Integrand size = 10, antiderivative size = 211 \[ \int x \arcsin (a+b x)^3 \, dx=\frac {6 a \sqrt {1-(a+b x)^2}}{b^2}-\frac {3 (a+b x) \sqrt {1-(a+b x)^2}}{8 b^2}+\frac {3 \arcsin (a+b x)}{8 b^2}+\frac {6 a (a+b x) \arcsin (a+b x)}{b^2}-\frac {3 (a+b x)^2 \arcsin (a+b x)}{4 b^2}-\frac {3 a \sqrt {1-(a+b x)^2} \arcsin (a+b x)^2}{b^2}+\frac {3 (a+b x) \sqrt {1-(a+b x)^2} \arcsin (a+b x)^2}{4 b^2}-\frac {\arcsin (a+b x)^3}{4 b^2}-\frac {a^2 \arcsin (a+b x)^3}{2 b^2}+\frac {1}{2} x^2 \arcsin (a+b x)^3 \]
3/8*arcsin(b*x+a)/b^2+6*a*(b*x+a)*arcsin(b*x+a)/b^2-3/4*(b*x+a)^2*arcsin(b *x+a)/b^2-1/4*arcsin(b*x+a)^3/b^2-1/2*a^2*arcsin(b*x+a)^3/b^2+1/2*x^2*arcs in(b*x+a)^3+6*a*(1-(b*x+a)^2)^(1/2)/b^2-3/8*(b*x+a)*(1-(b*x+a)^2)^(1/2)/b^ 2-3*a*arcsin(b*x+a)^2*(1-(b*x+a)^2)^(1/2)/b^2+3/4*(b*x+a)*arcsin(b*x+a)^2* (1-(b*x+a)^2)^(1/2)/b^2
Time = 0.35 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.64 \[ \int x \arcsin (a+b x)^3 \, dx=\frac {3 (15 a-b x) \sqrt {1-a^2-2 a b x-b^2 x^2}+\left (3+42 a^2+36 a b x-6 b^2 x^2\right ) \arcsin (a+b x)-6 (3 a-b x) \sqrt {1-a^2-2 a b x-b^2 x^2} \arcsin (a+b x)^2+\left (-2-4 a^2+4 b^2 x^2\right ) \arcsin (a+b x)^3}{8 b^2} \]
(3*(15*a - b*x)*Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2] + (3 + 42*a^2 + 36*a*b*x - 6*b^2*x^2)*ArcSin[a + b*x] - 6*(3*a - b*x)*Sqrt[1 - a^2 - 2*a*b*x - b^2 *x^2]*ArcSin[a + b*x]^2 + (-2 - 4*a^2 + 4*b^2*x^2)*ArcSin[a + b*x]^3)/(8*b ^2)
Time = 0.52 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.94, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {5304, 25, 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.
| \(\displaystyle \int x \arcsin (a+b x)^3 \, dx\) |
| \(\Big \downarrow \) 5304 |
| \(\displaystyle \frac {\int x \arcsin (a+b x)^3d(a+b x)}{b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -x \arcsin (a+b x)^3d(a+b x)}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int -b x \arcsin (a+b x)^3d(a+b x)}{b^2}\) |
| \(\Big \downarrow \) 5242 |
| \(\displaystyle -\frac {\frac {3}{2} \int \frac {b^2 x^2 \arcsin (a+b x)^2}{\sqrt {1-(a+b x)^2}}d(a+b x)-\frac {1}{2} b^2 x^2 \arcsin (a+b x)^3}{b^2}\) |
| \(\Big \downarrow \) 5272 |
| \(\displaystyle -\frac {\frac {3}{2} \int b^2 x^2 \arcsin (a+b x)^2d\arcsin (a+b x)-\frac {1}{2} b^2 x^2 \arcsin (a+b x)^3}{b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {3}{2} \int \arcsin (a+b x)^2 (a-\sin (\arcsin (a+b x)))^2d\arcsin (a+b x)-\frac {1}{2} b^2 x^2 \arcsin (a+b x)^3}{b^2}\) |
| \(\Big \downarrow \) 3798 |
| \(\displaystyle -\frac {\frac {3}{2} \int \left (a^2 \arcsin (a+b x)^2+(a+b x)^2 \arcsin (a+b x)^2-2 a (a+b x) \arcsin (a+b x)^2\right )d\arcsin (a+b x)-\frac {1}{2} b^2 x^2 \arcsin (a+b x)^3}{b^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\frac {3}{2} \left (\frac {1}{3} a^2 \arcsin (a+b x)^3+\frac {1}{6} \arcsin (a+b x)^3+2 a \sqrt {1-(a+b x)^2} \arcsin (a+b x)^2-\frac {1}{2} (a+b x) \sqrt {1-(a+b x)^2} \arcsin (a+b x)^2+\frac {1}{2} (a+b x)^2 \arcsin (a+b x)-4 a (a+b x) \arcsin (a+b x)+\frac {1}{4} (-a-b x)-4 a \sqrt {1-(a+b x)^2}+\frac {1}{4} (a+b x) \sqrt {1-(a+b x)^2}\right )-\frac {1}{2} b^2 x^2 \arcsin (a+b x)^3}{b^2}\) |
