3.227 \(\int x^2 (d-c^2 d x^2)^{5/2} (a+b \sin ^{-1}(c x))^2 \, dx\)

Optimal. Leaf size=556 \[ -\frac{b c^5 d^2 x^8 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{32 \sqrt{1-c^2 x^2}}+\frac{17 b c^3 d^2 x^6 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{144 \sqrt{1-c^2 x^2}}-\frac{59 b c d^2 x^4 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{384 \sqrt{1-c^2 x^2}}+\frac{5}{64} d^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{5 b d^2 x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{128 c \sqrt{1-c^2 x^2}}-\frac{5 d^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{128 c^2}+\frac{5 d^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{384 b c^3 \sqrt{1-c^2 x^2}}+\frac{1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{5}{48} d x^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{1}{256} b^2 c^4 d^2 x^7 \sqrt{d-c^2 d x^2}+\frac{209 b^2 c^2 d^2 x^5 \sqrt{d-c^2 d x^2}}{13824}-\frac{1079 b^2 d^2 x^3 \sqrt{d-c^2 d x^2}}{55296}-\frac{359 b^2 d^2 x \sqrt{d-c^2 d x^2}}{36864 c^2}+\frac{359 b^2 d^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{36864 c^3 \sqrt{1-c^2 x^2}} \]

[Out]

(-359*b^2*d^2*x*Sqrt[d - c^2*d*x^2])/(36864*c^2) - (1079*b^2*d^2*x^3*Sqrt[d - c^2*d*x^2])/55296 + (209*b^2*c^2
*d^2*x^5*Sqrt[d - c^2*d*x^2])/13824 - (b^2*c^4*d^2*x^7*Sqrt[d - c^2*d*x^2])/256 + (359*b^2*d^2*Sqrt[d - c^2*d*
x^2]*ArcSin[c*x])/(36864*c^3*Sqrt[1 - c^2*x^2]) + (5*b*d^2*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(128*c
*Sqrt[1 - c^2*x^2]) - (59*b*c*d^2*x^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(384*Sqrt[1 - c^2*x^2]) + (17*b
*c^3*d^2*x^6*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(144*Sqrt[1 - c^2*x^2]) - (b*c^5*d^2*x^8*Sqrt[d - c^2*d*
x^2]*(a + b*ArcSin[c*x]))/(32*Sqrt[1 - c^2*x^2]) - (5*d^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(128*c^
2) + (5*d^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/64 + (5*d*x^3*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c
*x])^2)/48 + (x^3*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2)/8 + (5*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*
x])^3)/(384*b*c^3*Sqrt[1 - c^2*x^2])

________________________________________________________________________________________

Rubi [A]  time = 1.10658, antiderivative size = 556, normalized size of antiderivative = 1., number of steps used = 25, number of rules used = 14, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.483, Rules used = {4699, 4697, 4707, 4641, 4627, 321, 216, 14, 4687, 12, 459, 266, 43, 1267} \[ -\frac{b c^5 d^2 x^8 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{32 \sqrt{1-c^2 x^2}}+\frac{17 b c^3 d^2 x^6 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{144 \sqrt{1-c^2 x^2}}-\frac{59 b c d^2 x^4 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{384 \sqrt{1-c^2 x^2}}+\frac{5}{64} d^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{5 b d^2 x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{128 c \sqrt{1-c^2 x^2}}-\frac{5 d^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{128 c^2}+\frac{5 d^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{384 b c^3 \sqrt{1-c^2 x^2}}+\frac{1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{5}{48} d x^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{1}{256} b^2 c^4 d^2 x^7 \sqrt{d-c^2 d x^2}+\frac{209 b^2 c^2 d^2 x^5 \sqrt{d-c^2 d x^2}}{13824}-\frac{1079 b^2 d^2 x^3 \sqrt{d-c^2 d x^2}}{55296}-\frac{359 b^2 d^2 x \sqrt{d-c^2 d x^2}}{36864 c^2}+\frac{359 b^2 d^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{36864 c^3 \sqrt{1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2,x]

