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

Optimal. Leaf size=273 \[ -\frac {6}{125} a^4 c^2 x^5 \sin ^{-1}(a x)+\frac {76}{225} a^2 c^2 x^3 \sin ^{-1}(a x)-\frac {6 c^2 \left (1-a^2 x^2\right )^{5/2}}{625 a}-\frac {272 c^2 \left (1-a^2 x^2\right )^{3/2}}{3375 a}-\frac {4144 c^2 \sqrt {1-a^2 x^2}}{1125 a}+\frac {1}{5} c^2 x \left (1-a^2 x^2\right )^2 \sin ^{-1}(a x)^3+\frac {4}{15} c^2 x \left (1-a^2 x^2\right ) \sin ^{-1}(a x)^3+\frac {3 c^2 \left (1-a^2 x^2\right )^{5/2} \sin ^{-1}(a x)^2}{25 a}+\frac {4 c^2 \left (1-a^2 x^2\right )^{3/2} \sin ^{-1}(a x)^2}{15 a}+\frac {8 c^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{5 a}+\frac {8}{15} c^2 x \sin ^{-1}(a x)^3-\frac {298}{75} c^2 x \sin ^{-1}(a x) \]

[Out]

-272/3375*c^2*(-a^2*x^2+1)^(3/2)/a-6/625*c^2*(-a^2*x^2+1)^(5/2)/a-298/75*c^2*x*arcsin(a*x)+76/225*a^2*c^2*x^3*
arcsin(a*x)-6/125*a^4*c^2*x^5*arcsin(a*x)+4/15*c^2*(-a^2*x^2+1)^(3/2)*arcsin(a*x)^2/a+3/25*c^2*(-a^2*x^2+1)^(5
/2)*arcsin(a*x)^2/a+8/15*c^2*x*arcsin(a*x)^3+4/15*c^2*x*(-a^2*x^2+1)*arcsin(a*x)^3+1/5*c^2*x*(-a^2*x^2+1)^2*ar
csin(a*x)^3-4144/1125*c^2*(-a^2*x^2+1)^(1/2)/a+8/5*c^2*arcsin(a*x)^2*(-a^2*x^2+1)^(1/2)/a

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Rubi [A]  time = 0.41, antiderivative size = 273, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 11, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {4649, 4619, 4677, 261, 4645, 444, 43, 194, 12, 1247, 698} \[ -\frac {6 c^2 \left (1-a^2 x^2\right )^{5/2}}{625 a}-\frac {272 c^2 \left (1-a^2 x^2\right )^{3/2}}{3375 a}-\frac {4144 c^2 \sqrt {1-a^2 x^2}}{1125 a}-\frac {6}{125} a^4 c^2 x^5 \sin ^{-1}(a x)+\frac {76}{225} a^2 c^2 x^3 \sin ^{-1}(a x)+\frac {1}{5} c^2 x \left (1-a^2 x^2\right )^2 \sin ^{-1}(a x)^3+\frac {4}{15} c^2 x \left (1-a^2 x^2\right ) \sin ^{-1}(a x)^3+\frac {3 c^2 \left (1-a^2 x^2\right )^{5/2} \sin ^{-1}(a x)^2}{25 a}+\frac {4 c^2 \left (1-a^2 x^2\right )^{3/2} \sin ^{-1}(a x)^2}{15 a}+\frac {8 c^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{5 a}+\frac {8}{15} c^2 x \sin ^{-1}(a x)^3-\frac {298}{75} c^2 x \sin ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4144*c^2*Sqrt[1 - a^2*x^2])/(1125*a) - (272*c^2*(1 - a^2*x^2)^(3/2))/(3375*a) - (6*c^2*(1 - a^2*x^2)^(5/2))/
(625*a) - (298*c^2*x*ArcSin[a*x])/75 + (76*a^2*c^2*x^3*ArcSin[a*x])/225 - (6*a^4*c^2*x^5*ArcSin[a*x])/125 + (8
*c^2*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2)/(5*a) + (4*c^2*(1 - a^2*x^2)^(3/2)*ArcSin[a*x]^2)/(15*a) + (3*c^2*(1 - a
^2*x^2)^(5/2)*ArcSin[a*x]^2)/(25*a) + (8*c^2*x*ArcSin[a*x]^3)/15 + (4*c^2*x*(1 - a^2*x^2)*ArcSin[a*x]^3)/15 +
(c^2*x*(1 - a^2*x^2)^2*ArcSin[a*x]^3)/5

