3*c*Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^(7/2))/(28*a*Sqrt[1 - a^2*x^2]) + (15*c*Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*
FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(4096*a*Sqrt[1 - a^2*x^2]) + (15*c*Sqrt[Pi]*Sqrt[c - a^2*c*x^2]*Fres
nelS[(2*Sqrt[ArcSin[a*x]])/Sqrt[Pi]])/(128*a*Sqrt[1 - a^2*x^2])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 3305

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 3351

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

Rule 4406

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(c + d*x)^m, Sin[a + b*x]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0]
&& IGtQ[p, 0]

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 4635

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[1/c^(m + 1), Subst[Int[(a + b*x)^n*S
in[x]^m*Cos[x], x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 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 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 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 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 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 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])

Rubi steps

\begin {align*} \int \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^{5/2} \, dx &=\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^{5/2}+\frac {1}{4} (3 c) \int \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2} \, dx-\frac {\left (5 a c \sqrt {c-a^2 c x^2}\right ) \int x \left (1-a^2 x^2\right ) \sin ^{-1}(a x)^{3/2} \, dx}{8 \sqrt {1-a^2 x^2}}\\ &=\frac {5 c \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{32 a}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^{5/2}-\frac {\left (15 c \sqrt {c-a^2 c x^2}\right ) \int \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)^{5/2}}{\sqrt {1-a^2 x^2}} \, dx}{8 \sqrt {1-a^2 x^2}}-\frac {\left (15 a c \sqrt {c-a^2 c x^2}\right ) \int x \sin ^{-1}(a x)^{3/2} \, dx}{16 \sqrt {1-a^2 x^2}}\\ &=-\frac {15}{256} c x \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}-\frac {15 a c x^2 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{32 \sqrt {1-a^2 x^2}}+\frac {5 c \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{32 a}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^{5/2}+\frac {3 c \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{7/2}}{28 a \sqrt {1-a^2 x^2}}-\frac {\left (45 c \sqrt {c-a^2 c x^2}\right ) \int \sqrt {1-a^2 x^2} \sqrt {\sin ^{-1}(a x)} \, dx}{256 \sqrt {1-a^2 x^2}}+\frac {\left (15 a c \sqrt {c-a^2 c x^2}\right ) \int \frac {x \left (1-a^2 x^2\right )}{\sqrt {\sin ^{-1}(a x)}} \, dx}{512 \sqrt {1-a^2 x^2}}+\frac {\left (45 a^2 c \sqrt {c-a^2 c x^2}\right ) \int \frac {x^2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {1-a^2 x^2}} \, dx}{64 \sqrt {1-a^2 x^2}}\\ &=-\frac {225}{512} c x \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}-\frac {15}{256} c x \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}-\frac {15 a c x^2 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{32 \sqrt {1-a^2 x^2}}+\frac {5 c \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{32 a}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^{5/2}+\frac {3 c \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{7/2}}{28 a \sqrt {1-a^2 x^2}}-\frac {\left (45 c \sqrt {c-a^2 c x^2}\right ) \int \frac {\sqrt {\sin ^{-1}(a x)}}{\sqrt {1-a^2 x^2}} \, dx}{512 \sqrt {1-a^2 x^2}}+\frac {\left (45 c \sqrt {c-a^2 c x^2}\right ) \int \frac {\sqrt {\sin ^{-1}(a x)}}{\sqrt {1-a^2 x^2}} \, dx}{128 \sqrt {1-a^2 x^2}}+\frac {\left (15 c \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {\cos ^3(x) \sin (x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{512 a \sqrt {1-a^2 x^2}}+\frac {\left (45 a c \sqrt {c-a^2 c x^2}\right ) \int \frac {x}{\sqrt {\sin ^{-1}(a x)}} \, dx}{1024 \sqrt {1-a^2 x^2}}+\frac {\left (45 a c \sqrt {c-a^2 c x^2}\right ) \int \frac {x}{\sqrt {\sin ^{-1}(a x)}} \, dx}{256 \sqrt {1-a^2 x^2}}\\ &=-\frac {225}{512} c x \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}-\frac {15}{256} c x \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}+\frac {45 c \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{256 a \sqrt {1-a^2 x^2}}-\frac {15 a c x^2 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{32 \sqrt {1-a^2 x^2}}+\frac {5 c \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{32 a}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^{5/2}+\frac {3 c \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{7/2}}{28 a \sqrt {1-a^2 x^2}}+\frac {\left (15 c \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {\sin (2 x)}{4 \sqrt {x}}+\frac {\sin (4 x)}{8 \sqrt {x}}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{512 a \sqrt {1-a^2 x^2}}+\frac {\left (45 c \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {\cos (x) \sin (x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{1024 a \sqrt {1-a^2 x^2}}+\frac {\left (45 c \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {\cos (x) \sin (x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{256 a \sqrt {1-a^2 x^2}}\\ &=-\frac {225}{512} c x \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}-\frac {15}{256} c x \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}+\frac {45 c \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{256 a \sqrt {1-a^2 x^2}}-\frac {15 a c x^2 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{32 \sqrt {1-a^2 x^2}}+\frac {5 c \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{32 