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

Optimal. Leaf size=391 \[ \frac{1}{9} d^3 x^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{2}{21} d^3 x^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{8}{105} d^3 x^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{32 b d^3 x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{945 c}-\frac{2 b d^3 \left (1-c^2 x^2\right )^{9/2} \left (a+b \sin ^{-1}(c x)\right )}{81 c^3}+\frac{2 b d^3 \left (1-c^2 x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{441 c^3}+\frac{4 b d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{525 c^3}+\frac{16 b d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{315 c^3}+\frac{64 b d^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{945 c^3}+\frac{16}{315} d^3 x^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{2}{729} b^2 c^6 d^3 x^9-\frac{374 b^2 c^4 d^3 x^7}{27783}+\frac{4198 b^2 c^2 d^3 x^5}{165375}-\frac{10516 b^2 d^3 x}{99225 c^2}-\frac{5258 b^2 d^3 x^3}{297675} \]

[Out]

(-10516*b^2*d^3*x)/(99225*c^2) - (5258*b^2*d^3*x^3)/297675 + (4198*b^2*c^2*d^3*x^5)/165375 - (374*b^2*c^4*d^3*
x^7)/27783 + (2*b^2*c^6*d^3*x^9)/729 + (64*b*d^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(945*c^3) + (32*b*d^3*
x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(945*c) + (16*b*d^3*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(315*c
^3) + (4*b*d^3*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(525*c^3) + (2*b*d^3*(1 - c^2*x^2)^(7/2)*(a + b*ArcSin
[c*x]))/(441*c^3) - (2*b*d^3*(1 - c^2*x^2)^(9/2)*(a + b*ArcSin[c*x]))/(81*c^3) + (16*d^3*x^3*(a + b*ArcSin[c*x
])^2)/315 + (8*d^3*x^3*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/105 + (2*d^3*x^3*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x
])^2)/21 + (d^3*x^3*(1 - c^2*x^2)^3*(a + b*ArcSin[c*x])^2)/9

________________________________________________________________________________________

Rubi [A]  time = 0.822623, antiderivative size = 391, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 11, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.407, Rules used = {4699, 4627, 4707, 4677, 8, 30, 266, 43, 4689, 12, 373} \[ \frac{1}{9} d^3 x^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{2}{21} d^3 x^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{8}{105} d^3 x^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{32 b d^3 x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{945 c}-\frac{2 b d^3 \left (1-c^2 x^2\right )^{9/2} \left (a+b \sin ^{-1}(c x)\right )}{81 c^3}+\frac{2 b d^3 \left (1-c^2 x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{441 c^3}+\frac{4 b d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{525 c^3}+\frac{16 b d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{315 c^3}+\frac{64 b d^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{945 c^3}+\frac{16}{315} d^3 x^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{2}{729} b^2 c^6 d^3 x^9-\frac{374 b^2 c^4 d^3 x^7}{27783}+\frac{4198 b^2 c^2 d^3 x^5}{165375}-\frac{10516 b^2 d^3 x}{99225 c^2}-\frac{5258 b^2 d^3 x^3}{297675} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-10516*b^2*d^3*x)/(99225*c^2) - (5258*b^2*d^3*x^3)/297675 + (4198*b^2*c^2*d^3*x^5)/165375 - (374*b^2*c^4*d^3*
x^7)/27783 + (2*b^2*c^6*d^3*x^9)/729 + (64*b*d^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(945*c^3) + (32*b*d^3*
x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(945*c) + (16*b*d^3*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(315*c
^3) + (4*b*d^3*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(525*c^3) + (2*b*d^3*(1 - c^2*x^2)^(7/2)*(a + b*ArcSin
[c*x]))/(441*c^3) - (2*b*d^3*(1 - c^2*x^2)^(9/2)*(a + b*ArcSin[c*x]))/(81*c^3) + (16*d^3*x^3*(a + b*ArcSin[c*x
])^2)/315 + (8*d^3*x^3*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/105 + (2*d^3*x^3*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x
])^2)/21 + (d^3*x^3*(1 - c^2*x^2)^3*(a + b*ArcSin[c*x])^2)/9

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 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 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 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && 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 4689

