Integrand size = 27, antiderivative size = 382 \[ \int x \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=\frac {32 b^2 d^2 \sqrt {d-c^2 d x^2}}{245 c^2}+\frac {16 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{735 c^2}+\frac {12 b^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{1225 c^2}+\frac {2 b^2 d^2 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{343 c^2}+\frac {2 b d^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{7 c \sqrt {1-c^2 x^2}}-\frac {2 b c d^2 x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{7 \sqrt {1-c^2 x^2}}+\frac {6 b c^3 d^2 x^5 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{35 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 x^7 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{49 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \arcsin (c x))^2}{7 c^2 d} \]
-1/7*(-c^2*d*x^2+d)^(7/2)*(a+b*arcsin(c*x))^2/c^2/d+32/245*b^2*d^2*(-c^2*d *x^2+d)^(1/2)/c^2+16/735*b^2*d^2*(-c^2*x^2+1)*(-c^2*d*x^2+d)^(1/2)/c^2+12/ 1225*b^2*d^2*(-c^2*x^2+1)^2*(-c^2*d*x^2+d)^(1/2)/c^2+2/343*b^2*d^2*(-c^2*x ^2+1)^3*(-c^2*d*x^2+d)^(1/2)/c^2+2/7*b*d^2*x*(a+b*arcsin(c*x))*(-c^2*d*x^2 +d)^(1/2)/c/(-c^2*x^2+1)^(1/2)-2/7*b*c*d^2*x^3*(a+b*arcsin(c*x))*(-c^2*d*x ^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+6/35*b*c^3*d^2*x^5*(a+b*arcsin(c*x))*(-c^2* d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-2/49*b*c^5*d^2*x^7*(a+b*arcsin(c*x))*(-c ^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)
Time = 0.21 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.46 \[ \int x \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=\frac {d^2 \sqrt {d-c^2 d x^2} \left (\left (-1+c^2 x^2\right )^3 (a+b \arcsin (c x))^2-\frac {2 b \left (105 a c x \left (-35+35 c^2 x^2-21 c^4 x^4+5 c^6 x^6\right )+b \sqrt {1-c^2 x^2} \left (-2161+757 c^2 x^2-351 c^4 x^4+75 c^6 x^6\right )+105 b c x \left (-35+35 c^2 x^2-21 c^4 x^4+5 c^6 x^6\right ) \arcsin (c x)\right )}{3675 \sqrt {1-c^2 x^2}}\right )}{7 c^2} \]
(d^2*Sqrt[d - c^2*d*x^2]*((-1 + c^2*x^2)^3*(a + b*ArcSin[c*x])^2 - (2*b*(1 05*a*c*x*(-35 + 35*c^2*x^2 - 21*c^4*x^4 + 5*c^6*x^6) + b*Sqrt[1 - c^2*x^2] *(-2161 + 757*c^2*x^2 - 351*c^4*x^4 + 75*c^6*x^6) + 105*b*c*x*(-35 + 35*c^ 2*x^2 - 21*c^4*x^4 + 5*c^6*x^6)*ArcSin[c*x]))/(3675*Sqrt[1 - c^2*x^2])))/( 7*c^2)
Time = 0.66 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.60, 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, 2331, 2389, 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.
