Integrand size = 18, antiderivative size = 317 \[ \int \left (d+e x^2\right )^4 (a+b \arccos (c x)) \, dx=-\frac {b \left (315 c^8 d^4+420 c^6 d^3 e+378 c^4 d^2 e^2+180 c^2 d e^3+35 e^4\right ) \sqrt {1-c^2 x^2}}{315 c^9}+\frac {4 b e \left (105 c^6 d^3+189 c^4 d^2 e+135 c^2 d e^2+35 e^3\right ) \left (1-c^2 x^2\right )^{3/2}}{945 c^9}-\frac {2 b e^2 \left (63 c^4 d^2+90 c^2 d e+35 e^2\right ) \left (1-c^2 x^2\right )^{5/2}}{525 c^9}+\frac {4 b e^3 \left (9 c^2 d+7 e\right ) \left (1-c^2 x^2\right )^{7/2}}{441 c^9}-\frac {b e^4 \left (1-c^2 x^2\right )^{9/2}}{81 c^9}+d^4 x (a+b \arccos (c x))+\frac {4}{3} d^3 e x^3 (a+b \arccos (c x))+\frac {6}{5} d^2 e^2 x^5 (a+b \arccos (c x))+\frac {4}{7} d e^3 x^7 (a+b \arccos (c x))+\frac {1}{9} e^4 x^9 (a+b \arccos (c x)) \] Output:
-1/315*b*(315*c^8*d^4+420*c^6*d^3*e+378*c^4*d^2*e^2+180*c^2*d*e^3+35*e^4)* (-c^2*x^2+1)^(1/2)/c^9+4/945*b*e*(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)^(3/2)/c^9-2/525*b*e^2*(63*c^4*d^2+90*c^2*d*e+35*e^2)* (-c^2*x^2+1)^(5/2)/c^9+4/441*b*e^3*(9*c^2*d+7*e)*(-c^2*x^2+1)^(7/2)/c^9-1/ 81*b*e^4*(-c^2*x^2+1)^(9/2)/c^9+d^4*x*(a+b*arccos(c*x))+4/3*d^3*e*x^3*(a+b *arccos(c*x))+6/5*d^2*e^2*x^5*(a+b*arccos(c*x))+4/7*d*e^3*x^7*(a+b*arccos( c*x))+1/9*e^4*x^9*(a+b*arccos(c*x))
Time = 0.18 (sec) , antiderivative size = 261, normalized size of antiderivative = 0.82 \[ \int \left (d+e x^2\right )^4 (a+b \arccos (c x)) \, dx=\frac {315 a x \left (315 d^4+420 d^3 e x^2+378 d^2 e^2 x^4+180 d e^3 x^6+35 e^4 x^8\right )-\frac {b \sqrt {1-c^2 x^2} \left (4480 e^4+320 c^2 e^3 \left (81 d+7 e x^2\right )+48 c^4 e^2 \left (1323 d^2+270 d e x^2+35 e^2 x^4\right )+8 c^6 e \left (11025 d^3+3969 d^2 e x^2+1215 d e^2 x^4+175 e^3 x^6\right )+c^8 \left (99225 d^4+44100 d^3 e x^2+23814 d^2 e^2 x^4+8100 d e^3 x^6+1225 e^4 x^8\right )\right )}{c^9}+315 b x \left (315 d^4+420 d^3 e x^2+378 d^2 e^2 x^4+180 d e^3 x^6+35 e^4 x^8\right ) \arccos (c x)}{99225} \] Input:
Integrate[(d + e*x^2)^4*(a + b*ArcCos[c*x]),x]
Output:
(315*a*x*(315*d^4 + 420*d^3*e*x^2 + 378*d^2*e^2*x^4 + 180*d*e^3*x^6 + 35*e ^4*x^8) - (b*Sqrt[1 - c^2*x^2]*(4480*e^4 + 320*c^2*e^3*(81*d + 7*e*x^2) + 48*c^4*e^2*(1323*d^2 + 270*d*e*x^2 + 35*e^2*x^4) + 8*c^6*e*(11025*d^3 + 39 69*d^2*e*x^2 + 1215*d*e^2*x^4 + 175*e^3*x^6) + c^8*(99225*d^4 + 44100*d^3* e*x^2 + 23814*d^2*e^2*x^4 + 8100*d*e^3*x^6 + 1225*e^4*x^8)))/c^9 + 315*b*x *(315*d^4 + 420*d^3*e*x^2 + 378*d^2*e^2*x^4 + 180*d*e^3*x^6 + 35*e^4*x^8)* ArcCos[c*x])/99225
Time = 0.74 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {5171, 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.