[Out]

(-359*b^2*d^2*x*Sqrt[d - c^2*d*x^2])/(36864*c^2) - (1079*b^2*d^2*x^3*Sqrt[d - c^2*d*x^2])/55296 + (209*b^2*c^2
*d^2*x^5*Sqrt[d - c^2*d*x^2])/13824 - (b^2*c^4*d^2*x^7*Sqrt[d - c^2*d*x^2])/256 + (359*b^2*d^2*Sqrt[d - c^2*d*
x^2]*ArcSin[c*x])/(36864*c^3*Sqrt[1 - c^2*x^2]) + (5*b*d^2*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(128*c
*Sqrt[1 - c^2*x^2]) - (59*b*c*d^2*x^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(384*Sqrt[1 - c^2*x^2]) + (17*b
*c^3*d^2*x^6*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(144*Sqrt[1 - c^2*x^2]) - (b*c^5*d^2*x^8*Sqrt[d - c^2*d*
x^2]*(a + b*ArcSin[c*x]))/(32*Sqrt[1 - c^2*x^2]) - (5*d^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(128*c^
2) + (5*d^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/64 + (5*d*x^3*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c
*x])^2)/48 + (x^3*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2)/8 + (5*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*
x])^3)/(384*b*c^3*Sqrt[1 - c^2*x^2])

Rule 4699

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

Rule 4697

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

Rule 4707

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

Rule 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 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 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 4687

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

Rule 12

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

Rule 459

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(d*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(b*e*(m + n*(p + 1) + 1)), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m +
n*(p + 1) + 1)), Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0]
&& NeQ[m + n*(p + 1) + 1, 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 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 1267

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Si
mp[(c^p*(f*x)^(m + 4*p - 1)*(d + e*x^2)^(q + 1))/(e*f^(4*p - 1)*(m + 4*p + 2*q + 1)), x] + Dist[1/(e*(m + 4*p
+ 2*q + 1)), Int[(f*x)^m*(d + e*x^2)^q*ExpandToSum[e*(m + 4*p + 2*q + 1)*((a + b*x^2 + c*x^4)^p - c^p*x^(4*p))
- d*c^p*(m + 4*p - 1)*x^(4*p - 2), x], x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[b^2 - 4*a*c, 0] &&
IGtQ[p, 0] &&  !IntegerQ[q] && NeQ[m + 4*p + 2*q + 1, 0]