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> With[{u = IntHide[x^
m*(1 - c^2*x^2)^p, x]}, Dist[d^p*(a + b*ArcSin[c*x]), u, x] - Dist[b*c*d^p, Int[SimplifyIntegrand[u/Sqrt[1 - c
^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2] && (IGtQ[(m + 1)/2
, 0] || ILtQ[(m + 2*p + 3)/2, 0]) && NeQ[p, -2^(-1)] && GtQ[d, 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 373

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

Rubi steps

\begin{align*} \int x^2 \left (d-c^2 d x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\frac{1}{9} d^3 x^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{3} (2 d) \int x^2 \left (d-c^2 d x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx-\frac{1}{9} \left (2 b c d^3\right ) \int x^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx\\ &=\frac{2 b d^3 \left (1-c^2 x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{63 c^3}-\frac{2 b d^3 \left (1-c^2 x^2\right )^{9/2} \left (a+b \sin ^{-1}(c x)\right )}{81 c^3}+\frac{2}{21} d^3 x^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{9} d^3 x^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{21} \left (8 d^2\right ) \int x^2 \left (d-c^2 d x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2 \, dx-\frac{1}{21} \left (4 b c d^3\right ) \int x^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx+\frac{1}{9} \left (2 b^2 c^2 d^3\right ) \int \frac{\left (-2-7 c^2 x^2\right ) \left (1-c^2 x^2\right )^3}{63 c^4} \, dx\\ &=\frac{4 b d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{105 c^3}+\frac{2 b d^3 \left (1-c^2 x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{441 c^3}-\frac{2 b d^3 \left (1-c^2 x^2\right )^{9/2} \left (a+b \sin ^{-1}(c x)\right )}{81 c^3}+\frac{8}{105} d^3 x^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{2}{21} d^3 x^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{9} d^3 x^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{105} \left (16 d^3\right ) \int x^2 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx+\frac{\left (2 b^2 d^3\right ) \int \left (-2-7 c^2 x^2\right ) \left (1-c^2 x^2\right )^3 \, dx}{567 c^2}-\frac{1}{105} \left (16 b c d^3\right ) \int x^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx+\frac{1}{21} \left (4 b^2 c^2 d^3\right ) \int \frac{\left (-2-5 c^2 x^2\right ) \left (1-c^2 x^2\right )^2}{35 c^4} \, dx\\ &=\frac{16 b d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{315 c^3}+\frac{4 b d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{525 c^3}+\frac{2 b d^3 \left (1-c^2 x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{441 c^3}-\frac{2 b d^3 \left (1-c^2 x^2\right )^{9/2} \left (a+b \sin ^{-1}(c x)\right )}{81 c^3}+\frac{16}{315} d^3 x^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{8}{105} d^3 x^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{2}{21} d^3 x^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{9} d^3 x^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{\left (2 b^2 d^3\right ) \int \left (-2-c^2 x^2+15 c^4 x^4-19 c^6 x^6+7 c^8 x^8\right ) \, dx}{567 c^2}+\frac{\left (4 b^2 d^3\right ) \int \left (-2-5 c^2 x^2\right ) \left (1-c^2 x^2\right )^2 \, dx}{735 c^2}-\frac{1}{315} \left (32 b c d^3\right ) \int \frac{x^3 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx+\frac{1}{105} \left (16 b^2 c^2 d^3\right ) \int \frac{-2-c^2 x^2+3 c^4 x^4}{15 c^4} \, dx\\ &=-\frac{4 b^2 d^3 x}{567 c^2}-\frac{2 b^2 d^3 x^3}{1701}+\frac{2}{189} b^2 c^2 d^3 x^5-\frac{38 b^2 c^4 d^3 x^7}{3969}+\frac{2}{729} b^2 c^6 d^3 x^9+\frac{32 b d^3 x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{945 c}+\frac{16 b d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{315 c^3}+\frac{4 b d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{525 c^3}+\frac{2 b d^3 \left (1-c^2 x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{441 c^3}-\frac{2 b d^3 \left (1-c^2 x^2\right )^{9/2} \left (a+b \sin ^{-1}(c x)\right )}{81 c^3}+\frac{16}{315} d^3 x^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{8}{105} d^3 x^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{2}{21} d^3 x^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{9} d^3 x^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2-\frac{1}{945} \left (32 b^2 d^3\right ) \int x^2 \, dx+\frac{\left (4 b^2 d^3\right ) \int \left (-2-c^2 x^2+8 c^4 x^4-5 c^6 x^6\right ) \, dx}{735 c^2}+\frac{\left (16 b^2 d^3\right ) \int \left (-2-c^2 x^2+3 c^4 x^4\right ) \, dx}{1575 c^2}-\frac{\left (64 b d^3\right ) \int \frac{x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx}{945 c}\\ &=-\frac{3796 b^2 d^3 x}{99225 c^2}-\frac{5258 b^2 d^3 x^3}{297675}+\frac{4198 b^2 c^2 d^3 x^5}{165375}-\frac{374 b^2 c^4 d^3 x^7}{27783}+\frac{2}{729} b^2 c^6 d^3 x^9+\frac{64 b d^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{945 c^3}+\frac{32 b d^3 x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{945 c}+\frac{16 b d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{315 c^3}+\frac{4 b d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{525 c^3}+\frac{2 b d^3 \left (1-c^2 x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{441 c^3}-\frac{2 b d^3 \left (1-c^2 x^2\right )^{9/2} \left (a+b \sin ^{-1}(c x)\right )}{81 c^3}+\frac{16}{315} d^3 x^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{8}{105} d^3 x^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{2}{21} d^3 x^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{9} d^3 x^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2-\frac{\left (64 b^2 d^3\right ) \int 1 \, dx}{945 c^2}\\ &=-\frac{10516 b^2 d^3 x}{99225 c^2}-\frac{5258 b^2 d^3 x^3}{297675}+\frac{4198 b^2 c^2 d^3 x^5}{165375}-\frac{374 b^2 c^4 d^3 x^7}{27783}+\frac{2}{729} b^2 c^6 d^3 x^9+\frac{64 b d^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{945 c^3}+\frac{32 b d^3 x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{945 c}+\frac{16 b d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{315 c^3}+\frac{4 b d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{525 c^3}+\frac{2 b d^3 \left (1-c^2 x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{441 c^3}-\frac{2 b d^3 \left (1-c^2 x^2\right )^{9/2} \left (a+b \sin ^{-1}(c x)\right )}{81 c^3}+\frac{16}{315} d^3 x^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{8}{105} d^3 x^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{2}{21} d^3 x^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{9} d^3 x^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2\\ \end{align*}

