3.132 \(\int x^2 \sin ^{-1}(a+b x)^2 \, dx\)

Optimal. Leaf size=220 \[ \frac {a^3 \sin ^{-1}(a+b x)^2}{3 b^3}+\frac {2 a^2 \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)}{b^3}-\frac {2 a^2 x}{b^2}+\frac {a (a+b x)^2}{2 b^3}-\frac {2 (a+b x)^3}{27 b^3}+\frac {a \sin ^{-1}(a+b x)^2}{2 b^3}-\frac {a (a+b x) \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)}{b^3}+\frac {2 (a+b x)^2 \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)}{9 b^3}+\frac {4 \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)}{9 b^3}+\frac {1}{3} x^3 \sin ^{-1}(a+b x)^2-\frac {4 x}{9 b^2} \]

[Out]

-4/9*x/b^2-2*a^2*x/b^2+1/2*a*(b*x+a)^2/b^3-2/27*(b*x+a)^3/b^3+1/2*a*arcsin(b*x+a)^2/b^3+1/3*a^3*arcsin(b*x+a)^
2/b^3+1/3*x^3*arcsin(b*x+a)^2+4/9*arcsin(b*x+a)*(1-(b*x+a)^2)^(1/2)/b^3+2*a^2*arcsin(b*x+a)*(1-(b*x+a)^2)^(1/2
)/b^3-a*(b*x+a)*arcsin(b*x+a)*(1-(b*x+a)^2)^(1/2)/b^3+2/9*(b*x+a)^2*arcsin(b*x+a)*(1-(b*x+a)^2)^(1/2)/b^3

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Rubi [A]  time = 0.39, antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {4805, 4743, 4763, 4641, 4677, 8, 4707, 30} \[ -\frac {2 a^2 x}{b^2}+\frac {a^3 \sin ^{-1}(a+b x)^2}{3 b^3}+\frac {2 a^2 \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)}{b^3}+\frac {a (a+b x)^2}{2 b^3}-\frac {2 (a+b x)^3}{27 b^3}+\frac {a \sin ^{-1}(a+b x)^2}{2 b^3}-\frac {a (a+b x) \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)}{b^3}+\frac {2 (a+b x)^2 \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)}{9 b^3}+\frac {4 \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)}{9 b^3}+\frac {1}{3} x^3 \sin ^{-1}(a+b x)^2-\frac {4 x}{9 b^2} \]

Antiderivative was successfully verified.

[In]

Int[x^2*ArcSin[a + b*x]^2,x]

[Out]

(-4*x)/(9*b^2) - (2*a^2*x)/b^2 + (a*(a + b*x)^2)/(2*b^3) - (2*(a + b*x)^3)/(27*b^3) + (4*Sqrt[1 - (a + b*x)^2]
*ArcSin[a + b*x])/(9*b^3) + (2*a^2*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x])/b^3 - (a*(a + b*x)*Sqrt[1 - (a + b*x
)^2]*ArcSin[a + b*x])/b^3 + (2*(a + b*x)^2*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x])/(9*b^3) + (a*ArcSin[a + b*x]
^2)/(2*b^3) + (a^3*ArcSin[a + b*x]^2)/(3*b^3) + (x^3*ArcSin[a + b*x]^2)/3

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && 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 4677

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*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*c*(p + 1
)*(1 - c^2*x^2)^FracPart[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 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 4743

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

Rule 4763

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, (f + g*x)^m, x], x] /; FreeQ[{a, b, c, d, e, f, g},
x] && EqQ[c^2*d + e, 0] && IGtQ[m, 0] && IntegerQ[p + 1/2] && GtQ[d, 0] && IGtQ[n, 0] && (m == 1 || p > 0 ||
(n == 1 && p > -1) || (m == 2 && p < -2))