| \(\displaystyle \int x \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx\) |
| \(\Big \downarrow \) 5182 |
| \(\displaystyle \frac {2 b d^2 \sqrt {d-c^2 d x^2} \int \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))dx}{7 c \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \arcsin (c x))^2}{7 c^2 d}\) |
| \(\Big \downarrow \) 5154 |
| \(\displaystyle \frac {2 b d^2 \sqrt {d-c^2 d x^2} \left (-b c \int \frac {x \left (-5 c^6 x^6+21 c^4 x^4-35 c^2 x^2+35\right )}{35 \sqrt {1-c^2 x^2}}dx-\frac {1}{7} c^6 x^7 (a+b \arcsin (c x))+\frac {3}{5} c^4 x^5 (a+b \arcsin (c x))-c^2 x^3 (a+b \arcsin (c x))+x (a+b \arcsin (c x))\right )}{7 c \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \arcsin (c x))^2}{7 c^2 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 b d^2 \sqrt {d-c^2 d x^2} \left (-\frac {1}{35} b c \int \frac {x \left (-5 c^6 x^6+21 c^4 x^4-35 c^2 x^2+35\right )}{\sqrt {1-c^2 x^2}}dx-\frac {1}{7} c^6 x^7 (a+b \arcsin (c x))+\frac {3}{5} c^4 x^5 (a+b \arcsin (c x))-c^2 x^3 (a+b \arcsin (c x))+x (a+b \arcsin (c x))\right )}{7 c \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \arcsin (c x))^2}{7 c^2 d}\) |
| \(\Big \downarrow \) 2331 |
| \(\displaystyle \frac {2 b d^2 \sqrt {d-c^2 d x^2} \left (-\frac {1}{70} b c \int \frac {-5 c^6 x^6+21 c^4 x^4-35 c^2 x^2+35}{\sqrt {1-c^2 x^2}}dx^2-\frac {1}{7} c^6 x^7 (a+b \arcsin (c x))+\frac {3}{5} c^4 x^5 (a+b \arcsin (c x))-c^2 x^3 (a+b \arcsin (c x))+x (a+b \arcsin (c x))\right )}{7 c \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \arcsin (c x))^2}{7 c^2 d}\) |
| \(\Big \downarrow \) 2389 |
| \(\displaystyle \frac {2 b d^2 \sqrt {d-c^2 d x^2} \left (-\frac {1}{70} b c \int \left (5 \left (1-c^2 x^2\right )^{5/2}+6 \left (1-c^2 x^2\right )^{3/2}+8 \sqrt {1-c^2 x^2}+\frac {16}{\sqrt {1-c^2 x^2}}\right )dx^2-\frac {1}{7} c^6 x^7 (a+b \arcsin (c x))+\frac {3}{5} c^4 x^5 (a+b \arcsin (c x))-c^2 x^3 (a+b \arcsin (c x))+x (a+b \arcsin (c x))\right )}{7 c \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \arcsin (c x))^2}{7 c^2 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 b d^2 \sqrt {d-c^2 d x^2} \left (-\frac {1}{7} c^6 x^7 (a+b \arcsin (c x))+\frac {3}{5} c^4 x^5 (a+b \arcsin (c x))-c^2 x^3 (a+b \arcsin (c x))+x (a+b \arcsin (c x))-\frac {1}{70} b c \left (-\frac {10 \left (1-c^2 x^2\right )^{7/2}}{7 c^2}-\frac {12 \left (1-c^2 x^2\right )^{5/2}}{5 c^2}-\frac {16 \left (1-c^2 x^2\right )^{3/2}}{3 c^2}-\frac {32 \sqrt {1-c^2 x^2}}{c^2}\right )\right )}{7 c \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \arcsin (c x))^2}{7 c^2 d}\) |
-1/7*((d - c^2*d*x^2)^(7/2)*(a + b*ArcSin[c*x])^2)/(c^2*d) + (2*b*d^2*Sqrt [d - c^2*d*x^2]*(-1/70*(b*c*((-32*Sqrt[1 - c^2*x^2])/c^2 - (16*(1 - c^2*x^ 2)^(3/2))/(3*c^2) - (12*(1 - c^2*x^2)^(5/2))/(5*c^2) - (10*(1 - c^2*x^2)^( 7/2))/(7*c^2))) + x*(a + b*ArcSin[c*x]) - c^2*x^3*(a + b*ArcSin[c*x]) + (3 *c^4*x^5*(a + b*ArcSin[c*x]))/5 - (c^6*x^7*(a + b*ArcSin[c*x]))/7))/(7*c*S qrt[1 - c^2*x^2])
