Integrand size = 21, antiderivative size = 287 \[ \int x^2 \left (d+e x^2\right )^3 (a+b \arccos (c x)) \, dx=\frac {b \left (105 c^6 d^3+189 c^4 d^2 e+135 c^2 d e^2+35 e^3\right ) \sqrt {1-c^2 x^2}}{315 c^9}-\frac {b \left (105 c^6 d^3+378 c^4 d^2 e+405 c^2 d e^2+140 e^3\right ) \left (1-c^2 x^2\right )^{3/2}}{945 c^9}+\frac {b e \left (63 c^4 d^2+135 c^2 d e+70 e^2\right ) \left (1-c^2 x^2\right )^{5/2}}{525 c^9}-\frac {b e^2 \left (27 c^2 d+28 e\right ) \left (1-c^2 x^2\right )^{7/2}}{441 c^9}+\frac {b e^3 \left (1-c^2 x^2\right )^{9/2}}{81 c^9}+\frac {1}{3} d^3 x^3 (a+b \arccos (c x))+\frac {3}{5} d^2 e x^5 (a+b \arccos (c x))+\frac {3}{7} d e^2 x^7 (a+b \arccos (c x))+\frac {1}{9} e^3 x^9 (a+b \arccos (c x)) \] Output:
1/315*b*(105*c^6*d^3+189*c^4*d^2*e+135*c^2*d*e^2+35*e^3)*(-c^2*x^2+1)^(1/2 )/c^9-1/945*b*(105*c^6*d^3+378*c^4*d^2*e+405*c^2*d*e^2+140*e^3)*(-c^2*x^2+ 1)^(3/2)/c^9+1/525*b*e*(63*c^4*d^2+135*c^2*d*e+70*e^2)*(-c^2*x^2+1)^(5/2)/ c^9-1/441*b*e^2*(27*c^2*d+28*e)*(-c^2*x^2+1)^(7/2)/c^9+1/81*b*e^3*(-c^2*x^ 2+1)^(9/2)/c^9+1/3*d^3*x^3*(a+b*arccos(c*x))+3/5*d^2*e*x^5*(a+b*arccos(c*x ))+3/7*d*e^2*x^7*(a+b*arccos(c*x))+1/9*e^3*x^9*(a+b*arccos(c*x))
Time = 0.19 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.81 \[ \int x^2 \left (d+e x^2\right )^3 (a+b \arccos (c x)) \, dx=\frac {315 a x^3 \left (105 d^3+189 d^2 e x^2+135 d e^2 x^4+35 e^3 x^6\right )-\frac {b \sqrt {1-c^2 x^2} \left (4480 e^3+80 c^2 e^2 \left (243 d+28 e x^2\right )+24 c^4 e \left (1323 d^2+405 d e x^2+70 e^2 x^4\right )+2 c^6 \left (11025 d^3+7938 d^2 e x^2+3645 d e^2 x^4+700 e^3 x^6\right )+c^8 \left (11025 d^3 x^2+11907 d^2 e x^4+6075 d e^2 x^6+1225 e^3 x^8\right )\right )}{c^9}+315 b x^3 \left (105 d^3+189 d^2 e x^2+135 d e^2 x^4+35 e^3 x^6\right ) \arccos (c x)}{99225} \] Input:
Integrate[x^2*(d + e*x^2)^3*(a + b*ArcCos[c*x]),x]
Output:
(315*a*x^3*(105*d^3 + 189*d^2*e*x^2 + 135*d*e^2*x^4 + 35*e^3*x^6) - (b*Sqr t[1 - c^2*x^2]*(4480*e^3 + 80*c^2*e^2*(243*d + 28*e*x^2) + 24*c^4*e*(1323* d^2 + 405*d*e*x^2 + 70*e^2*x^4) + 2*c^6*(11025*d^3 + 7938*d^2*e*x^2 + 3645 *d*e^2*x^4 + 700*e^3*x^6) + c^8*(11025*d^3*x^2 + 11907*d^2*e*x^4 + 6075*d* e^2*x^6 + 1225*e^3*x^8)))/c^9 + 315*b*x^3*(105*d^3 + 189*d^2*e*x^2 + 135*d *e^2*x^4 + 35*e^3*x^6)*ArcCos[c*x])/99225
Time = 0.72 (sec) , antiderivative size = 287, 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, 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.
