Integrand size = 27, antiderivative size = 188 \[ \int x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx=\frac {4 b^2 \sqrt {d-c^2 d x^2}}{9 c^2}+\frac {2 b^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{27 c^2}+\frac {2 b x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{3 c \sqrt {1-c^2 x^2}}-\frac {2 b c x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{9 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{3 c^2 d} \]
-1/3*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))^2/c^2/d+4/9*b^2*(-c^2*d*x^2+d) ^(1/2)/c^2+2/27*b^2*(-c^2*x^2+1)*(-c^2*d*x^2+d)^(1/2)/c^2+2/3*b*x*(a+b*arc sin(c*x))*(-c^2*d*x^2+d)^(1/2)/c/(-c^2*x^2+1)^(1/2)-2/9*b*c*x^3*(a+b*arcsi n(c*x))*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)
Time = 0.21 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.64 \[ \int x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx=\frac {\sqrt {d-c^2 d x^2} \left (\left (-1+c^2 x^2\right ) (a+b \arcsin (c x))^2-\frac {2 b \left (b \sqrt {1-c^2 x^2} \left (-7+c^2 x^2\right )+3 a c x \left (-3+c^2 x^2\right )+3 b c x \left (-3+c^2 x^2\right ) \arcsin (c x)\right )}{9 \sqrt {1-c^2 x^2}}\right )}{3 c^2} \]
(Sqrt[d - c^2*d*x^2]*((-1 + c^2*x^2)*(a + b*ArcSin[c*x])^2 - (2*b*(b*Sqrt[ 1 - c^2*x^2]*(-7 + c^2*x^2) + 3*a*c*x*(-3 + c^2*x^2) + 3*b*c*x*(-3 + c^2*x ^2)*ArcSin[c*x]))/(9*Sqrt[1 - c^2*x^2])))/(3*c^2)
Time = 0.43 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.79, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5182, 5154, 27, 353, 53, 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.
| \(\displaystyle \int x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx\) |
| \(\Big \downarrow \) 5182 |
| \(\displaystyle \frac {2 b \sqrt {d-c^2 d x^2} \int \left (1-c^2 x^2\right ) (a+b \arcsin (c x))dx}{3 c \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{3 c^2 d}\) |
| \(\Big \downarrow \) 5154 |
| \(\displaystyle \frac {2 b \sqrt {d-c^2 d x^2} \left (-b c \int \frac {x \left (3-c^2 x^2\right )}{3 \sqrt {1-c^2 x^2}}dx-\frac {1}{3} c^2 x^3 (a+b \arcsin (c x))+x (a+b \arcsin (c x))\right )}{3 c \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{3 c^2 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 b \sqrt {d-c^2 d x^2} \left (-\frac {1}{3} b c \int \frac {x \left (3-c^2 x^2\right )}{\sqrt {1-c^2 x^2}}dx-\frac {1}{3} c^2 x^3 (a+b \arcsin (c x))+x (a+b \arcsin (c x))\right )}{3 c \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{3 c^2 d}\) |
| \(\Big \downarrow \) 353 |
| \(\displaystyle \frac {2 b \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} b c \int \frac {3-c^2 x^2}{\sqrt {1-c^2 x^2}}dx^2-\frac {1}{3} c^2 x^3 (a+b \arcsin (c x))+x (a+b \arcsin (c x))\right )}{3 c \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{3 c^2 d}\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \frac {2 b \sqrt {d-c^2 d x^2} \left (-\frac {1}{6} b c \int \left (\sqrt {1-c^2 x^2}+\frac {2}{\sqrt {1-c^2 x^2}}\right )dx^2-\frac {1}{3} c^2 x^3 (a+b \arcsin (c x))+x (a+b \arcsin (c x))\right )}{3 c \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{3 c^2 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 b \sqrt {d-c^2 d x^2} \left (-\frac {1}{3} c^2 x^3 (a+b \arcsin (c x))+x (a+b \arcsin (c x))-\frac {1}{6} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^2}-\frac {4 \sqrt {1-c^2 x^2}}{c^2}\right )\right )}{3 c \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{3 c^2 d}\) |
-1/3*((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/(c^2*d) + (2*b*Sqrt[d - c^2*d*x^2]*(-1/6*(b*c*((-4*Sqrt[1 - c^2*x^2])/c^2 - (2*(1 - c^2*x^2)^(3/2 ))/(3*c^2))) + x*(a + b*ArcSin[c*x]) - (c^2*x^3*(a + b*ArcSin[c*x]))/3))/( 3*c*Sqrt[1 - c^2*x^2])
3.3.12.3.1 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_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] :> Simp[1/2 Subst[Int[(a + b*x)^p*(c + d*x)^q, x], x, x^2], x] /; FreeQ[ {a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbo l] :> With[{u = IntHide[(d + e*x^2)^p, x]}, Simp[(a + b*ArcSin[c*x]) u, x ] - Simp[b*c Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]] /; Fr eeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 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]
Result contains complex when optimal does not.
