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

Optimal. Leaf size=384 \[ -\frac {79 d^3 \left (a+b \sin ^{-1}(c x)\right )^2}{5120 c^4}-\frac {1}{50} b c d^3 x^5 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{32} b c d^3 x^5 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {31}{960} b c d^3 x^5 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{10} d^3 x^4 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {3}{40} d^3 x^4 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{20} d^3 x^4 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac {79 b d^3 x^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3840 c}+\frac {79 b d^3 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2560 c^3}+\frac {1}{40} d^3 x^4 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{500} b^2 c^6 d^3 x^{10}-\frac {57 b^2 c^4 d^3 x^8}{6400}+\frac {401 b^2 c^2 d^3 x^6}{28800}-\frac {79 b^2 d^3 x^2}{5120 c^2}-\frac {79 b^2 d^3 x^4}{15360} \]

[Out]

-79/5120*b^2*d^3*x^2/c^2-79/15360*b^2*d^3*x^4+401/28800*b^2*c^2*d^3*x^6-57/6400*b^2*c^4*d^3*x^8+1/500*b^2*c^6*
d^3*x^10-1/32*b*c*d^3*x^5*(-c^2*x^2+1)^(3/2)*(a+b*arcsin(c*x))-1/50*b*c*d^3*x^5*(-c^2*x^2+1)^(5/2)*(a+b*arcsin
(c*x))-79/5120*d^3*(a+b*arcsin(c*x))^2/c^4+1/40*d^3*x^4*(a+b*arcsin(c*x))^2+1/20*d^3*x^4*(-c^2*x^2+1)*(a+b*arc
sin(c*x))^2+3/40*d^3*x^4*(-c^2*x^2+1)^2*(a+b*arcsin(c*x))^2+1/10*d^3*x^4*(-c^2*x^2+1)^3*(a+b*arcsin(c*x))^2+79
/2560*b*d^3*x*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/c^3+79/3840*b*d^3*x^3*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/
c-31/960*b*c*d^3*x^5*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 1.59, antiderivative size = 384, normalized size of antiderivative = 1.00, number of steps used = 40, number of rules used = 9, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4699, 4627, 4707, 4641, 30, 4697, 14, 266, 43} \[ -\frac {1}{50} b c d^3 x^5 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{32} b c d^3 x^5 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {31}{960} b c d^3 x^5 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{10} d^3 x^4 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {3}{40} d^3 x^4 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{20} d^3 x^4 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac {79 b d^3 x^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3840 c}+\frac {79 b d^3 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2560 c^3}-\frac {79 d^3 \left (a+b \sin ^{-1}(c x)\right )^2}{5120 c^4}+\frac {1}{40} d^3 x^4 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{500} b^2 c^6 d^3 x^{10}-\frac {57 b^2 c^4 d^3 x^8}{6400}+\frac {401 b^2 c^2 d^3 x^6}{28800}-\frac {79 b^2 d^3 x^2}{5120 c^2}-\frac {79 b^2 d^3 x^4}{15360} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-79*b^2*d^3*x^2)/(5120*c^2) - (79*b^2*d^3*x^4)/15360 + (401*b^2*c^2*d^3*x^6)/28800 - (57*b^2*c^4*d^3*x^8)/640
0 + (b^2*c^6*d^3*x^10)/500 + (79*b*d^3*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(2560*c^3) + (79*b*d^3*x^3*Sqr
t[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(3840*c) - (31*b*c*d^3*x^5*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/960 - (b
*c*d^3*x^5*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/32 - (b*c*d^3*x^5*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x]))
/50 - (79*d^3*(a + b*ArcSin[c*x])^2)/(5120*c^4) + (d^3*x^4*(a + b*ArcSin[c*x])^2)/40 + (d^3*x^4*(1 - c^2*x^2)*
(a + b*ArcSin[c*x])^2)/20 + (3*d^3*x^4*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/40 + (d^3*x^4*(1 - c^2*x^2)^3*(a
+ b*ArcSin[c*x])^2)/10

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 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 4627

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcSi
n[c*x])^n)/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcSin[c*x])^(n - 1))/Sqrt[1
- c^2*x^2], x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 4641

