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

Optimal. Leaf size=503 \[ \frac{4 a b d x \sqrt{d-c^2 d x^2}}{35 c^3 \sqrt{1-c^2 x^2}}+\frac{2 b c^3 d x^7 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{49 \sqrt{1-c^2 x^2}}-\frac{16 b c d x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{175 \sqrt{1-c^2 x^2}}+\frac{1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{3}{35} d x^4 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{2 b d x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{105 c \sqrt{1-c^2 x^2}}-\frac{d x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{35 c^2}-\frac{2 d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{35 c^4}-\frac{2 b^2 d \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2}}{343 c^4}+\frac{38 b^2 d \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2}}{6125 c^4}+\frac{304 b^2 d \sqrt{d-c^2 d x^2}}{3675 c^4}+\frac{152 b^2 d \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{11025 c^4}+\frac{4 b^2 d x \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{35 c^3 \sqrt{1-c^2 x^2}} \]

[Out]

(304*b^2*d*Sqrt[d - c^2*d*x^2])/(3675*c^4) + (4*a*b*d*x*Sqrt[d - c^2*d*x^2])/(35*c^3*Sqrt[1 - c^2*x^2]) + (152
*b^2*d*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(11025*c^4) + (38*b^2*d*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2])/(6125*c
^4) - (2*b^2*d*(1 - c^2*x^2)^3*Sqrt[d - c^2*d*x^2])/(343*c^4) + (4*b^2*d*x*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(3
5*c^3*Sqrt[1 - c^2*x^2]) + (2*b*d*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(105*c*Sqrt[1 - c^2*x^2]) - (16
*b*c*d*x^5*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(175*Sqrt[1 - c^2*x^2]) + (2*b*c^3*d*x^7*Sqrt[d - c^2*d*x^
2]*(a + b*ArcSin[c*x]))/(49*Sqrt[1 - c^2*x^2]) - (2*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(35*c^4) - (d
*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(35*c^2) + (3*d*x^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)
/35 + (x^4*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/7

________________________________________________________________________________________

Rubi [A]  time = 0.779804, antiderivative size = 503, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 14, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.483, Rules used = {4699, 4697, 4707, 4677, 4619, 261, 4627, 266, 43, 14, 4687, 12, 446, 77} \[ \frac{4 a b d x \sqrt{d-c^2 d x^2}}{35 c^3 \sqrt{1-c^2 x^2}}+\frac{2 b c^3 d x^7 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{49 \sqrt{1-c^2 x^2}}-\frac{16 b c d x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{175 \sqrt{1-c^2 x^2}}+\frac{1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{3}{35} d x^4 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{2 b d x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{105 c \sqrt{1-c^2 x^2}}-\frac{d x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{35 c^2}-\frac{2 d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{35 c^4}-\frac{2 b^2 d \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2}}{343 c^4}+\frac{38 b^2 d \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2}}{6125 c^4}+\frac{304 b^2 d \sqrt{d-c^2 d x^2}}{3675 c^4}+\frac{152 b^2 d \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{11025 c^4}+\frac{4 b^2 d x \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{35 c^3 \sqrt{1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(304*b^2*d*Sqrt[d - c^2*d*x^2])/(3675*c^4) + (4*a*b*d*x*Sqrt[d - c^2*d*x^2])/(35*c^3*Sqrt[1 - c^2*x^2]) + (152
*b^2*d*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(11025*c^4) + (38*b^2*d*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2])/(6125*c
^4) - (2*b^2*d*(1 - c^2*x^2)^3*Sqrt[d - c^2*d*x^2])/(343*c^4) + (4*b^2*d*x*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(3
5*c^3*Sqrt[1 - c^2*x^2]) + (2*b*d*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(105*c*Sqrt[1 - c^2*x^2]) - (16
*b*c*d*x^5*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(175*Sqrt[1 - c^2*x^2]) + (2*b*c^3*d*x^7*Sqrt[d - c^2*d*x^
2]*(a + b*ArcSin[c*x]))/(49*Sqrt[1 - c^2*x^2]) - (2*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(35*c^4) - (d
*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(35*c^2) + (3*d*x^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)
/35 + (x^4*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/7