| \(\displaystyle \int x^2 \left (d+e x^2\right )^3 (a+b \arccos (c x)) \, dx\) |
| \(\Big \downarrow \) 5231 |
| \(\displaystyle b c \int \frac {x^3 \left (35 e^3 x^6+135 d e^2 x^4+189 d^2 e x^2+105 d^3\right )}{315 \sqrt {1-c^2 x^2}}dx+\frac {1}{3} d^3 x^3 (a+b \arccos (c x))+\frac {3}{5} d^2 e x^5 (a+b \arccos (c x))+\frac {3}{7} d e^2 x^7 (a+b \arccos (c x))+\frac {1}{9} e^3 x^9 (a+b \arccos (c x))\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{315} b c \int \frac {x^3 \left (35 e^3 x^6+135 d e^2 x^4+189 d^2 e x^2+105 d^3\right )}{\sqrt {1-c^2 x^2}}dx+\frac {1}{3} d^3 x^3 (a+b \arccos (c x))+\frac {3}{5} d^2 e x^5 (a+b \arccos (c x))+\frac {3}{7} d e^2 x^7 (a+b \arccos (c x))+\frac {1}{9} e^3 x^9 (a+b \arccos (c x))\) |
| \(\Big \downarrow \) 2331 |
| \(\displaystyle \frac {1}{630} b c \int \frac {x^2 \left (35 e^3 x^6+135 d e^2 x^4+189 d^2 e x^2+105 d^3\right )}{\sqrt {1-c^2 x^2}}dx^2+\frac {1}{3} d^3 x^3 (a+b \arccos (c x))+\frac {3}{5} d^2 e x^5 (a+b \arccos (c x))+\frac {3}{7} d e^2 x^7 (a+b \arccos (c x))+\frac {1}{9} e^3 x^9 (a+b \arccos (c x))\) |
| \(\Big \downarrow \) 2123 |
| \(\displaystyle \frac {1}{630} b c \int \left (\frac {35 e^3 \left (1-c^2 x^2\right )^{7/2}}{c^8}-\frac {5 e^2 \left (27 d c^2+28 e\right ) \left (1-c^2 x^2\right )^{5/2}}{c^8}+\frac {3 e \left (63 d^2 c^4+135 d e c^2+70 e^2\right ) \left (1-c^2 x^2\right )^{3/2}}{c^8}+\frac {\left (-105 d^3 c^6-378 d^2 e c^4-405 d e^2 c^2-140 e^3\right ) \sqrt {1-c^2 x^2}}{c^8}+\frac {105 d^3 c^6+189 d^2 e c^4+135 d e^2 c^2+35 e^3}{c^8 \sqrt {1-c^2 x^2}}\right )dx^2+\frac {1}{3} d^3 x^3 (a+b \arccos (c x))+\frac {3}{5} d^2 e x^5 (a+b \arccos (c x))+\frac {3}{7} d e^2 x^7 (a+b \arccos (c x))+\frac {1}{9} e^3 x^9 (a+b \arccos (c x))\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{3} d^3 x^3 (a+b \arccos (c x))+\frac {3}{5} d^2 e x^5 (a+b \arccos (c x))+\frac {3}{7} d e^2 x^7 (a+b \arccos (c x))+\frac {1}{9} e^3 x^9 (a+b \arccos (c x))+\frac {1}{630} b c \left (\frac {10 e^2 \left (1-c^2 x^2\right )^{7/2} \left (27 c^2 d+28 e\right )}{7 c^{10}}-\frac {70 e^3 \left (1-c^2 x^2\right )^{9/2}}{9 c^{10}}-\frac {6 e \left (1-c^2 x^2\right )^{5/2} \left (63 c^4 d^2+135 c^2 d e+70 e^2\right )}{5 c^{10}}+\frac {2 \left (1-c^2 x^2\right )^{3/2} \left (105 c^6 d^3+378 c^4 d^2 e+405 c^2 d e^2+140 e^3\right )}{3 c^{10}}-\frac {2 \sqrt {1-c^2 x^2} \left (105 c^6 d^3+189 c^4 d^2 e+135 c^2 d e^2+35 e^3\right )}{c^{10}}\right )\) |
Input:
Int[x^2*(d + e*x^2)^3*(a + b*ArcCos[c*x]),x]
Output:
(b*c*((-2*(105*c^6*d^3 + 189*c^4*d^2*e + 135*c^2*d*e^2 + 35*e^3)*Sqrt[1 - c^2*x^2])/c^10 + (2*(105*c^6*d^3 + 378*c^4*d^2*e + 405*c^2*d*e^2 + 140*e^3 )*(1 - c^2*x^2)^(3/2))/(3*c^10) - (6*e*(63*c^4*d^2 + 135*c^2*d*e + 70*e^2) *(1 - c^2*x^2)^(5/2))/(5*c^10) + (10*e^2*(27*c^2*d + 28*e)*(1 - c^2*x^2)^( 7/2))/(7*c^10) - (70*e^3*(1 - c^2*x^2)^(9/2))/(9*c^10)))/630 + (d^3*x^3*(a + b*ArcCos[c*x]))/3 + (3*d^2*e*x^5*(a + b*ArcCos[c*x]))/5 + (3*d*e^2*x^7* (a + b*ArcCos[c*x]))/7 + (e^3*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[(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_.) + 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 = 0.43 (sec) , antiderivative size = 404, normalized size of antiderivative = 1.41
| method | result | size |
| parts | \(a \left (\frac {1}{9} e^{3} x^{9}+\frac {3}{7} d \,e^{2} x^{7}+\frac {3}{5} d^{2} e \,x^{5}+\frac {1}{3} d^{3} x^{3}\right )+\frac {b \left (\frac {c^{3} \arccos \left (c x \right ) e^{3} x^{9}}{9}+\frac {3 c^{3} \arccos \left (c x \right ) d \,e^{2} x^{7}}{7}+\frac {3 c^{3} \arccos \left (c x \right ) d^{2} e \,x^{5}}{5}+\frac {\arccos \left (c x \right ) d^{3} c^{3} x^{3}}{3}+\frac {35 e^{3} \left (-\frac {c^{8} x^{8} \sqrt {-c^{2} x^{2}+1}}{9}-\frac {8 c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{63}-\frac {16 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{105}-\frac {64 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{315}-\frac {128 \sqrt {-c^{2} x^{2}+1}}{315}\right )+105 d^{3} c^{6} \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )+135 d \,c^{2} e^{2} \left (-\frac {c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{7}-\frac {6 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {8 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {16 \sqrt {-c^{2} x^{2}+1}}{35}\right )+189 d^{2} c^{4} e \left (-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-c^{2} x^{2}+1}}{15}\right )}{315 c^{6}}\right )}{c^{3}}\) | \(404\) |
| derivativedivides | \(\frac {\frac {a \left (\frac {1}{3} d^{3} c^{9} x^{3}+\frac {3}{5} d^{2} c^{9} e \,x^{5}+\frac {3}{7} d \,c^{9} e^{2} x^{7}+\frac {1}{9} e^{3} c^{9} x^{9}\right )}{c^{6}}+\frac {b \left (\frac {\arccos \left (c x \right ) d^{3} c^{9} x^{3}}{3}+\frac {3 \arccos \left (c x \right ) d^{2} c^{9} e \,x^{5}}{5}+\frac {3 \arccos \left (c x \right ) d \,c^{9} e^{2} x^{7}}{7}+\frac {\arccos \left (c x \right ) e^{3} c^{9} x^{9}}{9}+\frac {e^{3} \left (-\frac {c^{8} x^{8} \sqrt {-c^{2} x^{2}+1}}{9}-\frac {8 c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{63}-\frac {16 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{105}-\frac {64 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{315}-\frac {128 \sqrt {-c^{2} x^{2}+1}}{315}\right )}{9}+\frac {d^{3} c^{6} \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )}{3}+\frac {3 d \,c^{2} e^{2} \left (-\frac {c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{7}-\frac {6 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {8 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {16 \sqrt {-c^{2} x^{2}+1}}{35}\right )}{7}+\frac {3 d^{2} c^{4} e \left (-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-c^{2} x^{2}+1}}{15}\right )}{5}\right )}{c^{6}}}{c^{3}}\) | \(417\) |