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

Rule 4697

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

Rule 4699

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

Rule 4707

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

Rubi steps

\begin {align*} \int x^3 \left (d-c^2 d x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\frac {1}{10} d^3 x^4 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{5} (3 d) \int x^3 \left (d-c^2 d x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx-\frac {1}{5} \left (b c d^3\right ) \int x^4 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx\\ &=-\frac {1}{50} b c d^3 x^5 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {3}{40} d^3 x^4 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{10} d^3 x^4 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{10} \left (3 d^2\right ) \int x^3 \left (d-c^2 d x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2 \, dx-\frac {1}{10} \left (b c d^3\right ) \int x^4 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx-\frac {1}{20} \left (3 b c d^3\right ) \int x^4 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx+\frac {1}{50} \left (b^2 c^2 d^3\right ) \int x^5 \left (1-c^2 x^2\right )^2 \, dx\\ &=-\frac {1}{32} b c d^3 x^5 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{50} b c d^3 x^5 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{20} d^3 x^4 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac {3}{40} d^3 x^4 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{10} d^3 x^4 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{10} d^3 \int x^3 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx-\frac {1}{80} \left (3 b c d^3\right ) \int x^4 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx-\frac {1}{160} \left (9 b c d^3\right ) \int x^4 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx-\frac {1}{10} \left (b c d^3\right ) \int x^4 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx+\frac {1}{100} \left (b^2 c^2 d^3\right ) \operatorname {Subst}\left (\int x^2 \left (1-c^2 x\right )^2 \, dx,x,x^2\right )+\frac {1}{80} \left (b^2 c^2 d^3\right ) \int x^5 \left (1-c^2 x^2\right ) \, dx+\frac {1}{160} \left (3 b^2 c^2 d^3\right ) \int x^5 \left (1-c^2 x^2\right ) \, dx\\ &=-\frac {31}{960} b c d^3 x^5 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{32} b c d^3 x^5 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{50} b c d^3 x^5 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{40} d^3 x^4 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{20} d^3 x^4 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac {3}{40} d^3 x^4 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{10} d^3 x^4 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {1}{160} \left (b c d^3\right ) \int \frac {x^4 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx-\frac {1}{320} \left (3 b c d^3\right ) \int \frac {x^4 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx-\frac {1}{60} \left (b c d^3\right ) \int \frac {x^4 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx-\frac {1}{20} \left (b c d^3\right ) \int \frac {x^4 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx+\frac {1}{160} \left (b^2 c^2 d^3\right ) \int x^5 \, dx+\frac {1}{320} \left (3 b^2 c^2 d^3\right ) \int x^5 \, dx+\frac {1}{100} \left (b^2 c^2 d^3\right ) \operatorname {Subst}\left (\int \left (x^2-2 c^2 x^3+c^4 x^4\right ) \, dx,x,x^2\right )+\frac {1}{80} \left (b^2 c^2 d^3\right ) \int \left (x^5-c^2 x^7\right ) \, dx+\frac {1}{60} \left (b^2 c^2 d^3\right ) \int x^5 \, dx+\frac {1}{160} \left (3 b^2 c^2 d^3\right ) \int \left (x^5-c^2 x^7\right ) \, dx\\ &=\frac {401 b^2 c^2 d^3 x^6}{28800}-\frac {57 b^2 c^4 d^3 x^8}{6400}+\frac {1}{500} b^2 c^6 d^3 x^{10}+\frac {79 b d^3 x^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3840 c}-\frac {31}{960} b c d^3 x^5 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{32} b c d^3 x^5 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{50} b c d^3 x^5 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{40} d^3 x^4 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{20} d^3 x^4 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac {3}{40} d^3 x^4 