Integrand size = 12, antiderivative size = 371 \[ \int x^2 \arcsin (a+b x)^3 \, dx=-\frac {14 \sqrt {1-(a+b x)^2}}{9 b^3}-\frac {6 a^2 \sqrt {1-(a+b x)^2}}{b^3}+\frac {3 a (a+b x) \sqrt {1-(a+b x)^2}}{4 b^3}+\frac {2 \left (1-(a+b x)^2\right )^{3/2}}{27 b^3}-\frac {3 a \arcsin (a+b x)}{4 b^3}-\frac {4 (a+b x) \arcsin (a+b x)}{3 b^3}-\frac {6 a^2 (a+b x) \arcsin (a+b x)}{b^3}+\frac {3 a (a+b x)^2 \arcsin (a+b x)}{2 b^3}-\frac {2 (a+b x)^3 \arcsin (a+b x)}{9 b^3}+\frac {2 \sqrt {1-(a+b x)^2} \arcsin (a+b x)^2}{3 b^3}+\frac {3 a^2 \sqrt {1-(a+b x)^2} \arcsin (a+b x)^2}{b^3}-\frac {3 a (a+b x) \sqrt {1-(a+b x)^2} \arcsin (a+b x)^2}{2 b^3}+\frac {(a+b x)^2 \sqrt {1-(a+b x)^2} \arcsin (a+b x)^2}{3 b^3}+\frac {a \arcsin (a+b x)^3}{2 b^3}+\frac {a^3 \arcsin (a+b x)^3}{3 b^3}+\frac {1}{3} x^3 \arcsin (a+b x)^3 \]
2/27*(1-(b*x+a)^2)^(3/2)/b^3-3/4*a*arcsin(b*x+a)/b^3-4/3*(b*x+a)*arcsin(b* x+a)/b^3-6*a^2*(b*x+a)*arcsin(b*x+a)/b^3+3/2*a*(b*x+a)^2*arcsin(b*x+a)/b^3 -2/9*(b*x+a)^3*arcsin(b*x+a)/b^3+1/2*a*arcsin(b*x+a)^3/b^3+1/3*a^3*arcsin( b*x+a)^3/b^3+1/3*x^3*arcsin(b*x+a)^3-14/9*(1-(b*x+a)^2)^(1/2)/b^3-6*a^2*(1 -(b*x+a)^2)^(1/2)/b^3+3/4*a*(b*x+a)*(1-(b*x+a)^2)^(1/2)/b^3+2/3*arcsin(b*x +a)^2*(1-(b*x+a)^2)^(1/2)/b^3+3*a^2*arcsin(b*x+a)^2*(1-(b*x+a)^2)^(1/2)/b^ 3-3/2*a*(b*x+a)*arcsin(b*x+a)^2*(1-(b*x+a)^2)^(1/2)/b^3+1/3*(b*x+a)^2*arcs in(b*x+a)^2*(1-(b*x+a)^2)^(1/2)/b^3
Time = 0.54 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.49 \[ \int x^2 \arcsin (a+b x)^3 \, dx=\frac {-\sqrt {1-a^2-2 a b x-b^2 x^2} \left (160+575 a^2-65 a b x+8 b^2 x^2\right )-3 \left (170 a^3+132 a^2 b x+a \left (75-30 b^2 x^2\right )+8 b x \left (6+b^2 x^2\right )\right ) \arcsin (a+b x)+18 \sqrt {1-a^2-2 a b x-b^2 x^2} \left (4+11 a^2-5 a b x+2 b^2 x^2\right ) \arcsin (a+b x)^2+18 \left (3 a+2 a^3+2 b^3 x^3\right ) \arcsin (a+b x)^3}{108 b^3} \]
(-(Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]*(160 + 575*a^2 - 65*a*b*x + 8*b^2*x^2 )) - 3*(170*a^3 + 132*a^2*b*x + a*(75 - 30*b^2*x^2) + 8*b*x*(6 + b^2*x^2)) *ArcSin[a + b*x] + 18*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]^2 + 18*(3*a + 2*a^3 + 2*b^3*x^3)*ArcSin[a + b*x]^3)/(108*b^3)
Time = 0.61 (sec) , antiderivative size = 333, normalized size of antiderivative = 0.90, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {5304, 27, 5242, 5272, 3042, 3798, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int x^2 \arcsin (a+b x)^3 \, dx\) |
| \(\Big \downarrow \) 5304 |
| \(\displaystyle \frac {\int x^2 \arcsin (a+b x)^3d(a+b x)}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int b^2 x^2 \arcsin (a+b x)^3d(a+b x)}{b^3}\) |
| \(\Big \downarrow \) 5242 |
| \(\displaystyle \frac {\int -\frac {b^3 x^3 \arcsin (a+b x)^2}{\sqrt {1-(a+b x)^2}}d(a+b x)+\frac {1}{3} b^3 x^3 \arcsin (a+b x)^3}{b^3}\) |
| \(\Big \downarrow \) 5272 |
| \(\displaystyle \frac {\int -b^3 x^3 \arcsin (a+b x)^2d\arcsin (a+b x)+\frac {1}{3} b^3 x^3 \arcsin (a+b x)^3}{b^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \arcsin (a+b x)^2 (a-\sin (\arcsin (a+b x)))^3d\arcsin (a+b x)+\frac {1}{3} b^3 x^3 \arcsin (a+b x)^3}{b^3}\) |