a}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^{5/2}+\frac {3 c \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{7/2}}{28 a \sqrt {1-a^2 x^2}}+\frac {\left (15 c \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {\sin (4 x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{4096 a \sqrt {1-a^2 x^2}}+\frac {\left (15 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 )}{2048 a \sqrt {1-a^2 x^2}}+\frac {\left (45 c \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {\sin (2 x)}{2 \sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{1024 a \sqrt {1-a^2 x^2}}+\frac {\left (45 c \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {\sin (2 x)}{2 \sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{256 a \sqrt {1-a^2 x^2}}\\ &=-\frac {225}{512} c x \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}-\frac {15}{256} c x \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}+\frac {45 c \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{256 a \sqrt {1-a^2 x^2}}-\frac {15 a c x^2 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{32 \sqrt {1-a^2 x^2}}+\frac {5 c \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{32 a}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^{5/2}+\frac {3 c \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{7/2}}{28 a \sqrt {1-a^2 x^2}}+\frac {\left (15 c \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int \sin \left (4 x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{2048 a \sqrt {1-a^2 x^2}}+\frac {\left (15 c \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{1024 a \sqrt {1-a^2 x^2}}+\frac {\left (45 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 )}{2048 a \sqrt {1-a^2 x^2}}+\frac {\left (45 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 )}{512 a \sqrt {1-a^2 x^2}}\\ &=-\frac {225}{512} c x \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}-\frac {15}{256} c x \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}+\frac {45 c \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{256 a \sqrt {1-a^2 x^2}}-\frac {15 a c x^2 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{32 \sqrt {1-a^2 x^2}}+\frac {5 c \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{32 a}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^{5/2}+\frac {3 c \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{7/2}}{28 a \sqrt {1-a^2 x^2}}+\frac {15 c \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} S\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{4096 a \sqrt {1-a^2 x^2}}+\frac {15 c \sqrt {\pi } \sqrt {c-a^2 c x^2} S\left (\frac {2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {\pi }}\right )}{2048 a \sqrt {1-a^2 x^2}}+\frac {\left (45 c \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{1024 a \sqrt {1-a^2 x^2}}+\frac {\left (45 c \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{256 a \sqrt {1-a^2 x^2}}\\ &=-\frac {225}{512} c x \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}-\frac {15}{256} c x \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}+\frac {45 c \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{256 a \sqrt {1-a^2 x^2}}-\frac {15 a c x^2 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{32 \sqrt {1-a^2 x^2}}+\frac {5 c \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{32 a}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^{5/2}+\frac {3 c \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{7/2}}{28 a \sqrt {1-a^2 x^2}}+\frac {15 c \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} S\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{4096 a \sqrt {1-a^2 x^2}}+\frac {15 c \sqrt {\pi } \sqrt {c-a^2 c x^2} S\left (\frac {2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {\pi }}\right )}{128 a \sqrt {1-a^2 x^2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.40, size = 180, normalized size = 0.42 \[ \frac {c \sqrt {c-a^2 c x^2} \left (1680 \sqrt {\pi } \sqrt {\sin ^{-1}(a x)} S\left (\frac {2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {\pi }}\right )+1536 \sin ^{-1}(a x)^4+3584 \sin \left (2 \sin ^{-1}(a x)\right ) \sin ^{-1}(a x)^3-3360 \sin \left (2 \sin ^{-1}(a x)\right ) \sin ^{-1}(a x)+4480 \sin ^{-1}(a x)^2 \cos \left (2 \sin ^{-1}(a x)\right )-7 \sqrt {-i \sin ^{-1}(a x)} \Gamma \left (\frac {7}{2},-4 i \sin ^{-1}(a x)\right )-7 \sqrt {i \sin ^{-1}(a x)} \Gamma \left (\frac {7}{2},4 i \sin ^{-1}(a x)\right )\right )}{14336 a \sqrt {1-a^2 x^2} \sqrt {\sin ^{-1}(a x)}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c*Sqrt[c - a^2*c*x^2]*(1536*ArcSin[a*x]^4 + 4480*ArcSin[a*x]^2*Cos[2*ArcSin[a*x]] + 1680*Sqrt[Pi]*Sqrt[ArcSin
[a*x]]*FresnelS[(2*Sqrt[ArcSin[a*x]])/Sqrt[Pi]] - 7*Sqrt[(-I)*ArcSin[a*x]]*Gamma[7/2, (-4*I)*ArcSin[a*x]] - 7*
Sqrt[I*ArcSin[a*x]]*Gamma[7/2, (4*I)*ArcSin[a*x]] - 3360*ArcSin[a*x]*Sin[2*ArcSin[a*x]] + 3584*ArcSin[a*x]^3*S
in[2*ArcSin[a*x]]))/(14336*a*Sqrt[1 - a^2*x^2]*Sqrt[ArcSin[a*x]])

________________________________________________________________________________________

fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

________________________________________________________________________________________

maple [F]  time = 0.38, size = 0, normalized size = 0.00 \[ \int \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} \arcsin \left (a x \right )^{\frac {5}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int {\mathrm {asin}\left (a\,x\right )}^{5/2}\,{\left (c-a^2\,c\,x^2\right )}^{3/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________