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 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 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 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 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 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

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 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 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(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] && IntegerQ[Simplify[(m + 1)/n]]

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int x^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\frac{1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{7} (3 d) \int x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx-\frac{\left (2 b c d \sqrt{d-c^2 d x^2}\right ) \int x^4 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right ) \, dx}{7 \sqrt{1-c^2 x^2}}\\ &=-\frac{2 b c d x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{35 \sqrt{1-c^2 x^2}}+\frac{2 b c^3 d x^7 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{49 \sqrt{1-c^2 x^2}}+\frac{3}{35} d x^4 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{\left (3 d \sqrt{d-c^2 d x^2}\right ) \int \frac{x^3 \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{1-c^2 x^2}} \, dx}{35 \sqrt{1-c^2 x^2}}-\frac{\left (6 b c d \sqrt{d-c^2 d x^2}\right ) \int x^4 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{35 \sqrt{1-c^2 x^2}}+\frac{\left (2 b^2 c^2 d \sqrt{d-c^2 d x^2}\right ) \int \frac{x^5 \left (7-5 c^2 x^2\right )}{35 \sqrt{1-c^2 x^2}} \, dx}{7 \sqrt{1-c^2 x^2}}\\ &=-\frac{16 b c d x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{175 \sqrt{1-c^2 x^2}}+\frac{2 b c^3 d x^7 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{49 \sqrt{1-c^2 x^2}}-\frac{d x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{35 c^2}+\frac{3}{35} d x^4 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{\left (2 d \sqrt{d-c^2 d x^2}\right ) \int \frac{x \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{1-c^2 x^2}} \, dx}{35 c^2 \sqrt{1-c^2 x^2}}+\frac{\left (2 b d \sqrt{d-c^2 d x^2}\right ) \int x^2 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{35 c \sqrt{1-c^2 x^2}}+\frac{\left (2 b^2 c^2 d \sqrt{d-c^2 d x^2}\right ) \int \frac{x^5 \left (7-5 c^2 x^2\right )}{\sqrt{1-c^2 x^2}} \, dx}{245 \sqrt{1-c^2 x^2}}+\frac{\left (6 b^2 c^2 d \sqrt{d-c^2 d x^2}\right ) \int \frac{x^5}{\sqrt{1-c^2 x^2}} \, dx}{175 \sqrt{1-c^2 x^2}}\\ &=\frac{2 b d x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{105 c \sqrt{1-c^2 x^2}}-\frac{16 b c d x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{175 \sqrt{1-c^2 x^2}}+\frac{2 b c^3 d x^7 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{49 \sqrt{1-c^2 x^2}}-\frac{2 d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{35 c^4}-\frac{d x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{35 c^2}+\frac{3}{35} d x^4 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{\left (2 b^2 d \sqrt{d-c^2 d x^2}\right ) \int \frac{x^3}{\sqrt{1-c^2 x^2}} \, dx}{105 \sqrt{1-c^2 x^2}}+\frac{\left (4 b d \sqrt{d-c^2 d x^2}\right ) \int \left (a+b \sin ^{-1}(c x)\right ) \, dx}{35 c^3 \sqrt{1-c^2 x^2}}+\frac{\left (b^2 c^2 d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{x^2 \left (7-5 c^2 x\right )}{\sqrt{1-c^2 x}} \, dx,x,x^2\right )}{245 \sqrt{1-c^2 x^2}}+\frac{\left (3 b^2 c^2 d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1-c^2 