Integrand size = 25, antiderivative size = 232 \[ \int x^4 \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x)) \, dx=\frac {16 b d^3 \sqrt {1-c^2 x^2}}{1155 c^5}+\frac {8 b d^3 \left (1-c^2 x^2\right )^{3/2}}{3465 c^5}+\frac {2 b d^3 \left (1-c^2 x^2\right )^{5/2}}{1925 c^5}+\frac {b d^3 \left (1-c^2 x^2\right )^{7/2}}{1617 c^5}-\frac {4 b d^3 \left (1-c^2 x^2\right )^{9/2}}{297 c^5}+\frac {b d^3 \left (1-c^2 x^2\right )^{11/2}}{121 c^5}+\frac {1}{5} d^3 x^5 (a+b \arcsin (c x))-\frac {3}{7} c^2 d^3 x^7 (a+b \arcsin (c x))+\frac {1}{3} c^4 d^3 x^9 (a+b \arcsin (c x))-\frac {1}{11} c^6 d^3 x^{11} (a+b \arcsin (c x)) \] Output:
16/1155*b*d^3*(-c^2*x^2+1)^(1/2)/c^5+8/3465*b*d^3*(-c^2*x^2+1)^(3/2)/c^5+2 /1925*b*d^3*(-c^2*x^2+1)^(5/2)/c^5+1/1617*b*d^3*(-c^2*x^2+1)^(7/2)/c^5-4/2 97*b*d^3*(-c^2*x^2+1)^(9/2)/c^5+1/121*b*d^3*(-c^2*x^2+1)^(11/2)/c^5+1/5*d^ 3*x^5*(a+b*arcsin(c*x))-3/7*c^2*d^3*x^7*(a+b*arcsin(c*x))+1/3*c^4*d^3*x^9* (a+b*arcsin(c*x))-1/11*c^6*d^3*x^11*(a+b*arcsin(c*x))
Time = 0.12 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.62 \[ \int x^4 \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x)) \, dx=\frac {d^3 \left (-3465 a c^5 x^5 \left (-231+495 c^2 x^2-385 c^4 x^4+105 c^6 x^6\right )+b \sqrt {1-c^2 x^2} \left (50488+25244 c^2 x^2+18933 c^4 x^4-117625 c^6 x^6+111475 c^8 x^8-33075 c^{10} x^{10}\right )-3465 b c^5 x^5 \left (-231+495 c^2 x^2-385 c^4 x^4+105 c^6 x^6\right ) \arcsin (c x)\right )}{4002075 c^5} \] Input:
Integrate[x^4*(d - c^2*d*x^2)^3*(a + b*ArcSin[c*x]),x]
Output:
(d^3*(-3465*a*c^5*x^5*(-231 + 495*c^2*x^2 - 385*c^4*x^4 + 105*c^6*x^6) + b *Sqrt[1 - c^2*x^2]*(50488 + 25244*c^2*x^2 + 18933*c^4*x^4 - 117625*c^6*x^6 + 111475*c^8*x^8 - 33075*c^10*x^10) - 3465*b*c^5*x^5*(-231 + 495*c^2*x^2 - 385*c^4*x^4 + 105*c^6*x^6)*ArcSin[c*x]))/(4002075*c^5)
Time = 0.64 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.93, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5192, 27, 2331, 2123, 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.