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{10} d^3 x^4 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {1}{640} \left (b^2 d^3\right ) \int x^3 \, dx-\frac {\left (3 b^2 d^3\right ) \int x^3 \, dx}{1280}-\frac {1}{240} \left (b^2 d^3\right ) \int x^3 \, dx-\frac {1}{80} \left (b^2 d^3\right ) \int x^3 \, dx-\frac {\left (3 b d^3\right ) \int \frac {x^2 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx}{640 c}-\frac {\left (9 b d^3\right ) \int \frac {x^2 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx}{1280 c}-\frac {\left (b d^3\right ) \int \frac {x^2 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx}{80 c}-\frac {\left (3 b d^3\right ) \int \frac {x^2 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx}{80 c}\\ &=-\frac {79 b^2 d^3 x^4}{15360}+\frac {401 b^2 c^2 d^3 x^6}{28800}-\frac {57 b^2 c^4 d^3 x^8}{6400}+\frac {1}{500} b^2 c^6 d^3 x^{10}+\frac {79 b d^3 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2560 c^3}+\frac {79 b d^3 x^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3840 c}-\frac {31}{960} b c d^3 x^5 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{32} b c d^3 x^5 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{50} b c d^3 x^5 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{40} d^3 x^4 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{20} d^3 x^4 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac {3}{40} d^3 x^4 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{10} d^3 x^4 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {\left (3 b d^3\right ) \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{1280 c^3}-\frac {\left (9 b d^3\right ) \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{2560 c^3}-\frac {\left (b d^3\right ) \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{160 c^3}-\frac {\left (3 b d^3\right ) \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{160 c^3}-\frac {\left (3 b^2 d^3\right ) \int x \, dx}{1280 c^2}-\frac {\left (9 b^2 d^3\right ) \int x \, dx}{2560 c^2}-\frac {\left (b^2 d^3\right ) \int x \, dx}{160 c^2}-\frac {\left (3 b^2 d^3\right ) \int x \, dx}{160 c^2}\\ &=-\frac {79 b^2 d^3 x^2}{5120 c^2}-\frac {79 b^2 d^3 x^4}{15360}+\frac {401 b^2 c^2 d^3 x^6}{28800}-\frac {57 b^2 c^4 d^3 x^8}{6400}+\frac {1}{500} b^2 c^6 d^3 x^{10}+\frac {79 b d^3 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2560 c^3}+\frac {79 b d^3 x^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3840 c}-\frac {31}{960} b c d^3 x^5 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{32} b c d^3 x^5 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{50} b c d^3 x^5 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {79 d^3 \left (a+b \sin ^{-1}(c x)\right )^2}{5120 c^4}+\frac {1}{40} d^3 x^4 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{20} d^3 x^4 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac {3}{40} d^3 x^4 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{10} d^3 x^4 \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 = 0.45, size = 287, normalized size = 0.75 \[ -\frac {d^3 \left (c x \left (28800 a^2 c^3 x^3 \left (4 c^6 x^6-15 c^4 x^4+20 c^2 x^2-10\right )+30 a b \sqrt {1-c^2 x^2} \left (768 c^8 x^8-2736 c^6 x^6+3208 c^4 x^4-790 c^2 x^2-1185\right )+b^2 \left (-2304 c^9 x^9+10260 c^7 x^7-16040 c^5 x^5+5925 c^3 x^3+17775 c x\right )\right )+30 b \sin ^{-1}(c x) \left (15 a \left (512 c^{10} x^{10}-1920 c^8 x^8+2560 c^6 x^6-1280 c^4 x^4+79\right )+b c x \sqrt {1-c^2 x^2} \left (768 c^8 x^8-2736 c^6 x^6+3208 c^4 x^4-790 c^2 x^2-1185\right )\right )+225 b^2 \left (512 c^{10} x^{10}-1920 c^8 x^8+2560 c^6 x^6-1280 c^4 x^4+79\right ) \sin ^{-1}(c x)^2\right )}{1152000 c^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/1152000*(d^3*(c*x*(28800*a^2*c^3*x^3*(-10 + 20*c^2*x^2 - 15*c^4*x^4 + 4*c^6*x^6) + 30*a*b*Sqrt[1 - c^2*x^2]
*(-1185 - 790*c^2*x^2 + 3208*c^4*x^4 - 2736*c^6*x^6 + 768*c^8*x^8) + b^2*(17775*c*x + 5925*c^3*x^3 - 16040*c^5
*x^5 + 10260*c^7*x^7 - 2304*c^9*x^9)) + 30*b*(b*c*x*Sqrt[1 - c^2*x^2]*(-1185 - 790*c^2*x^2 + 3208*c^4*x^4 - 27
36*c^6*x^6 + 768*c^8*x^8) + 15*a*(79 - 1280*c^4*x^4 + 2560*c^6*x^6 - 1920*c^8*x^8 + 512*c^10*x^10))*ArcSin[c*x
] + 225*b^2*(79 - 1280*c^4*x^4 + 2560*c^6*x^6 - 1920*c^8*x^8 + 512*c^10*x^10)*ArcSin[c*x]^2))/c^4