| default | \(\frac {\frac {a \left (\frac {1}{3} d^{3} c^{9} x^{3}+\frac {3}{5} d^{2} c^{9} e \,x^{5}+\frac {3}{7} d \,c^{9} e^{2} x^{7}+\frac {1}{9} e^{3} c^{9} x^{9}\right )}{c^{6}}+\frac {b \left (\frac {\arccos \left (c x \right ) d^{3} c^{9} x^{3}}{3}+\frac {3 \arccos \left (c x \right ) d^{2} c^{9} e \,x^{5}}{5}+\frac {3 \arccos \left (c x \right ) d \,c^{9} e^{2} x^{7}}{7}+\frac {\arccos \left (c x \right ) e^{3} c^{9} x^{9}}{9}+\frac {e^{3} \left (-\frac {c^{8} x^{8} \sqrt {-c^{2} x^{2}+1}}{9}-\frac {8 c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{63}-\frac {16 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{105}-\frac {64 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{315}-\frac {128 \sqrt {-c^{2} x^{2}+1}}{315}\right )}{9}+\frac {d^{3} c^{6} \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )}{3}+\frac {3 d \,c^{2} e^{2} \left (-\frac {c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{7}-\frac {6 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {8 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {16 \sqrt {-c^{2} x^{2}+1}}{35}\right )}{7}+\frac {3 d^{2} c^{4} e \left (-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-c^{2} x^{2}+1}}{15}\right )}{5}\right )}{c^{6}}}{c^{3}}\) | \(417\) |
| orering | \(\frac {\left (20825 c^{10} e^{4} x^{12}+104600 c^{10} d \,e^{3} x^{10}+209466 c^{10} d^{2} e^{2} x^{8}+1400 c^{8} e^{4} x^{10}+204624 c^{10} d^{3} e \,x^{6}+10070 c^{8} d \,e^{3} x^{8}+55125 c^{10} d^{4} x^{4}+34182 c^{8} d^{2} e^{2} x^{6}+2240 c^{6} e^{4} x^{8}+96138 c^{8} d^{3} e \,x^{4}+20000 c^{6} d \,e^{3} x^{6}+22050 c^{8} d^{4} x^{2}+131868 c^{6} d^{2} e^{2} x^{4}+4480 c^{4} e^{4} x^{6}-144648 c^{6} d^{3} e \,x^{2}+78880 c^{4} d \,e^{3} x^{4}-44100 c^{6} d^{4}-234576 c^{4} d^{2} e^{2} x^{2}+17920 c^{2} e^{4} x^{4}-63504 c^{4} d^{3} e -151040 c^{2} d \,e^{3} x^{2}-38880 c^{2} d^{2} e^{2}-35840 e^{4} x^{2}-8960 d \,e^{3}\right ) \left (a +b \arccos \left (c x \right )\right )}{99225 \left (e \,x^{2}+d \right ) x \,c^{10}}-\frac {\left (1225 c^{8} e^{3} x^{8}+6075 c^{8} d \,e^{2} x^{6}+11907 c^{8} d^{2} e \,x^{4}+1400 e^{3} x^{6} c^{6}+11025 c^{8} d^{3} x^{2}+7290 c^{6} d \,e^{2} x^{4}+15876 c^{6} d^{2} e \,x^{2}+1680 c^{4} e^{3} x^{4}+22050 c^{6} d^{3}+9720 c^{4} d \,e^{2} x^{2}+31752 c^{4} d^{2} e +2240 c^{2} e^{3} x^{2}+19440 c^{2} d \,e^{2}+4480 e^{3}\right ) \left (c x -1\right ) \left (c x +1\right ) \left (2 x \left (e \,x^{2}+d \right )^{3} \left (a +b \arccos \left (c x \right )\right )+6 x^{3} \left (e \,x^{2}+d \right )^{2} \left (a +b \arccos \left (c x \right )\right ) e -\frac {x^{2} \left (e \,x^{2}+d \right )^{3} b c}{\sqrt {-c^{2} x^{2}+1}}\right )}{99225 c^{10} x^{2} \left (e \,x^{2}+d \right )^{3}}\) | \(545\) |