Rubi steps

\begin{align*} \int x^2 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\frac{1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{8} (5 d) \int x^2 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx-\frac{\left (b c d^2 \sqrt{d-c^2 d x^2}\right ) \int x^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{4 \sqrt{1-c^2 x^2}}\\ &=-\frac{b c d^2 x^4 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 \sqrt{1-c^2 x^2}}+\frac{b c^3 d^2 x^6 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{12 \sqrt{1-c^2 x^2}}-\frac{b c^5 d^2 x^8 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{32 \sqrt{1-c^2 x^2}}+\frac{5}{48} d x^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{16} \left (5 d^2\right ) \int x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx-\frac{\left (5 b c d^2 \sqrt{d-c^2 d x^2}\right ) \int x^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right ) \, dx}{24 \sqrt{1-c^2 x^2}}+\frac{\left (b^2 c^2 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^4 \left (6-8 c^2 x^2+3 c^4 x^4\right )}{24 \sqrt{1-c^2 x^2}} \, dx}{4 \sqrt{1-c^2 x^2}}\\ &=-\frac{11 b c d^2 x^4 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{96 \sqrt{1-c^2 x^2}}+\frac{17 b c^3 d^2 x^6 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{144 \sqrt{1-c^2 x^2}}-\frac{b c^5 d^2 x^8 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{32 \sqrt{1-c^2 x^2}}+\frac{5}{64} d^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{5}{48} d x^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{\left (5 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{1-c^2 x^2}} \, dx}{64 \sqrt{1-c^2 x^2}}-\frac{\left (5 b c d^2 \sqrt{d-c^2 d x^2}\right ) \int x^3 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{32 \sqrt{1-c^2 x^2}}+\frac{\left (b^2 c^2 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^4 \left (6-8 c^2 x^2+3 c^4 x^4\right )}{\sqrt{1-c^2 x^2}} \, dx}{96 \sqrt{1-c^2 x^2}}+\frac{\left (5 b^2 c^2 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^4 \left (3-2 c^2 x^2\right )}{12 \sqrt{1-c^2 x^2}} \, dx}{24 \sqrt{1-c^2 x^2}}\\ &=-\frac{1}{256} b^2 c^4 d^2 x^7 \sqrt{d-c^2 d x^2}-\frac{59 b c d^2 x^4 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{384 \sqrt{1-c^2 x^2}}+\frac{17 b c^3 d^2 x^6 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{144 \sqrt{1-c^2 x^2}}-\frac{b c^5 d^2 x^8 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{32 \sqrt{1-c^2 x^2}}-\frac{5 d^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{128 c^2}+\frac{5}{64} d^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{5}{48} d x^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{\left (b^2 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^4 \left (-48 c^2+43 c^4 x^2\right )}{\sqrt{1-c^2 x^2}} \, dx}{768 \sqrt{1-c^2 x^2}}+\frac{\left (5 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{1-c^2 x^2}} \, dx}{128 c^2 \sqrt{1-c^2 x^2}}+\frac{\left (5 b d^2 \sqrt{d-c^2 d x^2}\right ) \int x \left (a+b \sin ^{-1}(c x)\right ) \, dx}{64 c \sqrt{1-c^2 x^2}}+\frac{\left (5 b^2 c^2 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^4 \left (3-2 c^2 x^2\right )}{\sqrt{1-c^2 x^2}} \, dx}{288 \sqrt{1-c^2 x^2}}+\frac{\left (5 b^2 c^2 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^4}{\sqrt{1-c^2 x^2}} \, dx}{128 \sqrt{1-c^2 x^2}}\\ &=-\frac{5}{512} b^2 d^2 x^3 \sqrt{d-c^2 d x^2}+\frac{209 b^2 c^2 d^2 x^5 \sqrt{d-c^2 d x^2}}{13824}-\frac{1}{256} b^2 c^4 d^2 x^7 \sqrt{d-c^2 d x^2}+\frac{5 b d^2 x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{128 c \sqrt{1-c^2 x^2}}-\frac{59 b c d^2 x^4 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{384 \sqrt{1-c^2 x^2}}+\frac{17 b c^3 d^2 x^6 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{144 \sqrt{1-c^2 x^2}}-\frac{b c^5 d^2 x^8 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{32 \sqrt{1-c^2 x^2}}-\frac{5 d^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{128 c^2}+\frac{5}{64} d^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{5}{48} d x^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{5 d^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{384 b c^3 \sqrt{1-c^2 x^2}}+\frac{\left (15 b^2 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^2}{\sqrt{1-c^2 x^2}} \, dx}{512 \sqrt{1-c^2 x^2}}-\frac{\left (5 b^2 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^2}{\sqrt{1-c^2 x^2}} \, dx}{128 \sqrt{1-c^2 x^2}}+\frac{\left (73 b^2 c^2 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^4}{\sqrt{1-c^2 