3.4.20 \(\int (1-a^2-2 a b x-b^2 x^2)^{3/2} \text {ArcSin}(a+b x)^3 \, dx\) [320]

Optimal. Leaf size=245 \[ \frac {51 (a+b x)^2}{128 b}-\frac {3 (a+b x)^4}{128 b}-\frac {45 (a+b x) \sqrt {1-(a+b x)^2} \text {ArcSin}(a+b x)}{64 b}-\frac {3 (a+b x) \left (1-(a+b x)^2\right )^{3/2} \text {ArcSin}(a+b x)}{32 b}+\frac {27 \text {ArcSin}(a+b x)^2}{128 b}-\frac {9 (a+b x)^2 \text {ArcSin}(a+b x)^2}{16 b}+\frac {3 \left (1-(a+b x)^2\right )^2 \text {ArcSin}(a+b x)^2}{16 b}+\frac {3 (a+b x) \sqrt {1-(a+b x)^2} \text {ArcSin}(a+b x)^3}{8 b}+\frac {(a+b x) \left (1-(a+b x)^2\right )^{3/2} \text {ArcSin}(a+b x)^3}{4 b}+\frac {3 \text {ArcSin}(a+b x)^4}{32 b} \]

[Out]

51/128*(b*x+a)^2/b-3/128*(b*x+a)^4/b-3/32*(b*x+a)*(1-(b*x+a)^2)^(3/2)*arcsin(b*x+a)/b+27/128*arcsin(b*x+a)^2/b
-9/16*(b*x+a)^2*arcsin(b*x+a)^2/b+3/16*(1-(b*x+a)^2)^2*arcsin(b*x+a)^2/b+1/4*(b*x+a)*(1-(b*x+a)^2)^(3/2)*arcsi
n(b*x+a)^3/b+3/32*arcsin(b*x+a)^4/b-45/64*(b*x+a)*arcsin(b*x+a)*(1-(b*x+a)^2)^(1/2)/b+3/8*(b*x+a)*arcsin(b*x+a
)^3*(1-(b*x+a)^2)^(1/2)/b

________________________________________________________________________________________

Rubi [A]
time = 0.23, antiderivative size = 245, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 9, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {4891, 4743, 4741, 4737, 4723, 4795, 30, 4767, 14} \begin {gather*} -\frac {9 (a+b x)^2 \text {ArcSin}(a+b x)^2}{16 b}+\frac {\left (1-(a+b x)^2\right )^{3/2} (a+b x) \text {ArcSin}(a+b x)^3}{4 b}+\frac {3 \sqrt {1-(a+b x)^2} (a+b x) \text {ArcSin}(a+b x)^3}{8 b}-\frac {3 \left (1-(a+b x)^2\right )^{3/2} (a+b x) \text {ArcSin}(a+b x)}{32 b}-\frac {45 \sqrt {1-(a+b x)^2} (a+b x) \text {ArcSin}(a+b x)}{64 b}+\frac {3 \text {ArcSin}(a+b x)^4}{32 b}+\frac {3 \left (1-(a+b x)^2\right )^2 \text {ArcSin}(a+b x)^2}{16 b}+\frac {27 \text {ArcSin}(a+b x)^2}{128 b}-\frac {3 (a+b x)^4}{128 b}+\frac {51 (a+b x)^2}{128 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - a^2 - 2*a*b*x - b^2*x^2)^(3/2)*ArcSin[a + b*x]^3,x]

[Out]

(51*(a + b*x)^2)/(128*b) - (3*(a + b*x)^4)/(128*b) - (45*(a + b*x)*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x])/(64*
b) - (3*(a + b*x)*(1 - (a + b*x)^2)^(3/2)*ArcSin[a + b*x])/(32*b) + (27*ArcSin[a + b*x]^2)/(128*b) - (9*(a + b
*x)^2*ArcSin[a + b*x]^2)/(16*b) + (3*(1 - (a + b*x)^2)^2*ArcSin[a + b*x]^2)/(16*b) + (3*(a + b*x)*Sqrt[1 - (a
+ b*x)^2]*ArcSin[a + b*x]^3)/(8*b) + ((a + b*x)*(1 - (a + b*x)^2)^(3/2)*ArcSin[a + b*x]^3)/(4*b) + (3*ArcSin[a
+ b*x]^4)/(32*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 30