Rule 12

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 194

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

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]

Rule 444

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

Rule 698

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rule 1247

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[
Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]

Rule 4619

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

Rule 4645

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(d + e*x^2)
^p, x]}, Dist[a + b*ArcSin[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]] /; F
reeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 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]

Rubi steps

\begin {align*} \int \left (c-a^2 c x^2\right )^2 \sin ^{-1}(a x)^3 \, dx &=\frac {1}{5} c^2 x \left (1-a^2 x^2\right )^2 \sin ^{-1}(a x)^3+\frac {1}{5} (4 c) \int \left (c-a^2 c x^2\right ) \sin ^{-1}(a x)^3 \, dx-\frac {1}{5} \left (3 a c^2\right ) \int x \left (1-a^2 x^2\right )^{3/2} \sin ^{-1}(a x)^2 \, dx\\ &=\frac {3 c^2 \left (1-a^2 x^2\right )^{5/2} \sin ^{-1}(a x)^2}{25 a}+\frac {4}{15} c^2 x \left (1-a^2 x^2\right ) \sin ^{-1}(a x)^3+\frac {1}{5} c^2 x \left (1-a^2 x^2\right )^2 \sin ^{-1}(a x)^3-\frac {1}{25} \left (6 c^2\right ) \int \left (1-a^2 x^2\right )^2 \sin ^{-1}(a x) \, dx+\frac {1}{15} \left (8 c^2\right ) \int \sin ^{-1}(a x)^3 \, dx-\frac {1}{5} \left (4 a c^2\right ) \int x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2 \, dx\\ &=-\frac {6}{25} c^2 x \sin ^{-1}(a x)+\frac {4}{25} a^2 c^2 x^3 \sin ^{-1}(a x)-\frac {6}{125} a^4 c^2 x^5 \sin ^{-1}(a x)+\frac {4 c^2 \left (1-a^2 x^2\right )^{3/2} \sin ^{-1}(a x)^2}{15 a}+\frac {3 c^2 \left (1-a^2 x^2\right )^{5/2} \sin ^{-1}(a x)^2}{25 a}+\frac {8}{15} c^2 x \sin ^{-1}(a x)^3+\frac {4}{15} c^2 x \left (1-a^2 x^2\right ) \sin ^{-1}(a x)^3+\frac {1}{5} c^2 x \left (1-a^2 x^2\right )^2 \sin ^{-1}(a x)^3-\frac {1}{15} \left (8 c^2\right ) \int \left (1-a^2 x^2\right ) \sin ^{-1}(a x) \, dx+\frac {1}{25} \left (6 a c^2\right ) \int \frac {x \left (15-10 a^2 x^2+3 a^4 x^4\right )}{15 \sqrt {1-a^2 x^2}} \, dx-\frac {1}{5} \left (8 a c^2\right ) \int \frac {x \sin ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {58}{75} c^2 x \sin ^{-1}(a x)+\frac {76}{225} a^2 c^2 x^3 \sin ^{-1}(a x)-\frac {6}{125} a^4 c^2 x^5 \sin ^{-1}(a x)+\frac {8 c^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{5 a}+\frac {4 c^2 \left (1-a^2 x^2\right )^{3/2} \sin ^{-1}(a x)^2}{15 a}+\frac {3 c^2 \left (1-a^2 x^2\right )^{5/2} \sin ^{-1}(a x)^2}{25 a}+\frac {8}{15} c^2 x \sin ^{-1}(a x)^3+\frac {4}{15} c^2 x \left (1-a^2 x^2\right ) \sin ^{-1}(a x)^3+\frac {1}{5} c^2 x \left (1-a^2 