| \(\Big \downarrow \) 3798 |
| \(\displaystyle \frac {\int \left (\arcsin (a+b x)^2 a^3-3 (a+b x) \arcsin (a+b x)^2 a^2+3 (a+b x)^2 \arcsin (a+b x)^2 a-(a+b x)^3 \arcsin (a+b x)^2\right )d\arcsin (a+b x)+\frac {1}{3} b^3 x^3 \arcsin (a+b x)^3}{b^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {1}{3} a^3 \arcsin (a+b x)^3-6 a^2 (a+b x) \arcsin (a+b x)+3 a^2 \sqrt {1-(a+b x)^2} \arcsin (a+b x)^2-6 a^2 \sqrt {1-(a+b x)^2}+\frac {1}{3} b^3 x^3 \arcsin (a+b x)^3-\frac {2}{9} (a+b x)^3 \arcsin (a+b x)+\frac {1}{3} \sqrt {1-(a+b x)^2} (a+b x)^2 \arcsin (a+b x)^2+\frac {3}{2} a (a+b x)^2 \arcsin (a+b x)-\frac {3}{2} a \sqrt {1-(a+b x)^2} (a+b x) \arcsin (a+b x)^2-\frac {4}{3} (a+b x) \arcsin (a+b x)+\frac {1}{2} a \arcsin (a+b x)^3+\frac {2}{3} \sqrt {1-(a+b x)^2} \arcsin (a+b x)^2-\frac {3}{4} a \arcsin (a+b x)+\frac {3}{4} a \sqrt {1-(a+b x)^2} (a+b x)+\frac {2}{27} \left (1-(a+b x)^2\right )^{3/2}-\frac {14}{9} \sqrt {1-(a+b x)^2}}{b^3}\) |
((-14*Sqrt[1 - (a + b*x)^2])/9 - 6*a^2*Sqrt[1 - (a + b*x)^2] + (3*a*(a + b *x)*Sqrt[1 - (a + b*x)^2])/4 + (2*(1 - (a + b*x)^2)^(3/2))/27 - (3*a*ArcSi n[a + b*x])/4 - (4*(a + b*x)*ArcSin[a + b*x])/3 - 6*a^2*(a + b*x)*ArcSin[a + b*x] + (3*a*(a + b*x)^2*ArcSin[a + b*x])/2 - (2*(a + b*x)^3*ArcSin[a + b*x])/9 + (2*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x]^2)/3 + 3*a^2*Sqrt[1 - ( a + b*x)^2]*ArcSin[a + b*x]^2 - (3*a*(a + b*x)*Sqrt[1 - (a + b*x)^2]*ArcSi n[a + b*x]^2)/2 + ((a + b*x)^2*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x]^2)/3 + (a*ArcSin[a + b*x]^3)/2 + (a^3*ArcSin[a + b*x]^3)/3 + (b^3*x^3*ArcSin[a + b*x]^3)/3)/b^3
3.2.38.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.) , x_Symbol] :> Int[ExpandIntegrand[(c + d*x)^m, (a + b*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[n, 0] && (EqQ[n, 1] || IGtQ[ m, 0] || NeQ[a^2 - b^2, 0])
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_))^(m_.), x_S ymbol] :> Simp[(d + e*x)^(m + 1)*((a + b*ArcSin[c*x])^n/(e*(m + 1))), x] - Simp[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]
Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.))/Sq rt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[1/(c^(m + 1)*Sqrt[d]) Subst[In t[(a + b*x)^n*(c*f + g*Sin[x])^m, x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c , d, e, f, g, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && GtQ[d, 0] && (G tQ[m, 0] || IGtQ[n, 0])
Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m _.), x_Symbol] :> Simp[1/d Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b*A rcSin[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
Time = 0.92 (sec) , antiderivative size = 294, normalized size of antiderivative = 0.79
| method | result | size |