x^2}} \, dx}{4608 \sqrt{1-c^2 x^2}}+\frac{\left (5 b^2 c^2 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^4}{\sqrt{1-c^2 x^2}} \, dx}{216 \sqrt{1-c^2 x^2}}\\ &=\frac{5 b^2 d^2 x \sqrt{d-c^2 d x^2}}{1024 c^2}-\frac{1079 b^2 d^2 x^3 \sqrt{d-c^2 d x^2}}{55296}+\frac{209 b^2 c^2 d^2 x^5 \sqrt{d-c^2 d x^2}}{13824}-\frac{1}{256} b^2 c^4 d^2 x^7 \sqrt{d-c^2 d x^2}+\frac{5 b d^2 x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{128 c \sqrt{1-c^2 x^2}}-\frac{59 b c d^2 x^4 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{384 \sqrt{1-c^2 x^2}}+\frac{17 b c^3 d^2 x^6 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{144 \sqrt{1-c^2 x^2}}-\frac{b c^5 d^2 x^8 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{32 \sqrt{1-c^2 x^2}}-\frac{5 d^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{128 c^2}+\frac{5}{64} d^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{5}{48} d x^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{5 d^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{384 b c^3 \sqrt{1-c^2 x^2}}+\frac{\left (73 b^2 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^2}{\sqrt{1-c^2 x^2}} \, dx}{6144 \sqrt{1-c^2 x^2}}+\frac{\left (5 b^2 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x^2}{\sqrt{1-c^2 x^2}} \, dx}{288 \sqrt{1-c^2 x^2}}+\frac{\left (15 b^2 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{1024 c^2 \sqrt{1-c^2 x^2}}-\frac{\left (5 b^2 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{256 c^2 \sqrt{1-c^2 x^2}}\\ &=-\frac{359 b^2 d^2 x \sqrt{d-c^2 d x^2}}{36864 c^2}-\frac{1079 b^2 d^2 x^3 \sqrt{d-c^2 d x^2}}{55296}+\frac{209 b^2 c^2 d^2 x^5 \sqrt{d-c^2 d x^2}}{13824}-\frac{1}{256} b^2 c^4 d^2 x^7 \sqrt{d-c^2 d x^2}-\frac{5 b^2 d^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{1024 c^3 \sqrt{1-c^2 x^2}}+\frac{5 b d^2 x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{128 c \sqrt{1-c^2 x^2}}-\frac{59 b c d^2 x^4 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{384 \sqrt{1-c^2 x^2}}+\frac{17 b c^3 d^2 x^6 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{144 \sqrt{1-c^2 x^2}}-\frac{b c^5 d^2 x^8 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{32 \sqrt{1-c^2 x^2}}-\frac{5 d^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{128 c^2}+\frac{5}{64} d^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{5}{48} d x^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{5 d^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{384 b c^3 \sqrt{1-c^2 x^2}}+\frac{\left (73 b^2 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{12288 c^2 \sqrt{1-c^2 x^2}}+\frac{\left (5 b^2 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{576 c^2 \sqrt{1-c^2 x^2}}\\ &=-\frac{359 b^2 d^2 x \sqrt{d-c^2 d x^2}}{36864 c^2}-\frac{1079 b^2 d^2 x^3 \sqrt{d-c^2 d x^2}}{55296}+\frac{209 b^2 c^2 d^2 x^5 \sqrt{d-c^2 d x^2}}{13824}-\frac{1}{256} b^2 c^4 d^2 x^7 \sqrt{d-c^2 d x^2}+\frac{359 b^2 d^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{36864 c^3 \sqrt{1-c^2 x^2}}+\frac{5 b d^2 x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{128 c \sqrt{1-c^2 x^2}}-\frac{59 b c d^2 x^4 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{384 \sqrt{1-c^2 x^2}}+\frac{17 b c^3 d^2 x^6 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{144 \sqrt{1-c^2 x^2}}-\frac{b c^5 d^2 x^8 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{32 \sqrt{1-c^2 x^2}}-\frac{5 d^2 x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{128 c^2}+\frac{5}{64} d^2 x^3 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{5}{48} d x^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{5 d^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{384 b c^3 \sqrt{1-c^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.463001, size = 348, normalized size = 0.63 \[ \frac{d^2 \sqrt{d-c^2 d x^2} \left (3 b \sin ^{-1}(c x) \left (1440 a^2+192 a b c x \sqrt{1-c^2 x^2} \left (48 c^6 x^6-136 c^4 x^4+118 c^2 x^2-15\right )+b^2 \left (-1152 c^8 x^8+4352 c^6 x^6-5664 c^4 x^4+1440 c^2 x^2+359\right )\right )+288 a^2 b c x \sqrt{1-c^2 x^2} \left (48 c^6 x^6-136 c^4 x^4+118 c^2 x^2-15\right )+1440 a^3-96 a b^2 c^2 x^2 \left (36 c^6 x^6-136 c^4 x^4+177 c^2 x^2-45\right )+288 b^2 \sin ^{-1}(c x)^2 \left (15 a+b c x \sqrt{1-c^2 x^2} \left (48 c^6 x^6-136 c^4 x^4+118 c^2 x^2-15\right )\right )-b^3 c x \sqrt{1-c^2 x^2} \left (432 c^6 x^6-1672 c^4 x^4+2158 c^2 x^2+1077\right )+1440 b^3 \sin ^{-1}(c x)^3\right )}{110592 b c^3 \sqrt{1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2,x]