Mathematica [A]  time = 0.38232, size = 277, normalized size = 0.71 \[ -\frac{d^3 \left (99225 a^2 c^3 x^3 \left (35 c^6 x^6-135 c^4 x^4+189 c^2 x^2-105\right )+630 a b \sqrt{1-c^2 x^2} \left (1225 c^8 x^8-4675 c^6 x^6+6297 c^4 x^4-2629 c^2 x^2-5258\right )+630 b \sin ^{-1}(c x) \left (315 a c^3 x^3 \left (35 c^6 x^6-135 c^4 x^4+189 c^2 x^2-105\right )+b \sqrt{1-c^2 x^2} \left (1225 c^8 x^8-4675 c^6 x^6+6297 c^4 x^4-2629 c^2 x^2-5258\right )\right )+b^2 \left (-85750 c^9 x^9+420750 c^7 x^7-793422 c^5 x^5+552090 c^3 x^3+3312540 c x\right )+99225 b^2 c^3 x^3 \left (35 c^6 x^6-135 c^4 x^4+189 c^2 x^2-105\right ) \sin ^{-1}(c x)^2\right )}{31255875 c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(d^3*(99225*a^2*c^3*x^3*(-105 + 189*c^2*x^2 - 135*c^4*x^4 + 35*c^6*x^6) + 630*a*b*Sqrt[1 - c^2*x^2]*(-5258 -
2629*c^2*x^2 + 6297*c^4*x^4 - 4675*c^6*x^6 + 1225*c^8*x^8) + b^2*(3312540*c*x + 552090*c^3*x^3 - 793422*c^5*x^
5 + 420750*c^7*x^7 - 85750*c^9*x^9) + 630*b*(315*a*c^3*x^3*(-105 + 189*c^2*x^2 - 135*c^4*x^4 + 35*c^6*x^6) + b
*Sqrt[1 - c^2*x^2]*(-5258 - 2629*c^2*x^2 + 6297*c^4*x^4 - 4675*c^6*x^6 + 1225*c^8*x^8))*ArcSin[c*x] + 99225*b^
2*c^3*x^3*(-105 + 189*c^2*x^2 - 135*c^4*x^4 + 35*c^6*x^6)*ArcSin[c*x]^2))/(31255875*c^3)

