Integrand size = 21, antiderivative size = 241 \[ \int x^4 \left (d+e x^2\right )^2 (a+b \arccos (c x)) \, dx=\frac {b \left (63 c^4 d^2+90 c^2 d e+35 e^2\right ) \sqrt {1-c^2 x^2}}{315 c^9}-\frac {2 b \left (63 c^4 d^2+135 c^2 d e+70 e^2\right ) \left (1-c^2 x^2\right )^{3/2}}{945 c^9}+\frac {b \left (21 c^4 d^2+90 c^2 d e+70 e^2\right ) \left (1-c^2 x^2\right )^{5/2}}{525 c^9}-\frac {2 b e \left (9 c^2 d+14 e\right ) \left (1-c^2 x^2\right )^{7/2}}{441 c^9}+\frac {b e^2 \left (1-c^2 x^2\right )^{9/2}}{81 c^9}+\frac {1}{5} d^2 x^5 (a+b \arccos (c x))+\frac {2}{7} d e x^7 (a+b \arccos (c x))+\frac {1}{9} e^2 x^9 (a+b \arccos (c x)) \] Output:
1/315*b*(63*c^4*d^2+90*c^2*d*e+35*e^2)*(-c^2*x^2+1)^(1/2)/c^9-2/945*b*(63* c^4*d^2+135*c^2*d*e+70*e^2)*(-c^2*x^2+1)^(3/2)/c^9+1/525*b*(21*c^4*d^2+90* c^2*d*e+70*e^2)*(-c^2*x^2+1)^(5/2)/c^9-2/441*b*e*(9*c^2*d+14*e)*(-c^2*x^2+ 1)^(7/2)/c^9+1/81*b*e^2*(-c^2*x^2+1)^(9/2)/c^9+1/5*d^2*x^5*(a+b*arccos(c*x ))+2/7*d*e*x^7*(a+b*arccos(c*x))+1/9*e^2*x^9*(a+b*arccos(c*x))
Time = 0.14 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.78 \[ \int x^4 \left (d+e x^2\right )^2 (a+b \arccos (c x)) \, dx=\frac {315 a x^5 \left (63 d^2+90 d e x^2+35 e^2 x^4\right )-\frac {b \sqrt {1-c^2 x^2} \left (4480 e^2+160 c^2 e \left (81 d+14 e x^2\right )+24 c^4 \left (441 d^2+270 d e x^2+70 e^2 x^4\right )+4 c^6 \left (1323 d^2 x^2+1215 d e x^4+350 e^2 x^6\right )+c^8 \left (3969 d^2 x^4+4050 d e x^6+1225 e^2 x^8\right )\right )}{c^9}+315 b x^5 \left (63 d^2+90 d e x^2+35 e^2 x^4\right ) \arccos (c x)}{99225} \] Input:
Integrate[x^4*(d + e*x^2)^2*(a + b*ArcCos[c*x]),x]
Output:
(315*a*x^5*(63*d^2 + 90*d*e*x^2 + 35*e^2*x^4) - (b*Sqrt[1 - c^2*x^2]*(4480 *e^2 + 160*c^2*e*(81*d + 14*e*x^2) + 24*c^4*(441*d^2 + 270*d*e*x^2 + 70*e^ 2*x^4) + 4*c^6*(1323*d^2*x^2 + 1215*d*e*x^4 + 350*e^2*x^6) + c^8*(3969*d^2 *x^4 + 4050*d*e*x^6 + 1225*e^2*x^8)))/c^9 + 315*b*x^5*(63*d^2 + 90*d*e*x^2 + 35*e^2*x^4)*ArcCos[c*x])/99225
Time = 0.56 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {5231, 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.
