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

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

[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])

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Rubi [A]  time = 0.322146, antiderivative size = 365, normalized size of antiderivative = 1., 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 = {4649, 4647, 4641, 4627, 4707, 30, 4677, 14} \[ -\frac{3 a^3 c x^4 \sqrt{c-a^2 c x^2}}{128 \sqrt{1-a^2 x^2}}+\frac{51 a c x^2 \sqrt{c-a^2 c x^2}}{128 \sqrt{1-a^2 x^2}}-\frac{9 a c x^2 \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^2}{16 \sqrt{1-a^2 x^2}}+\frac{1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^3+\frac{3}{8} c x \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^3-\frac{45}{64} c x \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)-\frac{3}{32} c x \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)+\frac{3 c \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^4}{32 a \sqrt{1-a^2 x^2}}+\frac{3 c \left (1-a^2 x^2\right )^{3/2} \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^2}{16 a}+\frac{27 c \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^2}{128 a \sqrt{1-a^2 x^2}} \]

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 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]]

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*}

Mathematica [A]  time = 0.313282, size = 138, normalized size = 0.38 \[ \frac{c \sqrt{c-a^2 c x^2} \left (96 \sin ^{-1}(a x)^4+32 \left (8 \sin \left (2 \sin ^{-1}(a x)\right )+\sin \left (4 \sin ^{-1}(a x)\right )\right ) \sin ^{-1}(a x)^3-12 \left (32 \sin \left (2 \sin ^{-1}(a x)\right )+\sin \left (4 \sin ^{-1}(a x)\right )\right ) \sin ^{-1}(a x)+24 \sin ^{-1}(a x)^2 \left (16 \cos \left (2 \sin ^{-1}(a x)\right )+\cos \left (4 \sin ^{-1}(a x)\right )\right )-3 \left (64 \cos \left (2 \sin ^{-1}(a x)\right )+\cos \left (4 \sin ^{-1}(a x)\right )\right )\right )}{1024 a \sqrt{1-a^2 x^2}} \]

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]  time = 0.161, size = 533, normalized size = 1.5 \begin{align*} -{\frac{3\, \left ( \arcsin \left ( ax \right ) \right ) ^{4}c}{32\,a \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{ \left ( 24\,i \left ( \arcsin \left ( ax \right ) \right ) ^{2}+32\, \left ( \arcsin \left ( ax \right ) \right ) ^{3}-3\,i-12\,\arcsin \left ( ax \right ) \right ) c}{2048\,a \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) } \left ( -8\,i\sqrt{-{a}^{2}{x}^{2}+1}{x}^{4}{a}^{4}+8\,{a}^{5}{x}^{5}+8\,i\sqrt{-{a}^{2}{x}^{2}+1}{x}^{2}{a}^{2}-12\,{a}^{3}{x}^{3}-i\sqrt{-{a}^{2}{x}^{2}+1}+4\,ax \right ) }+{\frac{ \left ( 6\,i \left ( \arcsin \left ( ax \right ) \right ) ^{2}+4\, \left ( \arcsin \left ( ax \right ) \right ) ^{3}-3\,i-6\,\arcsin \left ( ax \right ) \right ) c}{32\,a \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) } \left ( -2\,i\sqrt{-{a}^{2}{x}^{2}+1}{x}^{2}{a}^{2}+2\,{a}^{3}{x}^{3}+i\sqrt{-{a}^{2}{x}^{2}+1}-2\,ax \right ) }+{\frac{ \left ( -6\,i \left ( \arcsin \left ( ax \right ) \right ) ^{2}+4\, \left ( \arcsin \left ( ax \right ) \right ) ^{3}+3\,i-6\,\arcsin \left ( ax \right ) \right ) c}{32\,a \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) } \left ( 2\,i\sqrt{-{a}^{2}{x}^{2}+1}{x}^{2}{a}^{2}+2\,{a}^{3}{x}^{3}-i\sqrt{-{a}^{2}{x}^{2}+1}-2\,ax \right ) }-{\frac{ \left ( -24\,i \left ( \arcsin \left ( ax \right ) \right ) ^{2}+32\, \left ( \arcsin \left ( ax \right ) \right ) ^{3}+3\,i-12\,\arcsin \left ( ax \right ) \right ) c}{2048\,a \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) } \left ( 8\,i\sqrt{-{a}^{2}{x}^{2}+1}{x}^{4}{a}^{4}+8\,{a}^{5}{x}^{5}-8\,i\sqrt{-{a}^{2}{x}^{2}+1}{x}^{2}{a}^{2}-12\,{a}^{3}{x}^{3}+i\sqrt{-{a}^{2}{x}^{2}+1}+4\,ax \right ) } \end{align*}

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)

[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)*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/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)*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/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)*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/a/(a^2*x^2-1)-1/2048*(-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/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)^(3/2)*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^{2} c x^{2} - c\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)^(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(-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)**(3/2)*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{3}{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)^(3/2)*arcsin(a*x)^3,x, algorithm="giac")

[Out]

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