x^2\right )^2 \sin ^{-1}(a x)^3-\frac {1}{5} \left (16 c^2\right ) \int \sin ^{-1}(a x) \, dx+\frac {1}{125} \left (2 a c^2\right ) \int \frac {x \left (15-10 a^2 x^2+3 a^4 x^4\right )}{\sqrt {1-a^2 x^2}} \, dx+\frac {1}{15} \left (8 a c^2\right ) \int \frac {x \left (1-\frac {a^2 x^2}{3}\right )}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {298}{75} c^2 x \sin ^{-1}(a x)+\frac {76}{225} a^2 c^2 x^3 \sin ^{-1}(a x)-\frac {6}{125} a^4 c^2 x^5 \sin ^{-1}(a x)+\frac {8 c^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{5 a}+\frac {4 c^2 \left (1-a^2 x^2\right )^{3/2} \sin ^{-1}(a x)^2}{15 a}+\frac {3 c^2 \left (1-a^2 x^2\right )^{5/2} \sin ^{-1}(a x)^2}{25 a}+\frac {8}{15} c^2 x \sin ^{-1}(a x)^3+\frac {4}{15} c^2 x \left (1-a^2 x^2\right ) \sin ^{-1}(a x)^3+\frac {1}{5} c^2 x \left (1-a^2 x^2\right )^2 \sin ^{-1}(a x)^3+\frac {1}{125} \left (a c^2\right ) \operatorname {Subst}\left (\int \frac {15-10 a^2 x+3 a^4 x^2}{\sqrt {1-a^2 x}} \, dx,x,x^2\right )+\frac {1}{15} \left (4 a c^2\right ) \operatorname {Subst}\left (\int \frac {1-\frac {a^2 x}{3}}{\sqrt {1-a^2 x}} \, dx,x,x^2\right )+\frac {1}{5} \left (16 a c^2\right ) \int \frac {x}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {16 c^2 \sqrt {1-a^2 x^2}}{5 a}-\frac {298}{75} c^2 x \sin ^{-1}(a x)+\frac {76}{225} a^2 c^2 x^3 \sin ^{-1}(a x)-\frac {6}{125} a^4 c^2 x^5 \sin ^{-1}(a x)+\frac {8 c^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{5 a}+\frac {4 c^2 \left (1-a^2 x^2\right )^{3/2} \sin ^{-1}(a x)^2}{15 a}+\frac {3 c^2 \left (1-a^2 x^2\right )^{5/2} \sin ^{-1}(a x)^2}{25 a}+\frac {8}{15} c^2 x \sin ^{-1}(a x)^3+\frac {4}{15} c^2 x \left (1-a^2 x^2\right ) \sin ^{-1}(a x)^3+\frac {1}{5} c^2 x \left (1-a^2 x^2\right )^2 \sin ^{-1}(a x)^3+\frac {1}{125} \left (a c^2\right ) \operatorname {Subst}\left (\int \left (\frac {8}{\sqrt {1-a^2 x}}+4 \sqrt {1-a^2 x}+3 \left (1-a^2 x\right )^{3/2}\right ) \, dx,x,x^2\right )+\frac {1}{15} \left (4 a c^2\right ) \operatorname {Subst}\left (\int \left (\frac {2}{3 \sqrt {1-a^2 x}}+\frac {1}{3} \sqrt {1-a^2 x}\right ) \, dx,x,x^2\right )\\ &=-\frac {4144 c^2 \sqrt {1-a^2 x^2}}{1125 a}-\frac {272 c^2 \left (1-a^2 x^2\right )^{3/2}}{3375 a}-\frac {6 c^2 \left (1-a^2 x^2\right )^{5/2}}{625 a}-\frac {298}{75} c^2 x \sin ^{-1}(a x)+\frac {76}{225} a^2 c^2 x^3 \sin ^{-1}(a x)-\frac {6}{125} a^4 c^2 x^5 \sin ^{-1}(a x)+\frac {8 c^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{5 a}+\frac {4 c^2 \left (1-a^2 x^2\right )^{3/2} \sin ^{-1}(a x)^2}{15 a}+\frac {3 c^2 \left (1-a^2 x^2\right )^{5/2} \sin ^{-1}(a x)^2}{25 a}+\frac {8}{15} c^2 x \sin ^{-1}(a x)^3+\frac {4}{15} c^2 x \left (1-a^2 x^2\right ) \sin ^{-1}(a x)^3+\frac {1}{5} c^2 x \left (1-a^2 x^2\right )^2 \sin ^{-1}(a x)^3\\ \end {align*}