| \(\displaystyle \int x^4 \left (d+e x^2\right )^2 (a+b \arccos (c x)) \, dx\) |
| \(\Big \downarrow \) 5231 |
| \(\displaystyle b c \int \frac {x^5 \left (35 e^2 x^4+90 d e x^2+63 d^2\right )}{315 \sqrt {1-c^2 x^2}}dx+\frac {1}{5} d^2 x^5 (a+b \arccos (c x))+\frac {2}{7} d e x^7 (a+b \arccos (c x))+\frac {1}{9} e^2 x^9 (a+b \arccos (c x))\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{315} b c \int \frac {x^5 \left (35 e^2 x^4+90 d e x^2+63 d^2\right )}{\sqrt {1-c^2 x^2}}dx+\frac {1}{5} d^2 x^5 (a+b \arccos (c x))+\frac {2}{7} d e x^7 (a+b \arccos (c x))+\frac {1}{9} e^2 x^9 (a+b \arccos (c x))\) |
| \(\Big \downarrow \) 1578 |
| \(\displaystyle \frac {1}{630} b c \int \frac {x^4 \left (35 e^2 x^4+90 d e x^2+63 d^2\right )}{\sqrt {1-c^2 x^2}}dx^2+\frac {1}{5} d^2 x^5 (a+b \arccos (c x))+\frac {2}{7} d e x^7 (a+b \arccos (c x))+\frac {1}{9} e^2 x^9 (a+b \arccos (c x))\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \frac {1}{630} b c \int \left (\frac {35 e^2 \left (1-c^2 x^2\right )^{7/2}}{c^8}-\frac {10 e \left (9 d c^2+14 e\right ) \left (1-c^2 x^2\right )^{5/2}}{c^8}+\frac {3 \left (21 d^2 c^4+90 d e c^2+70 e^2\right ) \left (1-c^2 x^2\right )^{3/2}}{c^8}-\frac {2 \left (63 d^2 c^4+135 d e c^2+70 e^2\right ) \sqrt {1-c^2 x^2}}{c^8}+\frac {63 d^2 c^4+90 d e c^2+35 e^2}{c^8 \sqrt {1-c^2 x^2}}\right )dx^2+\frac {1}{5} d^2 x^5 (a+b \arccos (c x))+\frac {2}{7} d e x^7 (a+b \arccos (c x))+\frac {1}{9} e^2 x^9 (a+b \arccos (c x))\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{5} d^2 x^5 (a+b \arccos (c x))+\frac {2}{7} d e x^7 (a+b \arccos (c x))+\frac {1}{9} e^2 x^9 (a+b \arccos (c x))+\frac {1}{630} b c \left (\frac {20 e \left (1-c^2 x^2\right )^{7/2} \left (9 c^2 d+14 e\right )}{7 c^{10}}-\frac {70 e^2 \left (1-c^2 x^2\right )^{9/2}}{9 c^{10}}-\frac {6 \left (1-c^2 x^2\right )^{5/2} \left (21 c^4 d^2+90 c^2 d e+70 e^2\right )}{5 c^{10}}+\frac {4 \left (1-c^2 x^2\right )^{3/2} \left (63 c^4 d^2+135 c^2 d e+70 e^2\right )}{3 c^{10}}-\frac {2 \sqrt {1-c^2 x^2} \left (63 c^4 d^2+90 c^2 d e+35 e^2\right )}{c^{10}}\right )\) |
Input:
Int[x^4*(d + e*x^2)^2*(a + b*ArcCos[c*x]),x]
Output:
(b*c*((-2*(63*c^4*d^2 + 90*c^2*d*e + 35*e^2)*Sqrt[1 - c^2*x^2])/c^10 + (4* (63*c^4*d^2 + 135*c^2*d*e + 70*e^2)*(1 - c^2*x^2)^(3/2))/(3*c^10) - (6*(21 *c^4*d^2 + 90*c^2*d*e + 70*e^2)*(1 - c^2*x^2)^(5/2))/(5*c^10) + (20*e*(9*c ^2*d + 14*e)*(1 - c^2*x^2)^(7/2))/(7*c^10) - (70*e^2*(1 - c^2*x^2)^(9/2))/ (9*c^10)))/630 + (d^2*x^5*(a + b*ArcCos[c*x]))/5 + (2*d*e*x^7*(a + b*ArcCo s[c*x]))/7 + (e^2*x^9*(a + b*ArcCos[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_.) + ArcCos[(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*ArcCos[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] && NeQ[c^2*d + e, 0] && IntegerQ[p] && (GtQ[p, 0] || (IGtQ[(m - 1)/2, 0] && LeQ[m + p, 0]))
Time = 0.41 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.37
| method | result | size |
| parts | \(a \left (\frac {1}{9} e^{2} x^{9}+\frac {2}{7} d e \,x^{7}+\frac {1}{5} d^{2} x^{5}\right )+\frac {b \left (\frac {c^{5} \arccos \left (c x \right ) e^{2} x^{9}}{9}+\frac {2 c^{5} \arccos \left (c x \right ) d e \,x^{7}}{7}+\frac {\arccos \left (c x \right ) c^{5} x^{5} d^{2}}{5}+\frac {35 e^{2} \left (-\frac {c^{8} x^{8} \sqrt {-c^{2} x^{2}+1}}{9}-\frac {8 c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{63}-\frac {16 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{105}-\frac {64 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{315}-\frac {128 \sqrt {-c^{2} x^{2}+1}}{315}\right )+63 d^{2} c^{4} \left (-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-c^{2} x^{2}+1}}{15}\right )+90 d \,c^{2} e \left (-\frac {c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{7}-\frac {6 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {8 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {16 \sqrt {-c^{2} x^{2}+1}}{35}\right )}{315 c^{4}}\right )}{c^{5}}\) | \(329\) |
| derivativedivides | \(\frac {\frac {a \left (\frac {1}{5} d^{2} c^{9} x^{5}+\frac {2}{7} d \,c^{9} e \,x^{7}+\frac {1}{9} e^{2} c^{9} x^{9}\right )}{c^{4}}+\frac {b \left (\frac {\arccos \left (c x \right ) d^{2} c^{9} x^{5}}{5}+\frac {2 \arccos \left (c x \right ) d \,c^{9} e \,x^{7}}{7}+\frac {\arccos \left (c x \right ) e^{2} c^{9} x^{9}}{9}+\frac {e^{2} \left (-\frac {c^{8} x^{8} \sqrt {-c^{2} x^{2}+1}}{9}-\frac {8 c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{63}-\frac {16 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{105}-\frac {64 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{315}-\frac {128 \sqrt {-c^{2} x^{2}+1}}{315}\right )}{9}+\frac {d^{2} c^{4} \left (-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-c^{2} x^{2}+1}}{15}\right )}{5}+\frac {2 d \,c^{2} e \left (-\frac {c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{7}-\frac {6 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {8 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {16 \sqrt {-c^{2} x^{2}+1}}{35}\right )}{7}\right )}{c^{4}}}{c^{5}}\) | \(339\) |