| derivativedivides | \(\frac {\frac {\arcsin \left (b x +a \right )^{3} \left (b x +a \right )^{3}}{3}+\frac {\arcsin \left (b x +a \right )^{2} \left (\left (b x +a \right )^{2}+2\right ) \sqrt {1-\left (b x +a \right )^{2}}}{3}-\frac {4 \sqrt {1-\left (b x +a \right )^{2}}}{3}-\frac {4 \arcsin \left (b x +a \right ) \left (b x +a \right )}{3}-\frac {2 \arcsin \left (b x +a \right ) \left (b x +a \right )^{3}}{9}-\frac {2 \left (\left (b x +a \right )^{2}+2\right ) \sqrt {1-\left (b x +a \right )^{2}}}{27}-\frac {a \left (4 \arcsin \left (b x +a \right )^{3} \left (b x +a \right )^{2}+6 \arcsin \left (b x +a \right )^{2} \sqrt {1-\left (b x +a \right )^{2}}\, \left (b x +a \right )-2 \arcsin \left (b x +a \right )^{3}-6 \arcsin \left (b x +a \right ) \left (b x +a \right )^{2}-3 \left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}+3 \arcsin \left (b x +a \right )\right )}{4}+a^{2} \left (\arcsin \left (b x +a \right )^{3} \left (b x +a \right )+3 \arcsin \left (b x +a \right )^{2} \sqrt {1-\left (b x +a \right )^{2}}-6 \sqrt {1-\left (b x +a \right )^{2}}-6 \arcsin \left (b x +a \right ) \left (b x +a \right )\right )}{b^{3}}\) | \(294\) |
| default | \(\frac {\frac {\arcsin \left (b x +a \right )^{3} \left (b x +a \right )^{3}}{3}+\frac {\arcsin \left (b x +a \right )^{2} \left (\left (b x +a \right )^{2}+2\right ) \sqrt {1-\left (b x +a \right )^{2}}}{3}-\frac {4 \sqrt {1-\left (b x +a \right )^{2}}}{3}-\frac {4 \arcsin \left (b x +a \right ) \left (b x +a \right )}{3}-\frac {2 \arcsin \left (b x +a \right ) \left (b x +a \right )^{3}}{9}-\frac {2 \left (\left (b x +a \right )^{2}+2\right ) \sqrt {1-\left (b x +a \right )^{2}}}{27}-\frac {a \left (4 \arcsin \left (b x +a \right )^{3} \left (b x +a \right )^{2}+6 \arcsin \left (b x +a \right )^{2} \sqrt {1-\left (b x +a \right )^{2}}\, \left (b x +a \right )-2 \arcsin \left (b x +a \right )^{3}-6 \arcsin \left (b x +a \right ) \left (b x +a \right )^{2}-3 \left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}+3 \arcsin \left (b x +a \right )\right )}{4}+a^{2} \left (\arcsin \left (b x +a \right )^{3} \left (b x +a \right )+3 \arcsin \left (b x +a \right )^{2} \sqrt {1-\left (b x +a \right )^{2}}-6 \sqrt {1-\left (b x +a \right )^{2}}-6 \arcsin \left (b x +a \right ) \left (b x +a \right )\right )}{b^{3}}\) | \(294\) |
1/b^3*(1/3*arcsin(b*x+a)^3*(b*x+a)^3+1/3*arcsin(b*x+a)^2*((b*x+a)^2+2)*(1- (b*x+a)^2)^(1/2)-4/3*(1-(b*x+a)^2)^(1/2)-4/3*arcsin(b*x+a)*(b*x+a)-2/9*arc sin(b*x+a)*(b*x+a)^3-2/27*((b*x+a)^2+2)*(1-(b*x+a)^2)^(1/2)-1/4*a*(4*arcsi n(b*x+a)^3*(b*x+a)^2+6*arcsin(b*x+a)^2*(1-(b*x+a)^2)^(1/2)*(b*x+a)-2*arcsi n(b*x+a)^3-6*arcsin(b*x+a)*(b*x+a)^2-3*(b*x+a)*(1-(b*x+a)^2)^(1/2)+3*arcsi n(b*x+a))+a^2*(arcsin(b*x+a)^3*(b*x+a)+3*arcsin(b*x+a)^2*(1-(b*x+a)^2)^(1/ 2)-6*(1-(b*x+a)^2)^(1/2)-6*arcsin(b*x+a)*(b*x+a)))
Time = 0.26 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.41 \[ \int x^2 \arcsin (a+b x)^3 \, dx=\frac {18 \, {\left (2 \, b^{3} x^{3} + 2 \, a^{3} + 3 \, a\right )} \arcsin \left (b x + a\right )^{3} - 3 \, {\left (8 \, b^{3} x^{3} - 30 \, a b^{2} x^{2} + 170 \, a^{3} + 12 \, {\left (11 \, a^{2} + 4\right )} b x + 75 \, a\right )} \arcsin \left (b x + a\right ) - {\left (8 \, b^{2} x^{2} - 65 \, a b x - 18 \, {\left (2 \, b^{2} x^{2} - 5 \, a b x + 11 \, a^{2} + 4\right )} \arcsin \left (b x + a\right )^{2} + 575 \, a^{2} + 160\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{108 \, b^{3}} \]
