Integrand size = 12, antiderivative size = 164 \[ \int (a+b \arcsin (c+d x))^5 \, dx=120 a b^4 x+\frac {120 b^5 \sqrt {1-(c+d x)^2}}{d}+\frac {120 b^5 (c+d x) \arcsin (c+d x)}{d}-\frac {60 b^3 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2}{d}-\frac {20 b^2 (c+d x) (a+b \arcsin (c+d x))^3}{d}+\frac {5 b \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^4}{d}+\frac {(c+d x) (a+b \arcsin (c+d x))^5}{d} \]
120*a*b^4*x+120*b^5*(d*x+c)*arcsin(d*x+c)/d-20*b^2*(d*x+c)*(a+b*arcsin(d*x +c))^3/d+(d*x+c)*(a+b*arcsin(d*x+c))^5/d+120*b^5*(1-(d*x+c)^2)^(1/2)/d-60* b^3*(a+b*arcsin(d*x+c))^2*(1-(d*x+c)^2)^(1/2)/d+5*b*(a+b*arcsin(d*x+c))^4* (1-(d*x+c)^2)^(1/2)/d
Time = 0.19 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.91 \[ \int (a+b \arcsin (c+d x))^5 \, dx=\frac {5 b \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^4+(c+d x) (a+b \arcsin (c+d x))^5-20 b^2 \left (3 b \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2+(c+d x) (a+b \arcsin (c+d x))^3-6 b^2 \left (a (c+d x)+b \sqrt {1-(c+d x)^2}+b (c+d x) \arcsin (c+d x)\right )\right )}{d} \]
(5*b*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^4 + (c + d*x)*(a + b*Ar cSin[c + d*x])^5 - 20*b^2*(3*b*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x ])^2 + (c + d*x)*(a + b*ArcSin[c + d*x])^3 - 6*b^2*(a*(c + d*x) + b*Sqrt[1 - (c + d*x)^2] + b*(c + d*x)*ArcSin[c + d*x])))/d
Time = 0.53 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.93, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5302, 5130, 5182, 5130, 5182, 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.
| \(\displaystyle \int (a+b \arcsin (c+d x))^5 \, dx\) |
| \(\Big \downarrow \) 5302 |
| \(\displaystyle \frac {\int (a+b \arcsin (c+d x))^5d(c+d x)}{d}\) |
| \(\Big \downarrow \) 5130 |
| \(\displaystyle \frac {(c+d x) (a+b \arcsin (c+d x))^5-5 b \int \frac {(c+d x) (a+b \arcsin (c+d x))^4}{\sqrt {1-(c+d x)^2}}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 5182 |
| \(\displaystyle \frac {(c+d x) (a+b \arcsin (c+d x))^5-5 b \left (4 b \int (a+b \arcsin (c+d x))^3d(c+d x)-\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^4\right )}{d}\) |
| \(\Big \downarrow \) 5130 |
| \(\displaystyle \frac {(c+d x) (a+b \arcsin (c+d x))^5-5 b \left (4 b \left ((c+d x) (a+b \arcsin (c+d x))^3-3 b \int \frac {(c+d x) (a+b \arcsin (c+d x))^2}{\sqrt {1-(c+d x)^2}}d(c+d x)\right )-\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^4\right )}{d}\) |
| \(\Big \downarrow \) 5182 |
| \(\displaystyle \frac {(c+d x) (a+b \arcsin (c+d x))^5-5 b \left (4 b \left ((c+d x) (a+b \arcsin (c+d x))^3-3 b \left (2 b \int (a+b \arcsin (c+d x))d(c+d x)-\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2\right )\right )-\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^4\right )}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(c+d x) (a+b \arcsin (c+d x))^5-5 b \left (4 b \left ((c+d x) (a+b \arcsin (c+d x))^3-3 b \left (2 b \left (a (c+d x)+b (c+d x) \arcsin (c+d x)+b \sqrt {1-(c+d x)^2}\right )-\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^2\right )\right )-\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^4\right )}{d}\) |
((c + d*x)*(a + b*ArcSin[c + d*x])^5 - 5*b*(-(Sqrt[1 - (c + d*x)^2]*(a + b *ArcSin[c + d*x])^4) + 4*b*((c + d*x)*(a + b*ArcSin[c + d*x])^3 - 3*b*(-(S qrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^2) + 2*b*(a*(c + d*x) + b*Sqr t[1 - (c + d*x)^2] + b*(c + d*x)*ArcSin[c + d*x])))))/d
3.3.14.3.1 Defintions of rubi rules used
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*Ar cSin[c*x])^n, x] - Simp[b*c*n Int[x*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]
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] + Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] I nt[(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]
Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[1/d Subst[Int[(a + b*ArcSin[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, n}, x]
Leaf count of result is larger than twice the leaf count of optimal. \(366\) vs. \(2(158)=316\).