| default | \(\frac {\frac {a \left (\frac {1}{5} d^{2} c^{9} x^{5}+\frac {2}{7} d \,c^{9} e \,x^{7}+\frac {1}{9} e^{2} c^{9} x^{9}\right )}{c^{4}}+\frac {b \left (\frac {\arccos \left (c x \right ) d^{2} c^{9} x^{5}}{5}+\frac {2 \arccos \left (c x \right ) d \,c^{9} e \,x^{7}}{7}+\frac {\arccos \left (c x \right ) e^{2} c^{9} x^{9}}{9}+\frac {e^{2} \left (-\frac {c^{8} x^{8} \sqrt {-c^{2} x^{2}+1}}{9}-\frac {8 c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{63}-\frac {16 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{105}-\frac {64 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{315}-\frac {128 \sqrt {-c^{2} x^{2}+1}}{315}\right )}{9}+\frac {d^{2} c^{4} \left (-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-c^{2} x^{2}+1}}{15}\right )}{5}+\frac {2 d \,c^{2} e \left (-\frac {c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{7}-\frac {6 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {8 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {16 \sqrt {-c^{2} x^{2}+1}}{35}\right )}{7}\right )}{c^{4}}}{c^{5}}\) | \(339\) |
| orering | \(\frac {\left (20825 c^{10} e^{3} x^{12}+76675 c^{10} d \,e^{2} x^{10}+96147 c^{10} d^{2} e \,x^{8}+1400 c^{8} e^{3} x^{10}+35721 c^{10} d^{3} x^{6}+7180 c^{8} d \,e^{2} x^{8}+13824 c^{8} d^{2} e \,x^{6}+2240 x^{8} e^{3} c^{6}+5292 c^{8} d^{3} x^{4}+14080 x^{6} e^{2} c^{6} d +48816 x^{4} e \,c^{6} d^{2}+4480 x^{6} e^{3} c^{4}+21168 c^{6} d^{3} x^{2}+54080 x^{4} e^{2} c^{4} d -58752 x^{2} e \,c^{4} d^{2}+17920 x^{4} e^{3} c^{2}-42336 c^{4} d^{3}-94720 x^{2} e^{2} c^{2} d -51840 c^{2} d^{2} e -35840 x^{2} e^{3}-17920 d \,e^{2}\right ) \left (a +b \arccos \left (c x \right )\right )}{99225 \left (e \,x^{2}+d \right ) x \,c^{10}}-\frac {\left (1225 e^{2} x^{8} c^{8}+4050 c^{8} d e \,x^{6}+3969 c^{8} d^{2} x^{4}+1400 x^{6} c^{6} e^{2}+4860 x^{4} c^{6} d e +5292 x^{2} c^{6} d^{2}+1680 e^{2} x^{4} c^{4}+6480 c^{4} d e \,x^{2}+10584 c^{4} d^{2}+2240 c^{2} e^{2} x^{2}+12960 c^{2} d e +4480 e^{2}\right ) \left (c x -1\right ) \left (c x +1\right ) \left (4 x^{3} \left (e \,x^{2}+d \right )^{2} \left (a +b \arccos \left (c x \right )\right )+4 x^{5} \left (e \,x^{2}+d \right ) \left (a +b \arccos \left (c x \right )\right ) e -\frac {x^{4} \left (e \,x^{2}+d \right )^{2} b c}{\sqrt {-c^{2} x^{2}+1}}\right )}{99225 c^{10} x^{4} \left (e \,x^{2}+d \right )^{2}}\) | \(471\) |
