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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.82, antiderivative size = 391, normalized size of antiderivative = 1.00, 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 8

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

Rule 12

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

Rule 30

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

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 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 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]

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 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 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 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 \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*}

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Mathematica [A]  time = 0.38, 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]

-1/31255875*(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 - 79
3422*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))/c^3

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fricas [A]  time = 0.64, size = 372, normalized size = 0.95 \[ -\frac {42875 \, {\left (81 \, a^{2} - 2 \, b^{2}\right )} c^{9} d^{3} x^{9} - 1125 \, {\left (11907 \, a^{2} - 374 \, b^{2}\right )} c^{7} d^{3} x^{7} + 189 \, {\left (99225 \, a^{2} - 4198 \, b^{2}\right )} c^{5} d^{3} x^{5} - 105 \, {\left (99225 \, a^{2} - 5258 \, b^{2}\right )} c^{3} d^{3} x^{3} + 3312540 \, b^{2} c d^{3} x + 99225 \, {\left (35 \, b^{2} c^{9} d^{3} x^{9} - 135 \, b^{2} c^{7} d^{3} x^{7} + 189 \, b^{2} c^{5} d^{3} x^{5} - 105 \, b^{2} c^{3} d^{3} x^{3}\right )} \arcsin \left (c x\right )^{2} + 198450 \, {\left (35 \, a b c^{9} d^{3} x^{9} - 135 \, a b c^{7} d^{3} x^{7} + 189 \, a b c^{5} d^{3} x^{5} - 105 \, a b c^{3} d^{3} x^{3}\right )} \arcsin \left (c x\right ) + 630 \, {\left (1225 \, a b c^{8} d^{3} x^{8} - 4675 \, a b c^{6} d^{3} x^{6} + 6297 \, a b c^{4} d^{3} x^{4} - 2629 \, a b c^{2} d^{3} x^{2} - 5258 \, a b d^{3} + {\left (1225 \, b^{2} c^{8} d^{3} x^{8} - 4675 \, b^{2} c^{6} d^{3} x^{6} + 6297 \, b^{2} c^{4} d^{3} x^{4} - 2629 \, b^{2} c^{2} d^{3} x^{2} - 5258 \, b^{2} d^{3}\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{31255875 \, c^{3}} \]

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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giac [B]  time = 0.53, size = 716, normalized size = 1.83 \[ -\frac {1}{9} \, a^{2} c^{6} d^{3} x^{9} + \frac {3}{7} \, a^{2} c^{4} d^{3} x^{7} - \frac {3}{5} \, a^{2} c^{2} d^{3} x^{5} - \frac {{\left (c^{2} x^{2} - 1\right )}^{4} b^{2} d^{3} x \arcsin \left (c x\right )^{2}}{9 \, c^{2}} - \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{4} a b d^{3} x \arcsin \left (c x\right )}{9 \, c^{2}} - \frac {{\left (c^{2} x^{2} - 1\right )}^{3} b^{2} d^{3} x \arcsin \left (c x\right )^{2}}{63 \, c^{2}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{4} b^{2} d^{3} x}{729 \, c^{2}} + \frac {1}{3} \, a^{2} d^{3} x^{3} - \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{3} a b d^{3} x \arcsin \left (c x\right )}{63 \, c^{2}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{2} b^{2} d^{3} x \arcsin \left (c x\right )^{2}}{105 \, c^{2}} - \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{4} \sqrt {-c^{2} x^{2} + 1} b^{2} d^{3} \arcsin \left (c x\right )}{81 \, c^{3}} - \frac {622 \, {\left (c^{2} x^{2} - 1\right )}^{3} b^{2} d^{3} x}{250047 \, c^{2}} + \frac {4 \, {\left (c^{2} x^{2} - 1\right )}^{2} a b d^{3} x \arcsin \left (c x\right )}{105 \, c^{2}} - \frac {8 \, {\left (c^{2} x^{2} - 1\right )} b^{2} d^{3} x \arcsin \left (c x\right )^{2}}{315 \, c^{2}} - \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{4} \sqrt {-c^{2} x^{2} + 1} a b d^{3}}{81 \, c^{3}} - \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{3} \sqrt {-c^{2} x^{2} + 1} b^{2} d^{3} \arcsin \left (c x\right )}{441 \, c^{3}} + \frac {15224 \, {\left (c^{2} x^{2} - 1\right )}^{2} b^{2} d^{3} x}{10418625 \, c^{2}} - \frac {16 \, {\left (c^{2} x^{2} - 1\right )} a b d^{3} x \arcsin \left (c x\right )}{315 \, c^{2}} + \frac {16 \, b^{2} d^{3} x \arcsin \left (c x\right )^{2}}{315 \, c^{2}} - \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{3} \sqrt {-c^{2} x^{2} + 1} a b d^{3}}{441 \, c^{3}} + \frac {4 \, {\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b^{2} d^{3} \arcsin \left (c x\right )}{525 \, c^{3}} + \frac {115504 \, {\left (c^{2} x^{2} - 1\right )} b^{2} d^{3} x}{31255875 \, c^{2}} + \frac {32 \, a b d^{3} x \arcsin \left (c x\right )}{315 \, c^{2}} + \frac {4 \, {\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} a b d^{3}}{525 \, c^{3}} + \frac {16 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b^{2} d^{3} \arcsin \left (c x\right )}{945 \, c^{3}} - \frac {3406208 \, b^{2} d^{3} x}{31255875 \, c^{2}} + \frac {16 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} a b d^{3}}{945 \, c^{3}} + \frac {32 \, \sqrt {-c^{2} x^{2} + 1} b^{2} d^{3} \arcsin \left (c x\right )}{315 \, c^{3}} + \frac {32 \, \sqrt {-c^{2} x^{2} + 1} a b d^{3}}{315 \, c^{3}} \]