1/108*(18*(2*b^3*x^3 + 2*a^3 + 3*a)*arcsin(b*x + a)^3 - 3*(8*b^3*x^3 - 30* a*b^2*x^2 + 170*a^3 + 12*(11*a^2 + 4)*b*x + 75*a)*arcsin(b*x + a) - (8*b^2 *x^2 - 65*a*b*x - 18*(2*b^2*x^2 - 5*a*b*x + 11*a^2 + 4)*arcsin(b*x + a)^2 + 575*a^2 + 160)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1))/b^3
Time = 0.44 (sec) , antiderivative size = 432, normalized size of antiderivative = 1.16 \[ \int x^2 \arcsin (a+b x)^3 \, dx=\begin {cases} \frac {a^{3} \operatorname {asin}^{3}{\left (a + b x \right )}}{3 b^{3}} - \frac {85 a^{3} \operatorname {asin}{\left (a + b x \right )}}{18 b^{3}} - \frac {11 a^{2} x \operatorname {asin}{\left (a + b x \right )}}{3 b^{2}} + \frac {11 a^{2} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a + b x \right )}}{6 b^{3}} - \frac {575 a^{2} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{108 b^{3}} + \frac {5 a x^{2} \operatorname {asin}{\left (a + b x \right )}}{6 b} - \frac {5 a x \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a + b x \right )}}{6 b^{2}} + \frac {65 a x \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{108 b^{2}} + \frac {a \operatorname {asin}^{3}{\left (a + b x \right )}}{2 b^{3}} - \frac {25 a \operatorname {asin}{\left (a + b x \right )}}{12 b^{3}} + \frac {x^{3} \operatorname {asin}^{3}{\left (a + b x \right )}}{3} - \frac {2 x^{3} \operatorname {asin}{\left (a + b x \right )}}{9} + \frac {x^{2} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a + b x \right )}}{3 b} - \frac {2 x^{2} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{27 b} - \frac {4 x \operatorname {asin}{\left (a + b x \right )}}{3 b^{2}} + \frac {2 \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a + b x \right )}}{3 b^{3}} - \frac {40 \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{27 b^{3}} & \text {for}\: b \neq 0 \\\frac {x^{3} \operatorname {asin}^{3}{\left (a \right )}}{3} & \text {otherwise} \end {cases} \]
Piecewise((a**3*asin(a + b*x)**3/(3*b**3) - 85*a**3*asin(a + b*x)/(18*b**3 ) - 11*a**2*x*asin(a + b*x)/(3*b**2) + 11*a**2*sqrt(-a**2 - 2*a*b*x - b**2 *x**2 + 1)*asin(a + b*x)**2/(6*b**3) - 575*a**2*sqrt(-a**2 - 2*a*b*x - b** 2*x**2 + 1)/(108*b**3) + 5*a*x**2*asin(a + b*x)/(6*b) - 5*a*x*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)*asin(a + b*x)**2/(6*b**2) + 65*a*x*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)/(108*b**2) + a*asin(a + b*x)**3/(2*b**3) - 25*a*a sin(a + b*x)/(12*b**3) + x**3*asin(a + b*x)**3/3 - 2*x**3*asin(a + b*x)/9 + x**2*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)*asin(a + b*x)**2/(3*b) - 2*x* *2*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)/(27*b) - 4*x*asin(a + b*x)/(3*b** 2) + 2*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)*asin(a + b*x)**2/(3*b**3) - 4 0*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)/(27*b**3), Ne(b, 0)), (x**3*asin(a )**3/3, True))
