3.219 \(\int x^2 (d-c^2 d x^2)^{3/2} (a+b \sin ^{-1}(c x))^2 \, dx\)

Optimal. Leaf size=421 \[ \frac {b d x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 c \sqrt {1-c^2 x^2}}-\frac {d x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 c^2}-\frac {7 b c d x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 \sqrt {1-c^2 x^2}}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{8} d x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{48 b c^3 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^6 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 \sqrt {1-c^2 x^2}}-\frac {7 b^2 d x \sqrt {d-c^2 d x^2}}{1152 c^2}+\frac {1}{108} b^2 c^2 d x^5 \sqrt {d-c^2 d x^2}-\frac {43 b^2 d x^3 \sqrt {d-c^2 d x^2}}{1728}+\frac {7 b^2 d \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{1152 c^3 \sqrt {1-c^2 x^2}} \]

[Out]

1/6*x^3*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))^2-7/1152*b^2*d*x*(-c^2*d*x^2+d)^(1/2)/c^2-43/1728*b^2*d*x^3*(-c
^2*d*x^2+d)^(1/2)+1/108*b^2*c^2*d*x^5*(-c^2*d*x^2+d)^(1/2)-1/16*d*x*(a+b*arcsin(c*x))^2*(-c^2*d*x^2+d)^(1/2)/c
^2+1/8*d*x^3*(a+b*arcsin(c*x))^2*(-c^2*d*x^2+d)^(1/2)+7/1152*b^2*d*arcsin(c*x)*(-c^2*d*x^2+d)^(1/2)/c^3/(-c^2*
x^2+1)^(1/2)+1/16*b*d*x^2*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/c/(-c^2*x^2+1)^(1/2)-7/48*b*c*d*x^4*(a+b*arcs
in(c*x))*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+1/18*b*c^3*d*x^6*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/(-c^2
*x^2+1)^(1/2)+1/48*d*(a+b*arcsin(c*x))^3*(-c^2*d*x^2+d)^(1/2)/b/c^3/(-c^2*x^2+1)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.71, antiderivative size = 421, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 11, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {4699, 4697, 4707, 4641, 4627, 321, 216, 14, 4687, 12, 459} \[ \frac {b c^3 d x^6 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 \sqrt {1-c^2 x^2}}-\frac {7 b c d x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 \sqrt {1-c^2 x^2}}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{8} d x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {b d x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 c \sqrt {1-c^2 x^2}}-\frac {d x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 c^2}+\frac {d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{48 b c^3 \sqrt {1-c^2 x^2}}+\frac {1}{108} b^2 c^2 d x^5 \sqrt {d-c^2 d x^2}-\frac {43 b^2 d x^3 \sqrt {d-c^2 d x^2}}{1728}-\frac {7 b^2 d x \sqrt {d-c^2 d x^2}}{1152 c^2}+\frac {7 b^2 d \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{1152 c^3 \sqrt {1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-7*b^2*d*x*Sqrt[d - c^2*d*x^2])/(1152*c^2) - (43*b^2*d*x^3*Sqrt[d - c^2*d*x^2])/1728 + (b^2*c^2*d*x^5*Sqrt[d
- c^2*d*x^2])/108 + (7*b^2*d*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(1152*c^3*Sqrt[1 - c^2*x^2]) + (b*d*x^2*Sqrt[d -
c^2*d*x^2]*(a + b*ArcSin[c*x]))/(16*c*Sqrt[1 - c^2*x^2]) - (7*b*c*d*x^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x
]))/(48*Sqrt[1 - c^2*x^2]) + (b*c^3*d*x^6*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(18*Sqrt[1 - c^2*x^2]) - (d
*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(16*c^2) + (d*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/8 +
(x^3*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/6 + (d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/(48*b*c^3
*Sqrt[1 - c^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 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 216

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

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 459

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

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 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 4687

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u = I
ntHide[(f*x)^m*(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]] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]