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

Rule 4723

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 4737

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

Rule 4741

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

Rule 4743

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

Rule 4767

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/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^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 4795

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[f
*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(e*(m + 2*p + 1))), x] + (Dist[f^2*((m - 1)/(c^2*(m
+ 2*p + 1))), Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] + Dist[b*f*(n/(c*(m + 2*p + 1)))*S
imp[(d + e*x^2)^p/(1 - c^2*x^2)^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, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && IGtQ[m, 1] && NeQ[m + 2*p + 1, 0]

Rule 4891

Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)^(p_.), x_Symbol] :> Di
st[1/d, Subst[Int[(-C/d^2 + (C/d^2)*x^2)^p*(a + b*ArcSin[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, A, B
, C, n, p}, x] && EqQ[B*(1 - c^2) + 2*A*c*d, 0] && EqQ[2*c*C - B*d, 0]

Rubi steps

\begin {align*} \int \left (1-a^2-2 a b x-b^2 x^2\right )^{3/2} \sin ^{-1}(a+b x)^3 \, dx &=\frac {\text {Subst}\left (\int \left (1-x^2\right )^{3/2} \sin ^{-1}(x)^3 \, dx,x,a+b x\right )}{b}\\ &=\frac {(a+b x) \left (1-(a+b x)^2\right )^{3/2} \sin ^{-1}(a+b x)^3}{4 b}-\frac {3 \text {Subst}\left (\int x \left (1-x^2\right ) \sin ^{-1}(x)^2 \, dx,x,a+b x\right )}{4 b}+\frac {3 \text {Subst}\left (\int \sqrt {1-x^2} \sin ^{-1}(x)^3 \, dx,x,a+b x\right )}{4 b}\\ &=\frac {3 \left (1-(a+b x)^2\right )^2 \sin ^{-1}(a+b x)^2}{16 b}+\frac {3 (a+b x) \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)^3}{8 b}+\frac {(a+b x) \left (1-(a+b x)^2\right )^{3/2} \sin ^{-1}(a+b x)^3}{4 b}-\frac {3 \text {Subst}\left (\int \left (1-x^2\right )^{3/2} \sin ^{-1}(x) \, dx,x,a+b x\right )}{8 b}+\frac {3 \text {Subst}\left (\int \frac {\sin ^{-1}(x)^3}{\sqrt {1-x^2}} \, dx,x,a+b x\right )}{8 b}-\frac {9 \text {Subst}\left (\int x \sin ^{-1}(x)^2 \, dx,x,a+b x\right )}{8 b}\\ &=-\frac {3 (a+b x) \left (1-(a+b x)^2\right )^{3/2} \sin ^{-1}(a+b x)}{32 b}-\frac {9 (a+b x)^2 \sin ^{-1}(a+b x)^2}{16 b}+\frac {3 \left (1-(a+b x)^2\right )^2 \sin ^{-1}(a+b x)^2}{16 b}+\frac {3 (a+b x) \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)^3}{8 b}+\frac {(a+b x) \left (1-(a+b