Integrand size = 29, antiderivative size = 400 \[ \int \frac {x^5 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\frac {16 a b x \sqrt {1-c^2 x^2}}{15 c^5 \sqrt {d-c^2 d x^2}}+\frac {298 b^2 \left (1-c^2 x^2\right )}{225 c^6 \sqrt {d-c^2 d x^2}}-\frac {76 b^2 \left (1-c^2 x^2\right )^2}{675 c^6 \sqrt {d-c^2 d x^2}}+\frac {2 b^2 \left (1-c^2 x^2\right )^3}{125 c^6 \sqrt {d-c^2 d x^2}}+\frac {16 b^2 x \sqrt {1-c^2 x^2} \arcsin (c x)}{15 c^5 \sqrt {d-c^2 d x^2}}+\frac {8 b x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{45 c^3 \sqrt {d-c^2 d x^2}}+\frac {2 b x^5 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{25 c \sqrt {d-c^2 d x^2}}-\frac {8 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{15 c^6 d}-\frac {4 x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{15 c^4 d}-\frac {x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{5 c^2 d} \]
298/225*b^2*(-c^2*x^2+1)/c^6/(-c^2*d*x^2+d)^(1/2)-76/675*b^2*(-c^2*x^2+1)^ 2/c^6/(-c^2*d*x^2+d)^(1/2)+2/125*b^2*(-c^2*x^2+1)^3/c^6/(-c^2*d*x^2+d)^(1/ 2)+16/15*a*b*x*(-c^2*x^2+1)^(1/2)/c^5/(-c^2*d*x^2+d)^(1/2)+16/15*b^2*x*arc sin(c*x)*(-c^2*x^2+1)^(1/2)/c^5/(-c^2*d*x^2+d)^(1/2)+8/45*b*x^3*(a+b*arcsi n(c*x))*(-c^2*x^2+1)^(1/2)/c^3/(-c^2*d*x^2+d)^(1/2)+2/25*b*x^5*(a+b*arcsin (c*x))*(-c^2*x^2+1)^(1/2)/c/(-c^2*d*x^2+d)^(1/2)-8/15*(a+b*arcsin(c*x))^2* (-c^2*d*x^2+d)^(1/2)/c^6/d-4/15*x^2*(a+b*arcsin(c*x))^2*(-c^2*d*x^2+d)^(1/ 2)/c^4/d-1/5*x^4*(a+b*arcsin(c*x))^2*(-c^2*d*x^2+d)^(1/2)/c^2/d
Time = 0.12 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.58 \[ \int \frac {x^5 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\frac {30 a b c x \sqrt {1-c^2 x^2} \left (120+20 c^2 x^2+9 c^4 x^4\right )+225 a^2 \left (-8+4 c^2 x^2+c^4 x^4+3 c^6 x^6\right )-2 b^2 \left (-2072+1936 c^2 x^2+109 c^4 x^4+27 c^6 x^6\right )+30 b \left (b c x \sqrt {1-c^2 x^2} \left (120+20 c^2 x^2+9 c^4 x^4\right )+15 a \left (-8+4 c^2 x^2+c^4 x^4+3 c^6 x^6\right )\right ) \arcsin (c x)+225 b^2 \left (-8+4 c^2 x^2+c^4 x^4+3 c^6 x^6\right ) \arcsin (c x)^2}{3375 c^6 \sqrt {d-c^2 d x^2}} \]
(30*a*b*c*x*Sqrt[1 - c^2*x^2]*(120 + 20*c^2*x^2 + 9*c^4*x^4) + 225*a^2*(-8 + 4*c^2*x^2 + c^4*x^4 + 3*c^6*x^6) - 2*b^2*(-2072 + 1936*c^2*x^2 + 109*c^ 4*x^4 + 27*c^6*x^6) + 30*b*(b*c*x*Sqrt[1 - c^2*x^2]*(120 + 20*c^2*x^2 + 9* c^4*x^4) + 15*a*(-8 + 4*c^2*x^2 + c^4*x^4 + 3*c^6*x^6))*ArcSin[c*x] + 225* b^2*(-8 + 4*c^2*x^2 + c^4*x^4 + 3*c^6*x^6)*ArcSin[c*x]^2)/(3375*c^6*Sqrt[d - c^2*d*x^2])
Time = 1.54 (sec) , antiderivative size = 412, normalized size of antiderivative = 1.03, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.414, Rules used = {5210, 5138, 243, 53, 2009, 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.