Rule 4805

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

Rubi steps

\begin {align*} \int x^2 \sin ^{-1}(a+b x)^2 \, dx &=\frac {\operatorname {Subst}\left (\int \left (-\frac {a}{b}+\frac {x}{b}\right )^2 \sin ^{-1}(x)^2 \, dx,x,a+b x\right )}{b}\\ &=\frac {1}{3} x^3 \sin ^{-1}(a+b x)^2-\frac {2}{3} \operatorname {Subst}\left (\int \frac {\left (-\frac {a}{b}+\frac {x}{b}\right )^3 \sin ^{-1}(x)}{\sqrt {1-x^2}} \, dx,x,a+b x\right )\\ &=\frac {1}{3} x^3 \sin ^{-1}(a+b x)^2-\frac {2}{3} \operatorname {Subst}\left (\int \left (-\frac {a^3 \sin ^{-1}(x)}{b^3 \sqrt {1-x^2}}+\frac {3 a^2 x \sin ^{-1}(x)}{b^3 \sqrt {1-x^2}}-\frac {3 a x^2 \sin ^{-1}(x)}{b^3 \sqrt {1-x^2}}+\frac {x^3 \sin ^{-1}(x)}{b^3 \sqrt {1-x^2}}\right ) \, dx,x,a+b x\right )\\ &=\frac {1}{3} x^3 \sin ^{-1}(a+b x)^2-\frac {2 \operatorname {Subst}\left (\int \frac {x^3 \sin ^{-1}(x)}{\sqrt {1-x^2}} \, dx,x,a+b x\right )}{3 b^3}+\frac {(2 a) \operatorname {Subst}\left (\int \frac {x^2 \sin ^{-1}(x)}{\sqrt {1-x^2}} \, dx,x,a+b x\right )}{b^3}-\frac {\left (2 a^2\right ) \operatorname {Subst}\left (\int \frac {x \sin ^{-1}(x)}{\sqrt {1-x^2}} \, dx,x,a+b x\right )}{b^3}+\frac {\left (2 a^3\right ) \operatorname {Subst}\left (\int \frac {\sin ^{-1}(x)}{\sqrt {1-x^2}} \, dx,x,a+b x\right )}{3 b^3}\\ &=\frac {2 a^2 \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)}{b^3}-\frac {a (a+b x) \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)}{b^3}+\frac {2 (a+b x)^2 \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)}{9 b^3}+\frac {a^3 \sin ^{-1}(a+b x)^2}{3 b^3}+\frac {1}{3} x^3 \sin ^{-1}(a+b x)^2-\frac {2 \operatorname {Subst}\left (\int x^2 \, dx,x,a+b x\right )}{9 b^3}-\frac {4 \operatorname {Subst}\left (\int \frac {x \sin ^{-1}(x)}{\sqrt {1-x^2}} \, dx,x,a+b x\right )}{9 b^3}+\frac {a \operatorname {Subst}(\int x \, dx,x,a+b x)}{b^3}+\frac {a \operatorname {Subst}\left (\int \frac {\sin ^{-1}(x)}{\sqrt {1-x^2}} \, dx,x,a+b x\right )}{b^3}-\frac {\left (2 a^2\right ) \operatorname {Subst}(\int 1 \, dx,x,a+b x)}{b^3}\\ &=-\frac {2 a^2 x}{b^2}+\frac {a (a+b x)^2}{2 b^3}-\frac {2 (a+b x)^3}{27 b^3}+\frac {4 \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)}{9 b^3}+\frac {2 a^2 \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)}{b^3}-\frac {a (a+b x) \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)}{b^3}+\frac {2 (a+b x)^2 \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)}{9 b^3}+\frac {a \sin ^{-1}(a+b x)^2}{2 b^3}+\frac {a^3 \sin ^{-1}(a+b x)^2}{3 b^3}+\frac {1}{3} x^3 \sin ^{-1}(a+b x)^2-\frac {4 \operatorname {Subst}(\int 1 \, dx,x,a+b x)}{9 b^3}\\ &=-\frac {4 x}{9 b^2}-\frac {2 a^2 x}{b^2}+\frac {a (a+b x)^2}{2 b^3}-\frac {2 (a+b x)^3}{27 b^3}+\frac {4 \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)}{9 b^3}+\frac {2 a^2 \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)}{b^3}-\frac {a (a+b x) \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)}{b^3}+\frac {2 (a+b x)^2 \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)}{9 b^3}+\frac {a \sin ^{-1}(a+b x)^2}{2 b^3}+\frac {a^3 \sin ^{-1}(a+b x)^2}{3 b^3}+\frac {1}{3} x^3 \sin ^{-1}(a+b x)^2\\ \end {align*}

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Mathematica [A]  time = 0.18, size = 111, normalized size = 0.50 \[ \frac {9 \left (2 a^3+3 a+2 b^3 x^3\right ) \sin ^{-1}(a+b x)^2-b x \left (66 a^2-15 a b x+4 b^2 x^2+24\right )+6 \sqrt {-a^2-2 a b x-b^2 x^2+1} \left (11 a^2-5 a b x+2 b^2 x^2+4\right ) \sin ^{-1}(a+b x)}{54 b^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*ArcSin[a + b*x]^2,x]

[Out]