x}} \, dx,x,x^2\right )}{175 \sqrt{1-c^2 x^2}}\\ &=\frac{4 a b d x \sqrt{d-c^2 d x^2}}{35 c^3 \sqrt{1-c^2 x^2}}+\frac{2 b d x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{105 c \sqrt{1-c^2 x^2}}-\frac{16 b c d x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{175 \sqrt{1-c^2 x^2}}+\frac{2 b c^3 d x^7 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{49 \sqrt{1-c^2 x^2}}-\frac{2 d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{35 c^4}-\frac{d x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{35 c^2}+\frac{3}{35} d x^4 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{\left (b^2 d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{1-c^2 x}} \, dx,x,x^2\right )}{105 \sqrt{1-c^2 x^2}}+\frac{\left (4 b^2 d \sqrt{d-c^2 d x^2}\right ) \int \sin ^{-1}(c x) \, dx}{35 c^3 \sqrt{1-c^2 x^2}}+\frac{\left (b^2 c^2 d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{2}{c^4 \sqrt{1-c^2 x}}+\frac{\sqrt{1-c^2 x}}{c^4}-\frac{8 \left (1-c^2 x\right )^{3/2}}{c^4}+\frac{5 \left (1-c^2 x\right )^{5/2}}{c^4}\right ) \, dx,x,x^2\right )}{245 \sqrt{1-c^2 x^2}}+\frac{\left (3 b^2 c^2 d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{1}{c^4 \sqrt{1-c^2 x}}-\frac{2 \sqrt{1-c^2 x}}{c^4}+\frac{\left (1-c^2 x\right )^{3/2}}{c^4}\right ) \, dx,x,x^2\right )}{175 \sqrt{1-c^2 x^2}}\\ &=-\frac{62 b^2 d \sqrt{d-c^2 d x^2}}{1225 c^4}+\frac{4 a b d x \sqrt{d-c^2 d x^2}}{35 c^3 \sqrt{1-c^2 x^2}}+\frac{74 b^2 d \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{3675 c^4}+\frac{38 b^2 d \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2}}{6125 c^4}-\frac{2 b^2 d \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2}}{343 c^4}+\frac{4 b^2 d x \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{35 c^3 \sqrt{1-c^2 x^2}}+\frac{2 b d x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{105 c \sqrt{1-c^2 x^2}}-\frac{16 b c d x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{175 \sqrt{1-c^2 x^2}}+\frac{2 b c^3 d x^7 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{49 \sqrt{1-c^2 x^2}}-\frac{2 d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{35 c^4}-\frac{d x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{35 c^2}+\frac{3}{35} d x^4 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{\left (b^2 d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{1}{c^2 \sqrt{1-c^2 x}}-\frac{\sqrt{1-c^2 x}}{c^2}\right ) \, dx,x,x^2\right )}{105 \sqrt{1-c^2 x^2}}-\frac{\left (4 b^2 d \sqrt{d-c^2 d x^2}\right ) \int \frac{x}{\sqrt{1-c^2 x^2}} \, dx}{35 c^2 \sqrt{1-c^2 x^2}}\\ &=\frac{304 b^2 d \sqrt{d-c^2 d x^2}}{3675 c^4}+\frac{4 a b d x \sqrt{d-c^2 d x^2}}{35 c^3 \sqrt{1-c^2 x^2}}+\frac{152 b^2 d \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{11025 c^4}+\frac{38 b^2 d \left (1-c^2 x^2\right )^2 \sqrt{d-c^2 d x^2}}{6125 c^4}-\frac{2 b^2 d \left (1-c^2 x^2\right )^3 \sqrt{d-c^2 d x^2}}{343 c^4}+\frac{4 b^2 d x \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{35 c^3 \sqrt{1-c^2 x^2}}+\frac{2 b d x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{105 c \sqrt{1-c^2 x^2}}-\frac{16 b c d x^5 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{175 \sqrt{1-c^2 x^2}}+\frac{2 b c^3 d x^7 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{49 \sqrt{1-c^2 x^2}}-\frac{2 d \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{35 c^4}-\frac{d x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{35 c^2}+\frac{3}{35} d x^4 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2\\ \end{align*}