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Mathematica [A]  time = 0.25, size = 139, normalized size = 0.51 \[ \frac {c^2 \left (-2 \sqrt {1-a^2 x^2} \left (81 a^4 x^4-842 a^2 x^2+31841\right )+1125 a x \left (3 a^4 x^4-10 a^2 x^2+15\right ) \sin ^{-1}(a x)^3+225 \sqrt {1-a^2 x^2} \left (9 a^4 x^4-38 a^2 x^2+149\right ) \sin ^{-1}(a x)^2-30 a x \left (27 a^4 x^4-190 a^2 x^2+2235\right ) \sin ^{-1}(a x)\right )}{16875 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c^2*(-2*Sqrt[1 - a^2*x^2]*(31841 - 842*a^2*x^2 + 81*a^4*x^4) - 30*a*x*(2235 - 190*a^2*x^2 + 27*a^4*x^4)*ArcSi
n[a*x] + 225*Sqrt[1 - a^2*x^2]*(149 - 38*a^2*x^2 + 9*a^4*x^4)*ArcSin[a*x]^2 + 1125*a*x*(15 - 10*a^2*x^2 + 3*a^
4*x^4)*ArcSin[a*x]^3))/(16875*a)

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fricas [A]  time = 0.48, size = 158, normalized size = 0.58 \[ \frac {1125 \, {\left (3 \, a^{5} c^{2} x^{5} - 10 \, a^{3} c^{2} x^{3} + 15 \, a c^{2} x\right )} \arcsin \left (a x\right )^{3} - 30 \, {\left (27 \, a^{5} c^{2} x^{5} - 190 \, a^{3} c^{2} x^{3} + 2235 \, a c^{2} x\right )} \arcsin \left (a x\right ) - {\left (162 \, a^{4} c^{2} x^{4} - 1684 \, a^{2} c^{2} x^{2} - 225 \, {\left (9 \, a^{4} c^{2} x^{4} - 38 \, a^{2} c^{2} x^{2} + 149 \, c^{2}\right )} \arcsin \left (a x\right )^{2} + 63682 \, c^{2}\right )} \sqrt {-a^{2} x^{2} + 1}}{16875 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16875*(1125*(3*a^5*c^2*x^5 - 10*a^3*c^2*x^3 + 15*a*c^2*x)*arcsin(a*x)^3 - 30*(27*a^5*c^2*x^5 - 190*a^3*c^2*x
^3 + 2235*a*c^2*x)*arcsin(a*x) - (162*a^4*c^2*x^4 - 1684*a^2*c^2*x^2 - 225*(9*a^4*c^2*x^4 - 38*a^2*c^2*x^2 + 1
49*c^2)*arcsin(a*x)^2 + 63682*c^2)*sqrt(-a^2*x^2 + 1))/a

