3.3.96 \(\int (c-a^2 c x^2)^{3/2} \text {ArcSin}(a x)^3 \, dx\) [296]

Optimal. Leaf size=365 \[ \frac {51 a c x^2 \sqrt {c-a^2 c x^2}}{128 \sqrt {1-a^2 x^2}}-\frac {3 a^3 c x^4 \sqrt {c-a^2 c x^2}}{128 \sqrt {1-a^2 x^2}}-\frac {45}{64} c x \sqrt {c-a^2 c x^2} \text {ArcSin}(a x)-\frac {3}{32} c x \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2} \text {ArcSin}(a x)+\frac {27 c \sqrt {c-a^2 c x^2} \text {ArcSin}(a x)^2}{128 a \sqrt {1-a^2 x^2}}-\frac {9 a c x^2 \sqrt {c-a^2 c x^2} \text {ArcSin}(a x)^2}{16 \sqrt {1-a^2 x^2}}+\frac {3 c \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2} \text {ArcSin}(a x)^2}{16 a}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \text {ArcSin}(a x)^3+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \text {ArcSin}(a x)^3+\frac {3 c \sqrt {c-a^2 c x^2} \text {ArcSin}(a x)^4}{32 a \sqrt {1-a^2 x^2}} \]

[Out]

1/4*x*(-a^2*c*x^2+c)^(3/2)*arcsin(a*x)^3-45/64*c*x*arcsin(a*x)*(-a^2*c*x^2+c)^(1/2)-3/32*c*x*(-a^2*x^2+1)*arcs
in(a*x)*(-a^2*c*x^2+c)^(1/2)+3/16*c*(-a^2*x^2+1)^(3/2)*arcsin(a*x)^2*(-a^2*c*x^2+c)^(1/2)/a+3/8*c*x*arcsin(a*x
)^3*(-a^2*c*x^2+c)^(1/2)+51/128*a*c*x^2*(-a^2*c*x^2+c)^(1/2)/(-a^2*x^2+1)^(1/2)-3/128*a^3*c*x^4*(-a^2*c*x^2+c)
^(1/2)/(-a^2*x^2+1)^(1/2)+27/128*c*arcsin(a*x)^2*(-a^2*c*x^2+c)^(1/2)/a/(-a^2*x^2+1)^(1/2)-9/16*a*c*x^2*arcsin
(a*x)^2*(-a^2*c*x^2+c)^(1/2)/(-a^2*x^2+1)^(1/2)+3/32*c*arcsin(a*x)^4*(-a^2*c*x^2+c)^(1/2)/a/(-a^2*x^2+1)^(1/2)

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Rubi [A]
time = 0.25, antiderivative size = 365, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {4743, 4741, 4737, 4723, 4795, 30, 4767, 14} \begin {gather*} -\frac {9 a c x^2 \text {ArcSin}(a x)^2 \sqrt {c-a^2 c x^2}}{16 \sqrt {1-a^2 x^2}}+\frac {1}{4} x \text {ArcSin}(a x)^3 \left (c-a^2 c x^2\right )^{3/2}+\frac {3}{8} c x \text {ArcSin}(a x)^3 \sqrt {c-a^2 c x^2}-\frac {45}{64} c x \text {ArcSin}(a x) \sqrt {c-a^2 c x^2}-\frac {3}{32} c x \left (1-a^2 x^2\right ) \text {ArcSin}(a x) \sqrt {c-a^2 c x^2}+\frac {3 c \text {ArcSin}(a x)^4 \sqrt {c-a^2 c x^2}}{32 a \sqrt {1-a^2 x^2}}+\frac {3 c \left (1-a^2 x^2\right )^{3/2} \text {ArcSin}(a x)^2 \sqrt {c-a^2 c x^2}}{16 a}+\frac {27 c \text {ArcSin}(a x)^2 \sqrt {c-a^2 c x^2}}{128 a \sqrt {1-a^2 x^2}}+\frac {51 a c x^2 \sqrt {c-a^2 c x^2}}{128 \sqrt {1-a^2 x^2}}-\frac {3 a^3 c x^4 \sqrt {c-a^2 c x^2}}{128 \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,x]

[Out]