[Out]

(d^2*Sqrt[d - c^2*d*x^2]*(1440*a^3 - 96*a*b^2*c^2*x^2*(-45 + 177*c^2*x^2 - 136*c^4*x^4 + 36*c^6*x^6) + 288*a^2
*b*c*x*Sqrt[1 - c^2*x^2]*(-15 + 118*c^2*x^2 - 136*c^4*x^4 + 48*c^6*x^6) - b^3*c*x*Sqrt[1 - c^2*x^2]*(1077 + 21
58*c^2*x^2 - 1672*c^4*x^4 + 432*c^6*x^6) + 3*b*(1440*a^2 + 192*a*b*c*x*Sqrt[1 - c^2*x^2]*(-15 + 118*c^2*x^2 -
136*c^4*x^4 + 48*c^6*x^6) + b^2*(359 + 1440*c^2*x^2 - 5664*c^4*x^4 + 4352*c^6*x^6 - 1152*c^8*x^8))*ArcSin[c*x]
+ 288*b^2*(15*a + b*c*x*Sqrt[1 - c^2*x^2]*(-15 + 118*c^2*x^2 - 136*c^4*x^4 + 48*c^6*x^6))*ArcSin[c*x]^2 + 144
0*b^3*ArcSin[c*x]^3))/(110592*b*c^3*Sqrt[1 - c^2*x^2])

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Maple [B]  time = 0.571, size = 1375, normalized size = 2.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))^2,x)

[Out]