Input:
int(x^2*(e*x^2+d)^3*(a+b*arccos(c*x)),x,method=_RETURNVERBOSE)
Output:
a*(1/9*e^3*x^9+3/7*d*e^2*x^7+3/5*d^2*e*x^5+1/3*d^3*x^3)+b/c^3*(1/9*c^3*arc cos(c*x)*e^3*x^9+3/7*c^3*arccos(c*x)*d*e^2*x^7+3/5*c^3*arccos(c*x)*d^2*e*x ^5+1/3*arccos(c*x)*d^3*c^3*x^3+1/315/c^6*(35*e^3*(-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))+105*d^3*c^6* (-1/3*c^2*x^2*(-c^2*x^2+1)^(1/2)-2/3*(-c^2*x^2+1)^(1/2))+135*d*c^2*e^2*(-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))+189*d^2*c^4*e*(-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) )))
Time = 0.11 (sec) , antiderivative size = 278, normalized size of antiderivative = 0.97 \[ \int x^2 \left (d+e x^2\right )^3 (a+b \arccos (c x)) \, dx=\frac {11025 \, a c^{9} e^{3} x^{9} + 42525 \, a c^{9} d e^{2} x^{7} + 59535 \, a c^{9} d^{2} e x^{5} + 33075 \, a c^{9} d^{3} x^{3} + 315 \, {\left (35 \, b c^{9} e^{3} x^{9} + 135 \, b c^{9} d e^{2} x^{7} + 189 \, b c^{9} d^{2} e x^{5} + 105 \, b c^{9} d^{3} x^{3}\right )} \arccos \left (c x\right ) - {\left (1225 \, b c^{8} e^{3} x^{8} + 22050 \, b c^{6} d^{3} + 31752 \, b c^{4} d^{2} e + 25 \, {\left (243 \, b c^{8} d e^{2} + 56 \, b c^{6} e^{3}\right )} x^{6} + 19440 \, b c^{2} d e^{2} + 3 \, {\left (3969 \, b c^{8} d^{2} e + 2430 \, b c^{6} d e^{2} + 560 \, b c^{4} e^{3}\right )} x^{4} + 4480 \, b e^{3} + {\left (11025 \, b c^{8} d^{3} + 15876 \, b c^{6} d^{2} e + 9720 \, b c^{4} d e^{2} + 2240 \, b c^{2} e^{3}\right )} x^{2}\right )} \sqrt {-c^{2} x^{2} + 1}}{99225 \, c^{9}} \] Input:
integrate(x^2*(e*x^2+d)^3*(a+b*arccos(c*x)),x, algorithm="fricas")
Output:
1/99225*(11025*a*c^9*e^3*x^9 + 42525*a*c^9*d*e^2*x^7 + 59535*a*c^9*d^2*e*x ^5 + 33075*a*c^9*d^3*x^3 + 315*(35*b*c^9*e^3*x^9 + 135*b*c^9*d*e^2*x^7 + 1 89*b*c^9*d^2*e*x^5 + 105*b*c^9*d^3*x^3)*arccos(c*x) - (1225*b*c^8*e^3*x^8 + 22050*b*c^6*d^3 + 31752*b*c^4*d^2*e + 25*(243*b*c^8*d*e^2 + 56*b*c^6*e^3 )*x^6 + 19440*b*c^2*d*e^2 + 3*(3969*b*c^8*d^2*e + 2430*b*c^6*d*e^2 + 560*b *c^4*e^3)*x^4 + 4480*b*e^3 + (11025*b*c^8*d^3 + 15876*b*c^6*d^2*e + 9720*b *c^4*d*e^2 + 2240*b*c^2*e^3)*x^2)*sqrt(-c^2*x^2 + 1))/c^9
Time = 1.30 (sec) , antiderivative size = 530, normalized size of antiderivative = 1.85 \[ \int x^2 \left (d+e x^2\right )^3 (a+b \arccos (c x)) \, dx=\begin {cases} \frac {a d^{3} x^{3}}{3} + \frac {3 a d^{2} e x^{5}}{5} + \frac {3 a d e^{2} x^{7}}{7} + \frac {a e^{3} x^{9}}{9} + \frac {b d^{3} x^{3} \operatorname {acos}{\left (c x \right )}}{3} + \frac {3 b d^{2} e x^{5} \operatorname {acos}{\left (c x \right )}}{5} + \frac {3 b d e^{2} x^{7} \operatorname {acos}{\left (c x \right )}}{7} + \frac {b e^{3} x^{9} \operatorname {acos}{\left (c x \right )}}{9} - \frac {b d^{3} x^{2} \sqrt {- c^{2} x^{2} + 1}}{9 c} - \frac {3 b d^{2} e x^{4} \sqrt {- c^{2} x^{2} + 1}}{25 c} - \frac {3 b d e^{2} x^{6} \sqrt {- c^{2} x^{2} + 1}}{49 c} - \frac {b e^{3} x^{8} \sqrt {- c^{2} x^{2} + 1}}{81 c} - \frac {2 b d^{3} \sqrt {- c^{2} x^{2} + 1}}{9 c^{3}} - \frac {4 b d^{2} e x^{2} \sqrt {- c^{2} x^{2} + 1}}{25 c^{3}} - \frac {18 b d e^{2} x^{4} \sqrt {- c^{2} x^{2} + 1}}{245 c^{3}} - \frac {8 b e^{3} x^{6} \sqrt {- c^{2} x^{2} + 1}}{567 c^{3}} - \frac {8 b d^{2} e \sqrt {- c^{2} x^{2} + 1}}{25 c^{5}} - \frac {24 b d e^{2} x^{2} \sqrt {- c^{2} x^{2} + 1}}{245 c^{5}} - \frac {16 b e^{3} x^{4} \sqrt {- c^{2} x^{2} + 1}}{945 c^{5}} - \frac {48 b d e^{2} \sqrt {- c^{2} x^{2} + 1}}{245 c^{7}} - \frac {64 b e^{3} x^{2} \sqrt {- c^{2} x^{2} + 1}}{2835 c^{7}} - \frac {128 b e^{3} \sqrt {- c^{2} x^{2} + 1}}{2835 c^{9}} & \text {for}\: c \neq 0 \\\left (a + \frac {\pi b}{2}\right ) \left (\frac {d^{3} x^{3}}{3} + \frac {3 d^{2} e x^{5}}{5} + \frac {3 d e^{2} x^{7}}{7} + \frac {e^{3} x^{9}}{9}\right ) & \text {otherwise} \end {cases} \] Input:
integrate(x**2*(e*x**2+d)**3*(a+b*acos(c*x)),x)
Output:
Piecewise((a*d**3*x**3/3 + 3*a*d**2*e*x**5/5 + 3*a*d*e**2*x**7/7 + a*e**3* x**9/9 + b*d**3*x**3*acos(c*x)/3 + 3*b*d**2*e*x**5*acos(c*x)/5 + 3*b*d*e** 2*x**7*acos(c*x)/7 + b*e**3*x**9*acos(c*x)/9 - b*d**3*x**2*sqrt(-c**2*x**2 + 1)/(9*c) - 3*b*d**2*e*x**4*sqrt(-c**2*x**2 + 1)/(25*c) - 3*b*d*e**2*x** 6*sqrt(-c**2*x**2 + 1)/(49*c) - b*e**3*x**8*sqrt(-c**2*x**2 + 1)/(81*c) - 2*b*d**3*sqrt(-c**2*x**2 + 1)/(9*c**3) - 4*b*d**2*e*x**2*sqrt(-c**2*x**2 + 1)/(25*c**3) - 18*b*d*e**2*x**4*sqrt(-c**2*x**2 + 1)/(245*c**3) - 8*b*e** 3*x**6*sqrt(-c**2*x**2 + 1)/(567*c**3) - 8*b*d**2*e*sqrt(-c**2*x**2 + 1)/( 25*c**5) - 24*b*d*e**2*x**2*sqrt(-c**2*x**2 + 1)/(245*c**5) - 16*b*e**3*x* *4*sqrt(-c**2*x**2 + 1)/(945*c**5) - 48*b*d*e**2*sqrt(-c**2*x**2 + 1)/(245 *c**7) - 64*b*e**3*x**2*sqrt(-c**2*x**2 + 1)/(2835*c**7) - 128*b*e**3*sqrt (-c**2*x**2 + 1)/(2835*c**9), Ne(c, 0)), ((a + pi*b/2)*(d**3*x**3/3 + 3*d* *2*e*x**5/5 + 3*d*e**2*x**7/7 + e**3*x**9/9), True))
Time = 0.14 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.35 \[ \int x^2 \left (d+e x^2\right )^3 (a+b \arccos (c x)) \, dx=\frac {1}{9} \, a e^{3} x^{9} + \frac {3}{7} \, a d e^{2} x^{7} + \frac {3}{5} \, a d^{2} e x^{5} + \frac {1}{3} \, a d^{3} x^{3} + \frac {1}{9} \, {\left (3 \, x^{3} \arccos \left (c x\right ) - c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b d^{3} + \frac {1}{25} \, {\left (15 \, x^{5} \arccos \left (c x\right ) - {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} b d^{2} e + \frac {3}{245} \, {\left (35 \, x^{7} \arccos \left (c x\right ) - {\left (\frac {5 \, \sqrt {-c^{2} x^{2} + 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {-c^{2} x^{2} + 1}}{c^{8}}\right )} c\right )} b d e^{2} + \frac {1}{2835} \, {\left (315 \, x^{9} \arccos \left (c x\right ) - {\left (\frac {35 \, \sqrt {-c^{2} x^{2} + 1} x^{8}}{c^{2}} + \frac {40 \, \sqrt {-c^{2} x^{2} + 1} x^{6}}{c^{4}} + \frac {48 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{6}} + \frac {64 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{8}} + \frac {128 \, \sqrt {-c^{2} x^{2} + 1}}{c^{10}}\right )} c\right )} b e^{3} \] Input:
integrate(x^2*(e*x^2+d)^3*(a+b*arccos(c*x)),x, algorithm="maxima")
Output:
1/9*a*e^3*x^9 + 3/7*a*d*e^2*x^7 + 3/5*a*d^2*e*x^5 + 1/3*a*d^3*x^3 + 1/9*(3 *x^3*arccos(c*x) - c*(sqrt(-c^2*x^2 + 1)*x^2/c^2 + 2*sqrt(-c^2*x^2 + 1)/c^ 4))*b*d^3 + 1/25*(15*x^5*arccos(c*x) - (3*sqrt(-c^2*x^2 + 1)*x^4/c^2 + 4*s qrt(-c^2*x^2 + 1)*x^2/c^4 + 8*sqrt(-c^2*x^2 + 1)/c^6)*c)*b*d^2*e + 3/245*( 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^2 + 1/2835*(315*x^9*arccos(c*x) - (35*sqrt(-c^2*x^2 + 1)*x^8/c^2 + 40*s qrt(-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^3
Time = 0.14 (sec) , antiderivative size = 424, normalized size of antiderivative = 1.48 \[ \int x^2 \left (d+e x^2\right )^3 (a+b \arccos (c x)) \, dx=\frac {1}{9} \, b e^{3} x^{9} \arccos \left (c x\right ) + \frac {1}{9} \, a e^{3} x^{9} + \frac {3}{7} \, b d e^{2} x^{7} \arccos \left (c x\right ) - \frac {\sqrt {-c^{2} x^{2} + 1} b e^{3} x^{8}}{81 \, c} + \frac {3}{7} \, a d e^{2} x^{7} + \frac {3}{5} \, b d^{2} e x^{5} \arccos \left (c x\right ) - \frac {3 \, \sqrt {-c^{2} x^{2} + 1} b d e^{2} x^{6}}{49 \, c} + \frac {3}{5} \, a d^{2} e x^{5} + \frac {1}{3} \, b d^{3} x^{3} \arccos \left (c x\right ) - \frac {3 \, \sqrt {-c^{2} x^{2} + 1} b d^{2} e x^{4}}{25 \, c} - \frac {8 \, \sqrt {-c^{2} x^{2} + 1} b e^{3} x^{6}}{567 \, c^{3}} + \frac {1}{3} \, a d^{3} x^{3} - \frac {\sqrt {-c^{2} x^{2} + 1} b d^{3} x^{2}}{9 \, c} - \frac {18 \, \sqrt {-c^{2} x^{2} + 1} b d e^{2} x^{4}}{245 \, c^{3}} - \frac {4 \, \sqrt {-c^{2} x^{2} + 1} b d^{2} e x^{2}}{25 \, c^{3}} - \frac {16 \, \sqrt {-c^{2} x^{2} + 1} b e^{3} x^{4}}{945 \, c^{5}} - \frac {2 \, \sqrt {-c^{2} x^{2} + 1} b d^{3}}{9 \, c^{3}} - \frac {24 \, \sqrt {-c^{2} x^{2} + 1} b d e^{2} x^{2}}{245 \, c^{5}} - \frac {8 \, \sqrt {-c^{2} x^{2} + 1} b d^{2} e}{25 \, c^{5}} - \frac {64 \, \sqrt {-c^{2} x^{2} + 1} b e^{3} x^{2}}{2835 \, c^{7}} - \frac {48 \, \sqrt {-c^{2} x^{2} + 1} b d e^{2}}{245 \, c^{7}} - \frac {128 \, \sqrt {-c^{2} x^{2} + 1} b e^{3}}{2835 \, c^{9}} \] Input:
integrate(x^2*(e*x^2+d)^3*(a+b*arccos(c*x)),x, algorithm="giac")
Output:
1/9*b*e^3*x^9*arccos(c*x) + 1/9*a*e^3*x^9 + 3/7*b*d*e^2*x^7*arccos(c*x) - 1/81*sqrt(-c^2*x^2 + 1)*b*e^3*x^8/c + 3/7*a*d*e^2*x^7 + 3/5*b*d^2*e*x^5*ar ccos(c*x) - 3/49*sqrt(-c^2*x^2 + 1)*b*d*e^2*x^6/c + 3/5*a*d^2*e*x^5 + 1/3* b*d^3*x^3*arccos(c*x) - 3/25*sqrt(-c^2*x^2 + 1)*b*d^2*e*x^4/c - 8/567*sqrt (-c^2*x^2 + 1)*b*e^3*x^6/c^3 + 1/3*a*d^3*x^3 - 1/9*sqrt(-c^2*x^2 + 1)*b*d^ 3*x^2/c - 18/245*sqrt(-c^2*x^2 + 1)*b*d*e^2*x^4/c^3 - 4/25*sqrt(-c^2*x^2 + 1)*b*d^2*e*x^2/c^3 - 16/945*sqrt(-c^2*x^2 + 1)*b*e^3*x^4/c^5 - 2/9*sqrt(- c^2*x^2 + 1)*b*d^3/c^3 - 24/245*sqrt(-c^2*x^2 + 1)*b*d*e^2*x^2/c^5 - 8/25* sqrt(-c^2*x^2 + 1)*b*d^2*e/c^5 - 64/2835*sqrt(-c^2*x^2 + 1)*b*e^3*x^2/c^7 - 48/245*sqrt(-c^2*x^2 + 1)*b*d*e^2/c^7 - 128/2835*sqrt(-c^2*x^2 + 1)*b*e^ 3/c^9