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

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maple [A]  time = 0.06, size = 525, normalized size = 1.34 \[ \frac {-d^{3} a^{2} \left (\frac {1}{9} c^{9} x^{9}-\frac {3}{7} c^{7} x^{7}+\frac {3}{5} c^{5} x^{5}-\frac {1}{3} c^{3} x^{3}\right )-d^{3} b^{2} \left (\frac {\arcsin \left (c x \right )^{2} \left (5 c^{6} x^{6}-21 c^{4} x^{4}+35 c^{2} x^{2}-35\right ) c x}{35}+\frac {32 c x}{315}-\frac {32 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{315}+\frac {2 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right )^{3} \sqrt {-c^{2} x^{2}+1}}{441}-\frac {2 \left (5 c^{6} x^{6}-21 c^{4} x^{4}+35 c^{2} x^{2}-35\right ) c x}{15435}-\frac {4 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right )^{2} \sqrt {-c^{2} x^{2}+1}}{525}+\frac {4 \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{7875}+\frac {16 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}}{945}-\frac {16 \left (c^{2} x^{2}-3\right ) c x}{2835}+\frac {\arcsin \left (c x \right )^{2} \left (35 c^{8} x^{8}-180 c^{6} x^{6}+378 c^{4} x^{4}-420 c^{2} x^{2}+315\right ) c x}{315}+\frac {2 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right )^{4} \sqrt {-c^{2} x^{2}+1}}{81}-\frac {2 \left (35 c^{8} x^{8}-180 c^{6} x^{6}+378 c^{4} x^{4}-420 c^{2} x^{2}+315\right ) c x}{25515}\right )-2 d^{3} a b \left (\frac {\arcsin \left (c x \right ) c^{9} x^{9}}{9}-\frac {3 \arcsin \left (c x \right ) c^{7} x^{7}}{7}+\frac {3 \arcsin \left (c x \right ) c^{5} x^{5}}{5}-\frac {c^{3} x^{3} \arcsin \left (c x \right )}{3}+\frac {c^{8} x^{8} \sqrt {-c^{2} x^{2}+1}}{81}-\frac {187 c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{3969}+\frac {2099 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{33075}-\frac {2629 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{99225}-\frac {5258 \sqrt {-c^{2} x^{2}+1}}{99225}\right )}{c^{3}} \]