Time = 0.66 (sec) , antiderivative size = 367, normalized size of antiderivative = 2.24
| method | result | size |
| derivativedivides | \(\frac {\left (d x +c \right ) a^{5}+b^{5} \left (\arcsin \left (d x +c \right )^{5} \left (d x +c \right )+5 \arcsin \left (d x +c \right )^{4} \sqrt {1-\left (d x +c \right )^{2}}-20 \arcsin \left (d x +c \right )^{3} \left (d x +c \right )-60 \arcsin \left (d x +c \right )^{2} \sqrt {1-\left (d x +c \right )^{2}}+120 \sqrt {1-\left (d x +c \right )^{2}}+120 \left (d x +c \right ) \arcsin \left (d x +c \right )\right )+5 a \,b^{4} \left (\arcsin \left (d x +c \right )^{4} \left (d x +c \right )+4 \arcsin \left (d x +c \right )^{3} \sqrt {1-\left (d x +c \right )^{2}}-12 \arcsin \left (d x +c \right )^{2} \left (d x +c \right )+24 d x +24 c -24 \arcsin \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}\right )+10 a^{2} b^{3} \left (\arcsin \left (d x +c \right )^{3} \left (d x +c \right )+3 \arcsin \left (d x +c \right )^{2} \sqrt {1-\left (d x +c \right )^{2}}-6 \sqrt {1-\left (d x +c \right )^{2}}-6 \left (d x +c \right ) \arcsin \left (d x +c \right )\right )+10 a^{3} b^{2} \left (\arcsin \left (d x +c \right )^{2} \left (d x +c \right )-2 d x -2 c +2 \arcsin \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}\right )+5 a^{4} b \left (\left (d x +c \right ) \arcsin \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )}{d}\) | \(367\) |
| default | \(\frac {\left (d x +c \right ) a^{5}+b^{5} \left (\arcsin \left (d x +c \right )^{5} \left (d x +c \right )+5 \arcsin \left (d x +c \right )^{4} \sqrt {1-\left (d x +c \right )^{2}}-20 \arcsin \left (d x +c \right )^{3} \left (d x +c \right )-60 \arcsin \left (d x +c \right )^{2} \sqrt {1-\left (d x +c \right )^{2}}+120 \sqrt {1-\left (d x +c \right )^{2}}+120 \left (d x +c \right ) \arcsin \left (d x +c \right )\right )+5 a \,b^{4} \left (\arcsin \left (d x +c \right )^{4} \left (d x +c \right )+4 \arcsin \left (d x +c \right )^{3} \sqrt {1-\left (d x +c \right )^{2}}-12 \arcsin \left (d x +c \right )^{2} \left (d x +c \right )+24 d x +24 c -24 \arcsin \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}\right )+10 a^{2} b^{3} \left (\arcsin \left (d x +c \right )^{3} \left (d x +c \right )+3 \arcsin \left (d x +c \right )^{2} \sqrt {1-\left (d x +c \right )^{2}}-6 \sqrt {1-\left (d x +c \right )^{2}}-6 \left (d x +c \right ) \arcsin \left (d x +c \right )\right )+10 a^{3} b^{2} \left (\arcsin \left (d x +c \right )^{2} \left (d x +c \right )-2 d x -2 c +2 \arcsin \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}\right )+5 a^{4} b \left (\left (d x +c \right ) \arcsin \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )}{d}\) | \(367\) |