x)^2\right )^{3/2} \sin ^{-1}(a+b x)^3}{4 b}+\frac {3 \sin ^{-1}(a+b x)^4}{32 b}+\frac {3 \text {Subst}\left (\int x \left (1-x^2\right ) \, dx,x,a+b x\right )}{32 b}-\frac {9 \text {Subst}\left (\int \sqrt {1-x^2} \sin ^{-1}(x) \, dx,x,a+b x\right )}{32 b}+\frac {9 \text {Subst}\left (\int \frac {x^2 \sin ^{-1}(x)}{\sqrt {1-x^2}} \, dx,x,a+b x\right )}{8 b}\\ &=-\frac {45 (a+b x) \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)}{64 b}-\frac {3 (a+b x) \left (1-(a+b x)^2\right )^{3/2} \sin ^{-1}(a+b x)}{32 b}-\frac {9 (a+b x)^2 \sin ^{-1}(a+b x)^2}{16 b}+\frac {3 \left (1-(a+b x)^2\right )^2 \sin ^{-1}(a+b x)^2}{16 b}+\frac {3 (a+b x) \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)^3}{8 b}+\frac {(a+b x) \left (1-(a+b x)^2\right )^{3/2} \sin ^{-1}(a+b x)^3}{4 b}+\frac {3 \sin ^{-1}(a+b x)^4}{32 b}+\frac {3 \text {Subst}\left (\int \left (x-x^3\right ) \, dx,x,a+b x\right )}{32 b}+\frac {9 \text {Subst}(\int x \, dx,x,a+b x)}{64 b}-\frac {9 \text {Subst}\left (\int \frac {\sin ^{-1}(x)}{\sqrt {1-x^2}} \, dx,x,a+b x\right )}{64 b}+\frac {9 \text {Subst}(\int x \, dx,x,a+b x)}{16 b}+\frac {9 \text {Subst}\left (\int \frac {\sin ^{-1}(x)}{\sqrt {1-x^2}} \, dx,x,a+b x\right )}{16 b}\\ &=\frac {51 (a+b x)^2}{128 b}-\frac {3 (a+b x)^4}{128 b}-\frac {45 (a+b x) \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)}{64 b}-\frac {3 (a+b x) \left (1-(a+b x)^2\right )^{3/2} \sin ^{-1}(a+b x)}{32 b}+\frac {27 \sin ^{-1}(a+b x)^2}{128 b}-\frac {9 (a+b x)^2 \sin ^{-1}(a+b x)^2}{16 b}+\frac {3 \left (1-(a+b x)^2\right )^2 \sin ^{-1}(a+b x)^2}{16 b}+\frac {3 (a+b x) \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)^3}{8 b}+\frac {(a+b x) \left (1-(a+b x)^2\right )^{3/2} \sin ^{-1}(a+b x)^3}{4 b}+\frac {3 \sin ^{-1}(a+b x)^4}{32 b}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.15, size = 272, normalized size = 1.11 \begin {gather*} \frac {6 a \left (17-2 a^2\right ) b x+3 \left (17-6 a^2\right ) b^2 x^2-12 a b^3 x^3-3 b^4 x^4+6 \sqrt {1-a^2-2 a b x-b^2 x^2} \left (-17 a+2 a^3-17 b x+6 a^2 b x+6 a b^2 x^2+2 b^3 x^3\right ) \text {ArcSin}(a+b x)+3 \left (17+8 a^4+32 a^3 b x-40 b^2 x^2+8 b^4 x^4+16 a b x \left (-5+2 b^2 x^2\right )+8 a^2 \left (-5+6 b^2 x^2\right )\right ) \text {ArcSin}(a+b x)^2-16 \sqrt {1-a^2-2 a b x-b^2 x^2} \left (-5 a+2 a^3-5 b x+6 a^2 b x+6 a b^2 x^2+2 b^3 x^3\right ) \text {ArcSin}(a+b x)^3+12 \text {ArcSin}(a+b x)^4}{128 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - a^2 - 2*a*b*x - b^2*x^2)^(3/2)*ArcSin[a + b*x]^3,x]