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fricas [A]  time = 0.74, size = 395, normalized size = 1.03 \[ -\frac {2304 \, {\left (50 \, a^{2} - b^{2}\right )} c^{10} d^{3} x^{10} - 540 \, {\left (800 \, a^{2} - 19 \, b^{2}\right )} c^{8} d^{3} x^{8} + 40 \, {\left (14400 \, a^{2} - 401 \, b^{2}\right )} c^{6} d^{3} x^{6} - 75 \, {\left (3840 \, a^{2} - 79 \, b^{2}\right )} c^{4} d^{3} x^{4} + 17775 \, b^{2} c^{2} d^{3} x^{2} + 225 \, {\left (512 \, b^{2} c^{10} d^{3} x^{10} - 1920 \, b^{2} c^{8} d^{3} x^{8} + 2560 \, b^{2} c^{6} d^{3} x^{6} - 1280 \, b^{2} c^{4} d^{3} x^{4} + 79 \, b^{2} d^{3}\right )} \arcsin \left (c x\right )^{2} + 450 \, {\left (512 \, a b c^{10} d^{3} x^{10} - 1920 \, a b c^{8} d^{3} x^{8} + 2560 \, a b c^{6} d^{3} x^{6} - 1280 \, a b c^{4} d^{3} x^{4} + 79 \, a b d^{3}\right )} \arcsin \left (c x\right ) + 30 \, {\left (768 \, a b c^{9} d^{3} x^{9} - 2736 \, a b c^{7} d^{3} x^{7} + 3208 \, a b c^{5} d^{3} x^{5} - 790 \, a b c^{3} d^{3} x^{3} - 1185 \, a b c d^{3} x + {\left (768 \, b^{2} c^{9} d^{3} x^{9} - 2736 \, b^{2} c^{7} d^{3} x^{7} + 3208 \, b^{2} c^{5} d^{3} x^{5} - 790 \, b^{2} c^{3} d^{3} x^{3} - 1185 \, b^{2} c d^{3} x\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{1152000 \, c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1152000*(2304*(50*a^2 - b^2)*c^10*d^3*x^10 - 540*(800*a^2 - 19*b^2)*c^8*d^3*x^8 + 40*(14400*a^2 - 401*b^2)*
c^6*d^3*x^6 - 75*(3840*a^2 - 79*b^2)*c^4*d^3*x^4 + 17775*b^2*c^2*d^3*x^2 + 225*(512*b^2*c^10*d^3*x^10 - 1920*b
^2*c^8*d^3*x^8 + 2560*b^2*c^6*d^3*x^6 - 1280*b^2*c^4*d^3*x^4 + 79*b^2*d^3)*arcsin(c*x)^2 + 450*(512*a*b*c^10*d
^3*x^10 - 1920*a*b*c^8*d^3*x^8 + 2560*a*b*c^6*d^3*x^6 - 1280*a*b*c^4*d^3*x^4 + 79*a*b*d^3)*arcsin(c*x) + 30*(7
68*a*b*c^9*d^3*x^9 - 2736*a*b*c^7*d^3*x^7 + 3208*a*b*c^5*d^3*x^5 - 790*a*b*c^3*d^3*x^3 - 1185*a*b*c*d^3*x + (7
68*b^2*c^9*d^3*x^9 - 2736*b^2*c^7*d^3*x^7 + 3208*b^2*c^5*d^3*x^5 - 790*b^2*c^3*d^3*x^3 - 1185*b^2*c*d^3*x)*arc
sin(c*x))*sqrt(-c^2*x^2 + 1))/c^4