| \(\displaystyle \int x^4 \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x)) \, dx\) |
| \(\Big \downarrow \) 5192 |
| \(\displaystyle -b c \int \frac {d^3 x^5 \left (-105 c^6 x^6+385 c^4 x^4-495 c^2 x^2+231\right )}{1155 \sqrt {1-c^2 x^2}}dx-\frac {1}{11} c^6 d^3 x^{11} (a+b \arcsin (c x))+\frac {1}{3} c^4 d^3 x^9 (a+b \arcsin (c x))-\frac {3}{7} c^2 d^3 x^7 (a+b \arcsin (c x))+\frac {1}{5} d^3 x^5 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {b c d^3 \int \frac {x^5 \left (-105 c^6 x^6+385 c^4 x^4-495 c^2 x^2+231\right )}{\sqrt {1-c^2 x^2}}dx}{1155}-\frac {1}{11} c^6 d^3 x^{11} (a+b \arcsin (c x))+\frac {1}{3} c^4 d^3 x^9 (a+b \arcsin (c x))-\frac {3}{7} c^2 d^3 x^7 (a+b \arcsin (c x))+\frac {1}{5} d^3 x^5 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 2331 |
| \(\displaystyle -\frac {b c d^3 \int \frac {x^4 \left (-105 c^6 x^6+385 c^4 x^4-495 c^2 x^2+231\right )}{\sqrt {1-c^2 x^2}}dx^2}{2310}-\frac {1}{11} c^6 d^3 x^{11} (a+b \arcsin (c x))+\frac {1}{3} c^4 d^3 x^9 (a+b \arcsin (c x))-\frac {3}{7} c^2 d^3 x^7 (a+b \arcsin (c x))+\frac {1}{5} d^3 x^5 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 2123 |
| \(\displaystyle -\frac {b c d^3 \int \left (\frac {105 \left (1-c^2 x^2\right )^{9/2}}{c^4}-\frac {140 \left (1-c^2 x^2\right )^{7/2}}{c^4}+\frac {5 \left (1-c^2 x^2\right )^{5/2}}{c^4}+\frac {6 \left (1-c^2 x^2\right )^{3/2}}{c^4}+\frac {8 \sqrt {1-c^2 x^2}}{c^4}+\frac {16}{c^4 \sqrt {1-c^2 x^2}}\right )dx^2}{2310}-\frac {1}{11} c^6 d^3 x^{11} (a+b \arcsin (c x))+\frac {1}{3} c^4 d^3 x^9 (a+b \arcsin (c x))-\frac {3}{7} c^2 d^3 x^7 (a+b \arcsin (c x))+\frac {1}{5} d^3 x^5 (a+b \arcsin (c x))\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {1}{11} c^6 d^3 x^{11} (a+b \arcsin (c x))+\frac {1}{3} c^4 d^3 x^9 (a+b \arcsin (c x))-\frac {3}{7} c^2 d^3 x^7 (a+b \arcsin (c x))+\frac {1}{5} d^3 x^5 (a+b \arcsin (c x))-\frac {b c d^3 \left (-\frac {210 \left (1-c^2 x^2\right )^{11/2}}{11 c^6}+\frac {280 \left (1-c^2 x^2\right )^{9/2}}{9 c^6}-\frac {10 \left (1-c^2 x^2\right )^{7/2}}{7 c^6}-\frac {12 \left (1-c^2 x^2\right )^{5/2}}{5 c^6}-\frac {16 \left (1-c^2 x^2\right )^{3/2}}{3 c^6}-\frac {32 \sqrt {1-c^2 x^2}}{c^6}\right )}{2310}\) |
Input:
Int[x^4*(d - c^2*d*x^2)^3*(a + b*ArcSin[c*x]),x]
Output:
-1/2310*(b*c*d^3*((-32*Sqrt[1 - c^2*x^2])/c^6 - (16*(1 - c^2*x^2)^(3/2))/( 3*c^6) - (12*(1 - c^2*x^2)^(5/2))/(5*c^6) - (10*(1 - c^2*x^2)^(7/2))/(7*c^ 6) + (280*(1 - c^2*x^2)^(9/2))/(9*c^6) - (210*(1 - c^2*x^2)^(11/2))/(11*c^ 6))) + (d^3*x^5*(a + b*ArcSin[c*x]))/5 - (3*c^2*d^3*x^7*(a + b*ArcSin[c*x] ))/7 + (c^4*d^3*x^9*(a + b*ArcSin[c*x]))/3 - (c^6*d^3*x^11*(a + b*ArcSin[c *x]))/11