(-(b*x*(24 + 66*a^2 - 15*a*b*x + 4*b^2*x^2)) + 6*Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]*(4 + 11*a^2 - 5*a*b*x + 2*b
^2*x^2)*ArcSin[a + b*x] + 9*(3*a + 2*a^3 + 2*b^3*x^3)*ArcSin[a + b*x]^2)/(54*b^3)

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fricas [A]  time = 0.46, size = 111, normalized size = 0.50 \[ -\frac {4 \, b^{3} x^{3} - 15 \, a b^{2} x^{2} + 6 \, {\left (11 \, a^{2} + 4\right )} b x - 9 \, {\left (2 \, b^{3} x^{3} + 2 \, a^{3} + 3 \, a\right )} \arcsin \left (b x + a\right )^{2} - 6 \, {\left (2 \, b^{2} x^{2} - 5 \, a b x + 11 \, a^{2} + 4\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} \arcsin \left (b x + a\right )}{54 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/54*(4*b^3*x^3 - 15*a*b^2*x^2 + 6*(11*a^2 + 4)*b*x - 9*(2*b^3*x^3 + 2*a^3 + 3*a)*arcsin(b*x + a)^2 - 6*(2*b^
2*x^2 - 5*a*b*x + 11*a^2 + 4)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*arcsin(b*x + a))/b^3

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giac [A]  time = 2.81, size = 271, normalized size = 1.23 \[ \frac {{\left (b x + a\right )} a^{2} \arcsin \left (b x + a\right )^{2}}{b^{3}} + \frac {{\left ({\left (b x + a\right )}^{2} - 1\right )} {\left (b x + a\right )} \arcsin \left (b x + a\right )^{2}}{3 \, b^{3}} - \frac {{\left ({\left (b x + a\right )}^{2} - 1\right )} a \arcsin \left (b x + a\right )^{2}}{b^{3}} - \frac {\sqrt {-{\left (b x + a\right )}^{2} + 1} {\left (b x + a\right )} a \arcsin \left (b x + a\right )}{b^{3}} + \frac {2 \, \sqrt {-{\left (b x + a\right )}^{2} + 1} a^{2} \arcsin \left (b x + a\right )}{b^{3}} - \frac {2 \, {\left (b x + a\right )} a^{2}}{b^{3}} + \frac {{\left (b x + a\right )} \arcsin \left (b x + a\right )^{2}}{3 \, b^{3}} - \frac {a \arcsin \left (b x + a\right )^{2}}{2 \, b^{3}} - \frac {2 \, {\left (-{\left (b x + a\right )}^{2} + 1\right )}^{\frac {3}{2}} \arcsin \left (b x + a\right )}{9 \, b^{3}} - \frac {2 \, {\left ({\left (b x + a\right )}^{2} - 1\right )} {\left (b x + a\right )}}{27 \, b^{3}} + \frac {{\left ({\left (b x + a\right )}^{2} - 1\right )} a}{2 \, b^{3}} + \frac {2 \, \sqrt {-{\left (b x + a\right )}^{2} + 1} \arcsin \left (b x + a\right )}{3 \, b^{3}} - \frac {14 \, {\left (b x + a\right )}}{27 \, b^{3}} + \frac {a}{4 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x + a)*a^2*arcsin(b*x + a)^2/b^3 + 1/3*((b*x + a)^2 - 1)*(b*x + a)*arcsin(b*x + a)^2/b^3 - ((b*x + a)^2 - 1
)*a*arcsin(b*x + a)^2/b^3 - sqrt(-(b*x + a)^2 + 1)*(b*x + a)*a*arcsin(b*x + a)/b^3 + 2*sqrt(-(b*x + a)^2 + 1)*
a^2*arcsin(b*x + a)/b^3 - 2*(b*x + a)*a^2/b^3 + 1/3*(b*x + a)*arcsin(b*x + a)^2/b^3 - 1/2*a*arcsin(b*x + a)^2/
b^3 - 2/9*(-(b*x + a)^2 + 1)^(3/2)*arcsin(b*x + a)/b^3 - 2/27*((b*x + a)^2 - 1)*(b*x + a)/b^3 + 1/2*((b*x + a)
^2 - 1)*a/b^3 + 2/3*sqrt(-(b*x + a)^2 + 1)*arcsin(b*x + a)/b^3 - 14/27*(b*x + a)/b^3 + 1/4*a/b^3