-1/8*a^2*x*(-c^2*d*x^2+d)^(7/2)/c^2/d+5/192*a^2/c^2*d*x*(-c^2*d*x^2+d)^(3/2)+5/128*a^2/c^2*d^2*x*(-c^2*d*x^2+d
)^(1/2)+1/48*a^2/c^2*x*(-c^2*d*x^2+d)^(5/2)-5/128*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2/c/(c^2*x^2-1)*arcsin(c*x)*(-c
^2*x^2+1)^(1/2)*x^2+59/384*a*b*(-d*(c^2*x^2-1))^(1/2)*d^2*c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^4-5/128*a*b*(-d*(
c^2*x^2-1))^(1/2)*d^2/c/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^2+1/4*a*b*(-d*(c^2*x^2-1))^(1/2)*d^2*c^6/(c^2*x^2-1)*
arcsin(c*x)*x^9-23/24*a*b*(-d*(c^2*x^2-1))^(1/2)*d^2*c^4/(c^2*x^2-1)*arcsin(c*x)*x^7+127/96*a*b*(-d*(c^2*x^2-1
))^(1/2)*d^2*c^2/(c^2*x^2-1)*arcsin(c*x)*x^5+5/64*a*b*(-d*(c^2*x^2-1))^(1/2)*d^2/c^2/(c^2*x^2-1)*arcsin(c*x)*x
-5/128*a*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^3/(c^2*x^2-1)*arcsin(c*x)^2*d^2+1/32*a*b*(-d*(c^2*x^2-1
))^(1/2)*d^2*c^5/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x^8-17/144*a*b*(-d*(c^2*x^2-1))^(1/2)*d^2*c^3/(c^2*x^2-1)*(-c^
2*x^2+1)^(1/2)*x^6+1/32*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2*c^5/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*x^8-17/1
44*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2*c^3/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*x^6+59/384*b^2*(-d*(c^2*x^2-1
))^(1/2)*d^2*c/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*x^4+1081/110592*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2/(c^2*
x^2-1)*x^3+5/128*a^2/c^2*d^3/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))-359/36864*b^2*(-d*(c^2
*x^2-1))^(1/2)*d^2/c^3/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)-5/384*b^2*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1
)^(1/2)/c^3/(c^2*x^2-1)*arcsin(c*x)^3*d^2+1/8*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2*c^6/(c^2*x^2-1)*arcsin(c*x)^2*x^9
-23/48*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2*c^4/(c^2*x^2-1)*arcsin(c*x)^2*x^7-133/192*a*b*(-d*(c^2*x^2-1))^(1/2)*d^2
/(c^2*x^2-1)*arcsin(c*x)*x^3-359/36864*a*b*(-d*(c^2*x^2-1))^(1/2)*d^2/c^3/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)+127/1
92*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2*c^2/(c^2*x^2-1)*arcsin(c*x)^2*x^5+5/128*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2/c^2/(
c^2*x^2-1)*arcsin(c*x)^2*x-133/384*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2/(c^2*x^2-1)*arcsin(c*x)^2*x^3-1/256*b^2*(-d*
(c^2*x^2-1))^(1/2)*d^2*c^6/(c^2*x^2-1)*x^9+263/13824*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2*c^4/(c^2*x^2-1)*x^7-1915/5
5296*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2*c^2/(c^2*x^2-1)*x^5+359/36864*b^2*(-d*(c^2*x^2-1))^(1/2)*d^2/c^2/(c^2*x^2-
1)*x

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a^{2} c^{4} d^{2} x^{6} - 2 \, a^{2} c^{2} d^{2} x^{4} + a^{2} d^{2} x^{2} +{\left (b^{2} c^{4} d^{2} x^{6} - 2 \, b^{2} c^{2} d^{2} x^{4} + b^{2} d^{2} x^{2}\right )} \arcsin \left (c x\right )^{2} + 2 \,{\left (a b c^{4} d^{2} x^{6} - 2 \, a b c^{2} d^{2} x^{4} + a b d^{2} x^{2}\right )} \arcsin \left (c x\right )\right )} \sqrt{-c^{2} d x^{2} + d}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))^2,x, algorithm="fricas")

[Out]

integral((a^2*c^4*d^2*x^6 - 2*a^2*c^2*d^2*x^4 + a^2*d^2*x^2 + (b^2*c^4*d^2*x^6 - 2*b^2*c^2*d^2*x^4 + b^2*d^2*x
^2)*arcsin(c*x)^2 + 2*(a*b*c^4*d^2*x^6 - 2*a*b*c^2*d^2*x^4 + a*b*d^2*x^2)*arcsin(c*x))*sqrt(-c^2*d*x^2 + d), x
)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(-c**2*d*x**2+d)**(5/2)*(a+b*asin(c*x))**2,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-c^{2} d x^{2} + d\right )}^{\frac{5}{2}}{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))^2,x, algorithm="giac")

[Out]

integrate((-c^2*d*x^2 + d)^(5/2)*(b*arcsin(c*x) + a)^2*x^2, x)