Input:
int(x^4*(e*x^2+d)^2*(a+b*arccos(c*x)),x,method=_RETURNVERBOSE)
Output:
a*(1/9*e^2*x^9+2/7*d*e*x^7+1/5*d^2*x^5)+b/c^5*(1/9*c^5*arccos(c*x)*e^2*x^9 +2/7*c^5*arccos(c*x)*d*e*x^7+1/5*arccos(c*x)*c^5*x^5*d^2+1/315/c^4*(35*e^2 *(-1/9*c^8*x^8*(-c^2*x^2+1)^(1/2)-8/63*c^6*x^6*(-c^2*x^2+1)^(1/2)-16/105*c ^4*x^4*(-c^2*x^2+1)^(1/2)-64/315*c^2*x^2*(-c^2*x^2+1)^(1/2)-128/315*(-c^2* x^2+1)^(1/2))+63*d^2*c^4*(-1/5*c^4*x^4*(-c^2*x^2+1)^(1/2)-4/15*c^2*x^2*(-c ^2*x^2+1)^(1/2)-8/15*(-c^2*x^2+1)^(1/2))+90*d*c^2*e*(-1/7*c^6*x^6*(-c^2*x^ 2+1)^(1/2)-6/35*c^4*x^4*(-c^2*x^2+1)^(1/2)-8/35*c^2*x^2*(-c^2*x^2+1)^(1/2) -16/35*(-c^2*x^2+1)^(1/2))))
Time = 0.12 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.91 \[ \int x^4 \left (d+e x^2\right )^2 (a+b \arccos (c x)) \, dx=\frac {11025 \, a c^{9} e^{2} x^{9} + 28350 \, a c^{9} d e x^{7} + 19845 \, a c^{9} d^{2} x^{5} + 315 \, {\left (35 \, b c^{9} e^{2} x^{9} + 90 \, b c^{9} d e x^{7} + 63 \, b c^{9} d^{2} x^{5}\right )} \arccos \left (c x\right ) - {\left (1225 \, b c^{8} e^{2} x^{8} + 10584 \, b c^{4} d^{2} + 50 \, {\left (81 \, b c^{8} d e + 28 \, b c^{6} e^{2}\right )} x^{6} + 12960 \, b c^{2} d e + 3 \, {\left (1323 \, b c^{8} d^{2} + 1620 \, b c^{6} d e + 560 \, b c^{4} e^{2}\right )} x^{4} + 4480 \, b e^{2} + 4 \, {\left (1323 \, b c^{6} d^{2} + 1620 \, b c^{4} d e + 560 \, b c^{2} e^{2}\right )} x^{2}\right )} \sqrt {-c^{2} x^{2} + 1}}{99225 \, c^{9}} \] Input:
integrate(x^4*(e*x^2+d)^2*(a+b*arccos(c*x)),x, algorithm="fricas")
Output:
1/99225*(11025*a*c^9*e^2*x^9 + 28350*a*c^9*d*e*x^7 + 19845*a*c^9*d^2*x^5 + 315*(35*b*c^9*e^2*x^9 + 90*b*c^9*d*e*x^7 + 63*b*c^9*d^2*x^5)*arccos(c*x) - (1225*b*c^8*e^2*x^8 + 10584*b*c^4*d^2 + 50*(81*b*c^8*d*e + 28*b*c^6*e^2) *x^6 + 12960*b*c^2*d*e + 3*(1323*b*c^8*d^2 + 1620*b*c^6*d*e + 560*b*c^4*e^ 2)*x^4 + 4480*b*e^2 + 4*(1323*b*c^6*d^2 + 1620*b*c^4*d*e + 560*b*c^2*e^2)* x^2)*sqrt(-c^2*x^2 + 1))/c^9
