Integrand size = 12, antiderivative size = 220 \[ \int x^2 \arcsin (a+b x)^2 \, dx=-\frac {4 x}{9 b^2}-\frac {2 a^2 x}{b^2}+\frac {a (a+b x)^2}{2 b^3}-\frac {2 (a+b x)^3}{27 b^3}+\frac {4 \sqrt {1-(a+b x)^2} \arcsin (a+b x)}{9 b^3}+\frac {2 a^2 \sqrt {1-(a+b x)^2} \arcsin (a+b x)}{b^3}-\frac {a (a+b x) \sqrt {1-(a+b x)^2} \arcsin (a+b x)}{b^3}+\frac {2 (a+b x)^2 \sqrt {1-(a+b x)^2} \arcsin (a+b x)}{9 b^3}+\frac {a \arcsin (a+b x)^2}{2 b^3}+\frac {a^3 \arcsin (a+b x)^2}{3 b^3}+\frac {1}{3} x^3 \arcsin (a+b x)^2 \]
-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*arcs in(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
Time = 0.44 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.50 \[ \int x^2 \arcsin (a+b x)^2 \, dx=\frac {-b x \left (24+66 a^2-15 a b x+4 b^2 x^2\right )+6 \sqrt {1-a^2-2 a b x-b^2 x^2} \left (4+11 a^2-5 a b x+2 b^2 x^2\right ) \arcsin (a+b x)+9 \left (3 a+2 a^3+2 b^3 x^3\right ) \arcsin (a+b x)^2}{54 b^3} \]
(-(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)
Time = 0.57 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.96, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {5304, 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.
| \(\displaystyle \int x^2 \arcsin (a+b x)^2 \, dx\) |
| \(\Big \downarrow \) 5304 |
| \(\displaystyle \frac {\int x^2 \arcsin (a+b x)^2d(a+b x)}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int b^2 x^2 \arcsin (a+b x)^2d(a+b x)}{b^3}\) |
| \(\Big \downarrow \) 5242 |
| \(\displaystyle \frac {\frac {2}{3} \int -\frac {b^3 x^3 \arcsin (a+b x)}{\sqrt {1-(a+b x)^2}}d(a+b x)+\frac {1}{3} b^3 x^3 \arcsin (a+b x)^2}{b^3}\) |
| \(\Big \downarrow \) 5262 |
| \(\displaystyle \frac {\frac {2}{3} \int \left (\frac {\arcsin (a+b x) a^3}{\sqrt {1-(a+b x)^2}}-\frac {3 (a+b x) \arcsin (a+b x) a^2}{\sqrt {1-(a+b x)^2}}+\frac {3 (a+b x)^2 \arcsin (a+b x) a}{\sqrt {1-(a+b x)^2}}-\frac {(a+b x)^3 \arcsin (a+b x)}{\sqrt {1-(a+b x)^2}}\right )d(a+b x)+\frac {1}{3} b^3 x^3 \arcsin (a+b x)^2}{b^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {2}{3} \left (\frac {1}{2} a^3 \arcsin (a+b x)^2+3 a^2 \sqrt {1-(a+b x)^2} \arcsin (a+b x)-3 a^2 (a+b x)+\frac {3}{4} a \arcsin (a+b x)^2-\frac {3}{2} a (a+b x) \sqrt {1-(a+b x)^2} \arcsin (a+b x)+\frac {1}{3} (a+b x)^2 \sqrt {1-(a+b x)^2} \arcsin (a+b x)+\frac {2}{3} \sqrt {1-(a+b x)^2} \arcsin (a+b x)+\frac {3}{4} a (a+b x)^2-\frac {1}{9} (a+b x)^3-\frac {2}{3} (a+b x)\right )+\frac {1}{3} b^3 x^3 \arcsin (a+b x)^2}{b^3}\) |
((b^3*x^3*ArcSin[a + b*x]^2)/3 + (2*((-2*(a + b*x))/3 - 3*a^2*(a + b*x) + (3*a*(a + b*x)^2)/4 - (a + b*x)^3/9 + (2*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x])/3 + 3*a^2*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x] - (3*a*(a + b*x)*Sqr t[1 - (a + b*x)^2]*ArcSin[a + b*x])/2 + ((a + b*x)^2*Sqrt[1 - (a + b*x)^2] *ArcSin[a + b*x])/3 + (3*a*ArcSin[a + b*x]^2)/4 + (a^3*ArcSin[a + b*x]^2)/ 2))/3)/b^3
3.2.32.3.1 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 = 0.68 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.83
| method | result | size |
| derivativedivides | \(\frac {\frac {\left (b x +a \right )^{3} \arcsin \left (b x +a \right )^{2}}{3}+\frac {2 \arcsin \left (b x +a \right ) \left (\left (b x +a \right )^{2}+2\right ) \sqrt {1-\left (b x +a \right )^{2}}}{9}-\frac {2 \left (b x +a \right )^{3}}{27}-\frac {4 b x}{9}-\frac {4 a}{9}-\frac {a \left (2 \arcsin \left (b x +a \right )^{2} \left (b x +a \right )^{2}+2 \arcsin \left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}\, \left (b x +a \right )-\arcsin \left (b x +a \right )^{2}-\left (b x +a \right )^{2}\right )}{2}+a^{2} \left (\arcsin \left (b x +a \right )^{2} \left (b x +a \right )-2 b x -2 a +2 \arcsin \left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}\right )}{b^{3}}\) | \(182\) |