Mathematica [A]  time = 0.304257, size = 244, normalized size = 0.49 \[ \frac{d \sqrt{d-c^2 d x^2} \left (-11025 a^2 \left (5 c^2 x^2+2\right ) \left (1-c^2 x^2\right )^{5/2}+210 a b c x \left (75 c^6 x^6-168 c^4 x^4+35 c^2 x^2+210\right )+210 b \sin ^{-1}(c x) \left (b c x \left (75 c^6 x^6-168 c^4 x^4+35 c^2 x^2+210\right )-105 a \left (1-c^2 x^2\right )^{5/2} \left (5 c^2 x^2+2\right )\right )+2 b^2 \left (1125 c^6 x^6-2178 c^4 x^4-1679 c^2 x^2+18692\right ) \sqrt{1-c^2 x^2}-11025 b^2 \left (5 c^2 x^2+2\right ) \left (1-c^2 x^2\right )^{5/2} \sin ^{-1}(c x)^2\right )}{385875 c^4 \sqrt{1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d*Sqrt[d - c^2*d*x^2]*(-11025*a^2*(1 - c^2*x^2)^(5/2)*(2 + 5*c^2*x^2) + 210*a*b*c*x*(210 + 35*c^2*x^2 - 168*c
^4*x^4 + 75*c^6*x^6) + 2*b^2*Sqrt[1 - c^2*x^2]*(18692 - 1679*c^2*x^2 - 2178*c^4*x^4 + 1125*c^6*x^6) + 210*b*(-
105*a*(1 - c^2*x^2)^(5/2)*(2 + 5*c^2*x^2) + b*c*x*(210 + 35*c^2*x^2 - 168*c^4*x^4 + 75*c^6*x^6))*ArcSin[c*x] -
11025*b^2*(1 - c^2*x^2)^(5/2)*(2 + 5*c^2*x^2)*ArcSin[c*x]^2))/(385875*c^4*Sqrt[1 - c^2*x^2])