| \(\displaystyle \int \frac {x^5 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx\) |
| \(\Big \downarrow \) 5210 |
| \(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \int x^4 (a+b \arcsin (c x))dx}{5 c \sqrt {d-c^2 d x^2}}+\frac {4 \int \frac {x^3 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}}dx}{5 c^2}-\frac {x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{5 c^2 d}\) |
| \(\Big \downarrow \) 5138 |
| \(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{5} b c \int \frac {x^5}{\sqrt {1-c^2 x^2}}dx\right )}{5 c \sqrt {d-c^2 d x^2}}+\frac {4 \int \frac {x^3 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}}dx}{5 c^2}-\frac {x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{5 c^2 d}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {4 \int \frac {x^3 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}}dx}{5 c^2}+\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{10} b c \int \frac {x^4}{\sqrt {1-c^2 x^2}}dx^2\right )}{5 c \sqrt {d-c^2 d x^2}}-\frac {x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{5 c^2 d}\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \frac {4 \int \frac {x^3 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}}dx}{5 c^2}+\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{10} b c \int \left (\frac {\left (1-c^2 x^2\right )^{3/2}}{c^4}-\frac {2 \sqrt {1-c^2 x^2}}{c^4}+\frac {1}{c^4 \sqrt {1-c^2 x^2}}\right )dx^2\right )}{5 c \sqrt {d-c^2 d x^2}}-\frac {x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{5 c^2 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {4 \int \frac {x^3 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}}dx}{5 c^2}-\frac {x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{5 c^2 d}+\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{10} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}+\frac {4 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {2 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{5 c \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 5210 |
| \(\displaystyle \frac {4 \left (\frac {2 b \sqrt {1-c^2 x^2} \int x^2 (a+b \arcsin (c x))dx}{3 c \sqrt {d-c^2 d x^2}}+\frac {2 \int \frac {x (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}}dx}{3 c^2}-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{3 c^2 d}\right )}{5 c^2}-\frac {x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{5 c^2 d}+\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{10} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}+\frac {4 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {2 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{5 c \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 5138 |
| \(\displaystyle \frac {4 \left (\frac {2 \int \frac {x (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}}dx}{3 c^2}+\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {1}{3} x^3 (a+b \arcsin (c x))-\frac {1}{3} b c \int \frac {x^3}{\sqrt {1-c^2 x^2}}dx\right )}{3 c \sqrt {d-c^2 d x^2}}-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{3 c^2 d}\right )}{5 c^2}-\frac {x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{5 c^2 d}+\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{10} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}+\frac {4 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {2 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{5 c \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {4 \left (\frac {2 \int \frac {x (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}}dx}{3 c^2}+\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {1}{3} x^3 (a+b \arcsin (c x))-\frac {1}{6} b c \int \frac {x^2}{\sqrt {1-c^2 x^2}}dx^2\right )}{3 c \sqrt {d-c^2 d x^2}}-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{3 c^2 d}\right )}{5 c^2}-\frac {x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{5 c^2 d}+\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{10} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}+\frac {4 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {2 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{5 c \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \frac {4 \left (\frac {2 \int \frac {x (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}}dx}{3 c^2}+\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {1}{3} x^3 (a+b \arcsin (c x))-\frac {1}{6} b c \int \left (\frac {1}{c^2 \sqrt {1-c^2 x^2}}-\frac {\sqrt {1-c^2 x^2}}{c^2}\right )dx^2\right )}{3 c \sqrt {d-c^2 d x^2}}-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{3 c^2 d}\right )}{5 c^2}-\frac {x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{5 c^2 d}+\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{10} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}+\frac {4 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {2 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{5 c \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {4 \left (\frac {2 \int \frac {x (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}}dx}{3 c^2}-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{3 c^2 d}+\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {1}{3} x^3 (a+b \arcsin (c x))-\frac {1}{6} b c \left (\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {1-c^2 x^2}}{c^4}\right )\right )}{3 c \sqrt {d-c^2 d x^2}}\right )}{5 c^2}-\frac {x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{5 c^2 d}+\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{10} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}+\frac {4 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {2 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{5 c \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 5182 |
| \(\displaystyle \frac {4 \left (\frac {2 \left (\frac {2 b \sqrt {1-c^2 x^2} \int (a+b \arcsin (c x))dx}{c \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{c^2 d}\right )}{3 c^2}-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{3 c^2 d}+\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {1}{3} x^3 (a+b \arcsin (c x))-\frac {1}{6} b c \left (\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {1-c^2 x^2}}{c^4}\right )\right )}{3 c \sqrt {d-c^2 d x^2}}\right )}{5 c^2}-\frac {x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{5 c^2 d}+\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{10} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}+\frac {4 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {2 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{5 c \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{5 c^2 d}+\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {1}{5} x^5 (a+b \arcsin (c x))-\frac {1}{10} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}+\frac {4 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {2 \sqrt {1-c^2 x^2}}{c^6}\right )\right )}{5 c \sqrt {d-c^2 d x^2}}+\frac {4 \left (-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{3 c^2 d}+\frac {2 \left (\frac {2 b \sqrt {1-c^2 x^2} \left (a x+b x \arcsin (c x)+\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{c \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{c^2 d}\right )}{3 c^2}+\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {1}{3} x^3 (a+b \arcsin (c x))-\frac {1}{6} b c \left (\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {1-c^2 x^2}}{c^4}\right )\right )}{3 c \sqrt {d-c^2 d x^2}}\right )}{5 c^2}\) |
-1/5*(x^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(c^2*d) + (2*b*Sqrt[1 - c^2*x^2]*(-1/10*(b*c*((-2*Sqrt[1 - c^2*x^2])/c^6 + (4*(1 - c^2*x^2)^(3/ 2))/(3*c^6) - (2*(1 - c^2*x^2)^(5/2))/(5*c^6))) + (x^5*(a + b*ArcSin[c*x]) )/5))/(5*c*Sqrt[d - c^2*d*x^2]) + (4*(-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*Arc Sin[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)))/(5*c^2)
3.3.34.3.1 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 = 0.33 (sec) , antiderivative size = 1020, normalized size of antiderivative = 2.55
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1020\) |
| parts | \(\text {Expression too large to display}\) | \(1020\) |