[Out]

(6*a*(17 - 2*a^2)*b*x + 3*(17 - 6*a^2)*b^2*x^2 - 12*a*b^3*x^3 - 3*b^4*x^4 + 6*Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2
]*(-17*a + 2*a^3 - 17*b*x + 6*a^2*b*x + 6*a*b^2*x^2 + 2*b^3*x^3)*ArcSin[a + b*x] + 3*(17 + 8*a^4 + 32*a^3*b*x
- 40*b^2*x^2 + 8*b^4*x^4 + 16*a*b*x*(-5 + 2*b^2*x^2) + 8*a^2*(-5 + 6*b^2*x^2))*ArcSin[a + b*x]^2 - 16*Sqrt[1 -
a^2 - 2*a*b*x - b^2*x^2]*(-5*a + 2*a^3 - 5*b*x + 6*a^2*b*x + 6*a*b^2*x^2 + 2*b^3*x^3)*ArcSin[a + b*x]^3 + 12*
ArcSin[a + b*x]^4)/(128*b)

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(627\) vs. \(2(217)=434\).
time = 0.50, size = 628, normalized size = 2.56

method	result	size
default	\(\frac {-75+408 a b x +48 \arcsin \left (b x +a \right )^{4}+320 \arcsin \left (b x +a \right )^{3} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, b x -960 \arcsin \left (b x +a \right )^{2} a b x -408 \arcsin \left (b x +a \right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, b x +204 a^{2}-12 b^{4} x^{4}+96 \arcsin \left (b x +a \right )^{2} a^{4}-128 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \arcsin \left (b x +a \right )^{3} b^{3} x^{3}+48 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \arcsin \left (b x +a \right ) b^{3} x^{3}+384 \arcsin \left (b x +a \right )^{2} a \,b^{3} x^{3}+576 \arcsin \left (b x +a \right )^{2} a^{2} b^{2} x^{2}+384 \arcsin \left (b x +a \right )^{2} a^{3} b x +96 \arcsin \left (b x +a \right )^{2} b^{4} x^{4}-128 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \arcsin \left (b x +a \right )^{3} a^{3}+48 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \arcsin \left (b x +a \right ) a^{3}-48 a \,b^{3} x^{3}-72 a^{2} b^{2} x^{2}-48 a^{3} b x -12 a^{4}-480 \arcsin \left (b x +a \right )^{2} b^{2} x^{2}+320 \arcsin \left (b x +a \right )^{3} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a -408 \arcsin \left (b x +a \right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a -480 \arcsin \left (b x +a \right )^{2} a^{2}+204 \arcsin \left (b x +a \right )^{2}+204 b^{2} x^{2}+144 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \arcsin \left (b x +a \right ) a^{2} b x -384 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \arcsin \left (b x +a \right )^{3} a \,b^{2} x^{2}-384 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \arcsin \left (b x +a \right )^{3} a^{2} b x +144 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \arcsin \left (b x +a \right ) a \,b^{2} x^{2}}{512 b}\)	\(628\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-b^2*x^2-2*a*b*x-a^2+1)^(3/2)*arcsin(b*x+a)^3,x,method=_RETURNVERBOSE)

[Out]

1/512*(-75+408*a*b*x+48*arcsin(b*x+a)^4+320*arcsin(b*x+a)^3*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*b*x-960*arcsin(b*x+
a)^2*a*b*x-408*arcsin(b*x+a)*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*b*x+204*a^2-12*b^4*x^4+96*arcsin(b*x+a)^2*a^4-128*
(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*arcsin(b*x+a)^3*b^3*x^3+48*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*arcsin(b*x+a)*b^3*x^3
+384*arcsin(b*x+a)^2*a*b^3*x^3+576*arcsin(b*x+a)^2*a^2*b^2*x^2+384*arcsin(b*x+a)^2*a^3*b*x+96*arcsin(b*x+a)^2*
b^4*x^4-128*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*arcsin(b*x+a)^3*a^3+48*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*arcsin(b*x+a)
*a^3-48*a*b^3*x^3-72*a^2*b^2*x^2-48*a^3*b*x-12*a^4-480*arcsin(b*x+a)^2*b^2*x^2+320*arcsin(b*x+a)^3*(-b^2*x^2-2
*a*b*x-a^2+1)^(1/2)*a-408*arcsin(b*x+a)*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*a-480*arcsin(b*x+a)^2*a^2+204*arcsin(b*
x+a)^2+204*b^2*x^2+144*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*arcsin(b*x+a)*a^2*b*x-384*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)
*arcsin(b*x+a)^3*a*b^2*x^2-384*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*arcsin(b*x+a)^3*a^2*b*x+144*(-b^2*x^2-2*a*b*x-a^
2+1)^(1/2)*arcsin(b*x+a)*a*b^2*x^2)/b

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b^2*x^2-2*a*b*x-a^2+1)^(3/2)*arcsin(b*x+a)^3,x, algorithm="maxima")

[Out]

integrate((-b^2*x^2 - 2*a*b*x - a^2 + 1)^(3/2)*arcsin(b*x + a)^3, x)

________________________________________________________________________________________

Fricas [A]
time = 2.62, size = 243, normalized size = 0.99 \begin {gather*} -\frac {3 \, b^{4} x^{4} + 12 \, a b^{3} x^{3} + 3 \, {\left (6 \, a^{2} - 17\right )} b^{2} x^{2} - 12 \, \arcsin \left (b x + a\right )^{4} + 6 \, {\left (2 \, a^{3} - 17 \, a\right )} b x - 3 \, {\left (8 \, b^{4} x^{4} + 32 \, a b^{3} x^{3} + 8 \, {\left (6 \, a^{2} - 5\right )} b^{2} x^{2} + 8 \, a^{4} + 16 \, {\left (2 \, a^{3} - 5 \, a\right )} b x - 40 \, a^{2} + 17\right )} \arcsin \left (b x + a\right )^{2} + 2 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (8 \, {\left (2 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} + 2 \, a^{3} + {\left (6 \, a^{2} - 5\right )} b x - 5 \, a\right )} \arcsin \left (b x + a\right )^{3} - 3 \, {\left (2 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} + 2 \, a^{3} + {\left (6 \, a^{2} - 17\right )} b x - 17 \, a\right )} \arcsin \left (b x + a\right )\right )}}{128 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b^2*x^2-2*a*b*x-a^2+1)^(3/2)*arcsin(b*x+a)^3,x, algorithm="fricas")

