3.4.21 \(\int (1-a^2-2 a b x-b^2 x^2)^{3/2} \text {ArcSin}(a+b x)^2 \, dx\) [321]

Optimal. Leaf size=199 \[ -\frac {15 (a+b x) \sqrt {1-(a+b x)^2}}{64 b}-\frac {(a+b x) \left (1-(a+b x)^2\right )^{3/2}}{32 b}+\frac {9 \text {ArcSin}(a+b x)}{64 b}-\frac {3 (a+b x)^2 \text {ArcSin}(a+b x)}{8 b}+\frac {\left (1-(a+b x)^2\right )^2 \text {ArcSin}(a+b x)}{8 b}+\frac {3 (a+b x) \sqrt {1-(a+b x)^2} \text {ArcSin}(a+b x)^2}{8 b}+\frac {(a+b x) \left (1-(a+b x)^2\right )^{3/2} \text {ArcSin}(a+b x)^2}{4 b}+\frac {\text {ArcSin}(a+b x)^3}{8 b} \]

[Out]

-1/32*(b*x+a)*(1-(b*x+a)^2)^(3/2)/b+9/64*arcsin(b*x+a)/b-3/8*(b*x+a)^2*arcsin(b*x+a)/b+1/8*(1-(b*x+a)^2)^2*arc
sin(b*x+a)/b+1/4*(b*x+a)*(1-(b*x+a)^2)^(3/2)*arcsin(b*x+a)^2/b+1/8*arcsin(b*x+a)^3/b-15/64*(b*x+a)*(1-(b*x+a)^
2)^(1/2)/b+3/8*(b*x+a)*arcsin(b*x+a)^2*(1-(b*x+a)^2)^(1/2)/b

________________________________________________________________________________________

Rubi [A]
time = 0.15, antiderivative size = 199, normalized size of antiderivative = 1.00, number of steps used = 11, 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, 327, 222, 4767, 201} \begin {gather*} \frac {\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)^2}{4 b}+\frac {3 (a+b x) \sqrt {1-(a+b x)^2} \text {ArcSin}(a+b x)^2}{8 b}-\frac {3 (a+b x)^2 \text {ArcSin}(a+b x)}{8 b}+\frac {\left (1-(a+b x)^2\right )^2 \text {ArcSin}(a+b x)}{8 b}+\frac {9 \text {ArcSin}(a+b x)}{64 b}-\frac {(a+b x) \left (1-(a+b x)^2\right )^{3/2}}{32 b}-\frac {15 (a+b x) \sqrt {1-(a+b x)^2}}{64 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]^2,x]

[Out]

(-15*(a + b*x)*Sqrt[1 - (a + b*x)^2])/(64*b) - ((a + b*x)*(1 - (a + b*x)^2)^(3/2))/(32*b) + (9*ArcSin[a + b*x]
)/(64*b) - (3*(a + b*x)^2*ArcSin[a + b*x])/(8*b) + ((1 - (a + b*x)^2)^2*ArcSin[a + b*x])/(8*b) + (3*(a + b*x)*
Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x]^2)/(8*b) + ((a + b*x)*(1 - (a + b*x)^2)^(3/2)*ArcSin[a + b*x]^2)/(4*b) +
ArcSin[a + b*x]^3/(8*b)

Rule 201

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(n*p + 1)), x] + Dist[a*n*(p/(n*p + 1)),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 222

