Integrand size = 29, antiderivative size = 277 \[ \int \frac {x^3 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\frac {4 a b x \sqrt {1-c^2 x^2}}{3 c^3 \sqrt {d-c^2 d x^2}}+\frac {14 b^2 \left (1-c^2 x^2\right )}{9 c^4 \sqrt {d-c^2 d x^2}}-\frac {2 b^2 \left (1-c^2 x^2\right )^2}{27 c^4 \sqrt {d-c^2 d x^2}}+\frac {4 b^2 x \sqrt {1-c^2 x^2} \arcsin (c x)}{3 c^3 \sqrt {d-c^2 d x^2}}+\frac {2 b x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{9 c \sqrt {d-c^2 d x^2}}-\frac {2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{3 c^4 d}-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{3 c^2 d} \]
14/9*b^2*(-c^2*x^2+1)/c^4/(-c^2*d*x^2+d)^(1/2)-2/27*b^2*(-c^2*x^2+1)^2/c^4 /(-c^2*d*x^2+d)^(1/2)+4/3*a*b*x*(-c^2*x^2+1)^(1/2)/c^3/(-c^2*d*x^2+d)^(1/2 )+4/3*b^2*x*arcsin(c*x)*(-c^2*x^2+1)^(1/2)/c^3/(-c^2*d*x^2+d)^(1/2)+2/9*b* x^3*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/c/(-c^2*d*x^2+d)^(1/2)-2/3*(a+b*a rcsin(c*x))^2*(-c^2*d*x^2+d)^(1/2)/c^4/d-1/3*x^2*(a+b*arcsin(c*x))^2*(-c^2 *d*x^2+d)^(1/2)/c^2/d
Time = 0.09 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.64 \[ \int \frac {x^3 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\frac {6 a b c x \sqrt {1-c^2 x^2} \left (6+c^2 x^2\right )+9 a^2 \left (-2+c^2 x^2+c^4 x^4\right )-2 b^2 \left (-20+19 c^2 x^2+c^4 x^4\right )+6 b \left (b c x \sqrt {1-c^2 x^2} \left (6+c^2 x^2\right )+3 a \left (-2+c^2 x^2+c^4 x^4\right )\right ) \arcsin (c x)+9 b^2 \left (-2+c^2 x^2+c^4 x^4\right ) \arcsin (c x)^2}{27 c^4 \sqrt {d-c^2 d x^2}} \]
(6*a*b*c*x*Sqrt[1 - c^2*x^2]*(6 + c^2*x^2) + 9*a^2*(-2 + c^2*x^2 + c^4*x^4 ) - 2*b^2*(-20 + 19*c^2*x^2 + c^4*x^4) + 6*b*(b*c*x*Sqrt[1 - c^2*x^2]*(6 + c^2*x^2) + 3*a*(-2 + c^2*x^2 + c^4*x^4))*ArcSin[c*x] + 9*b^2*(-2 + c^2*x^ 2 + c^4*x^4)*ArcSin[c*x]^2)/(27*c^4*Sqrt[d - c^2*d*x^2])
Time = 0.82 (sec) , antiderivative size = 245, normalized size of antiderivative = 0.88, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {5210, 5138, 243, 53, 2009, 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.
| \(\displaystyle \int \frac {x^3 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx\) |
| \(\Big \downarrow \) 5210 |
| \(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \int x^2 (a+b \arcsin (c x))dx}{3 c \sqrt {d-c^2 d x^2}}+\frac {2 \int \frac {x (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}}dx}{3 c^2}-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{3 c^2 d}\) |
| \(\Big \downarrow \) 5138 |
| \(\displaystyle \frac {2 \int \frac {x (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}}dx}{3 c^2}+\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {1}{3} x^3 (a+b \arcsin (c x))-\frac {1}{3} b c \int \frac {x^3}{\sqrt {1-c^2 x^2}}dx\right )}{3 c \sqrt {d-c^2 d x^2}}-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{3 c^2 d}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {2 \int \frac {x (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}}dx}{3 c^2}+\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {1}{3} x^3 (a+b \arcsin (c x))-\frac {1}{6} b c \int \frac {x^2}{\sqrt {1-c^2 x^2}}dx^2\right )}{3 c \sqrt {d-c^2 d x^2}}-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{3 c^2 d}\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \frac {2 \int \frac {x (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}}dx}{3 c^2}+\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {1}{3} x^3 (a+b \arcsin (c x))-\frac {1}{6} b c \int \left (\frac {1}{c^2 \sqrt {1-c^2 x^2}}-\frac {\sqrt {1-c^2 x^2}}{c^2}\right )dx^2\right )}{3 c \sqrt {d-c^2 d x^2}}-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{3 c^2 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \int \frac {x (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}}dx}{3 c^2}-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{3 c^2 d}+\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {1}{3} x^3 (a+b \arcsin (c x))-\frac {1}{6} b c \left (\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {1-c^2 x^2}}{c^4}\right )\right )}{3 c \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 5182 |
| \(\displaystyle \frac {2 \left (\frac {2 b \sqrt {1-c^2 x^2} \int (a+b \arcsin (c x))dx}{c \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{c^2 d}\right )}{3 c^2}-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{3 c^2 d}+\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {1}{3} x^3 (a+b \arcsin (c x))-\frac {1}{6} b c \left (\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {1-c^2 x^2}}{c^4}\right )\right )}{3 c \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{3 c^2 d}+\frac {2 \left (\frac {2 b \sqrt {1-c^2 x^2} \left (a x+b x \arcsin (c x)+\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{c \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{c^2 d}\right )}{3 c^2}+\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {1}{3} x^3 (a+b \arcsin (c x))-\frac {1}{6} b c \left (\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {1-c^2 x^2}}{c^4}\right )\right )}{3 c \sqrt {d-c^2 d x^2}}\) |
-1/3*(x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(c^2*d) + (2*b*Sqrt[1 - c^2*x^2]*(-1/6*(b*c*((-2*Sqrt[1 - c^2*x^2])/c^4 + (2*(1 - c^2*x^2)^(3/2 ))/(3*c^4))) + (x^3*(a + b*ArcSin[c*x]))/3))/(3*c*Sqrt[d - c^2*d*x^2]) + ( 2*(-((Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(c^2*d)) + (2*b*Sqrt[1 - c^2*x^2]*(a*x + (b*Sqrt[1 - c^2*x^2])/c + b*x*ArcSin[c*x]))/(c*Sqrt[d - c^ 2*d*x^2])))/(3*c^2)
3.3.36.3.1 Defintions of rubi rules used
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_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcSin[c*x])^n/(d*(m + 1))), x] - Simp[b*c*(n /(d*(m + 1))) Int[(d*x)^(m + 1)*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2 *x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
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_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(e*(m + 2*p + 1))), x] + (Simp[f^2*((m - 1)/(c^2*(m + 2*p + 1))) Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] + S imp[b*f*(n/(c*(m + 2*p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] Int[(f* x)^(m - 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && IGtQ[m , 1] && NeQ[m + 2*p + 1, 0]
Result contains complex when optimal does not.
Time = 0.36 (sec) , antiderivative size = 812, normalized size of antiderivative = 2.93
| method | result | size |
| default | \(a^{2} \left (-\frac {x^{2} \sqrt {-c^{2} d \,x^{2}+d}}{3 c^{2} d}-\frac {2 \sqrt {-c^{2} d \,x^{2}+d}}{3 d \,c^{4}}\right )+b^{2} \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{2} x^{2}-2 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 )}{432 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {3 \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^{4} d \left (c^{2} x^{2}-1\right )}-\frac {3 \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^{4} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 i c x \sqrt {-c^{2} x^{2}+1}+2 c^{2} x^{2}-1\right ) \left (-6 i \arcsin \left (c x \right )+9 \arcsin \left (c x \right )^{2}-2\right )}{432 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (9 \arcsin \left (c x \right )^{2}-2\right ) \cos \left (4 \arcsin \left (c x \right )\right )}{216 c^{4} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \sin \left (4 \arcsin \left (c x \right )\right )}{36 c^{4} d \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{2} x^{2}-2 i c x \sqrt {-c^{2} x^{2}+1}-1\right ) \left (i+3 \arcsin \left (c x \right )\right )}{144 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {3 \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^{4} d \left (c^{2} x^{2}-1\right )}-\frac {3 \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^{4} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 i c x \sqrt {-c^{2} x^{2}+1}+2 c^{2} x^{2}-1\right ) \left (-i+3 \arcsin \left (c x \right )\right )}{144 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \cos \left (4 \arcsin \left (c x \right )\right )}{24 c^{4} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sin \left (4 \arcsin \left (c x \right )\right )}{72 c^{4} d \left (c^{2} x^{2}-1\right )}\right )\) | \(812\) |
| parts | \(a^{2} \left (-\frac {x^{2} \sqrt {-c^{2} d \,x^{2}+d}}{3 c^{2} d}-\frac {2 \sqrt {-c^{2} d \,x^{2}+d}}{3 d \,c^{4}}\right )+b^{2} \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{2} x^{2}-2 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 )}{432 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {3 \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^{4} d \left (c^{2} x^{2}-1\right )}-\frac {3 \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^{4} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 i c x \sqrt {-c^{2} x^{2}+1}+2 c^{2} x^{2}-1\right ) \left (-6 i \arcsin \left (c x \right )+9 \arcsin \left (c x \right )^{2}-2\right )}{432 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (9 \arcsin \left (c x \right )^{2}-2\right ) \cos \left (4 \arcsin \left (c x \right )\right )}{216 c^{4} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \sin \left (4 \arcsin \left (c x \right )\right )}{36 c^{4} d \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{2} x^{2}-2 i c x \sqrt {-c^{2} x^{2}+1}-1\right ) \left (i+3 \arcsin \left (c x \right )\right )}{144 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {3 \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^{4} d \left (c^{2} x^{2}-1\right )}-\frac {3 \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^{4} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 i c x \sqrt {-c^{2} x^{2}+1}+2 c^{2} x^{2}-1\right ) \left (-i+3 \arcsin \left (c x \right )\right )}{144 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \cos \left (4 \arcsin \left (c x \right )\right )}{24 c^{4} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sin \left (4 \arcsin \left (c x \right )\right )}{72 c^{4} d \left (c^{2} x^{2}-1\right )}\right )\) | \(812\) |
a^2*(-1/3*x^2/c^2/d*(-c^2*d*x^2+d)^(1/2)-2/3/d/c^4*(-c^2*d*x^2+d)^(1/2))+b ^2*(1/432*(-d*(c^2*x^2-1))^(1/2)*(2*c^2*x^2-2*I*c*x*(-c^2*x^2+1)^(1/2)-1)* (6*I*arcsin(c*x)+9*arcsin(c*x)^2-2)/c^4/d/(c^2*x^2-1)-3/8*(-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^4/d/(c^2*x^2-1)-3/8*(-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^4/d/(c^2*x^2-1)+1/432*(-d* (c^2*x^2-1))^(1/2)*(2*I*c*x*(-c^2*x^2+1)^(1/2)+2*c^2*x^2-1)*(-6*I*arcsin(c *x)+9*arcsin(c*x)^2-2)/c^4/d/(c^2*x^2-1)-1/216*(-d*(c^2*x^2-1))^(1/2)/c^4/ d/(c^2*x^2-1)*(9*arcsin(c*x)^2-2)*cos(4*arcsin(c*x))+1/36*(-d*(c^2*x^2-1)) ^(1/2)/c^4/d/(c^2*x^2-1)*arcsin(c*x)*sin(4*arcsin(c*x)))+2*a*b*(1/144*(-d* (c^2*x^2-1))^(1/2)*(2*c^2*x^2-2*I*c*x*(-c^2*x^2+1)^(1/2)-1)*(I+3*arcsin(c* x))/c^4/d/(c^2*x^2-1)-3/8*(-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)+I)/c^4/d/(c^2*x^2-1)-3/8*(-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^4/d/(c^2*x^2-1)+1/14 4*(-d*(c^2*x^2-1))^(1/2)*(2*I*c*x*(-c^2*x^2+1)^(1/2)+2*c^2*x^2-1)*(-I+3*ar csin(c*x))/c^4/d/(c^2*x^2-1)-1/24*(-d*(c^2*x^2-1))^(1/2)/c^4/d/(c^2*x^2-1) *arcsin(c*x)*cos(4*arcsin(c*x))+1/72*(-d*(c^2*x^2-1))^(1/2)/c^4/d/(c^2*x^2 -1)*sin(4*arcsin(c*x)))
Time = 0.27 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.76 \[ \int \frac {x^3 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=-\frac {6 \, {\left (a b c^{3} x^{3} + 6 \, a b c x + {\left (b^{2} c^{3} x^{3} + 6 \, b^{2} c x\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} + {\left ({\left (9 \, a^{2} - 2 \, b^{2}\right )} c^{4} x^{4} + {\left (9 \, a^{2} - 38 \, b^{2}\right )} c^{2} x^{2} + 9 \, {\left (b^{2} c^{4} x^{4} + b^{2} c^{2} x^{2} - 2 \, b^{2}\right )} \arcsin \left (c x\right )^{2} - 18 \, a^{2} + 40 \, b^{2} + 18 \, {\left (a b c^{4} x^{4} + a b c^{2} x^{2} - 2 \, a b\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}}{27 \, {\left (c^{6} d x^{2} - c^{4} d\right )}} \]
-1/27*(6*(a*b*c^3*x^3 + 6*a*b*c*x + (b^2*c^3*x^3 + 6*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 + (9*a ^2 - 38*b^2)*c^2*x^2 + 9*(b^2*c^4*x^4 + b^2*c^2*x^2 - 2*b^2)*arcsin(c*x)^2 - 18*a^2 + 40*b^2 + 18*(a*b*c^4*x^4 + a*b*c^2*x^2 - 2*a*b)*arcsin(c*x))*s qrt(-c^2*d*x^2 + d))/(c^6*d*x^2 - c^4*d)
\[ \int \frac {x^3 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\int \frac {x^{3} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \]
Time = 0.29 (sec) , antiderivative size = 251, normalized size of antiderivative = 0.91 \[ \int \frac {x^3 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=-\frac {1}{3} \, b^{2} {\left (\frac {\sqrt {-c^{2} d x^{2} + d} x^{2}}{c^{2} d} + \frac {2 \, \sqrt {-c^{2} d x^{2} + d}}{c^{4} d}\right )} \arcsin \left (c x\right )^{2} - \frac {2}{3} \, a b {\left (\frac {\sqrt {-c^{2} d x^{2} + d} x^{2}}{c^{2} d} + \frac {2 \, \sqrt {-c^{2} d x^{2} + d}}{c^{4} d}\right )} \arcsin \left (c x\right ) - \frac {1}{3} \, a^{2} {\left (\frac {\sqrt {-c^{2} d x^{2} + d} x^{2}}{c^{2} d} + \frac {2 \, \sqrt {-c^{2} d x^{2} + d}}{c^{4} d}\right )} + \frac {2}{27} \, b^{2} {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2} + \frac {20 \, \sqrt {-c^{2} x^{2} + 1}}{c^{2}}}{c^{2} \sqrt {d}} + \frac {3 \, {\left (c^{2} x^{3} + 6 \, x\right )} \arcsin \left (c x\right )}{c^{3} \sqrt {d}}\right )} + \frac {2 \, {\left (c^{2} x^{3} + 6 \, x\right )} a b}{9 \, c^{3} \sqrt {d}} \]
-1/3*b^2*(sqrt(-c^2*d*x^2 + d)*x^2/(c^2*d) + 2*sqrt(-c^2*d*x^2 + d)/(c^4*d ))*arcsin(c*x)^2 - 2/3*a*b*(sqrt(-c^2*d*x^2 + d)*x^2/(c^2*d) + 2*sqrt(-c^2 *d*x^2 + d)/(c^4*d))*arcsin(c*x) - 1/3*a^2*(sqrt(-c^2*d*x^2 + d)*x^2/(c^2* d) + 2*sqrt(-c^2*d*x^2 + d)/(c^4*d)) + 2/27*b^2*((sqrt(-c^2*x^2 + 1)*x^2 + 20*sqrt(-c^2*x^2 + 1)/c^2)/(c^2*sqrt(d)) + 3*(c^2*x^3 + 6*x)*arcsin(c*x)/ (c^3*sqrt(d))) + 2/9*(c^2*x^3 + 6*x)*a*b/(c^3*sqrt(d))
Exception generated. \[ \int \frac {x^3 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {x^3 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\int \frac {x^3\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{\sqrt {d-c^2\,d\,x^2}} \,d x \]