Integrand size = 12, antiderivative size = 343 \[ \int x^3 \arcsin (a+b x)^2 \, dx=\frac {4 a x}{3 b^3}+\frac {2 a^3 x}{b^3}-\frac {3 (a+b x)^2}{32 b^4}-\frac {3 a^2 (a+b x)^2}{4 b^4}+\frac {2 a (a+b x)^3}{9 b^4}-\frac {(a+b x)^4}{32 b^4}-\frac {4 a \sqrt {1-(a+b x)^2} \arcsin (a+b x)}{3 b^4}-\frac {2 a^3 \sqrt {1-(a+b x)^2} \arcsin (a+b x)}{b^4}+\frac {3 (a+b x) \sqrt {1-(a+b x)^2} \arcsin (a+b x)}{16 b^4}+\frac {3 a^2 (a+b x) \sqrt {1-(a+b x)^2} \arcsin (a+b x)}{2 b^4}-\frac {2 a (a+b x)^2 \sqrt {1-(a+b x)^2} \arcsin (a+b x)}{3 b^4}+\frac {(a+b x)^3 \sqrt {1-(a+b x)^2} \arcsin (a+b x)}{8 b^4}-\frac {3 \arcsin (a+b x)^2}{32 b^4}-\frac {3 a^2 \arcsin (a+b x)^2}{4 b^4}-\frac {a^4 \arcsin (a+b x)^2}{4 b^4}+\frac {1}{4} x^4 \arcsin (a+b x)^2 \]
4/3*a*x/b^3+2*a^3*x/b^3-3/32*(b*x+a)^2/b^4-3/4*a^2*(b*x+a)^2/b^4+2/9*a*(b* x+a)^3/b^4-1/32*(b*x+a)^4/b^4-3/32*arcsin(b*x+a)^2/b^4-3/4*a^2*arcsin(b*x+ a)^2/b^4-1/4*a^4*arcsin(b*x+a)^2/b^4+1/4*x^4*arcsin(b*x+a)^2-4/3*a*arcsin( b*x+a)*(1-(b*x+a)^2)^(1/2)/b^4-2*a^3*arcsin(b*x+a)*(1-(b*x+a)^2)^(1/2)/b^4 +3/16*(b*x+a)*arcsin(b*x+a)*(1-(b*x+a)^2)^(1/2)/b^4+3/2*a^2*(b*x+a)*arcsin (b*x+a)*(1-(b*x+a)^2)^(1/2)/b^4-2/3*a*(b*x+a)^2*arcsin(b*x+a)*(1-(b*x+a)^2 )^(1/2)/b^4+1/8*(b*x+a)^3*arcsin(b*x+a)*(1-(b*x+a)^2)^(1/2)/b^4
Time = 0.61 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.43 \[ \int x^3 \arcsin (a+b x)^2 \, dx=\frac {b x \left (300 a^3-78 a^2 b x-9 b x \left (3+b^2 x^2\right )+a \left (330+28 b^2 x^2\right )\right )-6 \sqrt {1-a^2-2 a b x-b^2 x^2} \left (55 a+50 a^3-9 b x-26 a^2 b x+14 a b^2 x^2-6 b^3 x^3\right ) \arcsin (a+b x)-9 \left (3+24 a^2+8 a^4-8 b^4 x^4\right ) \arcsin (a+b x)^2}{288 b^4} \]
(b*x*(300*a^3 - 78*a^2*b*x - 9*b*x*(3 + b^2*x^2) + a*(330 + 28*b^2*x^2)) - 6*Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]*(55*a + 50*a^3 - 9*b*x - 26*a^2*b*x + 14*a*b^2*x^2 - 6*b^3*x^3)*ArcSin[a + b*x] - 9*(3 + 24*a^2 + 8*a^4 - 8*b^4 *x^4)*ArcSin[a + b*x]^2)/(288*b^4)
Time = 0.71 (sec) , antiderivative size = 317, normalized size of antiderivative = 0.92, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5304, 25, 27, 5242, 5262, 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.