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 327

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 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 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)^2 \, dx &=\frac {\text {Subst}\left (\int \left (1-x^2\right )^{3/2} \sin ^{-1}(x)^2 \, 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)^2}{4 b}-\frac {\text {Subst}\left (\int x \left (1-x^2\right ) \sin ^{-1}(x) \, dx,x,a+b x\right )}{2 b}+\frac {3 \text {Subst}\left (\int \sqrt {1-x^2} \sin ^{-1}(x)^2 \, dx,x,a+b x\right )}{4 b}\\ &=\frac {\left (1-(a+b x)^2\right )^2 \sin ^{-1}(a+b x)}{8 b}+\frac {3 (a+b x) \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{8 b}+\frac {(a+b x) \left (1-(a+b x)^2\right )^{3/2} \sin ^{-1}(a+b x)^2}{4 b}-\frac {\text {Subst}\left (\int \left (1-x^2\right )^{3/2} \, dx,x,a+b x\right )}{8 b}+\frac {3 \text {Subst}\left (\int \frac {\sin ^{-1}(x)^2}{\sqrt {1-x^2}} \, dx,x,a+b x\right )}{8 b}-\frac {3 \text {Subst}\left (\int x \sin ^{-1}(x) \, dx,x,a+b x\right )}{4 b}\\ &=-\frac {(a+b x) \left (1-(a+b x)^2\right )^{3/2}}{32 b}-\frac {3 (a+b x)^2 \sin ^{-1}(a+b x)}{8 b}+\frac {\left (1-(a+b x)^2\right )^2 \sin ^{-1}(a+b x)}{8 b}+\frac {3 (a+b x) \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{8 b}+\frac {(a+b x) \left (1-(a+b x)^2\right )^{3/2} \sin ^{-1}(a+b x)^2}{4 b}+\frac {\sin ^{-1}(a+b x)^3}{8 b}-\frac {3 \text {Subst}\left (\int \sqrt {1-x^2} \, dx,x,a+b x\right )}{32 b}+\frac {3 \text {Subst}\left (\int \frac {x^2}{\sqrt {1-x^2}} \, dx,x,a+b x\right )}{8 b}\\ &=-\frac {15 (a+b x) \sqrt {1-(a+b x)^2}}{64 b}-\frac {(a+b x) \left (1-(a+b x)^2\right )^{3/2}}{32 b}-\frac {3 (a+b x)^2 \sin ^{-1}(a+b x)}{8 b}+\frac {\left (1-(a+b x)^2\right )^2 \sin ^{-1}(a+b x)}{8 b}+\frac {3 (a+b x) \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{8 b}+\frac {(a+b x) \left (1-(a+b x)^2\right )^{3/2} \sin ^{-1}(a+b x)^2}{4 b}+\frac {\sin ^{-1}(a+b x)^3}{8 b}-\frac {3 \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,a+b x\right )}{64 b}+\frac {3 \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,a+b x\right )}{16 b}\\ &=-\frac {15 (a+b x) \sqrt {1-(a+b x)^2}}{64 b}-\frac {(a+b x) \left (1-(a+b x)^2\right )^{3/2}}{32 b}+\frac {9 \sin ^{-1}(a+b x)}{64 b}-\frac {3 (a+b x)^2 \sin ^{-1}(a+b x)}{8 b}+\frac {\left (1-(a+b x)^2\right )^2 \sin ^{-1}(a+b x)}{8 b}+\frac {3 (a+b x) \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{8 b}+\frac {(a+b x) \left (1-(a+b x)^2\right )^{3/2} \sin ^{-1}(a+b x)^2}{4 b}+\frac {\sin ^{-1}(a+b x)^3}{8 b}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.12, size = 216, normalized size = 1.09 \begin {gather*} \frac {\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 )+\left (17-40 a^2+8 a^4\right ) \text {ArcSin}(a+b x)+8 b x \left (-10 a+4 a^3-5 b x+6 a^2 b x+4 a b^2 x^2+b^3 x^3\right ) \text {ArcSin}(a+b x)-8 \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)^2+8 \text {ArcSin}(a+b x)^3}{64 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]^2,x]

[Out]

(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) + (17 - 40*a
^2 + 8*a^4)*ArcSin[a + b*x] + 8*b*x*(-10*a + 4*a^3 - 5*b*x + 6*a^2*b*x + 4*a*b^2*x^2 + b^3*x^3)*ArcSin[a + b*x
] - 8*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]^2 + 8*ArcSin[a + b*x]^3)/(64*b)

