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

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

[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

________________________________________________________________________________________

Rubi [A]  time = 1.59388, antiderivative size = 384, normalized size of antiderivative = 1., 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 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 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 30

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

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*}

Mathematica [A]  time = 0.437581, 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]

-(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 + 102
60*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 - 2736*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))/(1152000*c^4)

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Maple [A]  time = 0.105, size = 519, normalized size = 1.4 \begin{align*}{\frac{1}{{c}^{4}} \left ( -{d}^{3}{a}^{2} \left ({\frac{{c}^{10}{x}^{10}}{10}}-{\frac{3\,{c}^{8}{x}^{8}}{8}}+{\frac{{c}^{6}{x}^{6}}{2}}-{\frac{{c}^{4}{x}^{4}}{4}} \right ) -{d}^{3}{b}^{2} \left ({\frac{ \left ( \arcsin \left ( cx \right ) \right ) ^{2} \left ({c}^{2}{x}^{2}-1 \right ) ^{4}}{8}}-{\frac{\arcsin \left ( cx \right ) }{1536} \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\,cx\sqrt{-{c}^{2}{x}^{2}+1}+105\,\arcsin \left ( cx \right ) \right ) }+{\frac{49\, \left ( \arcsin \left ( cx \right ) \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{ \left ( \arcsin \left ( cx \right ) \right ) ^{2} \left ({c}^{2}{x}^{2}-1 \right ) ^{5}}{10}}+{\frac{\arcsin \left ( cx \right ) }{6400} \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\,cx\sqrt{-{c}^{2}{x}^{2}+1}+315\,\arcsin \left ( cx \right ) \right ) }-{\frac{ \left ({c}^{2}{x}^{2}-1 \right ) ^{5}}{500}} \right ) -2\,{d}^{3}ab \left ( 1/10\,\arcsin \left ( cx \right ){c}^{10}{x}^{10}-3/8\,\arcsin \left ( cx \right ){c}^{8}{x}^{8}+1/2\,\arcsin \left ( cx \right ){c}^{6}{x}^{6}-1/4\,{c}^{4}{x}^{4}\arcsin \left ( cx \right ) +{\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\,cx\sqrt{-{c}^{2}{x}^{2}+1}}{5120}}+{\frac{79\,\arcsin \left ( cx \right ) }{5120}} \right ) \right ) } \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

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^2*x/sqrt(c^2))/(sqrt(c^2)*c^10))*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^2*x/sqrt(c^2))/(sqrt(c^2)*c^8))*c)*a*b*c^4*d^3 + 1/
4*a^2*d^3*x^4 - 1/48*(48*x^6*arcsin(c*x) + (8*sqrt(-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^2*x/sqrt(c^2))/(sqrt(c^2)*c^6))*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*arcsin(c^2*x/sqrt(c^2))/(sqrt(c^2)*c^4))*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, s
qrt(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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Fricas [A]  time = 2.02756, size = 938, normalized size = 2.44 \begin{align*} -\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}} \end{align*}

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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Sympy [A]  time = 95.649, size = 654, normalized size = 1.7 \begin{align*} \text{result too large to display} \end{align*}

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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Giac [A]  time = 1.39333, size = 840, normalized size = 2.19 \begin{align*} \text{result too large to display} \end{align*}

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]

-1/50*(c^2*x^2 - 1)^4*sqrt(-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 - 1/10*(c^2*x^2 - 1)^5*a^2*d^3/c^4 + 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*sqrt(-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 - 1/8*(c^2*x^2 - 1)^4*a^2*d^3/c^4 + 7/6400*(c^2*x^2 - 1)^4*b^2*d^3/c^4 + 4
9/3840*(-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*arcsin(c*x)/c^4 - 2329
81/36864000*b^2*d^3/c^4