| \(\displaystyle \int x^3 \arcsin (a+b x)^2 \, dx\) |
| \(\Big \downarrow \) 5304 |
| \(\displaystyle \frac {\int x^3 \arcsin (a+b x)^2d(a+b x)}{b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -x^3 \arcsin (a+b x)^2d(a+b x)}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int -b^3 x^3 \arcsin (a+b x)^2d(a+b x)}{b^4}\) |
| \(\Big \downarrow \) 5242 |
| \(\displaystyle -\frac {\frac {1}{2} \int \frac {b^4 x^4 \arcsin (a+b x)}{\sqrt {1-(a+b x)^2}}d(a+b x)-\frac {1}{4} b^4 x^4 \arcsin (a+b x)^2}{b^4}\) |
| \(\Big \downarrow \) 5262 |
| \(\displaystyle -\frac {\frac {1}{2} \int \left (\frac {\arcsin (a+b x) a^4}{\sqrt {1-(a+b x)^2}}-\frac {4 (a+b x) \arcsin (a+b x) a^3}{\sqrt {1-(a+b x)^2}}+\frac {6 (a+b x)^2 \arcsin (a+b x) a^2}{\sqrt {1-(a+b x)^2}}-\frac {4 (a+b x)^3 \arcsin (a+b x) a}{\sqrt {1-(a+b x)^2}}+\frac {(a+b x)^4 \arcsin (a+b x)}{\sqrt {1-(a+b x)^2}}\right )d(a+b x)-\frac {1}{4} b^4 x^4 \arcsin (a+b x)^2}{b^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {1}{2} a^4 \arcsin (a+b x)^2+4 a^3 \sqrt {1-(a+b x)^2} \arcsin (a+b x)-4 a^3 (a+b x)+\frac {3}{2} a^2 \arcsin (a+b x)^2-3 a^2 (a+b x) \sqrt {1-(a+b x)^2} \arcsin (a+b x)+\frac {3}{2} a^2 (a+b x)^2+\frac {4}{3} a (a+b x)^2 \sqrt {1-(a+b x)^2} \arcsin (a+b x)+\frac {8}{3} a \sqrt {1-(a+b x)^2} \arcsin (a+b x)+\frac {3}{16} \arcsin (a+b x)^2-\frac {1}{4} (a+b x)^3 \sqrt {1-(a+b x)^2} \arcsin (a+b x)-\frac {3}{8} (a+b x) \sqrt {1-(a+b x)^2} \arcsin (a+b x)-\frac {4}{9} a (a+b x)^3-\frac {8}{3} a (a+b x)+\frac {1}{16} (a+b x)^4+\frac {3}{16} (a+b x)^2\right )-\frac {1}{4} b^4 x^4 \arcsin (a+b x)^2}{b^4}\) |
-((-1/4*(b^4*x^4*ArcSin[a + b*x]^2) + ((-8*a*(a + b*x))/3 - 4*a^3*(a + b*x ) + (3*(a + b*x)^2)/16 + (3*a^2*(a + b*x)^2)/2 - (4*a*(a + b*x)^3)/9 + (a + b*x)^4/16 + (8*a*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x])/3 + 4*a^3*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x] - (3*(a + b*x)*Sqrt[1 - (a + b*x)^2]*ArcSi n[a + b*x])/8 - 3*a^2*(a + b*x)*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x] + (4 *a*(a + b*x)^2*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x])/3 - ((a + b*x)^3*Sqr t[1 - (a + b*x)^2]*ArcSin[a + b*x])/4 + (3*ArcSin[a + b*x]^2)/16 + (3*a^2* ArcSin[a + b*x]^2)/2 + (a^4*ArcSin[a + b*x]^2)/2)/2)/b^4)
3.2.31.3.1 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[((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_.)*((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))
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 = 1.04 (sec) , antiderivative size = 309, normalized size of antiderivative = 0.90
| method | result | size |