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(514\) vs. \(2(175)=350\).
time = 0.47, size = 515, normalized size = 2.59

method	result	size
default	\(\frac {-16 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \arcsin \left (b x +a \right )^{2} b^{3} x^{3}+8 \arcsin \left (b x +a \right ) b^{4} x^{4}-48 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \arcsin \left (b x +a \right )^{2} a \,b^{2} x^{2}+32 \arcsin \left (b x +a \right ) a \,b^{3} x^{3}-48 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \arcsin \left (b x +a \right )^{2} a^{2} b x +48 \arcsin \left (b x +a \right ) a^{2} b^{2} x^{2}+2 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, b^{3} x^{3}-16 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \arcsin \left (b x +a \right )^{2} a^{3}+32 \arcsin \left (b x +a \right ) a^{3} b x +6 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a \,b^{2} x^{2}+40 \arcsin \left (b x +a \right )^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, b x +8 \arcsin \left (b x +a \right ) a^{4}-40 \arcsin \left (b x +a \right ) b^{2} x^{2}+6 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a^{2} b x +40 \arcsin \left (b x +a \right )^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a -80 \arcsin \left (b x +a \right ) a b x +2 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a^{3}+8 \arcsin \left (b x +a \right )^{3}-40 \arcsin \left (b x +a \right ) a^{2}-17 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, b x -17 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a +17 \arcsin \left (b x +a \right )}{64 b}\)	\(515\)

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)^2,x,method=_RETURNVERBOSE)

[Out]

1/64*(-16*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*arcsin(b*x+a)^2*b^3*x^3+8*arcsin(b*x+a)*b^4*x^4-48*(-b^2*x^2-2*a*b*x-
a^2+1)^(1/2)*arcsin(b*x+a)^2*a*b^2*x^2+32*arcsin(b*x+a)*a*b^3*x^3-48*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*arcsin(b*x
+a)^2*a^2*b*x+48*arcsin(b*x+a)*a^2*b^2*x^2+2*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*b^3*x^3-16*(-b^2*x^2-2*a*b*x-a^2+1
)^(1/2)*arcsin(b*x+a)^2*a^3+32*arcsin(b*x+a)*a^3*b*x+6*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*a*b^2*x^2+40*arcsin(b*x+
a)^2*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*b*x+8*arcsin(b*x+a)*a^4-40*arcsin(b*x+a)*b^2*x^2+6*(-b^2*x^2-2*a*b*x-a^2+1
)^(1/2)*a^2*b*x+40*arcsin(b*x+a)^2*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*a-80*arcsin(b*x+a)*a*b*x+2*(-b^2*x^2-2*a*b*x
-a^2+1)^(1/2)*a^3+8*arcsin(b*x+a)^3-40*arcsin(b*x+a)*a^2-17*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*b*x-17*(-b^2*x^2-2*
a*b*x-a^2+1)^(1/2)*a+17*arcsin(b*x+a))/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)^2,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 2.22, size = 185, normalized size = 0.93 \begin {gather*} \frac {8 \, \arcsin \left (b x + a\right )^{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 ) + {\left (2 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} + 2 \, a^{3} + {\left (6 \, a^{2} - 17\right )} b x - 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 )^{2} - 17 \, a\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{64 \, 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)^2,x, algorithm="fricas")

[Out]

1/64*(8*arcsin(b*x + a)^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*b^3*x^3 + 6*a*b^2*x^2 + 2*a^3 + (6*a^2 - 17)*b*x - 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)^2 - 17*a)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1))/b