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 = 0.94, size = 946, normalized size = 2.42 \[ -\frac {1}{9} \, b^{2} c^{6} d^{3} x^{9} \arcsin \left (c x\right )^{2} - \frac {1}{9} \, a^{2} c^{6} d^{3} x^{9} + \frac {3}{7} \, b^{2} c^{4} d^{3} x^{7} \arcsin \left (c x\right )^{2} + \frac {3}{7} \, a^{2} c^{4} d^{3} x^{7} - \frac {3}{5} \, b^{2} c^{2} d^{3} x^{5} \arcsin \left (c x\right )^{2} - \frac {2}{2835} \, {\left (315 \, x^{9} \arcsin \left (c x\right ) + {\left (\frac {35 \, \sqrt {-c^{2} x^{2} + 1} x^{8}}{c^{2}} + \frac {40 \, \sqrt {-c^{2} x^{2} + 1} x^{6}}{c^{4}} + \frac {48 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{6}} + \frac {64 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{8}} + \frac {128 \, \sqrt {-c^{2} x^{2} + 1}}{c^{10}}\right )} c\right )} a b c^{6} d^{3} - \frac {2}{893025} \, {\left (315 \, {\left (\frac {35 \, \sqrt {-c^{2} x^{2} + 1} x^{8}}{c^{2}} + \frac {40 \, \sqrt {-c^{2} x^{2} + 1} x^{6}}{c^{4}} + \frac {48 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{6}} + \frac {64 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{8}} + \frac {128 \, \sqrt {-c^{2} x^{2} + 1}}{c^{10}}\right )} c \arcsin \left (c x\right ) - \frac {1225 \, c^{8} x^{9} + 1800 \, c^{6} x^{7} + 3024 \, c^{4} x^{5} + 6720 \, c^{2} x^{3} + 40320 \, x}{c^{8}}\right )} b^{2} c^{6} d^{3} - \frac {3}{5} \, a^{2} c^{2} d^{3} x^{5} + \frac {6}{245} \, {\left (35 \, x^{7} \arcsin \left (c x\right ) + {\left (\frac {5 \, \sqrt {-c^{2} x^{2} + 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {-c^{2} x^{2} + 1}}{c^{8}}\right )} c\right )} a b c^{4} d^{3} + \frac {2}{8575} \, {\left (105 \, {\left (\frac {5 \, \sqrt {-c^{2} x^{2} + 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {-c^{2} x^{2} + 1}}{c^{8}}\right )} c \arcsin \left (c x\right ) - \frac {75 \, c^{6} x^{7} + 126 \, c^{4} x^{5} + 280 \, c^{2} x^{3} + 1680 \, x}{c^{6}}\right )} b^{2} c^{4} d^{3} + \frac {1}{3} \, b^{2} d^{3} x^{3} \arcsin \left (c x\right )^{2} - \frac {2}{25} \, {\left (15 \, x^{5} \arcsin \left (c x\right ) + {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} a b c^{2} d^{3} - \frac {2}{375} \, {\left (15 \, {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c \arcsin \left (c x\right ) - \frac {9 \, c^{4} x^{5} + 20 \, c^{2} x^{3} + 120 \, x}{c^{4}}\right )} b^{2} c^{2} d^{3} + \frac {1}{3} \, a^{2} d^{3} x^{3} + \frac {2}{9} \, {\left (3 \, x^{3} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} a b d^{3} + \frac {2}{27} \, {\left (3 \, c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )} \arcsin \left (c x\right ) - \frac {c^{2} x^{3} + 6 \, x}{c^{2}}\right )} b^{2} d^{3} \]

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

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,x)

[Out]

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

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sympy [A]  time = 30.10, size = 626, normalized size = 1.60 \[ \begin {cases} - \frac {a^{2} c^{6} d^{3} x^{9}}{9} + \frac {3 a^{2} c^{4} d^{3} x^{7}}{7} - \frac {3 a^{2} c^{2} d^{3} x^{5}}{5} + \frac {a^{2} d^{3} x^{3}}{3} - \frac {2 a b c^{6} d^{3} x^{9} \operatorname {asin}{\left (c x \right )}}{9} - \frac {2 a b c^{5} d^{3} x^{8} \sqrt {- c^{2} x^{2} + 1}}{81} + \frac {6 a b c^{4} d^{3} x^{7} \operatorname {asin}{\left (c x \right )}}{7} + \frac {374 a b c^{3} d^{3} x^{6} \sqrt {- c^{2} x^{2} + 1}}{3969} - \frac {6 a b c^{2} d^{3} x^{5} \operatorname {asin}{\left (c x \right )}}{5} - \frac {4198 a b c d^{3} x^{4} \sqrt {- c^{2} x^{2} + 1}}{33075} + \frac {2 a b d^{3} x^{3} \operatorname {asin}{\left (c x \right )}}{3} + \frac {5258 a b d^{3} x^{2} \sqrt {- c^{2} x^{2} + 1}}{99225 c} + \frac {10516 a b d^{3} \sqrt {- c^{2} x^{2} + 1}}{99225 c^{3}} - \frac {b^{2} c^{6} d^{3} x^{9} \operatorname {asin}^{2}{\left (c x \right )}}{9} + \frac {2 b^{2} c^{6} d^{3} x^{9}}{729} - \frac {2 b^{2} c^{5} d^{3} x^{8} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{81} + \frac {3 b^{2} c^{4} d^{3} x^{7} \operatorname {asin}^{2}{\left (c x \right )}}{7} - \frac {374 b^{2} c^{4} d^{3} x^{7}}{27783} + \frac {374 b^{2} c^{3} d^{3} x^{6} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{3969} - \frac {3 b^{2} c^{2} d^{3} x^{5} \operatorname {asin}^{2}{\left (c x \right )}}{5} + \frac {4198 b^{2} c^{2} d^{3} x^{5}}{165375} - \frac {4198 b^{2} c d^{3} x^{4} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{33075} + \frac {b^{2} d^{3} x^{3} \operatorname {asin}^{2}{\left (c x \right )}}{3} - \frac {5258 b^{2} d^{3} x^{3}}{297675} + \frac {5258 b^{2} d^{3} x^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{99225 c} - \frac {10516 b^{2} d^{3} x}{99225 c^{2}} + \frac {10516 b^{2} d^{3} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{99225 c^{3}} & \text {for}\: c \neq 0 \\\frac {a^{2} d^{3} x^{3}}{3} & \text {otherwise} \end {cases} \]

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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