Timed out. \[ \int x^2 \left (d+e x^2\right )^3 (a+b \arccos (c x)) \, dx=\int x^2\,\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^3 \,d x \] Input:
int(x^2*(a + b*acos(c*x))*(d + e*x^2)^3,x)
Output:
int(x^2*(a + b*acos(c*x))*(d + e*x^2)^3, x)
Time = 0.24 (sec) , antiderivative size = 436, normalized size of antiderivative = 1.52 \[ \int x^2 \left (d+e x^2\right )^3 (a+b \arccos (c x)) \, dx=\frac {33075 \mathit {acos} \left (c x \right ) b \,c^{9} d^{3} x^{3}+59535 \mathit {acos} \left (c x \right ) b \,c^{9} d^{2} e \,x^{5}+42525 \mathit {acos} \left (c x \right ) b \,c^{9} d \,e^{2} x^{7}+11025 \mathit {acos} \left (c x \right ) b \,c^{9} e^{3} x^{9}-11025 \sqrt {-c^{2} x^{2}+1}\, b \,c^{8} d^{3} x^{2}-11907 \sqrt {-c^{2} x^{2}+1}\, b \,c^{8} d^{2} e \,x^{4}-6075 \sqrt {-c^{2} x^{2}+1}\, b \,c^{8} d \,e^{2} x^{6}-1225 \sqrt {-c^{2} x^{2}+1}\, b \,c^{8} e^{3} x^{8}-22050 \sqrt {-c^{2} x^{2}+1}\, b \,c^{6} d^{3}-15876 \sqrt {-c^{2} x^{2}+1}\, b \,c^{6} d^{2} e \,x^{2}-7290 \sqrt {-c^{2} x^{2}+1}\, b \,c^{6} d \,e^{2} x^{4}-1400 \sqrt {-c^{2} x^{2}+1}\, b \,c^{6} e^{3} x^{6}-31752 \sqrt {-c^{2} x^{2}+1}\, b \,c^{4} d^{2} e -9720 \sqrt {-c^{2} x^{2}+1}\, b \,c^{4} d \,e^{2} x^{2}-1680 \sqrt {-c^{2} x^{2}+1}\, b \,c^{4} e^{3} x^{4}-19440 \sqrt {-c^{2} x^{2}+1}\, b \,c^{2} d \,e^{2}-2240 \sqrt {-c^{2} x^{2}+1}\, b \,c^{2} e^{3} x^{2}-4480 \sqrt {-c^{2} x^{2}+1}\, b \,e^{3}+33075 a \,c^{9} d^{3} x^{3}+59535 a \,c^{9} d^{2} e \,x^{5}+42525 a \,c^{9} d \,e^{2} x^{7}+11025 a \,c^{9} e^{3} x^{9}}{99225 c^{9}} \] Input:
int(x^2*(e*x^2+d)^3*(a+b*acos(c*x)),x)
Output:
(33075*acos(c*x)*b*c**9*d**3*x**3 + 59535*acos(c*x)*b*c**9*d**2*e*x**5 + 4 2525*acos(c*x)*b*c**9*d*e**2*x**7 + 11025*acos(c*x)*b*c**9*e**3*x**9 - 110 25*sqrt( - c**2*x**2 + 1)*b*c**8*d**3*x**2 - 11907*sqrt( - c**2*x**2 + 1)* b*c**8*d**2*e*x**4 - 6075*sqrt( - c**2*x**2 + 1)*b*c**8*d*e**2*x**6 - 1225 *sqrt( - c**2*x**2 + 1)*b*c**8*e**3*x**8 - 22050*sqrt( - c**2*x**2 + 1)*b* c**6*d**3 - 15876*sqrt( - c**2*x**2 + 1)*b*c**6*d**2*e*x**2 - 7290*sqrt( - c**2*x**2 + 1)*b*c**6*d*e**2*x**4 - 1400*sqrt( - c**2*x**2 + 1)*b*c**6*e* *3*x**6 - 31752*sqrt( - c**2*x**2 + 1)*b*c**4*d**2*e - 9720*sqrt( - c**2*x **2 + 1)*b*c**4*d*e**2*x**2 - 1680*sqrt( - c**2*x**2 + 1)*b*c**4*e**3*x**4 - 19440*sqrt( - c**2*x**2 + 1)*b*c**2*d*e**2 - 2240*sqrt( - c**2*x**2 + 1 )*b*c**2*e**3*x**2 - 4480*sqrt( - c**2*x**2 + 1)*b*e**3 + 33075*a*c**9*d** 3*x**3 + 59535*a*c**9*d**2*e*x**5 + 42525*a*c**9*d*e**2*x**7 + 11025*a*c** 9*e**3*x**9)/(99225*c**9)