-((-1/2*(b^2*x^2*ArcSin[a + b*x]^3) + (3*((-a - b*x)/4 - 4*a*Sqrt[1 - (a + b*x)^2] + ((a + b*x)*Sqrt[1 - (a + b*x)^2])/4 - 4*a*(a + b*x)*ArcSin[a + b*x] + ((a + b*x)^2*ArcSin[a + b*x])/2 + 2*a*Sqrt[1 - (a + b*x)^2]*ArcSin[ a + b*x]^2 - ((a + b*x)*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x]^2)/2 + ArcSi n[a + b*x]^3/6 + (a^2*ArcSin[a + b*x]^3)/3))/2)/b^2)
3.2.39.3.1 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 = 0.44 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.88
| method | result | size |
| derivativedivides | \(\frac {\frac {\left (-1+\left (b x +a \right )^{2}\right ) \arcsin \left (b x +a \right )^{3}}{2}+\frac {3 \arcsin \left (b x +a \right )^{2} \left (\left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}+\arcsin \left (b x +a \right )\right )}{4}-\frac {3 \left (-1+\left (b x +a \right )^{2}\right ) \arcsin \left (b x +a \right )}{4}-\frac {3 \left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}}{8}-\frac {3 \arcsin \left (b x +a \right )}{8}-\frac {\arcsin \left (b x +a \right )^{3}}{2}-a \left (\arcsin \left (b x +a \right )^{3} \left (b x +a \right )+3 \arcsin \left (b x +a \right )^{2} \sqrt {1-\left (b x +a \right )^{2}}-6 \sqrt {1-\left (b x +a \right )^{2}}-6 \arcsin \left (b x +a \right ) \left (b x +a \right )\right )}{b^{2}}\) | \(185\) |
| default | \(\frac {\frac {\left (-1+\left (b x +a \right )^{2}\right ) \arcsin \left (b x +a \right )^{3}}{2}+\frac {3 \arcsin \left (b x +a \right )^{2} \left (\left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}+\arcsin \left (b x +a \right )\right )}{4}-\frac {3 \left (-1+\left (b x +a \right )^{2}\right ) \arcsin \left (b x +a \right )}{4}-\frac {3 \left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}}{8}-\frac {3 \arcsin \left (b x +a \right )}{8}-\frac {\arcsin \left (b x +a \right )^{3}}{2}-a \left (\arcsin \left (b x +a \right )^{3} \left (b x +a \right )+3 \arcsin \left (b x +a \right )^{2} \sqrt {1-\left (b x +a \right )^{2}}-6 \sqrt {1-\left (b x +a \right )^{2}}-6 \arcsin \left (b x +a \right ) \left (b x +a \right )\right )}{b^{2}}\) | \(185\) |
1/b^2*(1/2*(-1+(b*x+a)^2)*arcsin(b*x+a)^3+3/4*arcsin(b*x+a)^2*((b*x+a)*(1- (b*x+a)^2)^(1/2)+arcsin(b*x+a))-3/4*(-1+(b*x+a)^2)*arcsin(b*x+a)-3/8*(b*x+ a)*(1-(b*x+a)^2)^(1/2)-3/8*arcsin(b*x+a)-1/2*arcsin(b*x+a)^3-a*(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 = 108, normalized size of antiderivative = 0.51 \[ \int x \arcsin (a+b x)^3 \, dx=\frac {2 \, {\left (2 \, b^{2} x^{2} - 2 \, a^{2} - 1\right )} \arcsin \left (b x + a\right )^{3} - 3 \, {\left (2 \, b^{2} x^{2} - 12 \, a b x - 14 \, a^{2} - 1\right )} \arcsin \left (b x + a\right ) + 3 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (2 \, {\left (b x - 3 \, a\right )} \arcsin \left (b x + a\right )^{2} - b x + 15 \, a\right )}}{8 \, b^{2}} \]
1/8*(2*(2*b^2*x^2 - 2*a^2 - 1)*arcsin(b*x + a)^3 - 3*(2*b^2*x^2 - 12*a*b*x - 14*a^2 - 1)*arcsin(b*x + a) + 3*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(2*( b*x - 3*a)*arcsin(b*x + a)^2 - b*x + 15*a))/b^2