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giac [A]  time = 0.61, size = 631, normalized size = 1.64 \[ -\frac {1}{10} \, a^{2} c^{6} d^{3} x^{10} + \frac {3}{8} \, a^{2} c^{4} d^{3} x^{8} - \frac {1}{2} \, a^{2} c^{2} d^{3} x^{6} + \frac {1}{4} \, a^{2} d^{3} x^{4} - \frac {{\left (c^{2} x^{2} - 1\right )}^{4} \sqrt {-c^{2} x^{2} + 1} b^{2} d^{3} x \arcsin \left (c x\right )}{50 \, c^{3}} - \frac {{\left (c^{2} x^{2} - 1\right )}^{5} b^{2} d^{3} \arcsin \left (c x\right )^{2}}{10 \, c^{4}} - \frac {{\left (c^{2} x^{2} - 1\right )}^{4} \sqrt {-c^{2} x^{2} + 1} a b d^{3} x}{50 \, c^{3}} - \frac {7 \, {\left (c^{2} x^{2} - 1\right )}^{3} \sqrt {-c^{2} x^{2} + 1} b^{2} d^{3} x \arcsin \left (c x\right )}{800 \, c^{3}} - \frac {{\left (c^{2} x^{2} - 1\right )}^{5} a b d^{3} \arcsin \left (c x\right )}{5 \, c^{4}} - \frac {{\left (c^{2} x^{2} - 1\right )}^{4} b^{2} d^{3} \arcsin \left (c x\right )^{2}}{8 \, c^{4}} - \frac {7 \, {\left (c^{2} x^{2} - 1\right )}^{3} \sqrt {-c^{2} x^{2} + 1} a b d^{3} x}{800 \, c^{3}} + \frac {49 \, {\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b^{2} d^{3} x \arcsin \left (c x\right )}{4800 \, c^{3}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{5} b^{2} d^{3}}{500 \, c^{4}} - \frac {{\left (c^{2} x^{2} - 1\right )}^{4} a b d^{3} \arcsin \left (c x\right )}{4 \, c^{4}} + \frac {49 \, {\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} a b d^{3} x}{4800 \, c^{3}} + \frac {49 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b^{2} d^{3} x \arcsin \left (c x\right )}{3840 \, c^{3}} + \frac {7 \, {\left (c^{2} x^{2} - 1\right )}^{4} b^{2} d^{3}}{6400 \, c^{4}} + \frac {49 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} a b d^{3} x}{3840 \, c^{3}} + \frac {49 \, \sqrt {-c^{2} x^{2} + 1} b^{2} d^{3} x \arcsin \left (c x\right )}{2560 \, c^{3}} - \frac {49 \, {\left (c^{2} x^{2} - 1\right )}^{3} b^{2} d^{3}}{28800 \, c^{4}} + \frac {49 \, \sqrt {-c^{2} x^{2} + 1} a b d^{3} x}{2560 \, c^{3}} + \frac {49 \, {\left (c^{2} x^{2} - 1\right )}^{2} b^{2} d^{3}}{15360 \, c^{4}} + \frac {49 \, b^{2} d^{3} \arcsin \left (c x\right )^{2}}{5120 \, c^{4}} - \frac {49 \, {\left (c^{2} x^{2} - 1\right )} b^{2} d^{3}}{5120 \, c^{4}} + \frac {49 \, a b d^{3} \arcsin \left (c x\right )}{2560 \, c^{4}} - \frac {232981 \, b^{2} d^{3}}{36864000 \, c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/10*a^2*c^6*d^3*x^10 + 3/8*a^2*c^4*d^3*x^8 - 1/2*a^2*c^2*d^3*x^6 + 1/4*a^2*d^3*x^4 - 1/50*(c^2*x^2 - 1)^4*sq
rt(-c^2*x^2 + 1)*b^2*d^3*x*arcsin(c*x)/c^3 - 1/10*(c^2*x^2 - 1)^5*b^2*d^3*arcsin(c*x)^2/c^4 - 1/50*(c^2*x^2 -
1)^4*sqrt(-c^2*x^2 + 1)*a*b*d^3*x/c^3 - 7/800*(c^2*x^2 - 1)^3*sqrt(-c^2*x^2 + 1)*b^2*d^3*x*arcsin(c*x)/c^3 - 1
/5*(c^2*x^2 - 1)^5*a*b*d^3*arcsin(c*x)/c^4 - 1/8*(c^2*x^2 - 1)^4*b^2*d^3*arcsin(c*x)^2/c^4 - 7/800*(c^2*x^2 -
1)^3*sqrt(-c^2*x^2 + 1)*a*b*d^3*x/c^3 + 49/4800*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b^2*d^3*x*arcsin(c*x)/c^3 +
1/500*(c^2*x^2 - 1)^5*b^2*d^3/c^4 - 1/4*(c^2*x^2 - 1)^4*a*b*d^3*arcsin(c*x)/c^4 + 49/4800*(c^2*x^2 - 1)^2*sqr
t(-c^2*x^2 + 1)*a*b*d^3*x/c^3 + 49/3840*(-c^2*x^2 + 1)^(3/2)*b^2*d^3*x*arcsin(c*x)/c^3 + 7/6400*(c^2*x^2 - 1)^
4*b^2*d^3/c^4 + 49/3840*(-c^2*x^2 + 1)^(3/2)*a*b*d^3*x/c^3 + 49/2560*sqrt(-c^2*x^2 + 1)*b^2*d^3*x*arcsin(c*x)/
c^3 - 49/28800*(c^2*x^2 - 1)^3*b^2*d^3/c^4 + 49/2560*sqrt(-c^2*x^2 + 1)*a*b*d^3*x/c^3 + 49/15360*(c^2*x^2 - 1)
^2*b^2*d^3/c^4 + 49/5120*b^2*d^3*arcsin(c*x)^2/c^4 - 49/5120*(c^2*x^2 - 1)*b^2*d^3/c^4 + 49/2560*a*b*d^3*arcsi
n(c*x)/c^4 - 232981/36864000*b^2*d^3/c^4