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c , d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2])
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[((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 = 0.19 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.91
| method | result | size |
| parts | \(-d^{3} a \left (\frac {1}{11} c^{6} x^{11}-\frac {1}{3} c^{4} x^{9}+\frac {3}{7} c^{2} x^{7}-\frac {1}{5} x^{5}\right )-\frac {d^{3} b \left (\frac {\arcsin \left (c x \right ) c^{11} x^{11}}{11}-\frac {\arcsin \left (c x \right ) c^{9} x^{9}}{3}+\frac {3 \arcsin \left (c x \right ) c^{7} x^{7}}{7}-\frac {c^{5} x^{5} \arcsin \left (c x \right )}{5}-\frac {6311 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{1334025}-\frac {25244 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{4002075}-\frac {50488 \sqrt {-c^{2} x^{2}+1}}{4002075}+\frac {4705 c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{160083}-\frac {91 c^{8} x^{8} \sqrt {-c^{2} x^{2}+1}}{3267}+\frac {c^{10} x^{10} \sqrt {-c^{2} x^{2}+1}}{121}\right )}{c^{5}}\) | \(210\) |
| derivativedivides | \(\frac {-d^{3} a \left (\frac {1}{11} c^{11} x^{11}-\frac {1}{3} c^{9} x^{9}+\frac {3}{7} c^{7} x^{7}-\frac {1}{5} c^{5} x^{5}\right )-d^{3} b \left (\frac {\arcsin \left (c x \right ) c^{11} x^{11}}{11}-\frac {\arcsin \left (c x \right ) c^{9} x^{9}}{3}+\frac {3 \arcsin \left (c x \right ) c^{7} x^{7}}{7}-\frac {c^{5} x^{5} \arcsin \left (c x \right )}{5}-\frac {6311 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{1334025}-\frac {25244 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{4002075}-\frac {50488 \sqrt {-c^{2} x^{2}+1}}{4002075}+\frac {4705 c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{160083}-\frac {91 c^{8} x^{8} \sqrt {-c^{2} x^{2}+1}}{3267}+\frac {c^{10} x^{10} \sqrt {-c^{2} x^{2}+1}}{121}\right )}{c^{5}}\) | \(214\) |
| default | \(\frac {-d^{3} a \left (\frac {1}{11} c^{11} x^{11}-\frac {1}{3} c^{9} x^{9}+\frac {3}{7} c^{7} x^{7}-\frac {1}{5} c^{5} x^{5}\right )-d^{3} b \left (\frac {\arcsin \left (c x \right ) c^{11} x^{11}}{11}-\frac {\arcsin \left (c x \right ) c^{9} x^{9}}{3}+\frac {3 \arcsin \left (c x \right ) c^{7} x^{7}}{7}-\frac {c^{5} x^{5} \arcsin \left (c x \right )}{5}-\frac {6311 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{1334025}-\frac {25244 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{4002075}-\frac {50488 \sqrt {-c^{2} x^{2}+1}}{4002075}+\frac {4705 c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{160083}-\frac {91 c^{8} x^{8} \sqrt {-c^{2} x^{2}+1}}{3267}+\frac {c^{10} x^{10} \sqrt {-c^{2} x^{2}+1}}{121}\right )}{c^{5}}\) | \(214\) |