[Out]

-1/128*(3*b^4*x^4 + 12*a*b^3*x^3 + 3*(6*a^2 - 17)*b^2*x^2 - 12*arcsin(b*x + a)^4 + 6*(2*a^3 - 17*a)*b*x - 3*(8
*b^4*x^4 + 32*a*b^3*x^3 + 8*(6*a^2 - 5)*b^2*x^2 + 8*a^4 + 16*(2*a^3 - 5*a)*b*x - 40*a^2 + 17)*arcsin(b*x + a)^
2 + 2*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(8*(2*b^3*x^3 + 6*a*b^2*x^2 + 2*a^3 + (6*a^2 - 5)*b*x - 5*a)*arcsin(b
*x + a)^3 - 3*(2*b^3*x^3 + 6*a*b^2*x^2 + 2*a^3 + (6*a^2 - 17)*b*x - 17*a)*arcsin(b*x + a)))/b

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 694 vs. \(2 (223) = 446\).
time = 1.47, size = 694, normalized size = 2.83 \begin {gather*} \begin {cases} \frac {3 a^{4} \operatorname {asin}^{2}{\left (a + b x \right )}}{16 b} + \frac {3 a^{3} x \operatorname {asin}^{2}{\left (a + b x \right )}}{4} - \frac {3 a^{3} x}{32} - \frac {a^{3} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}^{3}{\left (a + b x \right )}}{4 b} + \frac {3 a^{3} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}{\left (a + b x \right )}}{32 b} + \frac {9 a^{2} b x^{2} \operatorname {asin}^{2}{\left (a + b x \right )}}{8} - \frac {9 a^{2} b x^{2}}{64} - \frac {3 a^{2} x \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}^{3}{\left (a + b x \right )}}{4} + \frac {9 a^{2} x \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}{\left (a + b x \right )}}{32} - \frac {15 a^{2} \operatorname {asin}^{2}{\left (a + b x \right )}}{16 b} + \frac {3 a b^{2} x^{3} \operatorname {asin}^{2}{\left (a + b x \right )}}{4} - \frac {3 a b^{2} x^{3}}{32} - \frac {3 a b x^{2} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}^{3}{\left (a + b x \right )}}{4} + \frac {9 a b x^{2} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}{\left (a + b x \right )}}{32} - \frac {15 a x \operatorname {asin}^{2}{\left (a + b x \right )}}{8} + \frac {51 a x}{64} + \frac {5 a \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}^{3}{\left (a + b x \right )}}{8 b} - \frac {51 a \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}{\left (a + b x \right )}}{64 b} + \frac {3 b^{3} x^{4} \operatorname {asin}^{2}{\left (a + b x \right )}}{16} - \frac {3 b^{3} x^{4}}{128} - \frac {b^{2} x^{3} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}^{3}{\left (a + b x \right )}}{4} + \frac {3 b^{2} x^{3} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}{\left (a + b x \right )}}{32} - \frac {15 b x^{2} \operatorname {asin}^{2}{\left (a + b x \right )}}{16} + \frac {51 b x^{2}}{128} + \frac {5 x \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}^{3}{\left (a + b x \right )}}{8} - \frac {51 x \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}{\left (a + b x \right )}}{64} + \frac {3 \operatorname {asin}^{4}{\left (a + b x \right )}}{32 b} + \frac {51 \operatorname {asin}^{2}{\left (a + b x \right )}}{128 b} & \text {for}\: b \neq 0 \\x \left (1 - a^{2}\right )^{\frac {3}{2}} \operatorname {asin}^{3}{\left (a \right )} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b**2*x**2-2*a*b*x-a**2+1)**(3/2)*asin(b*x+a)**3,x)

[Out]