| parts | \(x \,a^{5}+\frac {b^{5} \left (\arcsin \left (d x +c \right )^{5} \left (d x +c \right )+5 \arcsin \left (d x +c \right )^{4} \sqrt {1-\left (d x +c \right )^{2}}-20 \arcsin \left (d x +c \right )^{3} \left (d x +c \right )-60 \arcsin \left (d x +c \right )^{2} \sqrt {1-\left (d x +c \right )^{2}}+120 \sqrt {1-\left (d x +c \right )^{2}}+120 \left (d x +c \right ) \arcsin \left (d x +c \right )\right )}{d}+\frac {5 a \,b^{4} \left (\arcsin \left (d x +c \right )^{4} \left (d x +c \right )+4 \arcsin \left (d x +c \right )^{3} \sqrt {1-\left (d x +c \right )^{2}}-12 \arcsin \left (d x +c \right )^{2} \left (d x +c \right )+24 d x +24 c -24 \arcsin \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}\right )}{d}+\frac {10 a^{2} b^{3} \left (\arcsin \left (d x +c \right )^{3} \left (d x +c \right )+3 \arcsin \left (d x +c \right )^{2} \sqrt {1-\left (d x +c \right )^{2}}-6 \sqrt {1-\left (d x +c \right )^{2}}-6 \left (d x +c \right ) \arcsin \left (d x +c \right )\right )}{d}+\frac {10 a^{3} b^{2} \left (\arcsin \left (d x +c \right )^{2} \left (d x +c \right )-2 d x -2 c +2 \arcsin \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}\right )}{d}+\frac {5 a^{4} b \left (\left (d x +c \right ) \arcsin \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )}{d}\) | \(374\) |
1/d*((d*x+c)*a^5+b^5*(arcsin(d*x+c)^5*(d*x+c)+5*arcsin(d*x+c)^4*(1-(d*x+c) ^2)^(1/2)-20*arcsin(d*x+c)^3*(d*x+c)-60*arcsin(d*x+c)^2*(1-(d*x+c)^2)^(1/2 )+120*(1-(d*x+c)^2)^(1/2)+120*(d*x+c)*arcsin(d*x+c))+5*a*b^4*(arcsin(d*x+c )^4*(d*x+c)+4*arcsin(d*x+c)^3*(1-(d*x+c)^2)^(1/2)-12*arcsin(d*x+c)^2*(d*x+ c)+24*d*x+24*c-24*arcsin(d*x+c)*(1-(d*x+c)^2)^(1/2))+10*a^2*b^3*(arcsin(d* x+c)^3*(d*x+c)+3*arcsin(d*x+c)^2*(1-(d*x+c)^2)^(1/2)-6*(1-(d*x+c)^2)^(1/2) -6*(d*x+c)*arcsin(d*x+c))+10*a^3*b^2*(arcsin(d*x+c)^2*(d*x+c)-2*d*x-2*c+2* arcsin(d*x+c)*(1-(d*x+c)^2)^(1/2))+5*a^4*b*((d*x+c)*arcsin(d*x+c)+(1-(d*x+ c)^2)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 323 vs. \(2 (158) = 316\).