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Maple [C]  time = 0.497, size = 1882, normalized size = 3.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^2*(-1/7*x^2*(-c^2*d*x^2+d)^(5/2)/c^2/d-2/35/d/c^4*(-c^2*d*x^2+d)^(5/2))+b^2*(-1/43904*(-d*(c^2*x^2-1))^(1/2)
*(64*c^8*x^8-144*c^6*x^6-64*I*(-c^2*x^2+1)^(1/2)*x^7*c^7+104*c^4*x^4+112*I*(-c^2*x^2+1)^(1/2)*x^5*c^5-25*c^2*x
^2-56*I*(-c^2*x^2+1)^(1/2)*x^3*c^3+7*I*(-c^2*x^2+1)^(1/2)*x*c+1)*(14*I*arcsin(c*x)+49*arcsin(c*x)^2-2)*d/c^4/(
c^2*x^2-1)+1/16000*(-d*(c^2*x^2-1))^(1/2)*(16*c^6*x^6-28*c^4*x^4-16*I*(-c^2*x^2+1)^(1/2)*x^5*c^5+13*c^2*x^2+20
*I*(-c^2*x^2+1)^(1/2)*x^3*c^3-5*I*(-c^2*x^2+1)^(1/2)*x*c-1)*(10*I*arcsin(c*x)+25*arcsin(c*x)^2-2)*d/c^4/(c^2*x
^2-1)+1/1152*(-d*(c^2*x^2-1))^(1/2)*(4*c^4*x^4-5*c^2*x^2-4*I*(-c^2*x^2+1)^(1/2)*x^3*c^3+3*I*(-c^2*x^2+1)^(1/2)
*x*c+1)*(6*I*arcsin(c*x)+9*arcsin(c*x)^2-2)*d/c^4/(c^2*x^2-1)-3/128*(-d*(c^2*x^2-1))^(1/2)*(c^2*x^2-I*(-c^2*x^
2+1)^(1/2)*x*c-1)*(arcsin(c*x)^2-2+2*I*arcsin(c*x))*d/c^4/(c^2*x^2-1)-3/128*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^
2+1)^(1/2)*x*c+c^2*x^2-1)*(arcsin(c*x)^2-2-2*I*arcsin(c*x))*d/c^4/(c^2*x^2-1)+1/1152*(-d*(c^2*x^2-1))^(1/2)*(4
*I*(-c^2*x^2+1)^(1/2)*x^3*c^3+4*c^4*x^4-3*I*(-c^2*x^2+1)^(1/2)*x*c-5*c^2*x^2+1)*(-6*I*arcsin(c*x)+9*arcsin(c*x
)^2-2)*d/c^4/(c^2*x^2-1)+1/16000*(-d*(c^2*x^2-1))^(1/2)*(16*I*(-c^2*x^2+1)^(1/2)*x^5*c^5+16*c^6*x^6-20*I*(-c^2
*x^2+1)^(1/2)*x^3*c^3-28*c^4*x^4+5*I*(-c^2*x^2+1)^(1/2)*x*c+13*c^2*x^2-1)*(-10*I*arcsin(c*x)+25*arcsin(c*x)^2-
2)*d/c^4/(c^2*x^2-1)-1/43904*(-d*(c^2*x^2-1))^(1/2)*(64*I*(-c^2*x^2+1)^(1/2)*x^7*c^7+64*c^8*x^8-112*I*(-c^2*x^
2+1)^(1/2)*x^5*c^5-144*c^6*x^6+56*I*(-c^2*x^2+1)^(1/2)*x^3*c^3+104*c^4*x^4-7*I*(-c^2*x^2+1)^(1/2)*x*c-25*c^2*x
^2+1)*(-14*I*arcsin(c*x)+49*arcsin(c*x)^2-2)*d/c^4/(c^2*x^2-1))+2*a*b*(-1/6272*(-d*(c^2*x^2-1))^(1/2)*(64*c^8*
x^8-144*c^6*x^6-64*I*(-c^2*x^2+1)^(1/2)*x^7*c^7+104*c^4*x^4+112*I*(-c^2*x^2+1)^(1/2)*x^5*c^5-25*c^2*x^2-56*I*(
-c^2*x^2+1)^(1/2)*x^3*c^3+7*I*(-c^2*x^2+1)^(1/2)*x*c+1)*(I+7*arcsin(c*x))*d/c^4/(c^2*x^2-1)+1/3200*(-d*(c^2*x^
2-1))^(1/2)*(16*c^6*x^6-28*c^4*x^4-16*I*(-c^2*x^2+1)^(1/2)*x^5*c^5+13*c^2*x^2+20*I*(-c^2*x^2+1)^(1/2)*x^3*c^3-
5*I*(-c^2*x^2+1)^(1/2)*x*c-1)*(I+5*arcsin(c*x))*d/c^4/(c^2*x^2-1)+1/384*(-d*(c^2*x^2-1))^(1/2)*(4*c^4*x^4-5*c^
2*x^2-4*I*(-c^2*x^2+1)^(1/2)*x^3*c^3+3*I*(-c^2*x^2+1)^(1/2)*x*c+1)*(I+3*arcsin(c*x))*d/c^4/(c^2*x^2-1)-3/128*(
-d*(c^2*x^2-1))^(1/2)*(c^2*x^2-I*(-c^2*x^2+1)^(1/2)*x*c-1)*(arcsin(c*x)+I)*d/c^4/(c^2*x^2-1)-3/128*(-d*(c^2*x^
2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*(arcsin(c*x)-I)*d/c^4/(c^2*x^2-1)+1/384*(-d*(c^2*x^2-1))^(1/2
)*(4*I*(-c^2*x^2+1)^(1/2)*x^3*c^3+4*c^4*x^4-3*I*(-c^2*x^2+1)^(1/2)*x*c-5*c^2*x^2+1)*(-I+3*arcsin(c*x))*d/c^4/(