(51*a*c*x^2*Sqrt[c - a^2*c*x^2])/(128*Sqrt[1 - a^2*x^2]) - (3*a^3*c*x^4*Sqrt[c - a^2*c*x^2])/(128*Sqrt[1 - a^2
*x^2]) - (45*c*x*Sqrt[c - a^2*c*x^2]*ArcSin[a*x])/64 - (3*c*x*(1 - a^2*x^2)*Sqrt[c - a^2*c*x^2]*ArcSin[a*x])/3
2 + (27*c*Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^2)/(128*a*Sqrt[1 - a^2*x^2]) - (9*a*c*x^2*Sqrt[c - a^2*c*x^2]*ArcSin
[a*x]^2)/(16*Sqrt[1 - a^2*x^2]) + (3*c*(1 - a^2*x^2)^(3/2)*Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^2)/(16*a) + (3*c*x*
Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^3)/8 + (x*(c - a^2*c*x^2)^(3/2)*ArcSin[a*x]^3)/4 + (3*c*Sqrt[c - a^2*c*x^2]*Ar
cSin[a*x]^4)/(32*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 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 )^{3/2} \sin ^{-1}(a x)^3 \, dx &=\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^3+\frac {1}{4} (3 c) \int \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^3 \, dx-\frac {\left (3 a c \sqrt {c-a^2 c x^2}\right ) \int x \left (1-a^2 x^2\right ) \sin ^{-1}(a x)^2 \, dx}{4 \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} \sin ^{-1}(a x)^2}{16 a}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^3+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^3-\frac {\left (3 c \sqrt {c-a^2 c x^2}\right ) \int \left (1-a^2 x^2\right )^{3/2} \sin ^{-1}(a x) \, dx}{8 \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}{\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 \sin ^{-1}(a x)^2 \, dx}{8 \sqrt {1-a^2 x^2}}\\ &=-\frac {3}{32} c x \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)-\frac {9 a c x^2 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^2}{16 \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} \sin ^{-1}(a x)^2}{16 a}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^3+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^3+\frac {3 c \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^4}{32 a \sqrt {1-a^2 x^2}}-\frac {\left (9 c \sqrt {c-a^2 c x^2}\right ) \int \sqrt {1-a^2 x^2} \sin ^{-1}(a x) \, dx}{32 \sqrt {1-a^2 x^2}}+\frac {\left (3 a c \sqrt {c-a^2 c x^2}\right ) \int x \left (1-a^2 x^2\right ) \, dx}{32 \sqrt {1-a^2 x^2}}+\frac {\left (9 a^2 c \sqrt {c-a^2 c x^2}\right ) \int \frac {x^2 \sin ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{8 \sqrt {1-a^2 x^2}}\\ &=-\frac {45}{64} c x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)-\frac {3}{32} c x \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)-\frac {9 a c x^2 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^2}{16 \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} \sin ^{-1}(a x)^2}{16 a}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^3+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^3+\frac {3 c \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^4}{32 a \sqrt {1-a^2 x^2}}-\frac {\left (9 c \sqrt {c-a^2 c x^2}\right ) \int \frac {\sin ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{64 \sqrt {1-a^2 x^2}}+\frac {\left (9 c \sqrt {c-a^2 c x^2}\right ) \int \frac {\sin ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{16 \sqrt {1-a^2 x^2}}+\frac {\left (3 a c \sqrt {c-a^2 c x^2}\right ) \int \left (x-a^2 x^3\right ) \, dx}{32 \sqrt {1-a^2 x^2}}+\frac {\left (9 a c \sqrt {c-a^2 c x^2}\right ) \int x \, dx}{64 \sqrt {1-a^2 x^2}}+\frac {\left (9 a c \sqrt {c-a^2 c x^2}\right ) \int x \, dx}{16 \sqrt {1-a^2 x^2}}\\ &=\frac {51 a c x^2 \sqrt {c-a^2 c x^2}}{128 \sqrt {1-a^2 x^2}}-\frac {3 a^3 c x^4 \sqrt {c-a^2 c x^2}}{128 \sqrt {1-a^2 x^2}}-\frac {45}{64} c x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)-\frac {3}{32} c x \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)+\frac {27 c \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^2}{128 a \sqrt {1-a^2 x^2}}-\frac {9 a c x^2 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^2}{16 \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} \sin ^{-1}(a x)^2}{16 a}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^3+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^3+\frac {3 c \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^4}{32 a \sqrt {1-a^2 x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.22, size = 138, normalized size = 0.38 \begin {gather*} \frac {c \sqrt {c-a^2 c x^2} \left (96 \text {ArcSin}(a x)^4+24 \text {ArcSin}(a x)^2 (16 \cos (2 \text {ArcSin}(a x))+\cos (4 \text {ArcSin}(a x)))-3 (64 \cos (2 \text {ArcSin}(a x))+\cos (4 \text {ArcSin}(a x)))+32 \text {ArcSin}(a x)^3 (8 \sin (2 \text {ArcSin}(a x))+\sin (4 \text {ArcSin}(a x)))-12 \text {ArcSin}(a x) (32 \sin (2 \text {ArcSin}(a x))+\sin (4 \text {ArcSin}(a x)))\right )}{1024 a \sqrt {1-a^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c*Sqrt[c - a^2*c*x^2]*(96*ArcSin[a*x]^4 + 24*ArcSin[a*x]^2*(16*Cos[2*ArcSin[a*x]] + Cos[4*ArcSin[a*x]]) - 3*(
64*Cos[2*ArcSin[a*x]] + Cos[4*ArcSin[a*x]]) + 32*ArcSin[a*x]^3*(8*Sin[2*ArcSin[a*x]] + Sin[4*ArcSin[a*x]]) - 1
2*ArcSin[a*x]*(32*Sin[2*ArcSin[a*x]] + Sin[4*ArcSin[a*x]])))/(1024*a*Sqrt[1 - a^2*x^2])