Time = 1.26 (sec) , antiderivative size = 420, normalized size of antiderivative = 1.74 \[ \int x^4 \left (d+e x^2\right )^2 (a+b \arccos (c x)) \, dx=\begin {cases} \frac {a d^{2} x^{5}}{5} + \frac {2 a d e x^{7}}{7} + \frac {a e^{2} x^{9}}{9} + \frac {b d^{2} x^{5} \operatorname {acos}{\left (c x \right )}}{5} + \frac {2 b d e x^{7} \operatorname {acos}{\left (c x \right )}}{7} + \frac {b e^{2} x^{9} \operatorname {acos}{\left (c x \right )}}{9} - \frac {b d^{2} x^{4} \sqrt {- c^{2} x^{2} + 1}}{25 c} - \frac {2 b d e x^{6} \sqrt {- c^{2} x^{2} + 1}}{49 c} - \frac {b e^{2} x^{8} \sqrt {- c^{2} x^{2} + 1}}{81 c} - \frac {4 b d^{2} x^{2} \sqrt {- c^{2} x^{2} + 1}}{75 c^{3}} - \frac {12 b d e x^{4} \sqrt {- c^{2} x^{2} + 1}}{245 c^{3}} - \frac {8 b e^{2} x^{6} \sqrt {- c^{2} x^{2} + 1}}{567 c^{3}} - \frac {8 b d^{2} \sqrt {- c^{2} x^{2} + 1}}{75 c^{5}} - \frac {16 b d e x^{2} \sqrt {- c^{2} x^{2} + 1}}{245 c^{5}} - \frac {16 b e^{2} x^{4} \sqrt {- c^{2} x^{2} + 1}}{945 c^{5}} - \frac {32 b d e \sqrt {- c^{2} x^{2} + 1}}{245 c^{7}} - \frac {64 b e^{2} x^{2} \sqrt {- c^{2} x^{2} + 1}}{2835 c^{7}} - \frac {128 b e^{2} \sqrt {- c^{2} x^{2} + 1}}{2835 c^{9}} & \text {for}\: c \neq 0 \\\left (a + \frac {\pi b}{2}\right ) \left (\frac {d^{2} x^{5}}{5} + \frac {2 d e x^{7}}{7} + \frac {e^{2} x^{9}}{9}\right ) & \text {otherwise} \end {cases} \] Input:
integrate(x**4*(e*x**2+d)**2*(a+b*acos(c*x)),x)
Output:
Piecewise((a*d**2*x**5/5 + 2*a*d*e*x**7/7 + a*e**2*x**9/9 + b*d**2*x**5*ac os(c*x)/5 + 2*b*d*e*x**7*acos(c*x)/7 + b*e**2*x**9*acos(c*x)/9 - b*d**2*x* *4*sqrt(-c**2*x**2 + 1)/(25*c) - 2*b*d*e*x**6*sqrt(-c**2*x**2 + 1)/(49*c) - b*e**2*x**8*sqrt(-c**2*x**2 + 1)/(81*c) - 4*b*d**2*x**2*sqrt(-c**2*x**2 + 1)/(75*c**3) - 12*b*d*e*x**4*sqrt(-c**2*x**2 + 1)/(245*c**3) - 8*b*e**2* x**6*sqrt(-c**2*x**2 + 1)/(567*c**3) - 8*b*d**2*sqrt(-c**2*x**2 + 1)/(75*c **5) - 16*b*d*e*x**2*sqrt(-c**2*x**2 + 1)/(245*c**5) - 16*b*e**2*x**4*sqrt (-c**2*x**2 + 1)/(945*c**5) - 32*b*d*e*sqrt(-c**2*x**2 + 1)/(245*c**7) - 6 4*b*e**2*x**2*sqrt(-c**2*x**2 + 1)/(2835*c**7) - 128*b*e**2*sqrt(-c**2*x** 2 + 1)/(2835*c**9), Ne(c, 0)), ((a + pi*b/2)*(d**2*x**5/5 + 2*d*e*x**7/7 + e**2*x**9/9), True))
Time = 0.14 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.32 \[ \int x^4 \left (d+e x^2\right )^2 (a+b \arccos (c x)) \, dx=\frac {1}{9} \, a e^{2} x^{9} + \frac {2}{7} \, a d e x^{7} + \frac {1}{5} \, a d^{2} x^{5} + \frac {1}{75} \, {\left (15 \, x^{5} \arccos \left (c x\right ) - {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} b d^{2} + \frac {2}{245} \, {\left (35 \, x^{7} \arccos \left (c x\right ) - {\left (\frac {5 \, \sqrt {-c^{2} x^{2} + 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {-c^{2} x^{2} + 