| orering | \(\frac {\left (694575 c^{12} x^{12}-2581075 c^{10} x^{10}+3337325 c^{8} x^{8}-1460245 c^{6} x^{6}-176708 c^{4} x^{4}-403904 c^{2} x^{2}+201952\right ) \left (-c^{2} d \,x^{2}+d \right )^{3} \left (a +b \arcsin \left (c x \right )\right )}{4002075 c^{6} \left (c x -1\right )^{2} x \left (c x +1\right )^{2} \left (c^{2} x^{2}-1\right )}-\frac {\left (33075 c^{10} x^{10}-111475 c^{8} x^{8}+117625 c^{6} x^{6}-18933 c^{4} x^{4}-25244 c^{2} x^{2}-50488\right ) \left (4 x^{3} \left (-c^{2} d \,x^{2}+d \right )^{3} \left (a +b \arcsin \left (c x \right )\right )-6 x^{5} \left (-c^{2} d \,x^{2}+d \right )^{2} \left (a +b \arcsin \left (c x \right )\right ) c^{2} d +\frac {x^{4} \left (-c^{2} d \,x^{2}+d \right )^{3} b c}{\sqrt {-c^{2} x^{2}+1}}\right )}{4002075 c^{6} \left (c x -1\right )^{2} x^{4} \left (c x +1\right )^{2}}\) | \(258\) |
Input:
int(x^4*(-c^2*d*x^2+d)^3*(a+b*arcsin(c*x)),x,method=_RETURNVERBOSE)
Output:
-d^3*a*(1/11*c^6*x^11-1/3*c^4*x^9+3/7*c^2*x^7-1/5*x^5)-d^3*b/c^5*(1/11*arc sin(c*x)*c^11*x^11-1/3*arcsin(c*x)*c^9*x^9+3/7*arcsin(c*x)*c^7*x^7-1/5*c^5 *x^5*arcsin(c*x)-6311/1334025*c^4*x^4*(-c^2*x^2+1)^(1/2)-25244/4002075*c^2 *x^2*(-c^2*x^2+1)^(1/2)-50488/4002075*(-c^2*x^2+1)^(1/2)+4705/160083*c^6*x ^6*(-c^2*x^2+1)^(1/2)-91/3267*c^8*x^8*(-c^2*x^2+1)^(1/2)+1/121*c^10*x^10*( -c^2*x^2+1)^(1/2))
Time = 0.11 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.81 \[ \int x^4 \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x)) \, dx=-\frac {363825 \, a c^{11} d^{3} x^{11} - 1334025 \, a c^{9} d^{3} x^{9} + 1715175 \, a c^{7} d^{3} x^{7} - 800415 \, a c^{5} d^{3} x^{5} + 3465 \, {\left (105 \, b c^{11} d^{3} x^{11} - 385 \, b c^{9} d^{3} x^{9} + 495 \, b c^{7} d^{3} x^{7} - 231 \, b c^{5} d^{3} x^{5}\right )} \arcsin \left (c x\right ) + {\left (33075 \, b c^{10} d^{3} x^{10} - 111475 \, b c^{8} d^{3} x^{8} + 117625 \, b c^{6} d^{3} x^{6} - 18933 \, b c^{4} d^{3} x^{4} - 25244 \, b c^{2} d^{3} x^{2} - 50488 \, b d^{3}\right )} \sqrt {-c^{2} x^{2} + 1}}{4002075 \, c^{5}} \] Input:
integrate(x^4*(-c^2*d*x^2+d)^3*(a+b*arcsin(c*x)),x, algorithm="fricas")
Output:
-1/4002075*(363825*a*c^11*d^3*x^11 - 1334025*a*c^9*d^3*x^9 + 1715175*a*c^7 *d^3*x^7 - 800415*a*c^5*d^3*x^5 + 3465*(105*b*c^11*d^3*x^11 - 385*b*c^9*d^ 3*x^9 + 495*b*c^7*d^3*x^7 - 231*b*c^5*d^3*x^5)*arcsin(c*x) + (33075*b*c^10 *d^3*x^10 - 111475*b*c^8*d^3*x^8 + 117625*b*c^6*d^3*x^6 - 18933*b*c^4*d^3* x^4 - 25244*b*c^2*d^3*x^2 - 50488*b*d^3)*sqrt(-c^2*x^2 + 1))/c^5
