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

[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])

________________________________________________________________________________________

Rubi [A]  time = 0.709541, antiderivative size = 421, normalized size of antiderivative = 1., 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 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 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 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 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 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 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 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 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 12

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

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*}

Mathematica [A]  time = 0.312958, size = 297, normalized size = 0.71 \[ \frac{d \sqrt{d-c^2 d x^2} \left (3 b \sin ^{-1}(c x) \left (72 a^2-48 a b c x \sqrt{1-c^2 x^2} \left (8 c^4 x^4-14 c^2 x^2+3\right )+b^2 \left (64 c^6 x^6-168 c^4 x^4+72 c^2 x^2+7\right )\right )-72 a^2 b c x \sqrt{1-c^2 x^2} \left (8 c^4 x^4-14 c^2 x^2+3\right )+72 a^3+24 a b^2 c^2 x^2 \left (8 c^4 x^4-21 c^2 x^2+9\right )+72 b^2 \sin ^{-1}(c x)^2 \left (3 a+b c x \sqrt{1-c^2 x^2} \left (-8 c^4 x^4+14 c^2 x^2-3\right )\right )+b^3 c x \sqrt{1-c^2 x^2} \left (32 c^4 x^4-86 c^2 x^2-21\right )+72 b^3 \sin ^{-1}(c x)^3\right )}{3456 b c^3 \sqrt{1-c^2 x^2}} \]

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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Maple [B]  time = 0.48, size = 1075, normalized size = 2.6 \begin{align*} \text{result too large to display} \end{align*}

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]

-1/18*b^2*(-d*(c^2*x^2-1))^(1/2)*d*c^3/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*x^6-1/16*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^2+7/48*b^2*(-d*(c^2*x^2-1))^(1/2)*d*c/(c^2*x^2-1)*arcsin(c*x)*(-c
^2*x^2+1)^(1/2)*x^4-1/16*b^2*(-d*(c^2*x^2-1))^(1/2)*d/c/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*x^2-1/18*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+7/48*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+1/8*a*b*(-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-1/3*a*b*(-d*(c^2*x^2-1))^(1/2)*d*c^4/(c^2*x^2-1
)*arcsin(c*x)*x^7+11/12*a*b*(-d*(c^2*x^2-1))^(1/2)*d*c^2/(c^2*x^2-1)*arcsin(c*x)*x^5-17/24*a*b*(-d*(c^2*x^2-1)
)^(1/2)*d/(c^2*x^2-1)*arcsin(c*x)*x^3-7/1152*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)-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-1/6*b^2*(-d*(c^2*x^2-1))^(1/
2)*d*c^4/(c^2*x^2-1)*arcsin(c*x)^2*x^7+11/24*b^2*(-d*(c^2*x^2-1))^(1/2)*d*c^2/(c^2*x^2-1)*arcsin(c*x)^2*x^5+1/
16*b^2*(-d*(c^2*x^2-1))^(1/2)*d/c^2/(c^2*x^2-1)*arcsin(c*x)^2*x-7/1152*b^2*(-d*(c^2*x^2-1))^(1/2)*d/c^3/(c^2*x
^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)-17/48*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(c^2*x^2-1)*arcsin(c*x)^2*x^3+1/108*b^
2*(-d*(c^2*x^2-1))^(1/2)*d*c^4/(c^2*x^2-1)*x^7-59/1728*b^2*(-d*(c^2*x^2-1))^(1/2)*d*c^2/(c^2*x^2-1)*x^5+7/1152
*b^2*(-d*(c^2*x^2-1))^(1/2)*d/c^2/(c^2*x^2-1)*x-1/6*a^2*x*(-c^2*d*x^2+d)^(5/2)/c^2/d+1/24*a^2/c^2*x*(-c^2*d*x^
2+d)^(3/2)+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))+65/3456*b^2*(-d*(c^2*x^2-1))^(1/2)*d/(c^2*x^2-1)*x^3

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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(x^2*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))^2,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^{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 ) \end{align*}

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

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)