| derivativedivides | \(\frac {\frac {\arcsin \left (b x +a \right )^{2} \left (b x +a \right )^{4}}{4}-\frac {\arcsin \left (b x +a \right ) \left (-2 \left (b x +a \right )^{3} \sqrt {1-\left (b x +a \right )^{2}}-3 \left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}+3 \arcsin \left (b x +a \right )\right )}{16}+\frac {3 \arcsin \left (b x +a \right )^{2}}{32}-\frac {\left (2 \left (b x +a \right )^{2}+3\right )^{2}}{128}-\frac {a \left (9 \left (b x +a \right )^{3} \arcsin \left (b x +a \right )^{2}+6 \arcsin \left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}\, \left (b x +a \right )^{2}-2 \left (b x +a \right )^{3}+12 \arcsin \left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}-12 b x -12 a \right )}{9}+\frac {3 a^{2} \left (2 \arcsin \left (b x +a \right )^{2} \left (b x +a \right )^{2}+2 \arcsin \left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}\, \left (b x +a \right )-\arcsin \left (b x +a \right )^{2}-\left (b x +a \right )^{2}\right )}{4}-a^{3} \left (\arcsin \left (b x +a \right )^{2} \left (b x +a \right )-2 b x -2 a +2 \arcsin \left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}\right )}{b^{4}}\) | \(309\) |
| default | \(\frac {\frac {\arcsin \left (b x +a \right )^{2} \left (b x +a \right )^{4}}{4}-\frac {\arcsin \left (b x +a \right ) \left (-2 \left (b x +a \right )^{3} \sqrt {1-\left (b x +a \right )^{2}}-3 \left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}+3 \arcsin \left (b x +a \right )\right )}{16}+\frac {3 \arcsin \left (b x +a \right )^{2}}{32}-\frac {\left (2 \left (b x +a \right )^{2}+3\right )^{2}}{128}-\frac {a \left (9 \left (b x +a \right )^{3} \arcsin \left (b x +a \right )^{2}+6 \arcsin \left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}\, \left (b x +a \right )^{2}-2 \left (b x +a \right )^{3}+12 \arcsin \left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}-12 b x -12 a \right )}{9}+\frac {3 a^{2} \left (2 \arcsin \left (b x +a \right )^{2} \left (b x +a \right )^{2}+2 \arcsin \left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}\, \left (b x +a \right )-\arcsin \left (b x +a \right )^{2}-\left (b x +a \right )^{2}\right )}{4}-a^{3} \left (\arcsin \left (b x +a \right )^{2} \left (b x +a \right )-2 b x -2 a +2 \arcsin \left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}\right )}{b^{4}}\) | \(309\) |
1/b^4*(1/4*arcsin(b*x+a)^2*(b*x+a)^4-1/16*arcsin(b*x+a)*(-2*(b*x+a)^3*(1-( b*x+a)^2)^(1/2)-3*(b*x+a)*(1-(b*x+a)^2)^(1/2)+3*arcsin(b*x+a))+3/32*arcsin (b*x+a)^2-1/128*(2*(b*x+a)^2+3)^2-1/9*a*(9*(b*x+a)^3*arcsin(b*x+a)^2+6*arc sin(b*x+a)*(1-(b*x+a)^2)^(1/2)*(b*x+a)^2-2*(b*x+a)^3+12*arcsin(b*x+a)*(1-( b*x+a)^2)^(1/2)-12*b*x-12*a)+3/4*a^2*(2*arcsin(b*x+a)^2*(b*x+a)^2+2*arcsin (b*x+a)*(1-(b*x+a)^2)^(1/2)*(b*x+a)-arcsin(b*x+a)^2-(b*x+a)^2)-a^3*(arcsin (b*x+a)^2*(b*x+a)-2*b*x-2*a+2*arcsin(b*x+a)*(1-(b*x+a)^2)^(1/2)))
Time = 0.25 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.43 \[ \int x^3 \arcsin (a+b x)^2 \, dx=-\frac {9 \, b^{4} x^{4} - 28 \, a b^{3} x^{3} + 3 \, {\left (26 \, a^{2} + 9\right )} b^{2} x^{2} - 30 \, {\left (10 \, a^{3} + 11 \, a\right )} b x - 9 \, {\left (8 \, b^{4} x^{4} - 8 \, a^{4} - 24 \, a^{2} - 3\right )} \arcsin \left (b x + a\right )^{2} - 6 \, {\left (6 \, b^{3} x^{3} - 14 \, a b^{2} x^{2} - 50 \, a^{3} + {\left (26 \, a^{2} + 9\right )} b x - 55 \, a\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} \arcsin \left (b x + a\right )}{288 \, b^{4}} \]