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 568 vs. \(2 (175) = 350\).
time = 0.95, size = 568, normalized size = 2.85 \begin {gather*} \begin {cases} \frac {a^{4} \operatorname {asin}{\left (a + b x \right )}}{8 b} + \frac {a^{3} x \operatorname {asin}{\left (a + b x \right )}}{2} - \frac {a^{3} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a + b x \right )}}{4 b} + \frac {a^{3} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{32 b} + \frac {3 a^{2} b x^{2} \operatorname {asin}{\left (a + b x \right )}}{4} - \frac {3 a^{2} x \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a + b x \right )}}{4} + \frac {3 a^{2} x \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{32} - \frac {5 a^{2} \operatorname {asin}{\left (a + b x \right )}}{8 b} + \frac {a b^{2} x^{3} \operatorname {asin}{\left (a + b x \right )}}{2} - \frac {3 a b x^{2} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a + b x \right )}}{4} + \frac {3 a b x^{2} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{32} - \frac {5 a x \operatorname {asin}{\left (a + b x \right )}}{4} + \frac {5 a \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a + b x \right )}}{8 b} - \frac {17 a \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{64 b} + \frac {b^{3} x^{4} \operatorname {asin}{\left (a + b x \right )}}{8} - \frac {b^{2} x^{3} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a + b x \right )}}{4} + \frac {b^{2} x^{3} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{32} - \frac {5 b x^{2} \operatorname {asin}{\left (a + b x \right )}}{8} + \frac {5 x \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a + b x \right )}}{8} - \frac {17 x \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{64} + \frac {\operatorname {asin}^{3}{\left (a + b x \right )}}{8 b} + \frac {17 \operatorname {asin}{\left (a + b x \right )}}{64 b} & \text {for}\: b \neq 0 \\x \left (1 - a^{2}\right )^{\frac {3}{2}} \operatorname {asin}^{2}{\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)**2,x)

[Out]

Piecewise((a**4*asin(a + b*x)/(8*b) + a**3*x*asin(a + b*x)/2 - a**3*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)*asin
(a + b*x)**2/(4*b) + a**3*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)/(32*b) + 3*a**2*b*x**2*asin(a + b*x)/4 - 3*a**
2*x*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)*asin(a + b*x)**2/4 + 3*a**2*x*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)/
32 - 5*a**2*asin(a + b*x)/(8*b) + a*b**2*x**3*asin(a + b*x)/2 - 3*a*b*x**2*sqrt(-a**2 - 2*a*b*x - b**2*x**2 +
1)*asin(a + b*x)**2/4 + 3*a*b*x**2*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)/32 - 5*a*x*asin(a + b*x)/4 + 5*a*sqrt
(-a**2 - 2*a*b*x - b**2*x**2 + 1)*asin(a + b*x)**2/(8*b) - 17*a*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)/(64*b) +
b**3*x**4*asin(a + b*x)/8 - b**2*x**3*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)*asin(a + b*x)**2/4 + b**2*x**3*sq
rt(-a**2 - 2*a*b*x - b**2*x**2 + 1)/32 - 5*b*x**2*asin(a + b*x)/8 + 5*x*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)*
asin(a + b*x)**2/8 - 17*x*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)/64 + asin(a + b*x)**3/(8*b) + 17*asin(a + b*x)
/(64*b), Ne(b, 0)), (x*(1 - a**2)**(3/2)*asin(a)**2, True))

________________________________________________________________________________________

Giac [A]
time = 0.49, size = 227, normalized size = 1.14 \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 )^{2}}{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 )^{2}}{8 \, b} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )}^{2} \arcsin \left (b x + a\right )}{8 \, b} + \frac {\arcsin \left (b x + a\right )^{3}}{8 \, b} - \frac {{\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac {3}{2}} {\left (b x + a\right )}}{32 \, b} - \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )} \arcsin \left (b x + a\right )}{8 \, b} - \frac {15 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (b x + a\right )}}{64 \, b} - \frac {15 \, \arcsin \left (b x + a\right )}{64 \, 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)^2,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)^2/b + 3/8*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1
)*(b*x + a)*arcsin(b*x + a)^2/b + 1/8*(b^2*x^2 + 2*a*b*x + a^2 - 1)^2*arcsin(b*x + a)/b + 1/8*arcsin(b*x + a)^
3/b - 1/32*(-b^2*x^2 - 2*a*b*x - a^2 + 1)^(3/2)*(b*x + a)/b - 3/8*(b^2*x^2 + 2*a*b*x + a^2 - 1)*arcsin(b*x + a
)/b - 15/64*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(b*x + a)/b - 15/64*arcsin(b*x + a)/b

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\mathrm {asin}\left (a+b\,x\right )}^2\,{\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)^2*(1 - b^2*x^2 - 2*a*b*x - a^2)^(3/2),x)

[Out]

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

________________________________________________________________________________________