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Maple [A]  time = 0.054, size = 525, normalized size = 1.3 \begin{align*}{\frac{1}{{c}^{3}} \left ( -{d}^{3}{a}^{2} \left ({\frac{{c}^{9}{x}^{9}}{9}}-{\frac{3\,{c}^{7}{x}^{7}}{7}}+{\frac{3\,{c}^{5}{x}^{5}}{5}}-{\frac{{c}^{3}{x}^{3}}{3}} \right ) -{d}^{3}{b}^{2} \left ({\frac{ \left ( \arcsin \left ( cx \right ) \right ) ^{2} \left ( 5\,{c}^{6}{x}^{6}-21\,{c}^{4}{x}^{4}+35\,{c}^{2}{x}^{2}-35 \right ) cx}{35}}+{\frac{32\,cx}{315}}-{\frac{32\,\arcsin \left ( cx \right ) }{315}\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{2\,\arcsin \left ( cx \right ) \left ({c}^{2}{x}^{2}-1 \right ) ^{3}}{441}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{ \left ( 10\,{c}^{6}{x}^{6}-42\,{c}^{4}{x}^{4}+70\,{c}^{2}{x}^{2}-70 \right ) cx}{15435}}-{\frac{4\,\arcsin \left ( cx \right ) \left ({c}^{2}{x}^{2}-1 \right ) ^{2}}{525}\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{ \left ( 12\,{c}^{4}{x}^{4}-40\,{c}^{2}{x}^{2}+60 \right ) cx}{7875}}+{\frac{16\, \left ({c}^{2}{x}^{2}-1 \right ) \arcsin \left ( cx \right ) }{945}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{ \left ( 16\,{c}^{2}{x}^{2}-48 \right ) cx}{2835}}+{\frac{ \left ( \arcsin \left ( cx \right ) \right ) ^{2} \left ( 35\,{c}^{8}{x}^{8}-180\,{c}^{6}{x}^{6}+378\,{c}^{4}{x}^{4}-420\,{c}^{2}{x}^{2}+315 \right ) cx}{315}}+{\frac{2\,\arcsin \left ( cx \right ) \left ({c}^{2}{x}^{2}-1 \right ) ^{4}}{81}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{ \left ( 70\,{c}^{8}{x}^{8}-360\,{c}^{6}{x}^{6}+756\,{c}^{4}{x}^{4}-840\,{c}^{2}{x}^{2}+630 \right ) cx}{25515}} \right ) -2\,{d}^{3}ab \left ( 1/9\,\arcsin \left ( cx \right ){c}^{9}{x}^{9}-3/7\,\arcsin \left ( cx \right ){c}^{7}{x}^{7}+3/5\,\arcsin \left ( cx \right ){c}^{5}{x}^{5}-1/3\,{c}^{3}{x}^{3}\arcsin \left ( cx \right ) +{\frac{{c}^{8}{x}^{8}\sqrt{-{c}^{2}{x}^{2}+1}}{81}}-{\frac{187\,{c}^{6}{x}^{6}\sqrt{-{c}^{2}{x}^{2}+1}}{3969}}+{\frac{2099\,{c}^{4}{x}^{4}\sqrt{-{c}^{2}{x}^{2}+1}}{33075}}-{\frac{2629\,{c}^{2}{x}^{2}\sqrt{-{c}^{2}{x}^{2}+1}}{99225}}-{\frac{5258\,\sqrt{-{c}^{2}{x}^{2}+1}}{99225}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^3*(-d^3*a^2*(1/9*c^9*x^9-3/7*c^7*x^7+3/5*c^5*x^5-1/3*c^3*x^3)-d^3*b^2*(1/35*arcsin(c*x)^2*(5*c^6*x^6-21*c^
4*x^4+35*c^2*x^2-35)*c*x+32/315*c*x-32/315*arcsin(c*x)*(-c^2*x^2+1)^(1/2)+2/441*arcsin(c*x)*(c^2*x^2-1)^3*(-c^
2*x^2+1)^(1/2)-2/15435*(5*c^6*x^6-21*c^4*x^4+35*c^2*x^2-35)*c*x-4/525*arcsin(c*x)*(c^2*x^2-1)^2*(-c^2*x^2+1)^(
1/2)+4/7875*(3*c^4*x^4-10*c^2*x^2+15)*c*x+16/945*arcsin(c*x)*(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)-16/2835*(c^2*x^2-3
)*c*x+1/315*arcsin(c*x)^2*(35*c^8*x^8-180*c^6*x^6+378*c^4*x^4-420*c^2*x^2+315)*c*x+2/81*arcsin(c*x)*(c^2*x^2-1
)^4*(-c^2*x^2+1)^(1/2)-2/25515*(35*c^8*x^8-180*c^6*x^6+378*c^4*x^4-420*c^2*x^2+315)*c*x)-2*d^3*a*b*(1/9*arcsin
(c*x)*c^9*x^9-3/7*arcsin(c*x)*c^7*x^7+3/5*arcsin(c*x)*c^5*x^5-1/3*c^3*x^3*arcsin(c*x)+1/81*c^8*x^8*(-c^2*x^2+1
)^(1/2)-187/3969*c^6*x^6*(-c^2*x^2+1)^(1/2)+2099/33075*c^4*x^4*(-c^2*x^2+1)^(1/2)-2629/99225*c^2*x^2*(-c^2*x^2
+1)^(1/2)-5258/99225*(-c^2*x^2+1)^(1/2)))