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giac [A]  time = 0.83, size = 267, normalized size = 0.98 \[ \frac {1}{5} \, {\left (a^{2} x^{2} - 1\right )}^{2} c^{2} x \arcsin \left (a x\right )^{3} - \frac {4}{15} \, {\left (a^{2} x^{2} - 1\right )} c^{2} x \arcsin \left (a x\right )^{3} - \frac {6}{125} \, {\left (a^{2} x^{2} - 1\right )}^{2} c^{2} x \arcsin \left (a x\right ) + \frac {8}{15} \, c^{2} x \arcsin \left (a x\right )^{3} + \frac {3 \, {\left (a^{2} x^{2} - 1\right )}^{2} \sqrt {-a^{2} x^{2} + 1} c^{2} \arcsin \left (a x\right )^{2}}{25 \, a} + \frac {272}{1125} \, {\left (a^{2} x^{2} - 1\right )} c^{2} x \arcsin \left (a x\right ) + \frac {4 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c^{2} \arcsin \left (a x\right )^{2}}{15 \, a} - \frac {4144}{1125} \, c^{2} x \arcsin \left (a x\right ) - \frac {6 \, {\left (a^{2} x^{2} - 1\right )}^{2} \sqrt {-a^{2} x^{2} + 1} c^{2}}{625 \, a} + \frac {8 \, \sqrt {-a^{2} x^{2} + 1} c^{2} \arcsin \left (a x\right )^{2}}{5 \, a} - \frac {272 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c^{2}}{3375 \, a} - \frac {4144 \, \sqrt {-a^{2} x^{2} + 1} c^{2}}{1125 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*(a^2*x^2 - 1)^2*c^2*x*arcsin(a*x)^3 - 4/15*(a^2*x^2 - 1)*c^2*x*arcsin(a*x)^3 - 6/125*(a^2*x^2 - 1)^2*c^2*x
*arcsin(a*x) + 8/15*c^2*x*arcsin(a*x)^3 + 3/25*(a^2*x^2 - 1)^2*sqrt(-a^2*x^2 + 1)*c^2*arcsin(a*x)^2/a + 272/11
25*(a^2*x^2 - 1)*c^2*x*arcsin(a*x) + 4/15*(-a^2*x^2 + 1)^(3/2)*c^2*arcsin(a*x)^2/a - 4144/1125*c^2*x*arcsin(a*
x) - 6/625*(a^2*x^2 - 1)^2*sqrt(-a^2*x^2 + 1)*c^2/a + 8/5*sqrt(-a^2*x^2 + 1)*c^2*arcsin(a*x)^2/a - 272/3375*(-
a^2*x^2 + 1)^(3/2)*c^2/a - 4144/1125*sqrt(-a^2*x^2 + 1)*c^2/a

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maple [A]  time = 0.09, size = 206, normalized size = 0.75 \[ \frac {c^{2} \left (3375 \arcsin \left (a x \right )^{3} a^{5} x^{5}+2025 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}-11250 a^{3} x^{3} \arcsin \left (a x \right )^{3}-810 \arcsin \left (a x \right ) a^{5} x^{5}-8550 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-162 a^{4} x^{4} \sqrt {-a^{2} x^{2}+1}+16875 a x \arcsin \left (a x \right )^{3}+5700 a^{3} x^{3} \arcsin \left (a x \right )+33525 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}+1684 a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}-67050 a x \arcsin \left (a x \right )-63682 \sqrt {-a^{2} x^{2}+1}\right )}{16875 a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16875/a*c^2*(3375*arcsin(a*x)^3*a^5*x^5+2025*arcsin(a*x)^2*(-a^2*x^2+1)^(1/2)*a^4*x^4-11250*a^3*x^3*arcsin(a
*x)^3-810*arcsin(a*x)*a^5*x^5-8550*arcsin(a*x)^2*(-a^2*x^2+1)^(1/2)*a^2*x^2-162*a^4*x^4*(-a^2*x^2+1)^(1/2)+168
75*a*x*arcsin(a*x)^3+5700*a^3*x^3*arcsin(a*x)+33525*arcsin(a*x)^2*(-a^2*x^2+1)^(1/2)+1684*a^2*x^2*(-a^2*x^2+1)
^(1/2)-67050*a*x*arcsin(a*x)-63682*(-a^2*x^2+1)^(1/2))