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maple [A]  time = 0.17, size = 519, normalized size = 1.35 \[ \frac {-d^{3} a^{2} \left (\frac {1}{10} c^{10} x^{10}-\frac {3}{8} c^{8} x^{8}+\frac {1}{2} c^{6} x^{6}-\frac {1}{4} c^{4} x^{4}\right )-d^{3} b^{2} \left (\frac {\arcsin \left (c x \right )^{2} \left (c^{2} x^{2}-1\right )^{4}}{8}-\frac {\arcsin \left (c x \right ) \left (-48 c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}+200 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}-326 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+279 c x \sqrt {-c^{2} x^{2}+1}+105 \arcsin \left (c x \right )\right )}{1536}+\frac {49 \arcsin \left (c x \right )^{2}}{5120}-\frac {7 \left (c^{2} x^{2}-1\right )^{4}}{6400}+\frac {49 \left (c^{2} x^{2}-1\right )^{3}}{28800}-\frac {49 \left (c^{2} x^{2}-1\right )^{2}}{15360}+\frac {49 c^{2} x^{2}}{5120}-\frac {49}{5120}+\frac {\arcsin \left (c x \right )^{2} \left (c^{2} x^{2}-1\right )^{5}}{10}+\frac {\arcsin \left (c x \right ) \left (128 c^{9} x^{9} \sqrt {-c^{2} x^{2}+1}-656 c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}+1368 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}-1490 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+965 c x \sqrt {-c^{2} x^{2}+1}+315 \arcsin \left (c x \right )\right )}{6400}-\frac {\left (c^{2} x^{2}-1\right )^{5}}{500}\right )-2 d^{3} a b \left (\frac {\arcsin \left (c x \right ) c^{10} x^{10}}{10}-\frac {3 \arcsin \left (c x \right ) c^{8} x^{8}}{8}+\frac {\arcsin \left (c x \right ) c^{6} x^{6}}{2}-\frac {c^{4} x^{4} \arcsin \left (c x \right )}{4}+\frac {c^{9} x^{9} \sqrt {-c^{2} x^{2}+1}}{100}-\frac {57 c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}}{1600}+\frac {401 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{9600}-\frac {79 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{7680}-\frac {79 c x \sqrt {-c^{2} x^{2}+1}}{5120}+\frac {79 \arcsin \left (c x \right )}{5120}\right )}{c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^4*(-d^3*a^2*(1/10*c^10*x^10-3/8*c^8*x^8+1/2*c^6*x^6-1/4*c^4*x^4)-d^3*b^2*(1/8*arcsin(c*x)^2*(c^2*x^2-1)^4-
1/1536*arcsin(c*x)*(-48*c^7*x^7*(-c^2*x^2+1)^(1/2)+200*c^5*x^5*(-c^2*x^2+1)^(1/2)-326*c^3*x^3*(-c^2*x^2+1)^(1/
2)+279*c*x*(-c^2*x^2+1)^(1/2)+105*arcsin(c*x))+49/5120*arcsin(c*x)^2-7/6400*(c^2*x^2-1)^4+49/28800*(c^2*x^2-1)
^3-49/15360*(c^2*x^2-1)^2+49/5120*c^2*x^2-49/5120+1/10*arcsin(c*x)^2*(c^2*x^2-1)^5+1/6400*arcsin(c*x)*(128*c^9
*x^9*(-c^2*x^2+1)^(1/2)-656*c^7*x^7*(-c^2*x^2+1)^(1/2)+1368*c^5*x^5*(-c^2*x^2+1)^(1/2)-1490*c^3*x^3*(-c^2*x^2+
1)^(1/2)+965*c*x*(-c^2*x^2+1)^(1/2)+315*arcsin(c*x))-1/500*(c^2*x^2-1)^5)-2*d^3*a*b*(1/10*arcsin(c*x)*c^10*x^1
0-3/8*arcsin(c*x)*c^8*x^8+1/2*arcsin(c*x)*c^6*x^6-1/4*c^4*x^4*arcsin(c*x)+1/100*c^9*x^9*(-c^2*x^2+1)^(1/2)-57/
1600*c^7*x^7*(-c^2*x^2+1)^(1/2)+401/9600*c^5*x^5*(-c^2*x^2+1)^(1/2)-79/7680*c^3*x^3*(-c^2*x^2+1)^(1/2)-79/5120