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Maxima [B]  time = 1.82218, size = 1277, normalized size = 3.27 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9*b^2*c^6*d^3*x^9*arcsin(c*x)^2 - 1/9*a^2*c^6*d^3*x^9 + 3/7*b^2*c^4*d^3*x^7*arcsin(c*x)^2 + 3/7*a^2*c^4*d^3
*x^7 - 3/5*b^2*c^2*d^3*x^5*arcsin(c*x)^2 - 2/2835*(315*x^9*arcsin(c*x) + (35*sqrt(-c^2*x^2 + 1)*x^8/c^2 + 40*s
qrt(-c^2*x^2 + 1)*x^6/c^4 + 48*sqrt(-c^2*x^2 + 1)*x^4/c^6 + 64*sqrt(-c^2*x^2 + 1)*x^2/c^8 + 128*sqrt(-c^2*x^2
+ 1)/c^10)*c)*a*b*c^6*d^3 - 2/893025*(315*(35*sqrt(-c^2*x^2 + 1)*x^8/c^2 + 40*sqrt(-c^2*x^2 + 1)*x^6/c^4 + 48*
sqrt(-c^2*x^2 + 1)*x^4/c^6 + 64*sqrt(-c^2*x^2 + 1)*x^2/c^8 + 128*sqrt(-c^2*x^2 + 1)/c^10)*c*arcsin(c*x) - (122
5*c^8*x^9 + 1800*c^6*x^7 + 3024*c^4*x^5 + 6720*c^2*x^3 + 40320*x)/c^8)*b^2*c^6*d^3 - 3/5*a^2*c^2*d^3*x^5 + 6/2
45*(35*x^7*arcsin(c*x) + (5*sqrt(-c^2*x^2 + 1)*x^6/c^2 + 6*sqrt(-c^2*x^2 + 1)*x^4/c^4 + 8*sqrt(-c^2*x^2 + 1)*x
^2/c^6 + 16*sqrt(-c^2*x^2 + 1)/c^8)*c)*a*b*c^4*d^3 + 2/8575*(105*(5*sqrt(-c^2*x^2 + 1)*x^6/c^2 + 6*sqrt(-c^2*x
^2 + 1)*x^4/c^4 + 8*sqrt(-c^2*x^2 + 1)*x^2/c^6 + 16*sqrt(-c^2*x^2 + 1)/c^8)*c*arcsin(c*x) - (75*c^6*x^7 + 126*
c^4*x^5 + 280*c^2*x^3 + 1680*x)/c^6)*b^2*c^4*d^3 + 1/3*b^2*d^3*x^3*arcsin(c*x)^2 - 2/25*(15*x^5*arcsin(c*x) +
(3*sqrt(-c^2*x^2 + 1)*x^4/c^2 + 4*sqrt(-c^2*x^2 + 1)*x^2/c^4 + 8*sqrt(-c^2*x^2 + 1)/c^6)*c)*a*b*c^2*d^3 - 2/37
5*(15*(3*sqrt(-c^2*x^2 + 1)*x^4/c^2 + 4*sqrt(-c^2*x^2 + 1)*x^2/c^4 + 8*sqrt(-c^2*x^2 + 1)/c^6)*c*arcsin(c*x) -
(9*c^4*x^5 + 20*c^2*x^3 + 120*x)/c^4)*b^2*c^2*d^3 + 1/3*a^2*d^3*x^3 + 2/9*(3*x^3*arcsin(c*x) + c*(sqrt(-c^2*x
^2 + 1)*x^2/c^2 + 2*sqrt(-c^2*x^2 + 1)/c^4))*a*b*d^3 + 2/27*(3*c*(sqrt(-c^2*x^2 + 1)*x^2/c^2 + 2*sqrt(-c^2*x^2
+ 1)/c^4)*arcsin(c*x) - (c^2*x^3 + 6*x)/c^2)*b^2*d^3