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maple [A]  time = 0.12, size = 231, normalized size = 1.05 \[ \frac {-\frac {a \left (2 \arcsin \left (b x +a \right )^{2} \left (b x +a \right )^{2}+2 \sqrt {1-\left (b x +a \right )^{2}}\, \arcsin \left (b x +a \right ) \left (b x +a \right )-\arcsin \left (b x +a \right )^{2}-\left (b x +a \right )^{2}\right )}{2}+\frac {\arcsin \left (b x +a \right )^{2} \left (\left (b x +a \right )^{2}-3\right ) \left (b x +a \right )}{3}-\frac {2 b x}{3}-\frac {2 a}{3}+\frac {2 \arcsin \left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}}{3}+\frac {2 \arcsin \left (b x +a \right ) \left (-1+\left (b x +a \right )^{2}\right ) \sqrt {1-\left (b x +a \right )^{2}}}{9}-\frac {2 \left (\left (b x +a \right )^{2}-3\right ) \left (b x +a \right )}{27}+a^{2} \left (\arcsin \left (b x +a \right )^{2} \left (b x +a \right )-2 b x -2 a +2 \arcsin \left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}\right )+\arcsin \left (b x +a \right )^{2} \left (b x +a \right )}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*arcsin(b*x+a)^2,x)

[Out]

1/b^3*(-1/2*a*(2*arcsin(b*x+a)^2*(b*x+a)^2+2*(1-(b*x+a)^2)^(1/2)*arcsin(b*x+a)*(b*x+a)-arcsin(b*x+a)^2-(b*x+a)
^2)+1/3*arcsin(b*x+a)^2*((b*x+a)^2-3)*(b*x+a)-2/3*b*x-2/3*a+2/3*arcsin(b*x+a)*(1-(b*x+a)^2)^(1/2)+2/9*arcsin(b
*x+a)*(-1+(b*x+a)^2)*(1-(b*x+a)^2)^(1/2)-2/27*((b*x+a)^2-3)*(b*x+a)+a^2*(arcsin(b*x+a)^2*(b*x+a)-2*b*x-2*a+2*a
rcsin(b*x+a)*(1-(b*x+a)^2)^(1/2))+arcsin(b*x+a)^2*(b*x+a))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{3} \, x^{3} \arctan \left (b x + a, \sqrt {b x + a + 1} \sqrt {-b x - a + 1}\right )^{2} + 2 \, b \int \frac {\sqrt {b x + a + 1} \sqrt {-b x - a + 1} x^{3} \arctan \left (b x + a, \sqrt {b x + a + 1} \sqrt {-b x - a + 1}\right )}{3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3*arctan2(b*x + a, sqrt(b*x + a + 1)*sqrt(-b*x - a + 1))^2 + 2*b*integrate(1/3*sqrt(b*x + a + 1)*sqrt(-b
*x - a + 1)*x^3*arctan2(b*x + a, sqrt(b*x + a + 1)*sqrt(-b*x - a + 1))/(b^2*x^2 + 2*a*b*x + a^2 - 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*asin(a + b*x)^2,x)

[Out]

int(x^2*asin(a + b*x)^2, x)

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sympy [A]  time = 1.28, size = 243, normalized size = 1.10 \[ \begin {cases} \frac {a^{3} \operatorname {asin}^{2}{\left (a + b x \right )}}{3 b^{3}} - \frac {11 a^{2} x}{9 b^{2}} + \frac {11 a^{2} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}{\left (a + b x \right )}}{9 b^{3}} + \frac {5 a x^{2}}{18 b} - \frac {5 a x \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}{\left (a + b x \right )}}{9 b^{2}} + \frac {a \operatorname {asin}^{2}{\left (a + b x \right )}}{2 b^{3}} + \frac {x^{3} \operatorname {asin}^{2}{\left (a + b x \right )}}{3} - \frac {2 x^{3}}{27} + \frac {2 x^{2} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}{\left (a + b x \right )}}{9 b} - \frac {4 x}{9 b^{2}} + \frac {4 \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}{\left (a + b x \right )}}{9 b^{3}} & \text {for}\: b \neq 0 \\\frac {x^{3} \operatorname {asin}^{2}{\relax (a )}}{3} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*asin(b*x+a)**2,x)

[Out]

Piecewise((a**3*asin(a + b*x)**2/(3*b**3) - 11*a**2*x/(9*b**2) + 11*a**2*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)
*asin(a + b*x)/(9*b**3) + 5*a*x**2/(18*b) - 5*a*x*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)*asin(a + b*x)/(9*b**2)
+ a*asin(a + b*x)**2/(2*b**3) + x**3*asin(a + b*x)**2/3 - 2*x**3/27 + 2*x**2*sqrt(-a**2 - 2*a*b*x - b**2*x**2
+ 1)*asin(a + b*x)/(9*b) - 4*x/(9*b**2) + 4*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)*asin(a + b*x)/(9*b**3), Ne(
b, 0)), (x**3*asin(a)**2/3, True))

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