\[ \int x^2 \arcsin (a+b x)^3 \, dx=\int { x^{2} \arcsin \left (b x + a\right )^{3} \,d x } \]
1/3*x^3*arctan2(b*x + a, sqrt(b*x + a + 1)*sqrt(-b*x - a + 1))^3 + b*integ rate(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))^2/(b^2*x^2 + 2*a*b*x + a^2 - 1), x)
Time = 0.30 (sec) , antiderivative size = 389, normalized size of antiderivative = 1.05 \[ \int x^2 \arcsin (a+b x)^3 \, dx=\frac {{\left (b x + a\right )} a^{2} \arcsin \left (b x + a\right )^{3}}{b^{3}} + \frac {{\left ({\left (b x + a\right )}^{2} - 1\right )} {\left (b x + a\right )} \arcsin \left (b x + a\right )^{3}}{3 \, b^{3}} - \frac {{\left ({\left (b x + a\right )}^{2} - 1\right )} a \arcsin \left (b x + a\right )^{3}}{b^{3}} - \frac {3 \, \sqrt {-{\left (b x + a\right )}^{2} + 1} {\left (b x + a\right )} a \arcsin \left (b x + a\right )^{2}}{2 \, b^{3}} + \frac {3 \, \sqrt {-{\left (b x + a\right )}^{2} + 1} a^{2} \arcsin \left (b x + a\right )^{2}}{b^{3}} - \frac {6 \, {\left (b x + a\right )} a^{2} \arcsin \left (b x + a\right )}{b^{3}} + \frac {{\left (b x + a\right )} \arcsin \left (b x + a\right )^{3}}{3 \, b^{3}} - \frac {a \arcsin \left (b x + a\right )^{3}}{2 \, b^{3}} - \frac {{\left (-{\left (b x + a\right )}^{2} + 1\right )}^{\frac {3}{2}} \arcsin \left (b x + a\right )^{2}}{3 \, b^{3}} - \frac {2 \, {\left ({\left (b x + a\right )}^{2} - 1\right )} {\left (b x + a\right )} \arcsin \left (b x + a\right )}{9 \, b^{3}} + \frac {3 \, {\left ({\left (b x + a\right )}^{2} - 1\right )} a \arcsin \left (b x + a\right )}{2 \, b^{3}} + \frac {3 \, \sqrt {-{\left (b x + a\right )}^{2} + 1} {\left (b x + a\right )} a}{4 \, b^{3}} - \frac {6 \, \sqrt {-{\left (b x + a\right )}^{2} + 1} a^{2}}{b^{3}} + \frac {\sqrt {-{\left (b x + a\right )}^{2} + 1} \arcsin \left (b x + a\right )^{2}}{b^{3}} - \frac {14 \, {\left (b x + a\right )} \arcsin \left (b x + a\right )}{9 \, b^{3}} + \frac {3 \, a \arcsin \left (b x + a\right )}{4 \, b^{3}} + \frac {2 \, {\left (-{\left (b x + a\right )}^{2} + 1\right )}^{\frac {3}{2}}}{27 \, b^{3}} - \frac {14 \, \sqrt {-{\left (b x + a\right )}^{2} + 1}}{9 \, b^{3}} \]
(b*x + a)*a^2*arcsin(b*x + a)^3/b^3 + 1/3*((b*x + a)^2 - 1)*(b*x + a)*arcs in(b*x + a)^3/b^3 - ((b*x + a)^2 - 1)*a*arcsin(b*x + a)^3/b^3 - 3/2*sqrt(- (b*x + a)^2 + 1)*(b*x + a)*a*arcsin(b*x + a)^2/b^3 + 3*sqrt(-(b*x + a)^2 + 1)*a^2*arcsin(b*x + a)^2/b^3 - 6*(b*x + a)*a^2*arcsin(b*x + a)/b^3 + 1/3* (b*x + a)*arcsin(b*x + a)^3/b^3 - 1/2*a*arcsin(b*x + a)^3/b^3 - 1/3*(-(b*x + a)^2 + 1)^(3/2)*arcsin(b*x + a)^2/b^3 - 2/9*((b*x + a)^2 - 1)*(b*x + a) *arcsin(b*x + a)/b^3 + 3/2*((b*x + a)^2 - 1)*a*arcsin(b*x + a)/b^3 + 3/4*s qrt(-(b*x + a)^2 + 1)*(b*x + a)*a/b^3 - 6*sqrt(-(b*x + a)^2 + 1)*a^2/b^3 + sqrt(-(b*x + a)^2 + 1)*arcsin(b*x + a)^2/b^3 - 14/9*(b*x + a)*arcsin(b*x + a)/b^3 + 3/4*a*arcsin(b*x + a)/b^3 + 2/27*(-(b*x + a)^2 + 1)^(3/2)/b^3 - 14/9*sqrt(-(b*x + a)^2 + 1)/b^3
Timed out. \[ \int x^2 \arcsin (a+b x)^3 \, dx=\int x^2\,{\mathrm {asin}\left (a+b\,x\right )}^3 \,d x \]