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Fricas [A]  time = 2.0016, size = 903, normalized size = 2.31 \begin{align*} -\frac{42875 \,{\left (81 \, a^{2} - 2 \, b^{2}\right )} c^{9} d^{3} x^{9} - 1125 \,{\left (11907 \, a^{2} - 374 \, b^{2}\right )} c^{7} d^{3} x^{7} + 189 \,{\left (99225 \, a^{2} - 4198 \, b^{2}\right )} c^{5} d^{3} x^{5} - 105 \,{\left (99225 \, a^{2} - 5258 \, b^{2}\right )} c^{3} d^{3} x^{3} + 3312540 \, b^{2} c d^{3} x + 99225 \,{\left (35 \, b^{2} c^{9} d^{3} x^{9} - 135 \, b^{2} c^{7} d^{3} x^{7} + 189 \, b^{2} c^{5} d^{3} x^{5} - 105 \, b^{2} c^{3} d^{3} x^{3}\right )} \arcsin \left (c x\right )^{2} + 198450 \,{\left (35 \, a b c^{9} d^{3} x^{9} - 135 \, a b c^{7} d^{3} x^{7} + 189 \, a b c^{5} d^{3} x^{5} - 105 \, a b c^{3} d^{3} x^{3}\right )} \arcsin \left (c x\right ) + 630 \,{\left (1225 \, a b c^{8} d^{3} x^{8} - 4675 \, a b c^{6} d^{3} x^{6} + 6297 \, a b c^{4} d^{3} x^{4} - 2629 \, a b c^{2} d^{3} x^{2} - 5258 \, a b d^{3} +{\left (1225 \, b^{2} c^{8} d^{3} x^{8} - 4675 \, b^{2} c^{6} d^{3} x^{6} + 6297 \, b^{2} c^{4} d^{3} x^{4} - 2629 \, b^{2} c^{2} d^{3} x^{2} - 5258 \, b^{2} d^{3}\right )} \arcsin \left (c x\right )\right )} \sqrt{-c^{2} x^{2} + 1}}{31255875 \, c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/31255875*(42875*(81*a^2 - 2*b^2)*c^9*d^3*x^9 - 1125*(11907*a^2 - 374*b^2)*c^7*d^3*x^7 + 189*(99225*a^2 - 41
98*b^2)*c^5*d^3*x^5 - 105*(99225*a^2 - 5258*b^2)*c^3*d^3*x^3 + 3312540*b^2*c*d^3*x + 99225*(35*b^2*c^9*d^3*x^9
- 135*b^2*c^7*d^3*x^7 + 189*b^2*c^5*d^3*x^5 - 105*b^2*c^3*d^3*x^3)*arcsin(c*x)^2 + 198450*(35*a*b*c^9*d^3*x^9
- 135*a*b*c^7*d^3*x^7 + 189*a*b*c^5*d^3*x^5 - 105*a*b*c^3*d^3*x^3)*arcsin(c*x) + 630*(1225*a*b*c^8*d^3*x^8 -
4675*a*b*c^6*d^3*x^6 + 6297*a*b*c^4*d^3*x^4 - 2629*a*b*c^2*d^3*x^2 - 5258*a*b*d^3 + (1225*b^2*c^8*d^3*x^8 - 46
75*b^2*c^6*d^3*x^6 + 6297*b^2*c^4*d^3*x^4 - 2629*b^2*c^2*d^3*x^2 - 5258*b^2*d^3)*arcsin(c*x))*sqrt(-c^2*x^2 +
1))/c^3