1}}{c^{8}}\right )} c\right )} b d e + \frac {1}{2835} \, {\left (315 \, x^{9} \arccos \left (c x\right ) - {\left (\frac {35 \, \sqrt {-c^{2} x^{2} + 1} x^{8}}{c^{2}} + \frac {40 \, \sqrt {-c^{2} x^{2} + 1} x^{6}}{c^{4}} + \frac {48 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{6}} + \frac {64 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{8}} + \frac {128 \, \sqrt {-c^{2} x^{2} + 1}}{c^{10}}\right )} c\right )} b e^{2} \] Input:
integrate(x^4*(e*x^2+d)^2*(a+b*arccos(c*x)),x, algorithm="maxima")
Output:
1/9*a*e^2*x^9 + 2/7*a*d*e*x^7 + 1/5*a*d^2*x^5 + 1/75*(15*x^5*arccos(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^2 + 2/245*(35*x^7*arccos(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*d*e + 1/2835*(315*x^9*arccos(c*x) - (35*s qrt(-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*e^2
Time = 0.14 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.41 \[ \int x^4 \left (d+e x^2\right )^2 (a+b \arccos (c x)) \, dx=\frac {1}{9} \, b e^{2} x^{9} \arccos \left (c x\right ) + \frac {1}{9} \, a e^{2} x^{9} + \frac {2}{7} \, b d e x^{7} \arccos \left (c x\right ) - \frac {\sqrt {-c^{2} x^{2} + 1} b e^{2} x^{8}}{81 \, c} + \frac {2}{7} \, a d e x^{7} + \frac {1}{5} \, b d^{2} x^{5} \arccos \left (c x\right ) - \frac {2 \, \sqrt {-c^{2} x^{2} + 1} b d e x^{6}}{49 \, c} + \frac {1}{5} \, a d^{2} x^{5} - \frac {\sqrt {-c^{2} x^{2} + 1} b d^{2} x^{4}}{25 \, c} - \frac {8 \, \sqrt {-c^{2} x^{2} + 1} b e^{2} x^{6}}{567 \, c^{3}} - \frac {12 \, \sqrt {-c^{2} x^{2} + 1} b d e x^{4}}{245 \, c^{3}} - \frac {4 \, \sqrt {-c^{2} x^{2} + 1} b d^{2} x^{2}}{75 \, c^{3}} - \frac {16 \, \sqrt {-c^{2} x^{2} + 1} b e^{2} x^{4}}{945 \, c^{5}} - \frac {16 \, \sqrt {-c^{2} x^{2} + 1} b d e x^{2}}{245 \, c^{5}} - \frac {8 \, \sqrt {-c^{2} x^{2} + 1} b d^{2}}{75 \, c^{5}} - \frac {64 \, \sqrt {-c^{2} x^{2} + 1} b e^{2} x^{2}}{2835 \, c^{7}} - \frac {32 \, \sqrt {-c^{2} x^{2} + 1} b d e}{245 \, c^{7}} - \frac {128 \, \sqrt {-c^{2} x^{2} + 1} b e^{2}}{2835 \, c^{9}} \] Input:
integrate(x^4*(e*x^2+d)^2*(a+b*arccos(c*x)),x, algorithm="giac")
Output:
1/9*b*e^2*x^9*arccos(c*x) + 1/9*a*e^2*x^9 + 2/7*b*d*e*x^7*arccos(c*x) - 1/ 81*sqrt(-c^2*x^2 + 1)*b*e^2*x^8/c + 2/7*a*d*e*x^7 + 1/5*b*d^2*x^5*arccos(c *x) - 2/49*sqrt(-c^2*x^2 + 1)*b*d*e*x^6/c + 1/5*a*d^2*x^5 - 1/25*sqrt(-c^2 *x^2 + 1)*b*d^2*x^4/c - 8/567*sqrt(-c^2*x^2 + 1)*b*e^2*x^6/c^3 - 12/245*sq rt(-c^2*x^2 + 1)*b*d*e*x^4/c^3 - 4/75*sqrt(-c^2*x^2 + 1)*b*d^2*x^2/c^3 - 1 6/945*sqrt(-c^2*x^2 + 1)*b*e^2*x^4/c^5 - 16/245*sqrt(-c^2*x^2 + 1)*b*d*e*x ^2/c^5 - 8/75*sqrt(-c^2*x^2 + 1)*b*d^2/c^5 - 64/2835*sqrt(-c^2*x^2 + 1)*b* e^2*x^2/c^7 - 32/245*sqrt(-c^2*x^2 + 1)*b*d*e/c^7 - 128/2835*sqrt(-c^2*x^2 + 1)*b*e^2/c^9