*c*x*(-c^2*x^2+1)^(1/2)+79/5120*arcsin(c*x)))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{10} \, a^{2} c^{6} d^{3} x^{10} + \frac {3}{8} \, a^{2} c^{4} d^{3} x^{8} - \frac {1}{2} \, a^{2} c^{2} d^{3} x^{6} - \frac {1}{6400} \, {\left (1280 \, x^{10} \arcsin \left (c x\right ) + {\left (\frac {128 \, \sqrt {-c^{2} x^{2} + 1} x^{9}}{c^{2}} + \frac {144 \, \sqrt {-c^{2} x^{2} + 1} x^{7}}{c^{4}} + \frac {168 \, \sqrt {-c^{2} x^{2} + 1} x^{5}}{c^{6}} + \frac {210 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{8}} + \frac {315 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{10}} - \frac {315 \, \arcsin \left (c x\right )}{c^{11}}\right )} c\right )} a b c^{6} d^{3} + \frac {1}{512} \, {\left (384 \, x^{8} \arcsin \left (c x\right ) + {\left (\frac {48 \, \sqrt {-c^{2} x^{2} + 1} x^{7}}{c^{2}} + \frac {56 \, \sqrt {-c^{2} x^{2} + 1} x^{5}}{c^{4}} + \frac {70 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{6}} + \frac {105 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{8}} - \frac {105 \, \arcsin \left (c x\right )}{c^{9}}\right )} c\right )} a b c^{4} d^{3} + \frac {1}{4} \, a^{2} d^{3} x^{4} - \frac {1}{48} \, {\left (48 \, x^{6} \arcsin \left (c x\right ) + {\left (\frac {8 \, \sqrt {-c^{2} x^{2} + 1} x^{5}}{c^{2}} + \frac {10 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{4}} + \frac {15 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{6}} - \frac {15 \, \arcsin \left (c x\right )}{c^{7}}\right )} c\right )} a b c^{2} d^{3} + \frac {1}{16} \, {\left (8 \, x^{4} \arcsin \left (c x\right ) + {\left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{4}} - \frac {3 \, \arcsin \left (c x\right )}{c^{5}}\right )} c\right )} a b d^{3} - \frac {1}{40} \, {\left (4 \, b^{2} c^{6} d^{3} x^{10} - 15 \, b^{2} c^{4} d^{3} x^{8} + 20 \, b^{2} c^{2} d^{3} x^{6} - 10 \, b^{2} d^{3} x^{4}\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} - \int \frac {{\left (4 \, b^{2} c^{7} d^{3} x^{10} - 15 \, b^{2} c^{5} d^{3} x^{8} + 20 \, b^{2} c^{3} d^{3} x^{6} - 10 \, b^{2} c d^{3} x^{4}\right )} \sqrt {c x + 1} \sqrt {-c x + 1} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )}{20 \, {\left (c^{2} x^{2} - 1\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/10*a^2*c^6*d^3*x^10 + 3/8*a^2*c^4*d^3*x^8 - 1/2*a^2*c^2*d^3*x^6 - 1/6400*(1280*x^10*arcsin(c*x) + (128*sqrt
(-c^2*x^2 + 1)*x^9/c^2 + 144*sqrt(-c^2*x^2 + 1)*x^7/c^4 + 168*sqrt(-c^2*x^2 + 1)*x^5/c^6 + 210*sqrt(-c^2*x^2 +
1)*x^3/c^8 + 315*sqrt(-c^2*x^2 + 1)*x/c^10 - 315*arcsin(c*x)/c^11)*c)*a*b*c^6*d^3 + 1/512*(384*x^8*arcsin(c*x
) + (48*sqrt(-c^2*x^2 + 1)*x^7/c^2 + 56*sqrt(-c^2*x^2 + 1)*x^5/c^4 + 70*sqrt(-c^2*x^2 + 1)*x^3/c^6 + 105*sqrt(
-c^2*x^2 + 1)*x/c^8 - 105*arcsin(c*x)/c^9)*c)*a*b*c^4*d^3 + 1/4*a^2*d^3*x^4 - 1/48*(48*x^6*arcsin(c*x) + (8*sq
rt(-c^2*x^2 + 1)*x^5/c^2 + 10*sqrt(-c^2*x^2 + 1)*x^3/c^4 + 15*sqrt(-c^2*x^2 + 1)*x/c^6 - 15*arcsin(c*x)/c^7)*c
)*a*b*c^2*d^3 + 1/16*(8*x^4*arcsin(c*x) + (2*sqrt(-c^2*x^2 + 1)*x^3/c^2 + 3*sqrt(-c^2*x^2 + 1)*x/c^4 - 3*arcsi
n(c*x)/c^5)*c)*a*b*d^3 - 1/40*(4*b^2*c^6*d^3*x^10 - 15*b^2*c^4*d^3*x^8 + 20*b^2*c^2*d^3*x^6 - 10*b^2*d^3*x^4)*
arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 - integrate(1/20*(4*b^2*c^7*d^3*x^10 - 15*b^2*c^5*d^3*x^8 + 20*b^
2*c^3*d^3*x^6 - 10*b^2*c*d^3*x^4)*sqrt(c*x + 1)*sqrt(-c*x + 1)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))/(c^2