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Sympy [A]  time = 55.6446, size = 626, normalized size = 1.6 \begin{align*} \begin{cases} - \frac{a^{2} c^{6} d^{3} x^{9}}{9} + \frac{3 a^{2} c^{4} d^{3} x^{7}}{7} - \frac{3 a^{2} c^{2} d^{3} x^{5}}{5} + \frac{a^{2} d^{3} x^{3}}{3} - \frac{2 a b c^{6} d^{3} x^{9} \operatorname{asin}{\left (c x \right )}}{9} - \frac{2 a b c^{5} d^{3} x^{8} \sqrt{- c^{2} x^{2} + 1}}{81} + \frac{6 a b c^{4} d^{3} x^{7} \operatorname{asin}{\left (c x \right )}}{7} + \frac{374 a b c^{3} d^{3} x^{6} \sqrt{- c^{2} x^{2} + 1}}{3969} - \frac{6 a b c^{2} d^{3} x^{5} \operatorname{asin}{\left (c x \right )}}{5} - \frac{4198 a b c d^{3} x^{4} \sqrt{- c^{2} x^{2} + 1}}{33075} + \frac{2 a b d^{3} x^{3} \operatorname{asin}{\left (c x \right )}}{3} + \frac{5258 a b d^{3} x^{2} \sqrt{- c^{2} x^{2} + 1}}{99225 c} + \frac{10516 a b d^{3} \sqrt{- c^{2} x^{2} + 1}}{99225 c^{3}} - \frac{b^{2} c^{6} d^{3} x^{9} \operatorname{asin}^{2}{\left (c x \right )}}{9} + \frac{2 b^{2} c^{6} d^{3} x^{9}}{729} - \frac{2 b^{2} c^{5} d^{3} x^{8} \sqrt{- c^{2} x^{2} + 1} \operatorname{asin}{\left (c x \right )}}{81} + \frac{3 b^{2} c^{4} d^{3} x^{7} \operatorname{asin}^{2}{\left (c x \right )}}{7} - \frac{374 b^{2} c^{4} d^{3} x^{7}}{27783} + \frac{374 b^{2} c^{3} d^{3} x^{6} \sqrt{- c^{2} x^{2} + 1} \operatorname{asin}{\left (c x \right )}}{3969} - \frac{3 b^{2} c^{2} d^{3} x^{5} \operatorname{asin}^{2}{\left (c x \right )}}{5} + \frac{4198 b^{2} c^{2} d^{3} x^{5}}{165375} - \frac{4198 b^{2} c d^{3} x^{4} \sqrt{- c^{2} x^{2} + 1} \operatorname{asin}{\left (c x \right )}}{33075} + \frac{b^{2} d^{3} x^{3} \operatorname{asin}^{2}{\left (c x \right )}}{3} - \frac{5258 b^{2} d^{3} x^{3}}{297675} + \frac{5258 b^{2} d^{3} x^{2} \sqrt{- c^{2} x^{2} + 1} \operatorname{asin}{\left (c x \right )}}{99225 c} - \frac{10516 b^{2} d^{3} x}{99225 c^{2}} + \frac{10516 b^{2} d^{3} \sqrt{- c^{2} x^{2} + 1} \operatorname{asin}{\left (c x \right )}}{99225 c^{3}} & \text{for}\: c \neq 0 \\\frac{a^{2} d^{3} x^{3}}{3} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-a**2*c**6*d**3*x**9/9 + 3*a**2*c**4*d**3*x**7/7 - 3*a**2*c**2*d**3*x**5/5 + a**2*d**3*x**3/3 - 2*a
*b*c**6*d**3*x**9*asin(c*x)/9 - 2*a*b*c**5*d**3*x**8*sqrt(-c**2*x**2 + 1)/81 + 6*a*b*c**4*d**3*x**7*asin(c*x)/
7 + 374*a*b*c**3*d**3*x**6*sqrt(-c**2*x**2 + 1)/3969 - 6*a*b*c**2*d**3*x**5*asin(c*x)/5 - 4198*a*b*c*d**3*x**4
*sqrt(-c**2*x**2 + 1)/33075 + 2*a*b*d**3*x**3*asin(c*x)/3 + 5258*a*b*d**3*x**2*sqrt(-c**2*x**2 + 1)/(99225*c)
+ 10516*a*b*d**3*sqrt(-c**2*x**2 + 1)/(99225*c**3) - b**2*c**6*d**3*x**9*asin(c*x)**2/9 + 2*b**2*c**6*d**3*x**
9/729 - 2*b**2*c**5*d**3*x**8*sqrt(-c**2*x**2 + 1)*asin(c*x)/81 + 3*b**2*c**4*d**3*x**7*asin(c*x)**2/7 - 374*b
**2*c**4*d**3*x**7/27783 + 374*b**2*c**3*d**3*x**6*sqrt(-c**2*x**2 + 1)*asin(c*x)/3969 - 3*b**2*c**2*d**3*x**5
*asin(c*x)**2/5 + 4198*b**2*c**2*d**3*x**5/165375 - 4198*b**2*c*d**3*x**4*sqrt(-c**2*x**2 + 1)*asin(c*x)/33075
+ b**2*d**3*x**3*asin(c*x)**2/3 - 5258*b**2*d**3*x**3/297675 + 5258*b**2*d**3*x**2*sqrt(-c**2*x**2 + 1)*asin(
c*x)/(99225*c) - 10516*b**2*d**3*x/(99225*c**2) + 10516*b**2*d**3*sqrt(-c**2*x**2 + 1)*asin(c*x)/(99225*c**3),
Ne(c, 0)), (a**2*d**3*x**3/3, True))