Timed out. \[ \int x^4 \left (d+e x^2\right )^2 (a+b \arccos (c x)) \, dx=\int x^4\,\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^2 \,d x \] Input:
int(x^4*(a + b*acos(c*x))*(d + e*x^2)^2,x)
Output:
int(x^4*(a + b*acos(c*x))*(d + e*x^2)^2, x)
Time = 0.24 (sec) , antiderivative size = 348, normalized size of antiderivative = 1.44 \[ \int x^4 \left (d+e x^2\right )^2 (a+b \arccos (c x)) \, dx=\frac {19845 \mathit {acos} \left (c x \right ) b \,c^{9} d^{2} x^{5}+28350 \mathit {acos} \left (c x \right ) b \,c^{9} d e \,x^{7}+11025 \mathit {acos} \left (c x \right ) b \,c^{9} e^{2} x^{9}-3969 \sqrt {-c^{2} x^{2}+1}\, b \,c^{8} d^{2} x^{4}-4050 \sqrt {-c^{2} x^{2}+1}\, b \,c^{8} d e \,x^{6}-1225 \sqrt {-c^{2} x^{2}+1}\, b \,c^{8} e^{2} x^{8}-5292 \sqrt {-c^{2} x^{2}+1}\, b \,c^{6} d^{2} x^{2}-4860 \sqrt {-c^{2} x^{2}+1}\, b \,c^{6} d e \,x^{4}-1400 \sqrt {-c^{2} x^{2}+1}\, b \,c^{6} e^{2} x^{6}-10584 \sqrt {-c^{2} x^{2}+1}\, b \,c^{4} d^{2}-6480 \sqrt {-c^{2} x^{2}+1}\, b \,c^{4} d e \,x^{2}-1680 \sqrt {-c^{2} x^{2}+1}\, b \,c^{4} e^{2} x^{4}-12960 \sqrt {-c^{2} x^{2}+1}\, b \,c^{2} d e -2240 \sqrt {-c^{2} x^{2}+1}\, b \,c^{2} e^{2} x^{2}-4480 \sqrt {-c^{2} x^{2}+1}\, b \,e^{2}+19845 a \,c^{9} d^{2} x^{5}+28350 a \,c^{9} d e \,x^{7}+11025 a \,c^{9} e^{2} x^{9}}{99225 c^{9}} \] Input:
int(x^4*(e*x^2+d)^2*(a+b*acos(c*x)),x)
Output:
(19845*acos(c*x)*b*c**9*d**2*x**5 + 28350*acos(c*x)*b*c**9*d*e*x**7 + 1102 5*acos(c*x)*b*c**9*e**2*x**9 - 3969*sqrt( - c**2*x**2 + 1)*b*c**8*d**2*x** 4 - 4050*sqrt( - c**2*x**2 + 1)*b*c**8*d*e*x**6 - 1225*sqrt( - c**2*x**2 + 1)*b*c**8*e**2*x**8 - 5292*sqrt( - c**2*x**2 + 1)*b*c**6*d**2*x**2 - 4860 *sqrt( - c**2*x**2 + 1)*b*c**6*d*e*x**4 - 1400*sqrt( - c**2*x**2 + 1)*b*c* *6*e**2*x**6 - 10584*sqrt( - c**2*x**2 + 1)*b*c**4*d**2 - 6480*sqrt( - c** 2*x**2 + 1)*b*c**4*d*e*x**2 - 1680*sqrt( - c**2*x**2 + 1)*b*c**4*e**2*x**4 - 12960*sqrt( - c**2*x**2 + 1)*b*c**2*d*e - 2240*sqrt( - c**2*x**2 + 1)*b *c**2*e**2*x**2 - 4480*sqrt( - c**2*x**2 + 1)*b*e**2 + 19845*a*c**9*d**2*x **5 + 28350*a*c**9*d*e*x**7 + 11025*a*c**9*e**2*x**9)/(99225*c**9)