Time = 2.23 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.25 \[ \int x^4 \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x)) \, dx=\begin {cases} - \frac {a c^{6} d^{3} x^{11}}{11} + \frac {a c^{4} d^{3} x^{9}}{3} - \frac {3 a c^{2} d^{3} x^{7}}{7} + \frac {a d^{3} x^{5}}{5} - \frac {b c^{6} d^{3} x^{11} \operatorname {asin}{\left (c x \right )}}{11} - \frac {b c^{5} d^{3} x^{10} \sqrt {- c^{2} x^{2} + 1}}{121} + \frac {b c^{4} d^{3} x^{9} \operatorname {asin}{\left (c x \right )}}{3} + \frac {91 b c^{3} d^{3} x^{8} \sqrt {- c^{2} x^{2} + 1}}{3267} - \frac {3 b c^{2} d^{3} x^{7} \operatorname {asin}{\left (c x \right )}}{7} - \frac {4705 b c d^{3} x^{6} \sqrt {- c^{2} x^{2} + 1}}{160083} + \frac {b d^{3} x^{5} \operatorname {asin}{\left (c x \right )}}{5} + \frac {6311 b d^{3} x^{4} \sqrt {- c^{2} x^{2} + 1}}{1334025 c} + \frac {25244 b d^{3} x^{2} \sqrt {- c^{2} x^{2} + 1}}{4002075 c^{3}} + \frac {50488 b d^{3} \sqrt {- c^{2} x^{2} + 1}}{4002075 c^{5}} & \text {for}\: c \neq 0 \\\frac {a d^{3} x^{5}}{5} & \text {otherwise} \end {cases} \] Input:
integrate(x**4*(-c**2*d*x**2+d)**3*(a+b*asin(c*x)),x)
Output:
Piecewise((-a*c**6*d**3*x**11/11 + a*c**4*d**3*x**9/3 - 3*a*c**2*d**3*x**7 /7 + a*d**3*x**5/5 - b*c**6*d**3*x**11*asin(c*x)/11 - b*c**5*d**3*x**10*sq rt(-c**2*x**2 + 1)/121 + b*c**4*d**3*x**9*asin(c*x)/3 + 91*b*c**3*d**3*x** 8*sqrt(-c**2*x**2 + 1)/3267 - 3*b*c**2*d**3*x**7*asin(c*x)/7 - 4705*b*c*d* *3*x**6*sqrt(-c**2*x**2 + 1)/160083 + b*d**3*x**5*asin(c*x)/5 + 6311*b*d** 3*x**4*sqrt(-c**2*x**2 + 1)/(1334025*c) + 25244*b*d**3*x**2*sqrt(-c**2*x** 2 + 1)/(4002075*c**3) + 50488*b*d**3*sqrt(-c**2*x**2 + 1)/(4002075*c**5), Ne(c, 0)), (a*d**3*x**5/5, True))
Leaf count of result is larger than twice the leaf count of optimal. 479 vs. \(2 (200) = 400\).
Time = 0.12 (sec) , antiderivative size = 479, normalized size of antiderivative = 2.06 \[ \int x^4 \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x)) \, dx=-\frac {1}{11} \, a c^{6} d^{3} x^{11} + \frac {1}{3} \, a c^{4} d^{3} x^{9} - \frac {3}{7} \, a c^{2} d^{3} x^{7} - \frac {1}{7623} \, {\left (693 \, x^{11} \arcsin \left (c x\right ) + {\left (\frac {63 \, \sqrt {-c^{2} x^{2} + 1} x^{10}}{c^{2}} + \frac {70 \, \sqrt {-c^{2} x^{2} + 1} x^{8}}{c^{4}} + \frac {80 \, \sqrt {-c^{2} x^{2} + 1} x^{6}}{c^{6}} + \frac {96 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{8}} + \frac {128 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{10}} + \frac {256 \, \sqrt {-c^{2} x^{2} + 1}}{c^{12}}\right )} c\right )} b c^{6} d^{3} + \frac {1}{945} \, {\left (315 \, x^{9} \arcsin \left (c