-1/288*(9*b^4*x^4 - 28*a*b^3*x^3 + 3*(26*a^2 + 9)*b^2*x^2 - 30*(10*a^3 + 1 1*a)*b*x - 9*(8*b^4*x^4 - 8*a^4 - 24*a^2 - 3)*arcsin(b*x + a)^2 - 6*(6*b^3 *x^3 - 14*a*b^2*x^2 - 50*a^3 + (26*a^2 + 9)*b*x - 55*a)*sqrt(-b^2*x^2 - 2* a*b*x - a^2 + 1)*arcsin(b*x + a))/b^4
Time = 0.46 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.07 \[ \int x^3 \arcsin (a+b x)^2 \, dx=\begin {cases} - \frac {a^{4} \operatorname {asin}^{2}{\left (a + b x \right )}}{4 b^{4}} + \frac {25 a^{3} x}{24 b^{3}} - \frac {25 a^{3} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}{\left (a + b x \right )}}{24 b^{4}} - \frac {13 a^{2} x^{2}}{48 b^{2}} + \frac {13 a^{2} x \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}{\left (a + b x \right )}}{24 b^{3}} - \frac {3 a^{2} \operatorname {asin}^{2}{\left (a + b x \right )}}{4 b^{4}} + \frac {7 a x^{3}}{72 b} - \frac {7 a x^{2} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}{\left (a + b x \right )}}{24 b^{2}} + \frac {55 a x}{48 b^{3}} - \frac {55 a \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}{\left (a + b x \right )}}{48 b^{4}} + \frac {x^{4} \operatorname {asin}^{2}{\left (a + b x \right )}}{4} - \frac {x^{4}}{32} + \frac {x^{3} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}{\left (a + b x \right )}}{8 b} - \frac {3 x^{2}}{32 b^{2}} + \frac {3 x \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}{\left (a + b x \right )}}{16 b^{3}} - \frac {3 \operatorname {asin}^{2}{\left (a + b x \right )}}{32 b^{4}} & \text {for}\: b \neq 0 \\\frac {x^{4} \operatorname {asin}^{2}{\left (a \right )}}{4} & \text {otherwise} \end {cases} \]
Piecewise((-a**4*asin(a + b*x)**2/(4*b**4) + 25*a**3*x/(24*b**3) - 25*a**3 *sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)*asin(a + b*x)/(24*b**4) - 13*a**2*x **2/(48*b**2) + 13*a**2*x*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)*asin(a + b *x)/(24*b**3) - 3*a**2*asin(a + b*x)**2/(4*b**4) + 7*a*x**3/(72*b) - 7*a*x **2*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)*asin(a + b*x)/(24*b**2) + 55*a*x /(48*b**3) - 55*a*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)*asin(a + b*x)/(48* b**4) + x**4*asin(a + b*x)**2/4 - x**4/32 + x**3*sqrt(-a**2 - 2*a*b*x - b* *2*x**2 + 1)*asin(a + b*x)/(8*b) - 3*x**2/(32*b**2) + 3*x*sqrt(-a**2 - 2*a *b*x - b**2*x**2 + 1)*asin(a + b*x)/(16*b**3) - 3*asin(a + b*x)**2/(32*b** 4), Ne(b, 0)), (x**4*asin(a)**2/4, True))
\[ \int x^3 \arcsin (a+b x)^2 \, dx=\int { x^{3} \arcsin \left (b x + a\right )^{2} \,d x } \]
1/4*x^4*arctan2(b*x + a, sqrt(b*x + a + 1)*sqrt(-b*x - a + 1))^2 + b*integ rate(1/2*sqrt(b*x + a + 1)*sqrt(-b*x - a + 1)*x^4*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)