Piecewise((3*a**4*asin(a + b*x)**2/(16*b) + 3*a**3*x*asin(a + b*x)**2/4 - 3*a**3*x/32 - a**3*sqrt(-a**2 - 2*a*
b*x - b**2*x**2 + 1)*asin(a + b*x)**3/(4*b) + 3*a**3*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)*asin(a + b*x)/(32*b
) + 9*a**2*b*x**2*asin(a + b*x)**2/8 - 9*a**2*b*x**2/64 - 3*a**2*x*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)*asin(
a + b*x)**3/4 + 9*a**2*x*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)*asin(a + b*x)/32 - 15*a**2*asin(a + b*x)**2/(16
*b) + 3*a*b**2*x**3*asin(a + b*x)**2/4 - 3*a*b**2*x**3/32 - 3*a*b*x**2*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)*a
sin(a + b*x)**3/4 + 9*a*b*x**2*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)*asin(a + b*x)/32 - 15*a*x*asin(a + b*x)**
2/8 + 51*a*x/64 + 5*a*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)*asin(a + b*x)**3/(8*b) - 51*a*sqrt(-a**2 - 2*a*b*x
- b**2*x**2 + 1)*asin(a + b*x)/(64*b) + 3*b**3*x**4*asin(a + b*x)**2/16 - 3*b**3*x**4/128 - b**2*x**3*sqrt(-a
**2 - 2*a*b*x - b**2*x**2 + 1)*asin(a + b*x)**3/4 + 3*b**2*x**3*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)*asin(a +
b*x)/32 - 15*b*x**2*asin(a + b*x)**2/16 + 51*b*x**2/128 + 5*x*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)*asin(a +
b*x)**3/8 - 51*x*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)*asin(a + b*x)/64 + 3*asin(a + b*x)**4/(32*b) + 51*asin(
a + b*x)**2/(128*b), Ne(b, 0)), (x*(1 - a**2)**(3/2)*asin(a)**3, True))

________________________________________________________________________________________

Giac [A]
time = 0.51, size = 296, normalized size = 1.21 \begin {gather*} \frac {{\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac {3}{2}} {\left (b x + a\right )} \arcsin \left (b x + a\right )^{3}}{4 \, b} + \frac {3 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (b x + a\right )} \arcsin \left (b x + a\right )^{3}}{8 \, b} + \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )}^{2} \arcsin \left (b x + a\right )^{2}}{16 \, b} + \frac {3 \, \arcsin \left (b x + a\right )^{4}}{32 \, b} - \frac {3 \, {\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac {3}{2}} {\left (b x + a\right )} \arcsin \left (b x + a\right )}{32 \, b} - \frac {9 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )} \arcsin \left (b x + a\right )^{2}}{16 \, b} - \frac {45 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (b x + a\right )} \arcsin \left (b x + a\right )}{64 \, b} - \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )}^{2}}{128 \, b} - \frac {45 \, \arcsin \left (b x + a\right )^{2}}{128 \, b} + \frac {45 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )}}{128 \, b} + \frac {189}{1024 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b^2*x^2-2*a*b*x-a^2+1)^(3/2)*arcsin(b*x+a)^3,x, algorithm="giac")

[Out]

1/4*(-b^2*x^2 - 2*a*b*x - a^2 + 1)^(3/2)*(b*x + a)*arcsin(b*x + a)^3/b + 3/8*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1
)*(b*x + a)*arcsin(b*x + a)^3/b + 3/16*(b^2*x^2 + 2*a*b*x + a^2 - 1)^2*arcsin(b*x + a)^2/b + 3/32*arcsin(b*x +
a)^4/b - 3/32*(-b^2*x^2 - 2*a*b*x - a^2 + 1)^(3/2)*(b*x + a)*arcsin(b*x + a)/b - 9/16*(b^2*x^2 + 2*a*b*x + a^
2 - 1)*arcsin(b*x + a)^2/b - 45/64*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(b*x + a)*arcsin(b*x + a)/b - 3/128*(b^2
*x^2 + 2*a*b*x + a^2 - 1)^2/b - 45/128*arcsin(b*x + a)^2/b + 45/128*(b^2*x^2 + 2*a*b*x + a^2 - 1)/b + 189/1024
/b

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\mathrm {asin}\left (a+b\,x\right )}^3\,{\left (-a^2-2\,a\,b\,x-b^2\,x^2+1\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(asin(a + b*x)^3*(1 - b^2*x^2 - 2*a*b*x - a^2)^(3/2),x)

[Out]

int(asin(a + b*x)^3*(1 - b^2*x^2 - 2*a*b*x - a^2)^(3/2), x)

________________________________________________________________________________________