c^2*x^2-1)+1/3200*(-d*(c^2*x^2-1))^(1/2)*(16*I*(-c^2*x^2+1)^(1/2)*x^5*c^5+16*c^6*x^6-20*I*(-c^2*x^2+1)^(1/2)*x
^3*c^3-28*c^4*x^4+5*I*(-c^2*x^2+1)^(1/2)*x*c+13*c^2*x^2-1)*(-I+5*arcsin(c*x))*d/c^4/(c^2*x^2-1)-1/6272*(-d*(c^
2*x^2-1))^(1/2)*(64*I*(-c^2*x^2+1)^(1/2)*x^7*c^7+64*c^8*x^8-112*I*(-c^2*x^2+1)^(1/2)*x^5*c^5-144*c^6*x^6+56*I*
(-c^2*x^2+1)^(1/2)*x^3*c^3+104*c^4*x^4-7*I*(-c^2*x^2+1)^(1/2)*x*c-25*c^2*x^2+1)*(-I+7*arcsin(c*x))*d/c^4/(c^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(x^3*(-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 [A]  time = 2.7937, size = 853, normalized size = 1.7 \begin{align*} -\frac{210 \,{\left (75 \, a b c^{7} d x^{7} - 168 \, a b c^{5} d x^{5} + 35 \, a b c^{3} d x^{3} + 210 \, a b c d x +{\left (75 \, b^{2} c^{7} d x^{7} - 168 \, b^{2} c^{5} d x^{5} + 35 \, b^{2} c^{3} d x^{3} + 210 \, b^{2} c d x\right )} \arcsin \left (c x\right )\right )} \sqrt{-c^{2} d x^{2} + d} \sqrt{-c^{2} x^{2} + 1} +{\left (1125 \,{\left (49 \, a^{2} - 2 \, b^{2}\right )} c^{8} d x^{8} - 9 \,{\left (15925 \, a^{2} - 734 \, b^{2}\right )} c^{6} d x^{6} +{\left (99225 \, a^{2} - 998 \, b^{2}\right )} c^{4} d x^{4} +{\left (11025 \, a^{2} - 40742 \, b^{2}\right )} c^{2} d x^{2} + 11025 \,{\left (5 \, b^{2} c^{8} d x^{8} - 13 \, b^{2} c^{6} d x^{6} + 9 \, b^{2} c^{4} d x^{4} + b^{2} c^{2} d x^{2} - 2 \, b^{2} d\right )} \arcsin \left (c x\right )^{2} - 2 \,{\left (11025 \, a^{2} - 18692 \, b^{2}\right )} d + 22050 \,{\left (5 \, a b c^{8} d x^{8} - 13 \, a b c^{6} d x^{6} + 9 \, a b c^{4} d x^{4} + a b c^{2} d x^{2} - 2 \, a b d\right )} \arcsin \left (c x\right )\right )} \sqrt{-c^{2} d x^{2} + d}}{385875 \,{\left (c^{6} x^{2} - c^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/385875*(210*(75*a*b*c^7*d*x^7 - 168*a*b*c^5*d*x^5 + 35*a*b*c^3*d*x^3 + 210*a*b*c*d*x + (75*b^2*c^7*d*x^7 -
168*b^2*c^5*d*x^5 + 35*b^2*c^3*d*x^3 + 210*b^2*c*d*x)*arcsin(c*x))*sqrt(-c^2*d*x^2 + d)*sqrt(-c^2*x^2 + 1) + (
1125*(49*a^2 - 2*b^2)*c^8*d*x^8 - 9*(15925*a^2 - 734*b^2)*c^6*d*x^6 + (99225*a^2 - 998*b^2)*c^4*d*x^4 + (11025
*a^2 - 40742*b^2)*c^2*d*x^2 + 11025*(5*b^2*c^8*d*x^8 - 13*b^2*c^6*d*x^6 + 9*b^2*c^4*d*x^4 + b^2*c^2*d*x^2 - 2*
b^2*d)*arcsin(c*x)^2 - 2*(11025*a^2 - 18692*b^2)*d + 22050*(5*a*b*c^8*d*x^8 - 13*a*b*c^6*d*x^6 + 9*a*b*c^4*d*x
^4 + a*b*c^2*d*x^2 - 2*a*b*d)*arcsin(c*x))*sqrt(-c^2*d*x^2 + d))/(c^6*x^2 - c^4)

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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**3*(-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^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(-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^3, x)