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maxima [A]  time = 0.51, size = 216, normalized size = 0.79 \[ \frac {1}{75} \, {\left (9 \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{2} x^{4} - 38 \, \sqrt {-a^{2} x^{2} + 1} c^{2} x^{2} + \frac {149 \, \sqrt {-a^{2} x^{2} + 1} c^{2}}{a^{2}}\right )} a \arcsin \left (a x\right )^{2} + \frac {1}{15} \, {\left (3 \, a^{4} c^{2} x^{5} - 10 \, a^{2} c^{2} x^{3} + 15 \, c^{2} x\right )} \arcsin \left (a x\right )^{3} - \frac {2}{16875} \, {\left (81 \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{2} x^{4} - 842 \, \sqrt {-a^{2} x^{2} + 1} c^{2} x^{2} + \frac {15 \, {\left (27 \, a^{4} c^{2} x^{5} - 190 \, a^{2} c^{2} x^{3} + 2235 \, c^{2} x\right )} \arcsin \left (a x\right )}{a} + \frac {31841 \, \sqrt {-a^{2} x^{2} + 1} c^{2}}{a^{2}}\right )} a \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/75*(9*sqrt(-a^2*x^2 + 1)*a^2*c^2*x^4 - 38*sqrt(-a^2*x^2 + 1)*c^2*x^2 + 149*sqrt(-a^2*x^2 + 1)*c^2/a^2)*a*arc
sin(a*x)^2 + 1/15*(3*a^4*c^2*x^5 - 10*a^2*c^2*x^3 + 15*c^2*x)*arcsin(a*x)^3 - 2/16875*(81*sqrt(-a^2*x^2 + 1)*a
^2*c^2*x^4 - 842*sqrt(-a^2*x^2 + 1)*c^2*x^2 + 15*(27*a^4*c^2*x^5 - 190*a^2*c^2*x^3 + 2235*c^2*x)*arcsin(a*x)/a
+ 31841*sqrt(-a^2*x^2 + 1)*c^2/a^2)*a

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(asin(a*x)^3*(c - a^2*c*x^2)^2,x)

[Out]

int(asin(a*x)^3*(c - a^2*c*x^2)^2, x)

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sympy [A]  time = 6.25, size = 262, normalized size = 0.96 \[ \begin {cases} \frac {a^{4} c^{2} x^{5} \operatorname {asin}^{3}{\left (a x \right )}}{5} - \frac {6 a^{4} c^{2} x^{5} \operatorname {asin}{\left (a x \right )}}{125} + \frac {3 a^{3} c^{2} x^{4} \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a x \right )}}{25} - \frac {6 a^{3} c^{2} x^{4} \sqrt {- a^{2} x^{2} + 1}}{625} - \frac {2 a^{2} c^{2} x^{3} \operatorname {asin}^{3}{\left (a x \right )}}{3} + \frac {76 a^{2} c^{2} x^{3} \operatorname {asin}{\left (a x \right )}}{225} - \frac {38 a c^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a x \right )}}{75} + \frac {1684 a c^{2} x^{2} \sqrt {- a^{2} x^{2} + 1}}{16875} + c^{2} x \operatorname {asin}^{3}{\left (a x \right )} - \frac {298 c^{2} x \operatorname {asin}{\left (a x \right )}}{75} + \frac {149 c^{2} \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a x \right )}}{75 a} - \frac {63682 c^{2} \sqrt {- a^{2} x^{2} + 1}}{16875 a} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**4*c**2*x**5*asin(a*x)**3/5 - 6*a**4*c**2*x**5*asin(a*x)/125 + 3*a**3*c**2*x**4*sqrt(-a**2*x**2 +
1)*asin(a*x)**2/25 - 6*a**3*c**2*x**4*sqrt(-a**2*x**2 + 1)/625 - 2*a**2*c**2*x**3*asin(a*x)**3/3 + 76*a**2*c*
*2*x**3*asin(a*x)/225 - 38*a*c**2*x**2*sqrt(-a**2*x**2 + 1)*asin(a*x)**2/75 + 1684*a*c**2*x**2*sqrt(-a**2*x**2
+ 1)/16875 + c**2*x*asin(a*x)**3 - 298*c**2*x*asin(a*x)/75 + 149*c**2*sqrt(-a**2*x**2 + 1)*asin(a*x)**2/(75*a
) - 63682*c**2*sqrt(-a**2*x**2 + 1)/(16875*a), Ne(a, 0)), (0, True))

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