Time = 0.33 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.18 \[ \int x \arcsin (a+b x)^3 \, dx=\begin {cases} - \frac {a^{2} \operatorname {asin}^{3}{\left (a + b x \right )}}{2 b^{2}} + \frac {21 a^{2} \operatorname {asin}{\left (a + b x \right )}}{4 b^{2}} + \frac {9 a x \operatorname {asin}{\left (a + b x \right )}}{2 b} - \frac {9 a \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a + b x \right )}}{4 b^{2}} + \frac {45 a \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{8 b^{2}} + \frac {x^{2} \operatorname {asin}^{3}{\left (a + b x \right )}}{2} - \frac {3 x^{2} \operatorname {asin}{\left (a + b x \right )}}{4} + \frac {3 x \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a + b x \right )}}{4 b} - \frac {3 x \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{8 b} - \frac {\operatorname {asin}^{3}{\left (a + b x \right )}}{4 b^{2}} + \frac {3 \operatorname {asin}{\left (a + b x \right )}}{8 b^{2}} & \text {for}\: b \neq 0 \\\frac {x^{2} \operatorname {asin}^{3}{\left (a \right )}}{2} & \text {otherwise} \end {cases} \]
Piecewise((-a**2*asin(a + b*x)**3/(2*b**2) + 21*a**2*asin(a + b*x)/(4*b**2 ) + 9*a*x*asin(a + b*x)/(2*b) - 9*a*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)* asin(a + b*x)**2/(4*b**2) + 45*a*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)/(8* b**2) + x**2*asin(a + b*x)**3/2 - 3*x**2*asin(a + b*x)/4 + 3*x*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)*asin(a + b*x)**2/(4*b) - 3*x*sqrt(-a**2 - 2*a*b *x - b**2*x**2 + 1)/(8*b) - asin(a + b*x)**3/(4*b**2) + 3*asin(a + b*x)/(8 *b**2), Ne(b, 0)), (x**2*asin(a)**3/2, True))
\[ \int x \arcsin (a+b x)^3 \, dx=\int { x \arcsin \left (b x + a\right )^{3} \,d x } \]
1/2*x^2*arctan2(b*x + a, sqrt(b*x + a + 1)*sqrt(-b*x - a + 1))^3 + 3*b*int egrate(1/2*sqrt(b*x + a + 1)*sqrt(-b*x - a + 1)*x^2*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 = 0.28 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.96 \[ \int x \arcsin (a+b x)^3 \, dx=-\frac {{\left (b x + a\right )} a \arcsin \left (b x + a\right )^{3}}{b^{2}} + \frac {{\left ({\left (b x + a\right )}^{2} - 1\right )} \arcsin \left (b x + a\right )^{3}}{2 \, b^{2}} + \frac {3 \, \sqrt {-{\left (b x + a\right )}^{2} + 1} {\left (b x + a\right )} \arcsin \left (b x + a\right )^{2}}{4 \, b^{2}} - \frac {3 \, \sqrt {-{\left (b x + a\right )}^{2} + 1} a \arcsin \left (b x + a\right )^{2}}{b^{2}} + \frac {6 \, {\left (b x + a\right )} a \arcsin \left (b x + a\right )}{b^{2}} + \frac {\arcsin \left (b x + a\right )^{3}}{4 \, b^{2}} - \frac {3 \, {\left ({\left (b x + a\right )}^{2} - 1\right )} \arcsin \left (b x + a\right )}{4 \, b^{2}} - \frac {3 \, \sqrt {-{\left (b x + a\right )}^{2} + 1} {\left (b x + a\right )}}{8 \, b^{2}} + \frac {6 \, \sqrt {-{\left (b x + a\right )}^{2} + 1} a}{b^{2}} - \frac {3 \, \arcsin \left (b x + a\right )}{8 \, b^{2}} \]
-(b*x + a)*a*arcsin(b*x + a)^3/b^2 + 1/2*((b*x + a)^2 - 1)*arcsin(b*x + a) ^3/b^2 + 3/4*sqrt(-(b*x + a)^2 + 1)*(b*x + a)*arcsin(b*x + a)^2/b^2 - 3*sq rt(-(b*x + a)^2 + 1)*a*arcsin(b*x + a)^2/b^2 + 6*(b*x + a)*a*arcsin(b*x + a)/b^2 + 1/4*arcsin(b*x + a)^3/b^2 - 3/4*((b*x + a)^2 - 1)*arcsin(b*x + a) /b^2 - 3/8*sqrt(-(b*x + a)^2 + 1)*(b*x + a)/b^2 + 6*sqrt(-(b*x + a)^2 + 1) *a/b^2 - 3/8*arcsin(b*x + a)/b^2
Timed out. \[ \int x \arcsin (a+b x)^3 \, dx=\int x\,{\mathrm {asin}\left (a+b\,x\right )}^3 \,d x \]