Rule 4697

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

Rule 4699

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[
((f*x)^(m + 1)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n)/(f*(m + 2*p + 1)), x] + (Dist[(2*d*p)/(m + 2*p + 1), Int[(
f*x)^m*(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])/(
f*(m + 2*p + 1)*(1 - c^2*x^2)^FracPart[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, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] &&  !LtQ[m, -1]
&& (RationalQ[m] || EqQ[n, 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]

Rubi steps

\begin {align*} \int x^2 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{2} d \int x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx-\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int x^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right ) \, dx}{3 \sqrt {1-c^2 x^2}}\\ &=-\frac {b c d x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{12 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^6 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 \sqrt {1-c^2 x^2}}+\frac {1}{8} d x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{8 \sqrt {1-c^2 x^2}}-\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int x^3 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{4 \sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4 \left (3-2 c^2 x^2\right )}{12 \sqrt {1-c^2 x^2}} \, dx}{3 \sqrt {1-c^2 x^2}}\\ &=-\frac {7 b c d x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^6 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 \sqrt {1-c^2 x^2}}-\frac {d x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 c^2}+\frac {1}{8} d x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{16 c^2 \sqrt {1-c^2 x^2}}+\frac {\left (b d \sqrt {d-c^2 d x^2}\right ) \int x \left (a+b \sin ^{-1}(c x)\right ) \, dx}{8 c \sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4 \left (3-2 c^2 x^2\right )}{\sqrt {1-c^2 x^2}} \, dx}{36 \sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4}{\sqrt {1-c^2 x^2}} \, dx}{16 \sqrt {1-c^2 x^2}}\\ &=-\frac {1}{64} b^2 d x^3 \sqrt {d-c^2 d x^2}+\frac {1}{108} b^2 c^2 d x^5 \sqrt {d-c^2 d x^2}+\frac {b d x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 c \sqrt {1-c^2 x^2}}-\frac {7 b c d x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^6 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 \sqrt {1-c^2 x^2}}-\frac {d x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 c^2}+\frac {1}{8} d x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{48 b c^3 \sqrt {1-c^2 x^2}}+\frac {\left (3 b^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{64 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{16 \sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4}{\sqrt {1-c^2 x^2}} \, dx}{27 \sqrt {1-c^2 x^2}}\\ &=\frac {b^2 d x \sqrt {d-c^2 d x^2}}{128 c^2}-\frac {43 b^2 d x^3 \sqrt {d-c^2 d x^2}}{1728}+\frac {1}{108} b^2 c^2 d x^5 \sqrt {d-c^2 d x^2}+\frac {b d x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 c \sqrt {1-c^2 x^2}}-\frac {7 b c d x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^6 