Time = 0.29 (sec) , antiderivative size = 440, normalized size of antiderivative = 1.28 \[ \int x^3 \arcsin (a+b x)^2 \, dx=-\frac {{\left (b x + a\right )} a^{3} \arcsin \left (b x + a\right )^{2}}{b^{4}} - \frac {{\left ({\left (b x + a\right )}^{2} - 1\right )} {\left (b x + a\right )} a \arcsin \left (b x + a\right )^{2}}{b^{4}} + \frac {3 \, {\left ({\left (b x + a\right )}^{2} - 1\right )} a^{2} \arcsin \left (b x + a\right )^{2}}{2 \, b^{4}} + \frac {3 \, \sqrt {-{\left (b x + a\right )}^{2} + 1} {\left (b x + a\right )} a^{2} \arcsin \left (b x + a\right )}{2 \, b^{4}} - \frac {2 \, \sqrt {-{\left (b x + a\right )}^{2} + 1} a^{3} \arcsin \left (b x + a\right )}{b^{4}} + \frac {2 \, {\left (b x + a\right )} a^{3}}{b^{4}} + \frac {{\left ({\left (b x + a\right )}^{2} - 1\right )}^{2} \arcsin \left (b x + a\right )^{2}}{4 \, b^{4}} - \frac {{\left (b x + a\right )} a \arcsin \left (b x + a\right )^{2}}{b^{4}} + \frac {3 \, a^{2} \arcsin \left (b x + a\right )^{2}}{4 \, b^{4}} - \frac {{\left (-{\left (b x + a\right )}^{2} + 1\right )}^{\frac {3}{2}} {\left (b x + a\right )} \arcsin \left (b x + a\right )}{8 \, b^{4}} + \frac {2 \, {\left (-{\left (b x + a\right )}^{2} + 1\right )}^{\frac {3}{2}} a \arcsin \left (b x + a\right )}{3 \, b^{4}} + \frac {2 \, {\left ({\left (b x + a\right )}^{2} - 1\right )} {\left (b x + a\right )} a}{9 \, b^{4}} - \frac {3 \, {\left ({\left (b x + a\right )}^{2} - 1\right )} a^{2}}{4 \, b^{4}} + \frac {{\left ({\left (b x + a\right )}^{2} - 1\right )} \arcsin \left (b x + a\right )^{2}}{2 \, b^{4}} + \frac {5 \, \sqrt {-{\left (b x + a\right )}^{2} + 1} {\left (b x + a\right )} \arcsin \left (b x + a\right )}{16 \, b^{4}} - \frac {2 \, \sqrt {-{\left (b x + a\right )}^{2} + 1} a \arcsin \left (b x + a\right )}{b^{4}} - \frac {{\left ({\left (b x + a\right )}^{2} - 1\right )}^{2}}{32 \, b^{4}} + \frac {14 \, {\left (b x + a\right )} a}{9 \, b^{4}} - \frac {3 \, a^{2}}{8 \, b^{4}} + \frac {5 \, \arcsin \left (b x + a\right )^{2}}{32 \, b^{4}} - \frac {5 \, {\left ({\left (b x + a\right )}^{2} - 1\right )}}{32 \, b^{4}} - \frac {17}{256 \, b^{4}} \]
-(b*x + a)*a^3*arcsin(b*x + a)^2/b^4 - ((b*x + a)^2 - 1)*(b*x + a)*a*arcsi n(b*x + a)^2/b^4 + 3/2*((b*x + a)^2 - 1)*a^2*arcsin(b*x + a)^2/b^4 + 3/2*s qrt(-(b*x + a)^2 + 1)*(b*x + a)*a^2*arcsin(b*x + a)/b^4 - 2*sqrt(-(b*x + a )^2 + 1)*a^3*arcsin(b*x + a)/b^4 + 2*(b*x + a)*a^3/b^4 + 1/4*((b*x + a)^2 - 1)^2*arcsin(b*x + a)^2/b^4 - (b*x + a)*a*arcsin(b*x + a)^2/b^4 + 3/4*a^2 *arcsin(b*x + a)^2/b^4 - 1/8*(-(b*x + a)^2 + 1)^(3/2)*(b*x + a)*arcsin(b*x + a)/b^4 + 2/3*(-(b*x + a)^2 + 1)^(3/2)*a*arcsin(b*x + a)/b^4 + 2/9*((b*x + a)^2 - 1)*(b*x + a)*a/b^4 - 3/4*((b*x + a)^2 - 1)*a^2/b^4 + 1/2*((b*x + a)^2 - 1)*arcsin(b*x + a)^2/b^4 + 5/16*sqrt(-(b*x + a)^2 + 1)*(b*x + a)*a rcsin(b*x + a)/b^4 - 2*sqrt(-(b*x + a)^2 + 1)*a*arcsin(b*x + a)/b^4 - 1/32 *((b*x + a)^2 - 1)^2/b^4 + 14/9*(b*x + a)*a/b^4 - 3/8*a^2/b^4 + 5/32*arcsi n(b*x + a)^2/b^4 - 5/32*((b*x + a)^2 - 1)/b^4 - 17/256/b^4
Timed out. \[ \int x^3 \arcsin (a+b x)^2 \, dx=\int x^3\,{\mathrm {asin}\left (a+b\,x\right )}^2 \,d x \]