3.3.28.3.1 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[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/2 S ubst[Int[x^((m - 1)/2)*SubstFor[x^2, Pq, x]*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand [Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p , 0] || EqQ[n, 1])
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 = 0.43 (sec) , antiderivative size = 1611, normalized size of antiderivative = 4.22
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1611\) |
| parts | \(\text {Expression too large to display}\) | \(1611\) |
-1/7*a^2*(-c^2*d*x^2+d)^(7/2)/c^2/d+b^2*(1/43904*(-d*(c^2*x^2-1))^(1/2)*(6 4*c^8*x^8-144*c^6*x^6-64*I*c^7*x^7*(-c^2*x^2+1)^(1/2)+104*c^4*x^4+112*I*(- c^2*x^2+1)^(1/2)*x^5*c^5-25*c^2*x^2-56*I*(-c^2*x^2+1)^(1/2)*x^3*c^3+7*I*(- c^2*x^2+1)^(1/2)*x*c+1)*(14*I*arcsin(c*x)+49*arcsin(c*x)^2-2)*d^2/c^2/(c^2 *x^2-1)-1/3200*(-d*(c^2*x^2-1))^(1/2)*(16*c^6*x^6-28*c^4*x^4-16*I*(-c^2*x^ 2+1)^(1/2)*x^5*c^5+13*c^2*x^2+20*I*(-c^2*x^2+1)^(1/2)*x^3*c^3-5*I*(-c^2*x^ 2+1)^(1/2)*x*c-1)*(10*I*arcsin(c*x)+25*arcsin(c*x)^2-2)*d^2/c^2/(c^2*x^2-1 )-5/128*(-d*(c^2*x^2-1))^(1/2)*(c^2*x^2-I*(-c^2*x^2+1)^(1/2)*x*c-1)*(arcsi n(c*x)^2-2+2*I*arcsin(c*x))*d^2/c^2/(c^2*x^2-1)-5/128*(-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))* d^2/c^2/(c^2*x^2-1)+1/384*(-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)*d^2/c^2/(c^2*x^2-1)+1/43904*(-d*(c^2*x^2-1))^(1/2)*(64 *I*c^7*x^7*(-c^2*x^2+1)^(1/2)+64*c^8*x^8-112*I*(-c^2*x^2+1)^(1/2)*x^5*c^5- 144*c^6*x^6+56*I*(-c^2*x^2+1)^(1/2)*x^3*c^3+104*c^4*x^4-7*I*(-c^2*x^2+1)^( 1/2)*x*c-25*c^2*x^2+1)*(-14*I*arcsin(c*x)+49*arcsin(c*x)^2-2)*d^2/c^2/(c^2 *x^2-1)-1/2400*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1) *(30*I*arcsin(c*x)+75*arcsin(c*x)^2-14)*cos(4*arcsin(c*x))*d^2/c^2/(c^2*x^ 2-1)-1/4800*(-d*(c^2*x^2-1))^(1/2)*(I*c^2*x^2-c*x*(-c^2*x^2+1)^(1/2)-I)*(9 0*I*arcsin(c*x)+75*arcsin(c*x)^2-22)*sin(4*arcsin(c*x))*d^2/c^2/(c^2*x^...
Time = 0.28 (sec) , antiderivative size = 405, normalized size of antiderivative = 1.06 \[ \int x \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=\frac {210 \, {\left (5 \, a b c^{7} d^{2} x^{7} - 21 \, a b c^{5} d^{2} x^{5} + 35 \, a b c^{3} d^{2} x^{3} - 35 \, a b c d^{2} x + {\left (5 \, b^{2} c^{7} d^{2} x^{7} - 21 \, b^{2} c^{5} d^{2} x^{5} + 35 \, b^{2} c^{3} d^{2} x^{3} - 35 \, b^{2} c d^{2} x\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} + {\left (75 \, {\left (49 \, a^{2} - 2 \, b^{2}\right )} c^{8} d^{2} x^{8} - 12 \, {\left (1225 \, a^{2} - 71 \, b^{2}\right )} c^{6} d^{2} x^{6} + 2 \, {\left (11025 \, a^{2} - 1108 \, b^{2}\right )} c^{4} d^{2} x^{4} - 4 \, {\left (3675 \, a^{2} - 1459 \, b^{2}\right )} c^{2} d^{2} x^{2} + {\left (3675 \, a^{2} - 4322 \, b^{2}\right )} d^{2} + 3675 \, {\left (b^{2} c^{8} d^{2} x^{8} - 4 \, b^{2} c^{6} d^{2} x^{6} + 6 \, b^{2} c^{4} d^{2} x^{4} - 4 \, b^{2} c^{2} d^{2} x^{2} + b^{2} d^{2}\right )} \arcsin \left (c x\right )^{2} + 7350 \, {\left (a b c^{8} d^{2} x^{8} - 4 \, a b c^{6} d^{2} x^{6} + 6 \, a b c^{4} d^{2} x^{4} - 4 \, a b c^{2} d^{2} x^{2} + a b d^{2}\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}}{25725 \, {\left (c^{4} x^{2} - c^{2}\right )}} \]