*x^2 - 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 52.75, size = 654, normalized size = 1.70 \[ \begin {cases} - \frac {a^{2} c^{6} d^{3} x^{10}}{10} + \frac {3 a^{2} c^{4} d^{3} x^{8}}{8} - \frac {a^{2} c^{2} d^{3} x^{6}}{2} + \frac {a^{2} d^{3} x^{4}}{4} - \frac {a b c^{6} d^{3} x^{10} \operatorname {asin}{\left (c x \right )}}{5} - \frac {a b c^{5} d^{3} x^{9} \sqrt {- c^{2} x^{2} + 1}}{50} + \frac {3 a b c^{4} d^{3} x^{8} \operatorname {asin}{\left (c x \right )}}{4} + \frac {57 a b c^{3} d^{3} x^{7} \sqrt {- c^{2} x^{2} + 1}}{800} - a b c^{2} d^{3} x^{6} \operatorname {asin}{\left (c x \right )} - \frac {401 a b c d^{3} x^{5} \sqrt {- c^{2} x^{2} + 1}}{4800} + \frac {a b d^{3} x^{4} \operatorname {asin}{\left (c x \right )}}{2} + \frac {79 a b d^{3} x^{3} \sqrt {- c^{2} x^{2} + 1}}{3840 c} + \frac {79 a b d^{3} x \sqrt {- c^{2} x^{2} + 1}}{2560 c^{3}} - \frac {79 a b d^{3} \operatorname {asin}{\left (c x \right )}}{2560 c^{4}} - \frac {b^{2} c^{6} d^{3} x^{10} \operatorname {asin}^{2}{\left (c x \right )}}{10} + \frac {b^{2} c^{6} d^{3} x^{10}}{500} - \frac {b^{2} c^{5} d^{3} x^{9} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{50} + \frac {3 b^{2} c^{4} d^{3} x^{8} \operatorname {asin}^{2}{\left (c x \right )}}{8} - \frac {57 b^{2} c^{4} d^{3} x^{8}}{6400} + \frac {57 b^{2} c^{3} d^{3} x^{7} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{800} - \frac {b^{2} c^{2} d^{3} x^{6} \operatorname {asin}^{2}{\left (c x \right )}}{2} + \frac {401 b^{2} c^{2} d^{3} x^{6}}{28800} - \frac {401 b^{2} c d^{3} x^{5} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{4800} + \frac {b^{2} d^{3} x^{4} \operatorname {asin}^{2}{\left (c x \right )}}{4} - \frac {79 b^{2} d^{3} x^{4}}{15360} + \frac {79 b^{2} d^{3} x^{3} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{3840 c} - \frac {79 b^{2} d^{3} x^{2}}{5120 c^{2}} + \frac {79 b^{2} d^{3} x \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{2560 c^{3}} - \frac {79 b^{2} d^{3} \operatorname {asin}^{2}{\left (c x \right )}}{5120 c^{4}} & \text {for}\: c \neq 0 \\\frac {a^{2} d^{3} x^{4}}{4} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-a**2*c**6*d**3*x**10/10 + 3*a**2*c**4*d**3*x**8/8 - a**2*c**2*d**3*x**6/2 + a**2*d**3*x**4/4 - a*b
*c**6*d**3*x**10*asin(c*x)/5 - a*b*c**5*d**3*x**9*sqrt(-c**2*x**2 + 1)/50 + 3*a*b*c**4*d**3*x**8*asin(c*x)/4 +
57*a*b*c**3*d**3*x**7*sqrt(-c**2*x**2 + 1)/800 - a*b*c**2*d**3*x**6*asin(c*x) - 401*a*b*c*d**3*x**5*sqrt(-c**
2*x**2 + 1)/4800 + a*b*d**3*x**4*asin(c*x)/2 + 79*a*b*d**3*x**3*sqrt(-c**2*x**2 + 1)/(3840*c) + 79*a*b*d**3*x*
sqrt(-c**2*x**2 + 1)/(2560*c**3) - 79*a*b*d**3*asin(c*x)/(2560*c**4) - b**2*c**6*d**3*x**10*asin(c*x)**2/10 +
b**2*c**6*d**3*x**10/500 - b**2*c**5*d**3*x**9*sqrt(-c**2*x**2 + 1)*asin(c*x)/50 + 3*b**2*c**4*d**3*x**8*asin(
c*x)**2/8 - 57*b**2*c**4*d**3*x**8/6400 + 57*b**2*c**3*d**3*x**7*sqrt(-c**2*x**2 + 1)*asin(c*x)/800 - b**2*c**
2*d**3*x**6*asin(c*x)**2/2 + 401*b**2*c**2*d**3*x**6/28800 - 401*b**2*c*d**3*x**5*sqrt(-c**2*x**2 + 1)*asin(c*
x)/4800 + b**2*d**3*x**4*asin(c*x)**2/4 - 79*b**2*d**3*x**4/15360 + 79*b**2*d**3*x**3*sqrt(-c**2*x**2 + 1)*asi
n(c*x)/(3840*c) - 79*b**2*d**3*x**2/(5120*c**2) + 79*b**2*d**3*x*sqrt(-c**2*x**2 + 1)*asin(c*x)/(2560*c**3) -
79*b**2*d**3*asin(c*x)**2/(5120*c**4), Ne(c, 0)), (a**2*d**3*x**4/4, True))

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