x\right ) + {\left (\frac {35 \, \sqrt {-c^{2} x^{2} + 1} x^{8}}{c^{2}} + \frac {40 \, \sqrt {-c^{2} x^{2} + 1} x^{6}}{c^{4}} + \frac {48 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{6}} + \frac {64 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{8}} + \frac {128 \, \sqrt {-c^{2} x^{2} + 1}}{c^{10}}\right )} c\right )} b c^{4} d^{3} + \frac {1}{5} \, a d^{3} x^{5} - \frac {3}{245} \, {\left (35 \, x^{7} \arcsin \left (c x\right ) + {\left (\frac {5 \, \sqrt {-c^{2} x^{2} + 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {-c^{2} x^{2} + 1}}{c^{8}}\right )} c\right )} b c^{2} d^{3} + \frac {1}{75} \, {\left (15 \, x^{5} \arcsin \left (c x\right ) + {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} b d^{3} \] Input:
integrate(x^4*(-c^2*d*x^2+d)^3*(a+b*arcsin(c*x)),x, algorithm="maxima")
Output:
-1/11*a*c^6*d^3*x^11 + 1/3*a*c^4*d^3*x^9 - 3/7*a*c^2*d^3*x^7 - 1/7623*(693 *x^11*arcsin(c*x) + (63*sqrt(-c^2*x^2 + 1)*x^10/c^2 + 70*sqrt(-c^2*x^2 + 1 )*x^8/c^4 + 80*sqrt(-c^2*x^2 + 1)*x^6/c^6 + 96*sqrt(-c^2*x^2 + 1)*x^4/c^8 + 128*sqrt(-c^2*x^2 + 1)*x^2/c^10 + 256*sqrt(-c^2*x^2 + 1)/c^12)*c)*b*c^6* d^3 + 1/945*(315*x^9*arcsin(c*x) + (35*sqrt(-c^2*x^2 + 1)*x^8/c^2 + 40*sqr t(-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^3 + 1/5*a*d^3*x^5 - 3/245*(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^3 + 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*sqrt(-c^2*x^2 + 1)/c^6)*c)*b*d^3
Time = 0.14 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.52 \[ \int x^4 \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x)) \, dx=-\frac {1}{11} \, a c^{6} d^{3} x^{11} + \frac {1}{3} \, a c^{4} d^{3} x^{9} - \frac {3}{7} \, a c^{2} d^{3} x^{7} + \frac {1}{5} \, a d^{3} x^{5} - \frac {{\left (c^{2} x^{2} - 1\right )}^{5} b d^{3} x \arcsin \left (c x\right )}{11 \, c^{4}} - \frac {4 \, {\left (c^{2} x^{2} - 1\right )}^{4} b d^{3} x \arcsin \left (c x\right )}{33 \, c^{4}} - \frac {{\left (c^{2} x^{2} - 1\right )}^{3} b d^{3} x \arcsin \left (c x\right )}{231 \, c^{4}} - \frac {{\left (c^{2} x^{2} - 1\right )}^{5} \sqrt {-c^{2} x^{2} + 1} b d^{3}}{121 \, c^{5}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{2} b d^{3} x \arcsin \left (c x\right )}{385 \, c^{4}} - \frac {4 \, {\left (c^{2} x^{2} - 1\right )}^{4} \sqrt {-c^{2} x^{2} + 1} b d^{3}}{297 \, c^{5}} - \frac {8 \, {\left (c^{2} x^{2} - 1\right )} b d^{3} x \arcsin \left (c x\right )}{1155 \, c^{4}} - \frac {{\left (c^{2} x^{2} - 1\right )}^{3} \sqrt {-c^{2} x^{2} + 1} b d^{3}}{1617 \, c^{5}} + \frac {16 \, b d^{3} x \arcsin \left (c x\right )}{1155 \, c^{4}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b d^{3}}{1925 \, c^{5}} + \frac {8 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b d^{3}}{3465 \, c^{5}} + \frac {16 \, \sqrt {-c^{2} x^{2} + 1} b d^{3}}{1155 \, c^{5}} \] Input:
integrate(x^4*(-c^2*d*x^2+d)^3*(a+b*arcsin(c*x)),x, algorithm="giac")
Output:
-1/11*a*c^6*d^3*x^11 + 1/3*a*c^4*d^3*x^9 - 3/7*a*c^2*d^3*x^7 + 1/5*a*d^3*x ^5 - 1/11*(c^2*x^2 - 1)^5*b*d^3*x*arcsin(c*x)/c^4 - 4/33*(c^2*x^2 - 1)^4*b *d^3*x*arcsin(c*x)/c^4 - 1/231*(c^2*x^2 - 1)^3*b*d^3*x*arcsin(c*x)/c^4 - 1 /121*(c^2*x^2 - 1)^5*sqrt(-c^2*x^2 + 1)*b*d^3/c^5 + 2/385*(c^2*x^2 - 1)^2* b*d^3*x*arcsin(c*x)/c^4 - 4/297*(c^2*x^2 - 1)^4*sqrt(-c^2*x^2 + 1)*b*d^3/c ^5 - 8/1155*(c^2*x^2 - 1)*b*d^3*x*arcsin(c*x)/c^4 - 1/1617*(c^2*x^2 - 1)^3 *sqrt(-c^2*x^2 + 1)*b*d^3/c^5 + 16/1155*b*d^3*x*arcsin(c*x)/c^4 + 2/1925*( c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b*d^3/c^5 + 8/3465*(-c^2*x^2 + 1)^(3/2)* b*d^3/c^5 + 16/1155*sqrt(-c^2*x^2 + 1)*b*d^3/c^5
Timed out. \[ \int x^4 \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x)) \, dx=\int x^4\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^3 \,d x \] Input:
int(x^4*(a + b*asin(c*x))*(d - c^2*d*x^2)^3,x)
Output:
int(x^4*(a + b*asin(c*x))*(d - c^2*d*x^2)^3, x)
Time = 0.20 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.91 \[ \int x^4 \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x)) \, dx=\frac {d^{3} \left (-363825 \mathit {asin} \left (c x \right ) b \,c^{11} x^{11}+1334025 \mathit {asin} \left (c x \right ) b \,c^{9} x^{9}-1715175 \mathit {asin} \left (c x \right ) b \,c^{7} x^{7}+800415 \mathit {asin} \left (c x \right ) b \,c^{5} x^{5}-33075 \sqrt {-c^{2} x^{2}+1}\, b \,c^{10} x^{10}+111475 \sqrt {-c^{2} x^{2}+1}\, b \,c^{8} x^{8}-117625 \sqrt {-c^{2} x^{2}+1}\, b \,c^{6} x^{6}+18933 \sqrt {-c^{2} x^{2}+1}\, b \,c^{4} x^{4}+25244 \sqrt {-c^{2} x^{2}+1}\, b \,c^{2} x^{2}+50488 \sqrt {-c^{2} x^{2}+1}\, b -363825 a \,c^{11} x^{11}+1334025 a \,c^{9} x^{9}-1715175 a \,c^{7} x^{7}+800415 a \,c^{5} x^{5}\right )}{4002075 c^{5}} \] Input:
int(x^4*(-c^2*d*x^2+d)^3*(a+b*asin(c*x)),x)
Output:
(d**3*( - 363825*asin(c*x)*b*c**11*x**11 + 1334025*asin(c*x)*b*c**9*x**9 - 1715175*asin(c*x)*b*c**7*x**7 + 800415*asin(c*x)*b*c**5*x**5 - 33075*sqrt ( - c**2*x**2 + 1)*b*c**10*x**10 + 111475*sqrt( - c**2*x**2 + 1)*b*c**8*x* *8 - 117625*sqrt( - c**2*x**2 + 1)*b*c**6*x**6 + 18933*sqrt( - c**2*x**2 + 1)*b*c**4*x**4 + 25244*sqrt( - c**2*x**2 + 1)*b*c**2*x**2 + 50488*sqrt( - c**2*x**2 + 1)*b - 363825*a*c**11*x**11 + 1334025*a*c**9*x**9 - 1715175*a *c**7*x**7 + 800415*a*c**5*x**5))/(4002075*c**5)