1/25725*(210*(5*a*b*c^7*d^2*x^7 - 21*a*b*c^5*d^2*x^5 + 35*a*b*c^3*d^2*x^3 - 35*a*b*c*d^2*x + (5*b^2*c^7*d^2*x^7 - 21*b^2*c^5*d^2*x^5 + 35*b^2*c^3*d^ 2*x^3 - 35*b^2*c*d^2*x)*arcsin(c*x))*sqrt(-c^2*d*x^2 + d)*sqrt(-c^2*x^2 + 1) + (75*(49*a^2 - 2*b^2)*c^8*d^2*x^8 - 12*(1225*a^2 - 71*b^2)*c^6*d^2*x^6 + 2*(11025*a^2 - 1108*b^2)*c^4*d^2*x^4 - 4*(3675*a^2 - 1459*b^2)*c^2*d^2* x^2 + (3675*a^2 - 4322*b^2)*d^2 + 3675*(b^2*c^8*d^2*x^8 - 4*b^2*c^6*d^2*x^ 6 + 6*b^2*c^4*d^2*x^4 - 4*b^2*c^2*d^2*x^2 + b^2*d^2)*arcsin(c*x)^2 + 7350* (a*b*c^8*d^2*x^8 - 4*a*b*c^6*d^2*x^6 + 6*a*b*c^4*d^2*x^4 - 4*a*b*c^2*d^2*x ^2 + a*b*d^2)*arcsin(c*x))*sqrt(-c^2*d*x^2 + d))/(c^4*x^2 - c^2)
\[ \int x \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=\int x \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}\, dx \]
Time = 0.30 (sec) , antiderivative size = 281, normalized size of antiderivative = 0.74 \[ \int x \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=-\frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} b^{2} \arcsin \left (c x\right )^{2}}{7 \, c^{2} d} - \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} a b \arcsin \left (c x\right )}{7 \, c^{2} d} - \frac {2}{25725} \, b^{2} {\left (\frac {75 \, \sqrt {-c^{2} x^{2} + 1} c^{4} d^{\frac {7}{2}} x^{6} - 351 \, \sqrt {-c^{2} x^{2} + 1} c^{2} d^{\frac {7}{2}} x^{4} + 757 \, \sqrt {-c^{2} x^{2} + 1} d^{\frac {7}{2}} x^{2} - \frac {2161 \, \sqrt {-c^{2} x^{2} + 1} d^{\frac {7}{2}}}{c^{2}}}{d} + \frac {105 \, {\left (5 \, c^{6} d^{\frac {7}{2}} x^{7} - 21 \, c^{4} d^{\frac {7}{2}} x^{5} + 35 \, c^{2} d^{\frac {7}{2}} x^{3} - 35 \, d^{\frac {7}{2}} x\right )} \arcsin \left (c x\right )}{c d}\right )} - \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} a^{2}}{7 \, c^{2} d} - \frac {2 \, {\left (5 \, c^{6} d^{\frac {7}{2}} x^{7} - 21 \, c^{4} d^{\frac {7}{2}} x^{5} + 35 \, c^{2} d^{\frac {7}{2}} x^{3} - 35 \, d^{\frac {7}{2}} x\right )} a b}{245 \, c d} \]
-1/7*(-c^2*d*x^2 + d)^(7/2)*b^2*arcsin(c*x)^2/(c^2*d) - 2/7*(-c^2*d*x^2 + d)^(7/2)*a*b*arcsin(c*x)/(c^2*d) - 2/25725*b^2*((75*sqrt(-c^2*x^2 + 1)*c^4 *d^(7/2)*x^6 - 351*sqrt(-c^2*x^2 + 1)*c^2*d^(7/2)*x^4 + 757*sqrt(-c^2*x^2 + 1)*d^(7/2)*x^2 - 2161*sqrt(-c^2*x^2 + 1)*d^(7/2)/c^2)/d + 105*(5*c^6*d^( 7/2)*x^7 - 21*c^4*d^(7/2)*x^5 + 35*c^2*d^(7/2)*x^3 - 35*d^(7/2)*x)*arcsin( c*x)/(c*d)) - 1/7*(-c^2*d*x^2 + d)^(7/2)*a^2/(c^2*d) - 2/245*(5*c^6*d^(7/2 )*x^7 - 21*c^4*d^(7/2)*x^5 + 35*c^2*d^(7/2)*x^3 - 35*d^(7/2)*x)*a*b/(c*d)
Exception generated. \[ \int x \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=\text {Exception raised: TypeError} \]
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. \[ \int x \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^2 \, dx=\int x\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \]