Integrand size = 25, antiderivative size = 186 \[ \int x^4 \left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x)) \, dx=\frac {8 b d^2 \sqrt {1-c^2 x^2}}{315 c^5}+\frac {4 b d^2 \left (1-c^2 x^2\right )^{3/2}}{945 c^5}+\frac {b d^2 \left (1-c^2 x^2\right )^{5/2}}{525 c^5}-\frac {10 b d^2 \left (1-c^2 x^2\right )^{7/2}}{441 c^5}+\frac {b d^2 \left (1-c^2 x^2\right )^{9/2}}{81 c^5}+\frac {1}{5} d^2 x^5 (a+b \arcsin (c x))-\frac {2}{7} c^2 d^2 x^7 (a+b \arcsin (c x))+\frac {1}{9} c^4 d^2 x^9 (a+b \arcsin (c x)) \] Output:
8/315*b*d^2*(-c^2*x^2+1)^(1/2)/c^5+4/945*b*d^2*(-c^2*x^2+1)^(3/2)/c^5+1/52 5*b*d^2*(-c^2*x^2+1)^(5/2)/c^5-10/441*b*d^2*(-c^2*x^2+1)^(7/2)/c^5+1/81*b* d^2*(-c^2*x^2+1)^(9/2)/c^5+1/5*d^2*x^5*(a+b*arcsin(c*x))-2/7*c^2*d^2*x^7*( a+b*arcsin(c*x))+1/9*c^4*d^2*x^9*(a+b*arcsin(c*x))
Time = 0.07 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.64 \[ \int x^4 \left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x)) \, dx=\frac {d^2 \left (315 a c^5 x^5 \left (63-90 c^2 x^2+35 c^4 x^4\right )+b \sqrt {1-c^2 x^2} \left (2104+1052 c^2 x^2+789 c^4 x^4-2650 c^6 x^6+1225 c^8 x^8\right )+315 b c^5 x^5 \left (63-90 c^2 x^2+35 c^4 x^4\right ) \arcsin (c x)\right )}{99225 c^5} \] Input:
Integrate[x^4*(d - c^2*d*x^2)^2*(a + b*ArcSin[c*x]),x]
Output:
(d^2*(315*a*c^5*x^5*(63 - 90*c^2*x^2 + 35*c^4*x^4) + b*Sqrt[1 - c^2*x^2]*( 2104 + 1052*c^2*x^2 + 789*c^4*x^4 - 2650*c^6*x^6 + 1225*c^8*x^8) + 315*b*c ^5*x^5*(63 - 90*c^2*x^2 + 35*c^4*x^4)*ArcSin[c*x]))/(99225*c^5)
Time = 0.45 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.94, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5192, 27, 1578, 1195, 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.
| \(\displaystyle \int x^4 \left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x)) \, dx\) |
| \(\Big \downarrow \) 5192 |
| \(\displaystyle -b c \int \frac {d^2 x^5 \left (35 c^4 x^4-90 c^2 x^2+63\right )}{315 \sqrt {1-c^2 x^2}}dx+\frac {1}{9} c^4 d^2 x^9 (a+b \arcsin (c x))-\frac {2}{7} c^2 d^2 x^7 (a+b \arcsin (c x))+\frac {1}{5} d^2 x^5 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{315} b c d^2 \int \frac {x^5 \left (35 c^4 x^4-90 c^2 x^2+63\right )}{\sqrt {1-c^2 x^2}}dx+\frac {1}{9} c^4 d^2 x^9 (a+b \arcsin (c x))-\frac {2}{7} c^2 d^2 x^7 (a+b \arcsin (c x))+\frac {1}{5} d^2 x^5 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 1578 |
| \(\displaystyle -\frac {1}{630} b c d^2 \int \frac {x^4 \left (35 c^4 x^4-90 c^2 x^2+63\right )}{\sqrt {1-c^2 x^2}}dx^2+\frac {1}{9} c^4 d^2 x^9 (a+b \arcsin (c x))-\frac {2}{7} c^2 d^2 x^7 (a+b \arcsin (c x))+\frac {1}{5} d^2 x^5 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle -\frac {1}{630} b c d^2 \int \left (\frac {35 \left (1-c^2 x^2\right )^{7/2}}{c^4}-\frac {50 \left (1-c^2 x^2\right )^{5/2}}{c^4}+\frac {3 \left (1-c^2 x^2\right )^{3/2}}{c^4}+\frac {4 \sqrt {1-c^2 x^2}}{c^4}+\frac {8}{c^4 \sqrt {1-c^2 x^2}}\right )dx^2+\frac {1}{9} c^4 d^2 x^9 (a+b \arcsin (c x))-\frac {2}{7} c^2 d^2 x^7 (a+b \arcsin (c x))+\frac {1}{5} d^2 x^5 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{9} c^4 d^2 x^9 (a+b \arcsin (c x))-\frac {2}{7} c^2 d^2 x^7 (a+b \arcsin (c x))+\frac {1}{5} d^2 x^5 (a+b \arcsin (c x))-\frac {1}{630} b c d^2 \left (-\frac {70 \left (1-c^2 x^2\right )^{9/2}}{9 c^6}+\frac {100 \left (1-c^2 x^2\right )^{7/2}}{7 c^6}-\frac {6 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}-\frac {8 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {16 \sqrt {1-c^2 x^2}}{c^6}\right )\) |
Input:
Int[x^4*(d - c^2*d*x^2)^2*(a + b*ArcSin[c*x]),x]
Output:
-1/630*(b*c*d^2*((-16*Sqrt[1 - c^2*x^2])/c^6 - (8*(1 - c^2*x^2)^(3/2))/(3* c^6) - (6*(1 - c^2*x^2)^(5/2))/(5*c^6) + (100*(1 - c^2*x^2)^(7/2))/(7*c^6) - (70*(1 - c^2*x^2)^(9/2))/(9*c^6))) + (d^2*x^5*(a + b*ArcSin[c*x]))/5 - (2*c^2*d^2*x^7*(a + b*ArcSin[c*x]))/7 + (c^4*d^2*x^9*(a + b*ArcSin[c*x]))/ 9
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x ] && IGtQ[p, 0]
Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_ )^4)^(p_.), x_Symbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && Int egerQ[(m - 1)/2]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_) ^2)^(p_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(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]] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0 ] && IGtQ[p, 0]
Time = 0.17 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.90
| method | result | size |
| parts | \(a \,d^{2} \left (\frac {1}{9} c^{4} x^{9}-\frac {2}{7} c^{2} x^{7}+\frac {1}{5} x^{5}\right )+\frac {d^{2} b \left (\frac {\arcsin \left (c x \right ) c^{9} x^{9}}{9}-\frac {2 \arcsin \left (c x \right ) c^{7} x^{7}}{7}+\frac {c^{5} x^{5} \arcsin \left (c x \right )}{5}+\frac {263 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{33075}+\frac {1052 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{99225}+\frac {2104 \sqrt {-c^{2} x^{2}+1}}{99225}-\frac {106 c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{3969}+\frac {c^{8} x^{8} \sqrt {-c^{2} x^{2}+1}}{81}\right )}{c^{5}}\) | \(168\) |
| derivativedivides | \(\frac {a \,d^{2} \left (\frac {1}{9} c^{9} x^{9}-\frac {2}{7} c^{7} x^{7}+\frac {1}{5} c^{5} x^{5}\right )+d^{2} b \left (\frac {\arcsin \left (c x \right ) c^{9} x^{9}}{9}-\frac {2 \arcsin \left (c x \right ) c^{7} x^{7}}{7}+\frac {c^{5} x^{5} \arcsin \left (c x \right )}{5}+\frac {263 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{33075}+\frac {1052 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{99225}+\frac {2104 \sqrt {-c^{2} x^{2}+1}}{99225}-\frac {106 c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{3969}+\frac {c^{8} x^{8} \sqrt {-c^{2} x^{2}+1}}{81}\right )}{c^{5}}\) | \(172\) |
| default | \(\frac {a \,d^{2} \left (\frac {1}{9} c^{9} x^{9}-\frac {2}{7} c^{7} x^{7}+\frac {1}{5} c^{5} x^{5}\right )+d^{2} b \left (\frac {\arcsin \left (c x \right ) c^{9} x^{9}}{9}-\frac {2 \arcsin \left (c x \right ) c^{7} x^{7}}{7}+\frac {c^{5} x^{5} \arcsin \left (c x \right )}{5}+\frac {263 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{33075}+\frac {1052 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{99225}+\frac {2104 \sqrt {-c^{2} x^{2}+1}}{99225}-\frac {106 c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{3969}+\frac {c^{8} x^{8} \sqrt {-c^{2} x^{2}+1}}{81}\right )}{c^{5}}\) | \(172\) |
| orering | \(\frac {\left (20825 c^{10} x^{10}-54450 c^{8} x^{8}+36757 c^{6} x^{6}+5260 c^{4} x^{4}+12624 c^{2} x^{2}-8416\right ) \left (-c^{2} d \,x^{2}+d \right )^{2} \left (a +b \arcsin \left (c x \right )\right )}{99225 c^{6} \left (c x -1\right ) x \left (c x +1\right ) \left (c^{2} x^{2}-1\right )}-\frac {\left (1225 c^{8} x^{8}-2650 c^{6} x^{6}+789 c^{4} x^{4}+1052 c^{2} x^{2}+2104\right ) \left (4 x^{3} \left (-c^{2} d \,x^{2}+d \right )^{2} \left (a +b \arcsin \left (c x \right )\right )-4 x^{5} \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \arcsin \left (c x \right )\right ) c^{2} d +\frac {x^{4} \left (-c^{2} d \,x^{2}+d \right )^{2} b c}{\sqrt {-c^{2} x^{2}+1}}\right )}{99225 c^{6} \left (c x -1\right ) x^{4} \left (c x +1\right )}\) | \(240\) |
Input:
int(x^4*(-c^2*d*x^2+d)^2*(a+b*arcsin(c*x)),x,method=_RETURNVERBOSE)
Output:
a*d^2*(1/9*c^4*x^9-2/7*c^2*x^7+1/5*x^5)+d^2*b/c^5*(1/9*arcsin(c*x)*c^9*x^9 -2/7*arcsin(c*x)*c^7*x^7+1/5*c^5*x^5*arcsin(c*x)+263/33075*c^4*x^4*(-c^2*x ^2+1)^(1/2)+1052/99225*c^2*x^2*(-c^2*x^2+1)^(1/2)+2104/99225*(-c^2*x^2+1)^ (1/2)-106/3969*c^6*x^6*(-c^2*x^2+1)^(1/2)+1/81*c^8*x^8*(-c^2*x^2+1)^(1/2))
Time = 0.11 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.82 \[ \int x^4 \left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x)) \, dx=\frac {11025 \, a c^{9} d^{2} x^{9} - 28350 \, a c^{7} d^{2} x^{7} + 19845 \, a c^{5} d^{2} x^{5} + 315 \, {\left (35 \, b c^{9} d^{2} x^{9} - 90 \, b c^{7} d^{2} x^{7} + 63 \, b c^{5} d^{2} x^{5}\right )} \arcsin \left (c x\right ) + {\left (1225 \, b c^{8} d^{2} x^{8} - 2650 \, b c^{6} d^{2} x^{6} + 789 \, b c^{4} d^{2} x^{4} + 1052 \, b c^{2} d^{2} x^{2} + 2104 \, b d^{2}\right )} \sqrt {-c^{2} x^{2} + 1}}{99225 \, c^{5}} \] Input:
integrate(x^4*(-c^2*d*x^2+d)^2*(a+b*arcsin(c*x)),x, algorithm="fricas")
Output:
1/99225*(11025*a*c^9*d^2*x^9 - 28350*a*c^7*d^2*x^7 + 19845*a*c^5*d^2*x^5 + 315*(35*b*c^9*d^2*x^9 - 90*b*c^7*d^2*x^7 + 63*b*c^5*d^2*x^5)*arcsin(c*x) + (1225*b*c^8*d^2*x^8 - 2650*b*c^6*d^2*x^6 + 789*b*c^4*d^2*x^4 + 1052*b*c^ 2*d^2*x^2 + 2104*b*d^2)*sqrt(-c^2*x^2 + 1))/c^5
Time = 1.19 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.24 \[ \int x^4 \left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x)) \, dx=\begin {cases} \frac {a c^{4} d^{2} x^{9}}{9} - \frac {2 a c^{2} d^{2} x^{7}}{7} + \frac {a d^{2} x^{5}}{5} + \frac {b c^{4} d^{2} x^{9} \operatorname {asin}{\left (c x \right )}}{9} + \frac {b c^{3} d^{2} x^{8} \sqrt {- c^{2} x^{2} + 1}}{81} - \frac {2 b c^{2} d^{2} x^{7} \operatorname {asin}{\left (c x \right )}}{7} - \frac {106 b c d^{2} x^{6} \sqrt {- c^{2} x^{2} + 1}}{3969} + \frac {b d^{2} x^{5} \operatorname {asin}{\left (c x \right )}}{5} + \frac {263 b d^{2} x^{4} \sqrt {- c^{2} x^{2} + 1}}{33075 c} + \frac {1052 b d^{2} x^{2} \sqrt {- c^{2} x^{2} + 1}}{99225 c^{3}} + \frac {2104 b d^{2} \sqrt {- c^{2} x^{2} + 1}}{99225 c^{5}} & \text {for}\: c \neq 0 \\\frac {a d^{2} x^{5}}{5} & \text {otherwise} \end {cases} \] Input:
integrate(x**4*(-c**2*d*x**2+d)**2*(a+b*asin(c*x)),x)
Output:
Piecewise((a*c**4*d**2*x**9/9 - 2*a*c**2*d**2*x**7/7 + a*d**2*x**5/5 + b*c **4*d**2*x**9*asin(c*x)/9 + b*c**3*d**2*x**8*sqrt(-c**2*x**2 + 1)/81 - 2*b *c**2*d**2*x**7*asin(c*x)/7 - 106*b*c*d**2*x**6*sqrt(-c**2*x**2 + 1)/3969 + b*d**2*x**5*asin(c*x)/5 + 263*b*d**2*x**4*sqrt(-c**2*x**2 + 1)/(33075*c) + 1052*b*d**2*x**2*sqrt(-c**2*x**2 + 1)/(99225*c**3) + 2104*b*d**2*sqrt(- c**2*x**2 + 1)/(99225*c**5), Ne(c, 0)), (a*d**2*x**5/5, True))
Leaf count of result is larger than twice the leaf count of optimal. 328 vs. \(2 (160) = 320\).
Time = 0.12 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.76 \[ \int x^4 \left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x)) \, dx=\frac {1}{9} \, a c^{4} d^{2} x^{9} - \frac {2}{7} \, a c^{2} d^{2} x^{7} + \frac {1}{2835} \, {\left (315 \, x^{9} \arcsin \left (c x\right ) + {\left (\frac {35 \, \sqrt {-c^{2} x^{2} + 1} x^{8}}{c^{2}} + \frac {40 \, \sqrt {-c^{2} x^{2} + 1} x^{6}}{c^{4}} + \frac {48 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{6}} + \frac {64 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{8}} + \frac {128 \, \sqrt {-c^{2} x^{2} + 1}}{c^{10}}\right )} c\right )} b c^{4} d^{2} + \frac {1}{5} \, a d^{2} x^{5} - \frac {2}{245} \, {\left (35 \, x^{7} \arcsin \left (c x\right ) + {\left (\frac {5 \, \sqrt {-c^{2} x^{2} + 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {-c^{2} x^{2} + 1}}{c^{8}}\right )} c\right )} b c^{2} d^{2} + \frac {1}{75} \, {\left (15 \, x^{5} \arcsin \left (c x\right ) + {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} b d^{2} \] Input:
integrate(x^4*(-c^2*d*x^2+d)^2*(a+b*arcsin(c*x)),x, algorithm="maxima")
Output:
1/9*a*c^4*d^2*x^9 - 2/7*a*c^2*d^2*x^7 + 1/2835*(315*x^9*arcsin(c*x) + (35* sqrt(-c^2*x^2 + 1)*x^8/c^2 + 40*sqrt(-c^2*x^2 + 1)*x^6/c^4 + 48*sqrt(-c^2* x^2 + 1)*x^4/c^6 + 64*sqrt(-c^2*x^2 + 1)*x^2/c^8 + 128*sqrt(-c^2*x^2 + 1)/ c^10)*c)*b*c^4*d^2 + 1/5*a*d^2*x^5 - 2/245*(35*x^7*arcsin(c*x) + (5*sqrt(- c^2*x^2 + 1)*x^6/c^2 + 6*sqrt(-c^2*x^2 + 1)*x^4/c^4 + 8*sqrt(-c^2*x^2 + 1) *x^2/c^6 + 16*sqrt(-c^2*x^2 + 1)/c^8)*c)*b*c^2*d^2 + 1/75*(15*x^5*arcsin(c *x) + (3*sqrt(-c^2*x^2 + 1)*x^4/c^2 + 4*sqrt(-c^2*x^2 + 1)*x^2/c^4 + 8*sqr t(-c^2*x^2 + 1)/c^6)*c)*b*d^2
Time = 0.13 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.53 \[ \int x^4 \left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x)) \, dx=\frac {1}{9} \, a c^{4} d^{2} x^{9} - \frac {2}{7} \, a c^{2} d^{2} x^{7} + \frac {1}{5} \, a d^{2} x^{5} + \frac {{\left (c^{2} x^{2} - 1\right )}^{4} b d^{2} x \arcsin \left (c x\right )}{9 \, c^{4}} + \frac {10 \, {\left (c^{2} x^{2} - 1\right )}^{3} b d^{2} x \arcsin \left (c x\right )}{63 \, c^{4}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} b d^{2} x \arcsin \left (c x\right )}{105 \, c^{4}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{4} \sqrt {-c^{2} x^{2} + 1} b d^{2}}{81 \, c^{5}} - \frac {4 \, {\left (c^{2} x^{2} - 1\right )} b d^{2} x \arcsin \left (c x\right )}{315 \, c^{4}} + \frac {10 \, {\left (c^{2} x^{2} - 1\right )}^{3} \sqrt {-c^{2} x^{2} + 1} b d^{2}}{441 \, c^{5}} + \frac {8 \, b d^{2} x \arcsin \left (c x\right )}{315 \, c^{4}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b d^{2}}{525 \, c^{5}} + \frac {4 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b d^{2}}{945 \, c^{5}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1} b d^{2}}{315 \, c^{5}} \] Input:
integrate(x^4*(-c^2*d*x^2+d)^2*(a+b*arcsin(c*x)),x, algorithm="giac")
Output:
1/9*a*c^4*d^2*x^9 - 2/7*a*c^2*d^2*x^7 + 1/5*a*d^2*x^5 + 1/9*(c^2*x^2 - 1)^ 4*b*d^2*x*arcsin(c*x)/c^4 + 10/63*(c^2*x^2 - 1)^3*b*d^2*x*arcsin(c*x)/c^4 + 1/105*(c^2*x^2 - 1)^2*b*d^2*x*arcsin(c*x)/c^4 + 1/81*(c^2*x^2 - 1)^4*sqr t(-c^2*x^2 + 1)*b*d^2/c^5 - 4/315*(c^2*x^2 - 1)*b*d^2*x*arcsin(c*x)/c^4 + 10/441*(c^2*x^2 - 1)^3*sqrt(-c^2*x^2 + 1)*b*d^2/c^5 + 8/315*b*d^2*x*arcsin (c*x)/c^4 + 1/525*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b*d^2/c^5 + 4/945*(-c ^2*x^2 + 1)^(3/2)*b*d^2/c^5 + 8/315*sqrt(-c^2*x^2 + 1)*b*d^2/c^5
Timed out. \[ \int x^4 \left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x)) \, dx=\int x^4\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^2 \,d x \] Input:
int(x^4*(a + b*asin(c*x))*(d - c^2*d*x^2)^2,x)
Output:
int(x^4*(a + b*asin(c*x))*(d - c^2*d*x^2)^2, x)
Time = 0.22 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.91 \[ \int x^4 \left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x)) \, dx=\frac {d^{2} \left (11025 \mathit {asin} \left (c x \right ) b \,c^{9} x^{9}-28350 \mathit {asin} \left (c x \right ) b \,c^{7} x^{7}+19845 \mathit {asin} \left (c x \right ) b \,c^{5} x^{5}+1225 \sqrt {-c^{2} x^{2}+1}\, b \,c^{8} x^{8}-2650 \sqrt {-c^{2} x^{2}+1}\, b \,c^{6} x^{6}+789 \sqrt {-c^{2} x^{2}+1}\, b \,c^{4} x^{4}+1052 \sqrt {-c^{2} x^{2}+1}\, b \,c^{2} x^{2}+2104 \sqrt {-c^{2} x^{2}+1}\, b +11025 a \,c^{9} x^{9}-28350 a \,c^{7} x^{7}+19845 a \,c^{5} x^{5}\right )}{99225 c^{5}} \] Input:
int(x^4*(-c^2*d*x^2+d)^2*(a+b*asin(c*x)),x)
Output:
(d**2*(11025*asin(c*x)*b*c**9*x**9 - 28350*asin(c*x)*b*c**7*x**7 + 19845*a sin(c*x)*b*c**5*x**5 + 1225*sqrt( - c**2*x**2 + 1)*b*c**8*x**8 - 2650*sqrt ( - c**2*x**2 + 1)*b*c**6*x**6 + 789*sqrt( - c**2*x**2 + 1)*b*c**4*x**4 + 1052*sqrt( - c**2*x**2 + 1)*b*c**2*x**2 + 2104*sqrt( - c**2*x**2 + 1)*b + 11025*a*c**9*x**9 - 28350*a*c**7*x**7 + 19845*a*c**5*x**5))/(99225*c**5)