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 \sqrt {1-c^2 x^2}}-\frac {d x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 c^2}+\frac {1}{8} d x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{48 b c^3 \sqrt {1-c^2 x^2}}+\frac {\left (b^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{36 \sqrt {1-c^2 x^2}}+\frac {\left (3 b^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{128 c^2 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{32 c^2 \sqrt {1-c^2 x^2}}\\ &=-\frac {7 b^2 d x \sqrt {d-c^2 d x^2}}{1152 c^2}-\frac {43 b^2 d x^3 \sqrt {d-c^2 d x^2}}{1728}+\frac {1}{108} b^2 c^2 d x^5 \sqrt {d-c^2 d x^2}-\frac {b^2 d \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{128 c^3 \sqrt {1-c^2 x^2}}+\frac {b d x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 c \sqrt {1-c^2 x^2}}-\frac {7 b c d x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^6 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 \sqrt {1-c^2 x^2}}-\frac {d x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 c^2}+\frac {1}{8} d x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{48 b c^3 \sqrt {1-c^2 x^2}}+\frac {\left (b^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{72 c^2 \sqrt {1-c^2 x^2}}\\ &=-\frac {7 b^2 d x \sqrt {d-c^2 d x^2}}{1152 c^2}-\frac {43 b^2 d x^3 \sqrt {d-c^2 d x^2}}{1728}+\frac {1}{108} b^2 c^2 d x^5 \sqrt {d-c^2 d x^2}+\frac {7 b^2 d \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{1152 c^3 \sqrt {1-c^2 x^2}}+\frac {b d x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 c \sqrt {1-c^2 x^2}}-\frac {7 b c d x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^6 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 \sqrt {1-c^2 x^2}}-\frac {d x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 c^2}+\frac {1}{8} d x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{48 b c^3 \sqrt {1-c^2 x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.33, size = 297, normalized size = 0.71 \[ \frac {d \sqrt {d-c^2 d x^2} \left (72 a^3+3 b \sin ^{-1}(c x) \left (72 a^2-48 a b c x \sqrt {1-c^2 x^2} \left (8 c^4 x^4-14 c^2 x^2+3\right )+b^2 \left (64 c^6 x^6-168 c^4 x^4+72 c^2 x^2+7\right )\right )-72 a^2 b c x \sqrt {1-c^2 x^2} \left (8 c^4 x^4-14 c^2 x^2+3\right )+24 a b^2 c^2 x^2 \left (8 c^4 x^4-21 c^2 x^2+9\right )+72 b^2 \sin ^{-1}(c x)^2 \left (3 a+b c x \sqrt {1-c^2 x^2} \left (-8 c^4 x^4+14 c^2 x^2-3\right )\right )+b^3 c x \sqrt {1-c^2 x^2} \left (32 c^4 x^4-86 c^2 x^2-21\right )+72 b^3 \sin ^{-1}(c x)^3\right )}{3456 b c^3 \sqrt {1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d*Sqrt[d - c^2*d*x^2]*(72*a^3 + 24*a*b^2*c^2*x^2*(9 - 21*c^2*x^2 + 8*c^4*x^4) - 72*a^2*b*c*x*Sqrt[1 - c^2*x^2
]*(3 - 14*c^2*x^2 + 8*c^4*x^4) + b^3*c*x*Sqrt[1 - c^2*x^2]*(-21 - 86*c^2*x^2 + 32*c^4*x^4) + 3*b*(72*a^2 - 48*
a*b*c*x*Sqrt[1 - c^2*x^2]*(3 - 14*c^2*x^2 + 8*c^4*x^4) + b^2*(7 + 72*c^2*x^2 - 168*c^4*x^4 + 64*c^6*x^6))*ArcS
in[c*x] + 72*b^2*(3*a + b*c*x*Sqrt[1 - c^2*x^2]*(-3 + 14*c^2*x^2 - 8*c^4*x^4))*ArcSin[c*x]^2 + 72*b^3*ArcSin[c
*x]^3))/(3456*b*c^3*Sqrt[1 - c^2*x^2])

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fricas [F]  time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (a^{2} c^{2} d x^{4} - a^{2} d x^{2} + {\left (b^{2} c^{2} d x^{4} - b^{2} d x^{2}\right )} \arcsin \left (c x\right )^{2} + 2 \, {\left (a b c^{2} d x^{4} - a b d x^{2}\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(a^2*c^2*d*x^4 - a^2*d*x^2 + (b^2*c^2*d*x^4 - b^2*d*x^2)*arcsin(c*x)^2 + 2*(a*b*c^2*d*x^4 - a*b*d*x^
2)*arcsin(c*x))*sqrt(-c^2*d*x^2 + d), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.64, size = 4469, normalized size = 10.62 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-31/27648*b^2*(-d*(c^2*x^2-1))^(1/2)*sin(5*arcsin(c*x))*d/c^3/(c^2*x^2-1)-1/12*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(c
^2*x^2-1)*arcsin(c*x)^2*x^3+1/216*b^2*(-d*(c^2*x^2-1))^(1/2)*d*c^4/(c^2*x^2-1)*x^7-1/108*b^2*(-d*(c^2*x^2-1))^
(1/2)*d*c^2/(c^2*x^2-1)*x^5+1/144*b^2*(-d*(c^2*x^2-1))^(1/2)*d/c^2/(c^2*x^2-1)*x+3/1024*b^2*(-d*(c^2*x^2-1))^(
1/2)*sin(3*arcsin(c*x))*d/c^3/(c^2*x^2-1)+1/24*a^2/c^2*x*(-c^2*d*x^2+d)^(3/2)+1/12*I*b^2*(-d*(c^2*x^2-1))^(1/2
)*d*c^3/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)^2*x^6-1/8*I*b^2*(-d*(c^2*x^2-1))^(1/2)*d*c/(c^2*x^2-1)*(-c^
2*x^2+1)^(1/2)*arcsin(c*x)^2*x^4+1/16*I*b^2*(-d*(c^2*x^2-1))^(1/2)*d/c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c
*x)^2*x^2+1/192*I*b^2*(-d*(c^2*x^2-1))^(1/2)*cos(5*arcsin(c*x))*d/c/(c^2*x^2-1)*arcsin(c*x)^2*x^2-7/2304*I*b^2
*(-d*(c^2*x^2-1))^(1/2)*sin(5*arcsin(c*x))*d/c/(c^2*x^2-1)*arcsin(c*x)*x^2+31/27648*I*b^2*(-d*(c^2*x^2-1))^(1/
2)*sin(5*arcsin(c*x))*d/c^2/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x-1/64*I*b^2*(-d*(c^2*x^2-1))^(1/2)*cos(3*arcsin(c*
x))*d/c/(c^2*x^2-1)*arcsin(c*x)^2*x^2+3/256*I*b^2*(-d*(c^2*x^2-1))^(1/2)*sin(3*arcsin(c*x))*d/c/(c^2*x^2-1)*ar
csin(c*x)*x^2-3/1024*I*b^2*(-d*(c^2*x^2-1))^(1/2)*sin(3*arcsin(c*x))*d/c^2/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x-1/
48*a*b*(-d*(c^2*x^2-1))^(1/2)*sin(5*arcsin(c*x))*d/c/(c^2*x^2-1)*arcsin(c*x)*x^2-3/256*a*b*(-d*(c^2*x^2-1))^(1
/2)*sin(3*arcsin(c*x))*d/c^2/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x+7/2304*a*b*(-d*(c^2*x^2-1))^(1/2)*sin(5*arcsin(c
*x))*d/c^2/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x-1/48*I*a*b*(-d*(c^2*x^2-1))^(1/2)*d/c^3/(c^2*x^2-1)*(-c^2*x^2+1)^(
1/2)*arcsin(c*x)-1/96*I*a*b*(-d*(c^2*x^2-1))^(1/2)*cos(5*arcsin(c*x))*d/c^3/(c^2*x^2-1)*arcsin(c*x)-7/2304*I*a
*b*(-d*(c^2*x^2-1))^(1/2)*sin(5*arcsin(c*x))*d/c/(c^2*x^2-1)*x^2+1/32*I*a*b*(-d*(c^2*x^2-1))^(1/2)*cos(3*arcsi
n(c*x))*d/c^3/(c^2*x^2-1)*arcsin(c*x)+3/256*I*a*b*(-d*(c^2*x^2-1))^(1/2)*sin(3*arcsin(c*x))*d/c/(c^2*x^2-1)*x^
2-1/192*b^2*(-d*(c^2*x^2-1))^(1/2)*cos(5*arcsin(c*x))*d/c^2/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)^2*x+1/6
4*b^2*(-d*(c^2*x^2-1))^(1/2)*cos(3*arcsin(c*x))*d/c^2/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)^2*x-3/256*b^2
*(-d*(c^2*x^2-1))^(1/2)*sin(3*arcsin(c*x))*d/c^2/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x+7/2304*b^2*(-d*(
c^2*x^2-1))^(1/2)*sin(5*arcsin(c*x))*d/c^2/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x-1/32*I*a*b*(-d*(c^2*x^
2-1))^(1/2)*cos(3*arcsin(c*x))*d/c/(c^2*x^2-1)*arcsin(c*x)*x^2+1/256*I*a*b*(-d*(c^2*x^2-1))^(1/2)*cos(3*arcsin
(c*x))*d/c^2/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x-11/2304*I*b^2*(-d*(c^2*x^2-1))^(1/2)*cos(5*arcsin(c*x))*d/c^2/(c
^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x-1/96*I*b^2*(-d*(c^2*x^2-1))^(1/2)*sin(5*arcsin(c*x))*d/c^2/(c^2*x^2
-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)^2*x+1/256*I*b^2*(-d*(c^2*x^2-1))^(1/2)*cos(3*arcsin(c*x))*d/c^2/(c^2*x^2-1)
*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x-1/48*I*a*b*(-d*(c^2*x^2-1))^(1/2)*sin(5*arcsin(c*x))*d/c^2/(c^2*x^2-1)*(-c^2
*x^2+1)^(1/2)*arcsin(c*x)*x-1/6*a^2*x*(-c^2*d*x^2+d)^(5/2)/c^2/d+1/16*a^2/c^2*d*x*(-c^2*d*x^2+d)^(1/2)+1/16*a^
2/c^2*d^2/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))-1/432*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(c^2*x
^2-1)*x^3-1/12*b^2*(-d*(c^2*x^2-1))^(1/2)*d*c^4/(c^2*x^2-1)*arcsin(c*x)^2*x^7+1/6*b^2*(-d*(c^2*x^2-1))^(1/2)*d
*c^2/(c^2*x^2-1)*arcsin(c*x)^2*x^5+31/27648*b^2*(-d*(c^2*x^2-1))^(1/2)*sin(5*arcsin(c*x))*d/c/(c^2*x^2-1)*x^2+
1/96*b^2*(-d*(c^2*x^2-1))^(1/2)*sin(5*arcsin(c*x))*d/c^3/(c^2*x^2-1)*arcsin(c*x)^2-1/48*b^2*(-d*(c^2*x^2-1))^(
1/2)*(-c^2*x^2+1)^(1/2)/c^3/(c^2*x^2-1)*arcsin(c*x)^3*d-1/256*b^2*(-d*(c^2*x^2-1))^(1/2)*cos(3*arcsin(c*x))*d/
c^3/(c^2*x^2-1)*arcsin(c*x)-3/1024*b^2*(-d*(c^2*x^2-1))^(1/2)*sin(3*arcsin(c*x))*d/c/(c^2*x^2-1)*x^2+11/2304*b
^2*(-d*(c^2*x^2-1))^(1/2)*cos(5*arcsin(c*x))*d/c^3/(c^2*x^2-1)*arcsin(c*x)-1/144*b^2*(-d*(c^2*x^2-1))^(1/2)*d/
c^3/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)-7/144*I*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(c^2*x^2-1)*arcsin(c*x)*x^
3+7/1728*I*b^2*(-d*(c^2*x^2-1))^(1/2)*d/c^3/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)+23/27648*I*b^2*(-d*(c^2*x^2-1))^(1/
2)*cos(5*arcsin(c*x))*d/c^3/(c^2*x^2-1)-5/1024*I*b^2*(-d*(c^2*x^2-1))^(1/2)*cos(3*arcsin(c*x))*d/c^3/(c^2*x^2-
1)-1/144*a*b*(-d*(c^2*x^2-1))^(1/2)*d/c^3/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)+11/2304*a*b*(-d*(c^2*x^2-1))^(1/2)*co
s(5*arcsin(c*x))*d/c^3/(c^2*x^2-1)-7/144*I*a*b*(-d*(c^2*x^2-1))^(1/2)*d/(c^2*x^2-1)*x^3-1/6*a*b*(-d*(c^2*x^2-1
))^(1/2)*d/(c^2*x^2-1)*arcsin(c*x)*x^3-1/256*a*b*(-d*(c^2*x^2-1))^(1/2)*cos(3*arcsin(c*x))*d/c^3/(c^2*x^2-1)-1
/216*I*b^2*(-d*(c^2*x^2-1))^(1/2)*d*c^3/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^6+1/18*I*b^2*(-d*(c^2*x^2-1))^(1/2)*d
*c^2/(c^2*x^2-1)*arcsin(c*x)*x^5+1/144*I*b^2*(-d*(c^2*x^2-1))^(1/2)*d*c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^4-1/9
6*I*b^2*(-d*(c^2*x^2-1))^(1/2)*d/c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^2-1/96*I*b^2*(-d*(c^2*x^2-1))^(1/2)*d/c^3/
(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)^2-1/36*a*b*(-d*(c^2*x^2-1))^(1/2)*d*c^3/(c^2*x^2-1)*(-c^2*x^2+1)^(1
/2)*x^6+1/24*a*b*(-d*(c^2*x^2-1))^(1/2)*d*c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^4-3/256*I*b^2*(-d*(c^2*x^2-1))^(1
/2)*sin(3*arcsin(c*x))*d/c^3/(c^2*x^2-1)*arcsin(c*x)-1/36*I*b^2*(-d*(c^2*x^2-1))^(1/2)*d*c^4/(c^2*x^2-1)*arcsi
n(c*x)*x^7-23/27648*I*b^2*(-d*(c^2*x^2-1))^(1/2)*cos(5*arcsin(c*x))*d/c/(c^2*x^2-1)*x^2-1/192*I*b^2*(-d*(c^2*x
^2-1))^(1/2)*cos(5*arcsin(c*x))*d/c^3/(c^2*x^2-1)*arcsin(c*x)^2+7/2304*I*b^2*(-d*(c^2*x^2-1))^(1/2)*sin(5*arcs
in(c*x))*d/c^3/(c^2*x^2-1)*arcsin(c*x)+5/1024*I*b^2*(-d*(c^2*x^2-1))^(1/2)*cos(3*arcsin(c*x))*d/c/(c^2*x^2-1)*
x^2+1/48*I*b^2*(-d*(c^2*x^2-1))^(1/2)*d/c^2/(c^2*x^2-1)*arcsin(c*x)*x-1/16*a*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^
2+1)^(1/2)/c^3/(c^2*x^2-1)*arcsin(c*x)^2*d+7/2304*I*a*b*(-d*(c^2*x^2-1))^(1/2)*sin(5*arcsin(c*x))*d/c^3/(c^2*x
^2-1)-3/256*I*a*b*(-d*(c^2*x^2-1))^(1/2)*sin(3*arcsin(c*x))*d/c^3/(c^2*x^2-1)-1/36*b^2*(-d*(c^2*x^2-1))^(1/2)*
d*c^3/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x^6+1/24*b^2*(-d*(c^2*x^2-1))^(1/2)*d*c/(c^2*x^2-1)*(-c^2*x^2
+1)^(1/2)*arcsin(c*x)*x^4-11/2304*b^2*(-d*(c^2*x^2-1))^(1/2)*cos(5*arcsin(c*x))*d/c/(c^2*x^2-1)*arcsin(c*x)*x^
2+1/256*b^2*(-d*(c^2*x^2-1))^(1/2)*cos(3*arcsin(c*x))*d/c/(c^2*x^2-1)*arcsin(c*x)*x^2-5/1024*b^2*(-d*(c^2*x^2-
1))^(1/2)*cos(3*arcsin(c*x))*d/c^2/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x+23/27648*b^2*(-d*(c^2*x^2-1))^(1/2)*cos(5*
arcsin(c*x))*d/c^2/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x-1/96*b^2*(-d*(c^2*x^2-1))^(1/2)*sin(5*arcsin(c*x))*d/c/(c^
2*x^2-1)*arcsin(c*x)^2*x^2+1/64*I*b^2*(-d*(c^2*x^2-1))^(1/2)*cos(3*arcsin(c*x))*d/c^3/(c^2*x^2-1)*arcsin(c*x)^
2-1/36*I*a*b*(-d*(c^2*x^2-1))^(1/2)*d*c^4/(c^2*x^2-1)*x^7+1/18*I*a*b*(-d*(c^2*x^2-1))^(1/2)*d*c^2/(c^2*x^2-1)*
x^5+1/48*I*a*b*(-d*(c^2*x^2-1))^(1/2)*d/c^2/(c^2*x^2-1)*x-11/2304*a*b*(-d*(c^2*x^2-1))^(1/2)*cos(5*arcsin(c*x)
)*d/c/(c^2*x^2-1)*x^2+1/256*a*b*(-d*(c^2*x^2-1))^(1/2)*cos(3*arcsin(c*x))*d/c/(c^2*x^2-1)*x^2+1/48*a*b*(-d*(c^
2*x^2-1))^(1/2)*sin(5*arcsin(c*x))*d/c^3/(c^2*x^2-1)*arcsin(c*x)-1/6*a*b*(-d*(c^2*x^2-1))^(1/2)*d*c^4/(c^2*x^2
-1)*arcsin(c*x)*x^7+1/3*a*b*(-d*(c^2*x^2-1))^(1/2)*d*c^2/(c^2*x^2-1)*arcsin(c*x)*x^5-1/96*a*b*(-d*(c^2*x^2-1))
^(1/2)*cos(5*arcsin(c*x))*d/c^2/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x+1/32*a*b*(-d*(c^2*x^2-1))^(1/2)*c
os(3*arcsin(c*x))*d/c^2/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x+1/6*I*a*b*(-d*(c^2*x^2-1))^(1/2)*d*c^3/(c
^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x^6-1/4*I*a*b*(-d*(c^2*x^2-1))^(1/2)*d*c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/
2)*arcsin(c*x)*x^4+1/8*I*a*b*(-d*(c^2*x^2-1))^(1/2)*d/c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x^2+1/96*I*
a*b*(-d*(c^2*x^2-1))^(1/2)*cos(5*arcsin(c*x))*d/c/(c^2*x^2-1)*arcsin(c*x)*x^2-11/2304*I*a*b*(-d*(c^2*x^2-1))^(
1/2)*cos(5*arcsin(c*x))*d/c^2/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{48} \, a^{2} {\left (\frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x}{c^{2}} - \frac {8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x}{c^{2} d} + \frac {3 \, \sqrt {-c^{2} d x^{2} + d} d x}{c^{2}} + \frac {3 \, d^{\frac {3}{2}} \arcsin \left (c x\right )}{c^{3}}\right )} + \sqrt {d} \int -{\left ({\left (b^{2} c^{2} d x^{4} - b^{2} d x^{2}\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} + 2 \, {\left (a b c^{2} d x^{4} - a b d x^{2}\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )\right )} \sqrt {c x + 1} \sqrt {-c x + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/48*a^2*(2*(-c^2*d*x^2 + d)^(3/2)*x/c^2 - 8*(-c^2*d*x^2 + d)^(5/2)*x/(c^2*d) + 3*sqrt(-c^2*d*x^2 + d)*d*x/c^2
+ 3*d^(3/2)*arcsin(c*x)/c^3) + sqrt(d)*integrate(-((b^2*c^2*d*x^4 - b^2*d*x^2)*arctan2(c*x, sqrt(c*x + 1)*sqr
t(-c*x + 1))^2 + 2*(a*b*c^2*d*x^4 - a*b*d*x^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))*sqrt(c*x + 1)*sqrt(
-c*x + 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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