Time = 0.21 (sec) , antiderivative size = 700, normalized size of antiderivative = 3.72
| method | result | size |
| default | \(-\frac {a^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3 c^{2} d}+b^{2} \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 c^{4} x^{4}-5 c^{2} x^{2}-4 i \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}+3 i c x \sqrt {-c^{2} x^{2}+1}+1\right ) \left (6 i \arcsin \left (c x \right )+9 \arcsin \left (c x \right )^{2}-2\right )}{216 c^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i c x \sqrt {-c^{2} x^{2}+1}-1\right ) \left (\arcsin \left (c x \right )^{2}-2+2 i \arcsin \left (c x \right )\right )}{8 c^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) \left (\arcsin \left (c x \right )^{2}-2-2 i \arcsin \left (c x \right )\right )}{8 c^{2} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 i \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}+4 c^{4} x^{4}-3 i c x \sqrt {-c^{2} x^{2}+1}-5 c^{2} x^{2}+1\right ) \left (-6 i \arcsin \left (c x \right )+9 \arcsin \left (c x \right )^{2}-2\right )}{216 c^{2} \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 c^{4} x^{4}-5 c^{2} x^{2}-4 i \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}+3 i c x \sqrt {-c^{2} x^{2}+1}+1\right ) \left (i+3 \arcsin \left (c x \right )\right )}{72 c^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i c x \sqrt {-c^{2} x^{2}+1}-1\right ) \left (\arcsin \left (c x \right )+i\right )}{8 c^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) \left (\arcsin \left (c x \right )-i\right )}{8 c^{2} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 i \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}+4 c^{4} x^{4}-3 i c x \sqrt {-c^{2} x^{2}+1}-5 c^{2} x^{2}+1\right ) \left (-i+3 \arcsin \left (c x \right )\right )}{72 c^{2} \left (c^{2} x^{2}-1\right )}\right )\) | \(700\) |
| parts | \(-\frac {a^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3 c^{2} d}+b^{2} \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 c^{4} x^{4}-5 c^{2} x^{2}-4 i \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}+3 i c x \sqrt {-c^{2} x^{2}+1}+1\right ) \left (6 i \arcsin \left (c x \right )+9 \arcsin \left (c x \right )^{2}-2\right )}{216 c^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i c x \sqrt {-c^{2} x^{2}+1}-1\right ) \left (\arcsin \left (c x \right )^{2}-2+2 i \arcsin \left (c x \right )\right )}{8 c^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) \left (\arcsin \left (c x \right )^{2}-2-2 i \arcsin \left (c x \right )\right )}{8 c^{2} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 i \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}+4 c^{4} x^{4}-3 i c x \sqrt {-c^{2} x^{2}+1}-5 c^{2} x^{2}+1\right ) \left (-6 i \arcsin \left (c x \right )+9 \arcsin \left (c x \right )^{2}-2\right )}{216 c^{2} \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 c^{4} x^{4}-5 c^{2} x^{2}-4 i \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}+3 i c x \sqrt {-c^{2} x^{2}+1}+1\right ) \left (i+3 \arcsin \left (c x \right )\right )}{72 c^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i c x \sqrt {-c^{2} x^{2}+1}-1\right ) \left (\arcsin \left (c x \right )+i\right )}{8 c^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) \left (\arcsin \left (c x \right )-i\right )}{8 c^{2} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 i \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}+4 c^{4} x^{4}-3 i c x \sqrt {-c^{2} x^{2}+1}-5 c^{2} x^{2}+1\right ) \left (-i+3 \arcsin \left (c x \right )\right )}{72 c^{2} \left (c^{2} x^{2}-1\right )}\right )\) | \(700\) |
-1/3*a^2*(-c^2*d*x^2+d)^(3/2)/c^2/d+b^2*(1/216*(-d*(c^2*x^2-1))^(1/2)*(4*c ^4*x^4-5*c^2*x^2-4*I*c^3*x^3*(-c^2*x^2+1)^(1/2)+3*I*(-c^2*x^2+1)^(1/2)*x*c +1)*(6*I*arcsin(c*x)+9*arcsin(c*x)^2-2)/c^2/(c^2*x^2-1)-1/8*(-d*(c^2*x^2-1 ))^(1/2)*(c^2*x^2-I*(-c^2*x^2+1)^(1/2)*x*c-1)*(arcsin(c*x)^2-2+2*I*arcsin( c*x))/c^2/(c^2*x^2-1)-1/8*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*x*c +c^2*x^2-1)*(arcsin(c*x)^2-2-2*I*arcsin(c*x))/c^2/(c^2*x^2-1)+1/216*(-d*(c ^2*x^2-1))^(1/2)*(4*I*c^3*x^3*(-c^2*x^2+1)^(1/2)+4*c^4*x^4-3*I*(-c^2*x^2+1 )^(1/2)*x*c-5*c^2*x^2+1)*(-6*I*arcsin(c*x)+9*arcsin(c*x)^2-2)/c^2/(c^2*x^2 -1))+2*a*b*(1/72*(-d*(c^2*x^2-1))^(1/2)*(4*c^4*x^4-5*c^2*x^2-4*I*c^3*x^3*( -c^2*x^2+1)^(1/2)+3*I*(-c^2*x^2+1)^(1/2)*x*c+1)*(I+3*arcsin(c*x))/c^2/(c^2 *x^2-1)-1/8*(-d*(c^2*x^2-1))^(1/2)*(c^2*x^2-I*(-c^2*x^2+1)^(1/2)*x*c-1)*(a rcsin(c*x)+I)/c^2/(c^2*x^2-1)-1/8*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^( 1/2)*x*c+c^2*x^2-1)*(arcsin(c*x)-I)/c^2/(c^2*x^2-1)+1/72*(-d*(c^2*x^2-1))^ (1/2)*(4*I*c^3*x^3*(-c^2*x^2+1)^(1/2)+4*c^4*x^4-3*I*(-c^2*x^2+1)^(1/2)*x*c -5*c^2*x^2+1)*(-I+3*arcsin(c*x))/c^2/(c^2*x^2-1))
Time = 0.27 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.11 \[ \int x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx=\frac {6 \, {\left (a b c^{3} x^{3} - 3 \, a b c x + {\left (b^{2} c^{3} x^{3} - 3 \, b^{2} c x\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} + {\left ({\left (9 \, a^{2} - 2 \, b^{2}\right )} c^{4} x^{4} - 2 \, {\left (9 \, a^{2} - 8 \, b^{2}\right )} c^{2} x^{2} + 9 \, {\left (b^{2} c^{4} x^{4} - 2 \, b^{2} c^{2} x^{2} + b^{2}\right )} \arcsin \left (c x\right )^{2} + 9 \, a^{2} - 14 \, b^{2} + 18 \, {\left (a b c^{4} x^{4} - 2 \, a b c^{2} x^{2} + a b\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}}{27 \, {\left (c^{4} x^{2} - c^{2}\right )}} \]
1/27*(6*(a*b*c^3*x^3 - 3*a*b*c*x + (b^2*c^3*x^3 - 3*b^2*c*x)*arcsin(c*x))* sqrt(-c^2*d*x^2 + d)*sqrt(-c^2*x^2 + 1) + ((9*a^2 - 2*b^2)*c^4*x^4 - 2*(9* a^2 - 8*b^2)*c^2*x^2 + 9*(b^2*c^4*x^4 - 2*b^2*c^2*x^2 + b^2)*arcsin(c*x)^2 + 9*a^2 - 14*b^2 + 18*(a*b*c^4*x^4 - 2*a*b*c^2*x^2 + a*b)*arcsin(c*x))*sq rt(-c^2*d*x^2 + d))/(c^4*x^2 - c^2)
\[ \int x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx=\int x \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}\, dx \]
Time = 0.31 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.00 \[ \int x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx=-\frac {2}{27} \, b^{2} {\left (\frac {\sqrt {-c^{2} x^{2} + 1} d^{\frac {3}{2}} x^{2} - \frac {7 \, \sqrt {-c^{2} x^{2} + 1} d^{\frac {3}{2}}}{c^{2}}}{d} + \frac {3 \, {\left (c^{2} d^{\frac {3}{2}} x^{3} - 3 \, d^{\frac {3}{2}} x\right )} \arcsin \left (c x\right )}{c d}\right )} - \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} b^{2} \arcsin \left (c x\right )^{2}}{3 \, c^{2} d} - \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} a b \arcsin \left (c x\right )}{3 \, c^{2} d} - \frac {2 \, {\left (c^{2} d^{\frac {3}{2}} x^{3} - 3 \, d^{\frac {3}{2}} x\right )} a b}{9 \, c d} - \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} a^{2}}{3 \, c^{2} d} \]
-2/27*b^2*((sqrt(-c^2*x^2 + 1)*d^(3/2)*x^2 - 7*sqrt(-c^2*x^2 + 1)*d^(3/2)/ c^2)/d + 3*(c^2*d^(3/2)*x^3 - 3*d^(3/2)*x)*arcsin(c*x)/(c*d)) - 1/3*(-c^2* d*x^2 + d)^(3/2)*b^2*arcsin(c*x)^2/(c^2*d) - 2/3*(-c^2*d*x^2 + d)^(3/2)*a* b*arcsin(c*x)/(c^2*d) - 2/9*(c^2*d^(3/2)*x^3 - 3*d^(3/2)*x)*a*b/(c*d) - 1/ 3*(-c^2*d*x^2 + d)^(3/2)*a^2/(c^2*d)
Exception generated. \[ \int x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \, dx=\int x\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,\sqrt {d-c^2\,d\,x^2} \,d x \]