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Maple [C] Result contains complex when optimal does not.
time = 0.18, size = 474, normalized size = 1.30

method	result	size
default	\(-\frac {3 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \arcsin \left (a x \right )^{4} c}{32 a \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (-8 i \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}+8 a^{5} x^{5}+8 i \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-12 a^{3} x^{3}-i \sqrt {-a^{2} x^{2}+1}+4 a x \right ) \left (24 i \arcsin \left (a x \right )^{2}+32 \arcsin \left (a x \right )^{3}-3 i-12 \arcsin \left (a x \right )\right ) c}{2048 a \left (a^{2} x^{2}-1\right )}+\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (2 i \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+2 a^{3} x^{3}-i \sqrt {-a^{2} x^{2}+1}-2 a x \right ) \left (-6 i \arcsin \left (a x \right )^{2}+4 \arcsin \left (a x \right )^{3}+3 i-6 \arcsin \left (a x \right )\right ) c}{32 a \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (i a^{2} x^{2}-a x \sqrt {-a^{2} x^{2}+1}-i\right ) \left (408 i \arcsin \left (a x \right )^{2}+224 \arcsin \left (a x \right )^{3}-195 i-372 \arcsin \left (a x \right )\right ) \cos \left (3 \arcsin \left (a x \right )\right ) c}{2048 a \left (a^{2} x^{2}-1\right )}+\frac {9 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (i \sqrt {-a^{2} x^{2}+1}\, a x +a^{2} x^{2}-1\right ) \left (40 i \arcsin \left (a x \right )^{2}+32 \arcsin \left (a x \right )^{3}-21 i-44 \arcsin \left (a x \right )\right ) \sin \left (3 \arcsin \left (a x \right )\right ) c}{2048 a \left (a^{2} x^{2}-1\right )}\)	\(474\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/32*(-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-1/2048*(-c*(a^2*x^2-1))^(1/2)*(-
8*I*(-a^2*x^2+1)^(1/2)*a^4*x^4+8*a^5*x^5+8*I*(-a^2*x^2+1)^(1/2)*a^2*x^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/a/(a^2*x^2-1)+1/32*(-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/a/(a^2*x^2-1)-1/2048*(-c*(a^2*x^2-1))^(1/2)*(I*a^2*x^2-a*x*(-a^2*x^2+1)^(1/2)-I)*(408*I*arcsin(a*x)^2
+224*arcsin(a*x)^3-195*I-372*arcsin(a*x))*cos(3*arcsin(a*x))*c/a/(a^2*x^2-1)+9/2048*(-c*(a^2*x^2-1))^(1/2)*(I*
(-a^2*x^2+1)^(1/2)*a*x+a^2*x^2-1)*(40*I*arcsin(a*x)^2+32*arcsin(a*x)^3-21*I-44*arcsin(a*x))*sin(3*arcsin(a*x))
*c/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)^(3/2)*arcsin(a*x)^3,x, algorithm="maxima")

[Out]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \operatorname {asin}^{3}{\left (a x \right )}\, dx \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,x)

[Out]

Integral((-c*(a*x - 1)*(a*x + 1))**(3/2)*asin(a*x)**3, x)

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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,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 )}^{3/2} \,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)^(3/2),x)

[Out]

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

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