Time = 0.26 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.97 \[ \int (a+b \arcsin (c+d x))^5 \, dx=\frac {{\left (b^{5} d x + b^{5} c\right )} \arcsin \left (d x + c\right )^{5} + 5 \, {\left (a b^{4} d x + a b^{4} c\right )} \arcsin \left (d x + c\right )^{4} + 10 \, {\left ({\left (a^{2} b^{3} - 2 \, b^{5}\right )} d x + {\left (a^{2} b^{3} - 2 \, b^{5}\right )} c\right )} \arcsin \left (d x + c\right )^{3} + {\left (a^{5} - 20 \, a^{3} b^{2} + 120 \, a b^{4}\right )} d x + 10 \, {\left ({\left (a^{3} b^{2} - 6 \, a b^{4}\right )} d x + {\left (a^{3} b^{2} - 6 \, a b^{4}\right )} c\right )} \arcsin \left (d x + c\right )^{2} + 5 \, {\left ({\left (a^{4} b - 12 \, a^{2} b^{3} + 24 \, b^{5}\right )} d x + {\left (a^{4} b - 12 \, a^{2} b^{3} + 24 \, b^{5}\right )} c\right )} \arcsin \left (d x + c\right ) + 5 \, {\left (b^{5} \arcsin \left (d x + c\right )^{4} + 4 \, a b^{4} \arcsin \left (d x + c\right )^{3} + a^{4} b - 12 \, a^{2} b^{3} + 24 \, b^{5} + 6 \, {\left (a^{2} b^{3} - 2 \, b^{5}\right )} \arcsin \left (d x + c\right )^{2} + 4 \, {\left (a^{3} b^{2} - 6 \, a b^{4}\right )} \arcsin \left (d x + c\right )\right )} \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}{d} \]
((b^5*d*x + b^5*c)*arcsin(d*x + c)^5 + 5*(a*b^4*d*x + a*b^4*c)*arcsin(d*x + c)^4 + 10*((a^2*b^3 - 2*b^5)*d*x + (a^2*b^3 - 2*b^5)*c)*arcsin(d*x + c)^ 3 + (a^5 - 20*a^3*b^2 + 120*a*b^4)*d*x + 10*((a^3*b^2 - 6*a*b^4)*d*x + (a^ 3*b^2 - 6*a*b^4)*c)*arcsin(d*x + c)^2 + 5*((a^4*b - 12*a^2*b^3 + 24*b^5)*d *x + (a^4*b - 12*a^2*b^3 + 24*b^5)*c)*arcsin(d*x + c) + 5*(b^5*arcsin(d*x + c)^4 + 4*a*b^4*arcsin(d*x + c)^3 + a^4*b - 12*a^2*b^3 + 24*b^5 + 6*(a^2* b^3 - 2*b^5)*arcsin(d*x + c)^2 + 4*(a^3*b^2 - 6*a*b^4)*arcsin(d*x + c))*sq rt(-d^2*x^2 - 2*c*d*x - c^2 + 1))/d
Leaf count of result is larger than twice the leaf count of optimal. 663 vs. \(2 (146) = 292\).
Time = 0.41 (sec) , antiderivative size = 663, normalized size of antiderivative = 4.04 \[ \int (a+b \arcsin (c+d x))^5 \, dx=\begin {cases} a^{5} x + \frac {5 a^{4} b c \operatorname {asin}{\left (c + d x \right )}}{d} + 5 a^{4} b x \operatorname {asin}{\left (c + d x \right )} + \frac {5 a^{4} b \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{d} + \frac {10 a^{3} b^{2} c \operatorname {asin}^{2}{\left (c + d x \right )}}{d} + 10 a^{3} b^{2} x \operatorname {asin}^{2}{\left (c + d x \right )} - 20 a^{3} b^{2} x + \frac {20 a^{3} b^{2} \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1} \operatorname {asin}{\left (c + d x \right )}}{d} + \frac {10 a^{2} b^{3} c \operatorname {asin}^{3}{\left (c + d x \right )}}{d} - \frac {60 a^{2} b^{3} c \operatorname {asin}{\left (c + d x \right )}}{d} + 10 a^{2} b^{3} x \operatorname {asin}^{3}{\left (c + d x \right )} - 60 a^{2} b^{3} x \operatorname {asin}{\left (c + d x \right )} + \frac {30 a^{2} b^{3} \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (c + d x \right )}}{d} - \frac {60 a^{2} b^{3} \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{d} + \frac {5 a b^{4} c \operatorname {asin}^{4}{\left (c + d x \right )}}{d} - \frac {60 a b^{4} c \operatorname {asin}^{2}{\left (c + d x \right )}}{d} + 5 a b^{4} x \operatorname {asin}^{4}{\left (c + d x \right )} - 60 a b^{4} x \operatorname {asin}^{2}{\left (c + d x \right )} + 120 a b^{4} x + \frac {20 a b^{4} \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1} \operatorname {asin}^{3}{\left (c + d x \right )}}{d} - \frac {120 a b^{4} \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1} \operatorname {asin}{\left (c + d x \right )}}{d} + \frac {b^{5} c \operatorname {asin}^{5}{\left (c + d x \right )}}{d} - \frac {20 b^{5} c \operatorname {asin}^{3}{\left (c + d x \right )}}{d} + \frac {120 b^{5} c \operatorname {asin}{\left (c + d x \right )}}{d} + b^{5} x \operatorname {asin}^{5}{\left (c + d x \right )} - 20 b^{5} x \operatorname {asin}^{3}{\left (c + d x \right )} + 120 b^{5} x \operatorname {asin}{\left (c + d x \right )} + \frac {5 b^{5} \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1} \operatorname {asin}^{4}{\left (c + d x \right )}}{d} - \frac {60 b^{5} \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (c + d x \right )}}{d} + \frac {120 b^{5} \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \operatorname {asin}{\left (c \right )}\right )^{5} & \text {otherwise} \end {cases} \]
Piecewise((a**5*x + 5*a**4*b*c*asin(c + d*x)/d + 5*a**4*b*x*asin(c + d*x) + 5*a**4*b*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/d + 10*a**3*b**2*c*asin(c + d*x)**2/d + 10*a**3*b**2*x*asin(c + d*x)**2 - 20*a**3*b**2*x + 20*a**3* b**2*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)/d + 10*a**2*b**3* c*asin(c + d*x)**3/d - 60*a**2*b**3*c*asin(c + d*x)/d + 10*a**2*b**3*x*asi n(c + d*x)**3 - 60*a**2*b**3*x*asin(c + d*x) + 30*a**2*b**3*sqrt(-c**2 - 2 *c*d*x - d**2*x**2 + 1)*asin(c + d*x)**2/d - 60*a**2*b**3*sqrt(-c**2 - 2*c *d*x - d**2*x**2 + 1)/d + 5*a*b**4*c*asin(c + d*x)**4/d - 60*a*b**4*c*asin (c + d*x)**2/d + 5*a*b**4*x*asin(c + d*x)**4 - 60*a*b**4*x*asin(c + d*x)** 2 + 120*a*b**4*x + 20*a*b**4*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)**3/d - 120*a*b**4*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d* x)/d + b**5*c*asin(c + d*x)**5/d - 20*b**5*c*asin(c + d*x)**3/d + 120*b**5 *c*asin(c + d*x)/d + b**5*x*asin(c + d*x)**5 - 20*b**5*x*asin(c + d*x)**3 + 120*b**5*x*asin(c + d*x) + 5*b**5*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)* asin(c + d*x)**4/d - 60*b**5*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)**2/d + 120*b**5*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/d, Ne(d, 0)), (x*(a + b*asin(c))**5, True))
\[ \int (a+b \arcsin (c+d x))^5 \, dx=\int { {\left (b \arcsin \left (d x + c\right ) + a\right )}^{5} \,d x } \]
b^5*x*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1))^5 + a^5*x + 5 *((d*x + c)*arcsin(d*x + c) + sqrt(-(d*x + c)^2 + 1))*a^4*b/d + integrate( 5*(sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)*b^5*d*x*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1))^4 + (a*b^4*d^2*x^2 + 2*a*b^4*c*d*x + a*b^4*c^ 2 - a*b^4)*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1))^4 + 2*(a ^2*b^3*d^2*x^2 + 2*a^2*b^3*c*d*x + a^2*b^3*c^2 - a^2*b^3)*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1))^3 + 2*(a^3*b^2*d^2*x^2 + 2*a^3*b^2* c*d*x + a^3*b^2*c^2 - a^3*b^2)*arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d* x - c + 1))^2)/(d^2*x^2 + 2*c*d*x + c^2 - 1), x)
Leaf count of result is larger than twice the leaf count of optimal. 482 vs. \(2 (158) = 316\).