| default | \(\frac {\frac {\left (b x +a \right )^{3} \arcsin \left (b x +a \right )^{2}}{3}+\frac {2 \arcsin \left (b x +a \right ) \left (\left (b x +a \right )^{2}+2\right ) \sqrt {1-\left (b x +a \right )^{2}}}{9}-\frac {2 \left (b x +a \right )^{3}}{27}-\frac {4 b x}{9}-\frac {4 a}{9}-\frac {a \left (2 \arcsin \left (b x +a \right )^{2} \left (b x +a \right )^{2}+2 \arcsin \left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}\, \left (b x +a \right )-\arcsin \left (b x +a \right )^{2}-\left (b x +a \right )^{2}\right )}{2}+a^{2} \left (\arcsin \left (b x +a \right )^{2} \left (b x +a \right )-2 b x -2 a +2 \arcsin \left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}\right )}{b^{3}}\) | \(182\) |
1/b^3*(1/3*(b*x+a)^3*arcsin(b*x+a)^2+2/9*arcsin(b*x+a)*((b*x+a)^2+2)*(1-(b *x+a)^2)^(1/2)-2/27*(b*x+a)^3-4/9*b*x-4/9*a-1/2*a*(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^2*(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 = 111, normalized size of antiderivative = 0.50 \[ \int x^2 \arcsin (a+b x)^2 \, dx=-\frac {4 \, b^{3} x^{3} - 15 \, a b^{2} x^{2} + 6 \, {\left (11 \, a^{2} + 4\right )} b x - 9 \, {\left (2 \, b^{3} x^{3} + 2 \, a^{3} + 3 \, a\right )} \arcsin \left (b x + a\right )^{2} - 6 \, {\left (2 \, b^{2} x^{2} - 5 \, a b x + 11 \, a^{2} + 4\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} \arcsin \left (b x + a\right )}{54 \, b^{3}} \]
-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
Time = 0.31 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.10 \[ \int x^2 \arcsin (a+b x)^2 \, dx=\begin {cases} \frac {a^{3} \operatorname {asin}^{2}{\left (a + b x \right )}}{3 b^{3}} - \frac {11 a^{2} x}{9 b^{2}} + \frac {11 a^{2} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}{\left (a + b x \right )}}{9 b^{3}} + \frac {5 a x^{2}}{18 b} - \frac {5 a x \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}{\left (a + b x \right )}}{9 b^{2}} + \frac {a \operatorname {asin}^{2}{\left (a + b x \right )}}{2 b^{3}} + \frac {x^{3} \operatorname {asin}^{2}{\left (a + b x \right )}}{3} - \frac {2 x^{3}}{27} + \frac {2 x^{2} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}{\left (a + b x \right )}}{9 b} - \frac {4 x}{9 b^{2}} + \frac {4 \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}{\left (a + b x \right )}}{9 b^{3}} & \text {for}\: b \neq 0 \\\frac {x^{3} \operatorname {asin}^{2}{\left (a \right )}}{3} & \text {otherwise} \end {cases} \]
Piecewise((a**3*asin(a + b*x)**2/(3*b**3) - 11*a**2*x/(9*b**2) + 11*a**2*s qrt(-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))
\[ \int x^2 \arcsin (a+b x)^2 \, dx=\int { x^{2} \arcsin \left (b x + a\right )^{2} \,d x } \]
1/3*x^3*arctan2(b*x + a, sqrt(b*x + a + 1)*sqrt(-b*x - a + 1))^2 + 2*b*int egrate(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)
Time = 0.28 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.23 \[ \int x^2 \arcsin (a+b x)^2 \, dx=\frac {{\left (b x + a\right )} a^{2} \arcsin \left (b x + a\right )^{2}}{b^{3}} + \frac {{\left ({\left (b x + a\right )}^{2} - 1\right )} {\left (b x + a\right )} \arcsin \left (b x + a\right )^{2}}{3 \, b^{3}} - \frac {{\left ({\left (b x + a\right )}^{2} - 1\right )} a \arcsin \left (b x + a\right )^{2}}{b^{3}} - \frac {\sqrt {-{\left (b x + a\right )}^{2} + 1} {\left (b x + a\right )} a \arcsin \left (b x + a\right )}{b^{3}} + \frac {2 \, \sqrt {-{\left (b x + a\right )}^{2} + 1} a^{2} \arcsin \left (b x + a\right )}{b^{3}} - \frac {2 \, {\left (b x + a\right )} a^{2}}{b^{3}} + \frac {{\left (b x + a\right )} \arcsin \left (b x + a\right )^{2}}{3 \, b^{3}} - \frac {a \arcsin \left (b x + a\right )^{2}}{2 \, b^{3}} - \frac {2 \, {\left (-{\left (b x + a\right )}^{2} + 1\right )}^{\frac {3}{2}} \arcsin \left (b x + a\right )}{9 \, b^{3}} - \frac {2 \, {\left ({\left (b x + a\right )}^{2} - 1\right )} {\left (b x + a\right )}}{27 \, b^{3}} + \frac {{\left ({\left (b x + a\right )}^{2} - 1\right )} a}{2 \, b^{3}} + \frac {2 \, \sqrt {-{\left (b x + a\right )}^{2} + 1} \arcsin \left (b x + a\right )}{3 \, b^{3}} - \frac {14 \, {\left (b x + a\right )}}{27 \, b^{3}} + \frac {a}{4 \, b^{3}} \]
(b*x + a)*a^2*arcsin(b*x + a)^2/b^3 + 1/3*((b*x + a)^2 - 1)*(b*x + a)*arcs in(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)*arcsi n(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
Timed out. \[ \int x^2 \arcsin (a+b x)^2 \, dx=\int x^2\,{\mathrm {asin}\left (a+b\,x\right )}^2 \,d x \]