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Giac [B]  time = 1.41748, size = 967, normalized size = 2.47 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9*a^2*c^6*d^3*x^9 + 3/7*a^2*c^4*d^3*x^7 - 3/5*a^2*c^2*d^3*x^5 - 1/9*(c^2*x^2 - 1)^4*b^2*d^3*x*arcsin(c*x)^2
/c^2 - 2/9*(c^2*x^2 - 1)^4*a*b*d^3*x*arcsin(c*x)/c^2 - 1/63*(c^2*x^2 - 1)^3*b^2*d^3*x*arcsin(c*x)^2/c^2 + 2/72
9*(c^2*x^2 - 1)^4*b^2*d^3*x/c^2 + 1/3*a^2*d^3*x^3 - 2/63*(c^2*x^2 - 1)^3*a*b*d^3*x*arcsin(c*x)/c^2 + 2/105*(c^
2*x^2 - 1)^2*b^2*d^3*x*arcsin(c*x)^2/c^2 - 2/81*(c^2*x^2 - 1)^4*sqrt(-c^2*x^2 + 1)*b^2*d^3*arcsin(c*x)/c^3 - 6
22/250047*(c^2*x^2 - 1)^3*b^2*d^3*x/c^2 + 4/105*(c^2*x^2 - 1)^2*a*b*d^3*x*arcsin(c*x)/c^2 - 8/315*(c^2*x^2 - 1
)*b^2*d^3*x*arcsin(c*x)^2/c^2 - 2/81*(c^2*x^2 - 1)^4*sqrt(-c^2*x^2 + 1)*a*b*d^3/c^3 - 2/441*(c^2*x^2 - 1)^3*sq
rt(-c^2*x^2 + 1)*b^2*d^3*arcsin(c*x)/c^3 + 15224/10418625*(c^2*x^2 - 1)^2*b^2*d^3*x/c^2 - 16/315*(c^2*x^2 - 1)
*a*b*d^3*x*arcsin(c*x)/c^2 + 16/315*b^2*d^3*x*arcsin(c*x)^2/c^2 - 2/441*(c^2*x^2 - 1)^3*sqrt(-c^2*x^2 + 1)*a*b
*d^3/c^3 + 4/525*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b^2*d^3*arcsin(c*x)/c^3 + 115504/31255875*(c^2*x^2 - 1)*b^
2*d^3*x/c^2 + 32/315*a*b*d^3*x*arcsin(c*x)/c^2 + 4/525*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*a*b*d^3/c^3 + 16/945
*(-c^2*x^2 + 1)^(3/2)*b^2*d^3*arcsin(c*x)/c^3 - 3406208/31255875*b^2*d^3*x/c^2 + 16/945*(-c^2*x^2 + 1)^(3/2)*a
*b*d^3/c^3 + 32/315*sqrt(-c^2*x^2 + 1)*b^2*d^3*arcsin(c*x)/c^3 + 32/315*sqrt(-c^2*x^2 + 1)*a*b*d^3/c^3