a^2*(-1/5*x^4/c^2/d*(-c^2*d*x^2+d)^(1/2)+4/5/c^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*(5/1728*(-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^6/d/(c^2*x^2-1)-5/16*(-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^6/d/(c^2*x^2-1)-5/16 *(-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^6/d/(c^2*x^2-1)+5/1728*(-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 ^6/d/(c^2*x^2-1)+1/4000*(-d*(c^2*x^2-1))^(1/2)/c^6/d/(c^2*x^2-1)*(25*arcsi n(c*x)^2-2)*cos(6*arcsin(c*x))-1/400*(-d*(c^2*x^2-1))^(1/2)/c^6/d/(c^2*x^2 -1)*arcsin(c*x)*sin(6*arcsin(c*x))-1/54000*(-d*(c^2*x^2-1))^(1/2)/c^6/d/(c ^2*x^2-1)*(2475*arcsin(c*x)^2-598)*cos(4*arcsin(c*x))+29/900*(-d*(c^2*x^2- 1))^(1/2)/c^6/d/(c^2*x^2-1)*arcsin(c*x)*sin(4*arcsin(c*x)))+2*a*b*(5/576*( -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^6/d/(c^2*x^2-1)-5/16*(-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^6/d/(c^2*x^2-1)-5/16*(-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^6/d/(c^2*x^2-1) +5/576*(-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*arcsin(c*x))/c^6/d/(c^2*x^2-1)+1/160*(-d*(c^2*x^2-1))^(1/2)/c^6/d/(c^2* x^2-1)*arcsin(c*x)*cos(6*arcsin(c*x))-1/800*(-d*(c^2*x^2-1))^(1/2)/c^6/...
Time = 0.28 (sec) , antiderivative size = 276, normalized size of antiderivative = 0.69 \[ \int \frac {x^5 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=-\frac {30 \, {\left (9 \, a b c^{5} x^{5} + 20 \, a b c^{3} x^{3} + 120 \, a b c x + {\left (9 \, b^{2} c^{5} x^{5} + 20 \, b^{2} c^{3} x^{3} + 120 \, b^{2} c x\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} + {\left (27 \, {\left (25 \, a^{2} - 2 \, b^{2}\right )} c^{6} x^{6} + {\left (225 \, a^{2} - 218 \, b^{2}\right )} c^{4} x^{4} + 4 \, {\left (225 \, a^{2} - 968 \, b^{2}\right )} c^{2} x^{2} + 225 \, {\left (3 \, b^{2} c^{6} x^{6} + b^{2} c^{4} x^{4} + 4 \, b^{2} c^{2} x^{2} - 8 \, b^{2}\right )} \arcsin \left (c x\right )^{2} - 1800 \, a^{2} + 4144 \, b^{2} + 450 \, {\left (3 \, a b c^{6} x^{6} + a b c^{4} x^{4} + 4 \, a b c^{2} x^{2} - 8 \, a b\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}}{3375 \, {\left (c^{8} d x^{2} - c^{6} d\right )}} \]
-1/3375*(30*(9*a*b*c^5*x^5 + 20*a*b*c^3*x^3 + 120*a*b*c*x + (9*b^2*c^5*x^5 + 20*b^2*c^3*x^3 + 120*b^2*c*x)*arcsin(c*x))*sqrt(-c^2*d*x^2 + d)*sqrt(-c ^2*x^2 + 1) + (27*(25*a^2 - 2*b^2)*c^6*x^6 + (225*a^2 - 218*b^2)*c^4*x^4 + 4*(225*a^2 - 968*b^2)*c^2*x^2 + 225*(3*b^2*c^6*x^6 + b^2*c^4*x^4 + 4*b^2* c^2*x^2 - 8*b^2)*arcsin(c*x)^2 - 1800*a^2 + 4144*b^2 + 450*(3*a*b*c^6*x^6 + a*b*c^4*x^4 + 4*a*b*c^2*x^2 - 8*a*b)*arcsin(c*x))*sqrt(-c^2*d*x^2 + d))/ (c^8*d*x^2 - c^6*d)
\[ \int \frac {x^5 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\int \frac {x^{5} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \]
Time = 0.31 (sec) , antiderivative size = 365, normalized size of antiderivative = 0.91 \[ \int \frac {x^5 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=-\frac {1}{15} \, {\left (\frac {3 \, \sqrt {-c^{2} d x^{2} + d} x^{4}}{c^{2} d} + \frac {4 \, \sqrt {-c^{2} d x^{2} + d} x^{2}}{c^{4} d} + \frac {8 \, \sqrt {-c^{2} d x^{2} + d}}{c^{6} d}\right )} b^{2} \arcsin \left (c x\right )^{2} - \frac {2}{15} \, {\left (\frac {3 \, \sqrt {-c^{2} d x^{2} + d} x^{4}}{c^{2} d} + \frac {4 \, \sqrt {-c^{2} d x^{2} + d} x^{2}}{c^{4} d} + \frac {8 \, \sqrt {-c^{2} d x^{2} + d}}{c^{6} d}\right )} a b \arcsin \left (c x\right ) - \frac {1}{15} \, {\left (\frac {3 \, \sqrt {-c^{2} d x^{2} + d} x^{4}}{c^{2} d} + \frac {4 \, \sqrt {-c^{2} d x^{2} + d} x^{2}}{c^{4} d} + \frac {8 \, \sqrt {-c^{2} d x^{2} + d}}{c^{6} d}\right )} a^{2} + \frac {2}{3375} \, b^{2} {\left (\frac {27 \, \sqrt {-c^{2} x^{2} + 1} c^{2} x^{4} + 136 \, \sqrt {-c^{2} x^{2} + 1} x^{2} + \frac {2072 \, \sqrt {-c^{2} x^{2} + 1}}{c^{2}}}{c^{4} \sqrt {d}} + \frac {15 \, {\left (9 \, c^{4} x^{5} + 20 \, c^{2} x^{3} + 120 \, x\right )} \arcsin \left (c x\right )}{c^{5} \sqrt {d}}\right )} + \frac {2 \, {\left (9 \, c^{4} x^{5} + 20 \, c^{2} x^{3} + 120 \, x\right )} a b}{225 \, c^{5} \sqrt {d}} \]
-1/15*(3*sqrt(-c^2*d*x^2 + d)*x^4/(c^2*d) + 4*sqrt(-c^2*d*x^2 + d)*x^2/(c^ 4*d) + 8*sqrt(-c^2*d*x^2 + d)/(c^6*d))*b^2*arcsin(c*x)^2 - 2/15*(3*sqrt(-c ^2*d*x^2 + d)*x^4/(c^2*d) + 4*sqrt(-c^2*d*x^2 + d)*x^2/(c^4*d) + 8*sqrt(-c ^2*d*x^2 + d)/(c^6*d))*a*b*arcsin(c*x) - 1/15*(3*sqrt(-c^2*d*x^2 + d)*x^4/ (c^2*d) + 4*sqrt(-c^2*d*x^2 + d)*x^2/(c^4*d) + 8*sqrt(-c^2*d*x^2 + d)/(c^6 *d))*a^2 + 2/3375*b^2*((27*sqrt(-c^2*x^2 + 1)*c^2*x^4 + 136*sqrt(-c^2*x^2 + 1)*x^2 + 2072*sqrt(-c^2*x^2 + 1)/c^2)/(c^4*sqrt(d)) + 15*(9*c^4*x^5 + 20 *c^2*x^3 + 120*x)*arcsin(c*x)/(c^5*sqrt(d))) + 2/225*(9*c^4*x^5 + 20*c^2*x ^3 + 120*x)*a*b/(c^5*sqrt(d))
Exception generated. \[ \int \frac {x^5 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {x^5 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\int \frac {x^5\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{\sqrt {d-c^2\,d\,x^2}} \,d x \]