Time = 0.30 (sec) , antiderivative size = 482, normalized size of antiderivative = 2.94 \[ \int (a+b \arcsin (c+d x))^5 \, dx=\frac {{\left (d x + c\right )} b^{5} \arcsin \left (d x + c\right )^{5}}{d} + \frac {5 \, {\left (d x + c\right )} a b^{4} \arcsin \left (d x + c\right )^{4}}{d} + \frac {5 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} b^{5} \arcsin \left (d x + c\right )^{4}}{d} + \frac {10 \, {\left (d x + c\right )} a^{2} b^{3} \arcsin \left (d x + c\right )^{3}}{d} - \frac {20 \, {\left (d x + c\right )} b^{5} \arcsin \left (d x + c\right )^{3}}{d} + \frac {20 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} a b^{4} \arcsin \left (d x + c\right )^{3}}{d} + \frac {10 \, {\left (d x + c\right )} a^{3} b^{2} \arcsin \left (d x + c\right )^{2}}{d} - \frac {60 \, {\left (d x + c\right )} a b^{4} \arcsin \left (d x + c\right )^{2}}{d} + \frac {30 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} a^{2} b^{3} \arcsin \left (d x + c\right )^{2}}{d} - \frac {60 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} b^{5} \arcsin \left (d x + c\right )^{2}}{d} + \frac {5 \, {\left (d x + c\right )} a^{4} b \arcsin \left (d x + c\right )}{d} - \frac {60 \, {\left (d x + c\right )} a^{2} b^{3} \arcsin \left (d x + c\right )}{d} + \frac {120 \, {\left (d x + c\right )} b^{5} \arcsin \left (d x + c\right )}{d} + \frac {20 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} a^{3} b^{2} \arcsin \left (d x + c\right )}{d} - \frac {120 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} a b^{4} \arcsin \left (d x + c\right )}{d} + \frac {{\left (d x + c\right )} a^{5}}{d} - \frac {20 \, {\left (d x + c\right )} a^{3} b^{2}}{d} + \frac {120 \, {\left (d x + c\right )} a b^{4}}{d} + \frac {5 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} a^{4} b}{d} - \frac {60 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} a^{2} b^{3}}{d} + \frac {120 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} b^{5}}{d} \]
(d*x + c)*b^5*arcsin(d*x + c)^5/d + 5*(d*x + c)*a*b^4*arcsin(d*x + c)^4/d + 5*sqrt(-(d*x + c)^2 + 1)*b^5*arcsin(d*x + c)^4/d + 10*(d*x + c)*a^2*b^3* arcsin(d*x + c)^3/d - 20*(d*x + c)*b^5*arcsin(d*x + c)^3/d + 20*sqrt(-(d*x + c)^2 + 1)*a*b^4*arcsin(d*x + c)^3/d + 10*(d*x + c)*a^3*b^2*arcsin(d*x + c)^2/d - 60*(d*x + c)*a*b^4*arcsin(d*x + c)^2/d + 30*sqrt(-(d*x + c)^2 + 1)*a^2*b^3*arcsin(d*x + c)^2/d - 60*sqrt(-(d*x + c)^2 + 1)*b^5*arcsin(d*x + c)^2/d + 5*(d*x + c)*a^4*b*arcsin(d*x + c)/d - 60*(d*x + c)*a^2*b^3*arcs in(d*x + c)/d + 120*(d*x + c)*b^5*arcsin(d*x + c)/d + 20*sqrt(-(d*x + c)^2 + 1)*a^3*b^2*arcsin(d*x + c)/d - 120*sqrt(-(d*x + c)^2 + 1)*a*b^4*arcsin( d*x + c)/d + (d*x + c)*a^5/d - 20*(d*x + c)*a^3*b^2/d + 120*(d*x + c)*a*b^ 4/d + 5*sqrt(-(d*x + c)^2 + 1)*a^4*b/d - 60*sqrt(-(d*x + c)^2 + 1)*a^2*b^3 /d + 120*sqrt(-(d*x + c)^2 + 1)*b^5/d
Time = 0.70 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.93 \[ \int (a+b \arcsin (c+d x))^5 \, dx=a^5\,x+\frac {10\,a^3\,b^2\,\left (2\,\mathrm {asin}\left (c+d\,x\right )\,\sqrt {1-{\left (c+d\,x\right )}^2}+\left ({\mathrm {asin}\left (c+d\,x\right )}^2-2\right )\,\left (c+d\,x\right )\right )}{d}+\frac {5\,a^4\,b\,\left (\sqrt {1-{\left (c+d\,x\right )}^2}+\mathrm {asin}\left (c+d\,x\right )\,\left (c+d\,x\right )\right )}{d}+\frac {b^5\,\left (c+d\,x\right )\,\left ({\mathrm {asin}\left (c+d\,x\right )}^5-20\,{\mathrm {asin}\left (c+d\,x\right )}^3+120\,\mathrm {asin}\left (c+d\,x\right )\right )}{d}+\frac {b^5\,\sqrt {1-{\left (c+d\,x\right )}^2}\,\left (5\,{\mathrm {asin}\left (c+d\,x\right )}^4-60\,{\mathrm {asin}\left (c+d\,x\right )}^2+120\right )}{d}+\frac {5\,a\,b^4\,\left (c+d\,x\right )\,\left ({\mathrm {asin}\left (c+d\,x\right )}^4-12\,{\mathrm {asin}\left (c+d\,x\right )}^2+24\right )}{d}+\frac {10\,a^2\,b^3\,\left (3\,{\mathrm {asin}\left (c+d\,x\right )}^2-6\right )\,\sqrt {1-{\left (c+d\,x\right )}^2}}{d}-\frac {10\,a^2\,b^3\,\left (6\,\mathrm {asin}\left (c+d\,x\right )-{\mathrm {asin}\left (c+d\,x\right )}^3\right )\,\left (c+d\,x\right )}{d}-\frac {5\,a\,b^4\,\left (24\,\mathrm {asin}\left (c+d\,x\right )-4\,{\mathrm {asin}\left (c+d\,x\right )}^3\right )\,\sqrt {1-{\left (c+d\,x\right )}^2}}{d} \]
a^5*x + (10*a^3*b^2*(2*asin(c + d*x)*(1 - (c + d*x)^2)^(1/2) + (asin(c + d *x)^2 - 2)*(c + d*x)))/d + (5*a^4*b*((1 - (c + d*x)^2)^(1/2) + asin(c + d* x)*(c + d*x)))/d + (b^5*(c + d*x)*(120*asin(c + d*x) - 20*asin(c + d*x)^3 + asin(c + d*x)^5))/d + (b^5*(1 - (c + d*x)^2)^(1/2)*(5*asin(c + d*x)^4 - 60*asin(c + d*x)^2 + 120))/d + (5*a*b^4*(c + d*x)*(asin(c + d*x)^4 - 12*as in(c + d*x)^2 + 24))/d + (10*a^2*b^3*(3*asin(c + d*x)^2 - 6)*(1 - (c + d*x )^2)^(1/2))/d - (10*a^2*b^3*(6*asin(c + d*x) - asin(c + d*x)^3)*(c + d*x)) /d - (5*a*b^4*(24*asin(c + d*x) - 4*asin(c + d*x)^3)*(1 - (c + d*x)^2)^(1/ 2))/d