| \(\displaystyle \int \left (d+e x^2\right )^4 (a+b \arccos (c x)) \, dx\) |
| \(\Big \downarrow \) 5171 |
| \(\displaystyle b c \int \frac {x \left (35 e^4 x^8+180 d e^3 x^6+378 d^2 e^2 x^4+420 d^3 e x^2+315 d^4\right )}{315 \sqrt {1-c^2 x^2}}dx+d^4 x (a+b \arccos (c x))+\frac {4}{3} d^3 e x^3 (a+b \arccos (c x))+\frac {6}{5} d^2 e^2 x^5 (a+b \arccos (c x))+\frac {4}{7} d e^3 x^7 (a+b \arccos (c x))+\frac {1}{9} e^4 x^9 (a+b \arccos (c x))\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{315} b c \int \frac {x \left (35 e^4 x^8+180 d e^3 x^6+378 d^2 e^2 x^4+420 d^3 e x^2+315 d^4\right )}{\sqrt {1-c^2 x^2}}dx+d^4 x (a+b \arccos (c x))+\frac {4}{3} d^3 e x^3 (a+b \arccos (c x))+\frac {6}{5} d^2 e^2 x^5 (a+b \arccos (c x))+\frac {4}{7} d e^3 x^7 (a+b \arccos (c x))+\frac {1}{9} e^4 x^9 (a+b \arccos (c x))\) |
| \(\Big \downarrow \) 2331 |
| \(\displaystyle \frac {1}{630} b c \int \frac {35 e^4 x^8+180 d e^3 x^6+378 d^2 e^2 x^4+420 d^3 e x^2+315 d^4}{\sqrt {1-c^2 x^2}}dx^2+d^4 x (a+b \arccos (c x))+\frac {4}{3} d^3 e x^3 (a+b \arccos (c x))+\frac {6}{5} d^2 e^2 x^5 (a+b \arccos (c x))+\frac {4}{7} d e^3 x^7 (a+b \arccos (c x))+\frac {1}{9} e^4 x^9 (a+b \arccos (c x))\) |
| \(\Big \downarrow \) 2389 |
| \(\displaystyle \frac {1}{630} b c \int \left (\frac {35 \left (1-c^2 x^2\right )^{7/2} e^4}{c^8}-\frac {20 \left (9 d c^2+7 e\right ) \left (1-c^2 x^2\right )^{5/2} e^3}{c^8}+\frac {6 \left (63 d^2 c^4+90 d e c^2+35 e^2\right ) \left (1-c^2 x^2\right )^{3/2} e^2}{c^8}-\frac {4 \left (105 d^3 c^6+189 d^2 e c^4+135 d e^2 c^2+35 e^3\right ) \sqrt {1-c^2 x^2} e}{c^8}+\frac {315 d^4 c^8+420 d^3 e c^6+378 d^2 e^2 c^4+180 d e^3 c^2+35 e^4}{c^8 \sqrt {1-c^2 x^2}}\right )dx^2+d^4 x (a+b \arccos (c x))+\frac {4}{3} d^3 e x^3 (a+b \arccos (c x))+\frac {6}{5} d^2 e^2 x^5 (a+b \arccos (c x))+\frac {4}{7} d e^3 x^7 (a+b \arccos (c x))+\frac {1}{9} e^4 x^9 (a+b \arccos (c x))\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle d^4 x (a+b \arccos (c x))+\frac {4}{3} d^3 e x^3 (a+b \arccos (c x))+\frac {6}{5} d^2 e^2 x^5 (a+b \arccos (c x))+\frac {4}{7} d e^3 x^7 (a+b \arccos (c x))+\frac {1}{9} e^4 x^9 (a+b \arccos (c x))+\frac {1}{630} b c \left (\frac {40 e^3 \left (1-c^2 x^2\right )^{7/2} \left (9 c^2 d+7 e\right )}{7 c^{10}}-\frac {70 e^4 \left (1-c^2 x^2\right )^{9/2}}{9 c^{10}}-\frac {12 e^2 \left (1-c^2 x^2\right )^{5/2} \left (63 c^4 d^2+90 c^2 d e+35 e^2\right )}{5 c^{10}}+\frac {8 e \left (1-c^2 x^2\right )^{3/2} \left (105 c^6 d^3+189 c^4 d^2 e+135 c^2 d e^2+35 e^3\right )}{3 c^{10}}-\frac {2 \sqrt {1-c^2 x^2} \left (315 c^8 d^4+420 c^6 d^3 e+378 c^4 d^2 e^2+180 c^2 d e^3+35 e^4\right )}{c^{10}}\right )\) |
Input:
Int[(d + e*x^2)^4*(a + b*ArcCos[c*x]),x]
Output:
(b*c*((-2*(315*c^8*d^4 + 420*c^6*d^3*e + 378*c^4*d^2*e^2 + 180*c^2*d*e^3 + 35*e^4)*Sqrt[1 - c^2*x^2])/c^10 + (8*e*(105*c^6*d^3 + 189*c^4*d^2*e + 135 *c^2*d*e^2 + 35*e^3)*(1 - c^2*x^2)^(3/2))/(3*c^10) - (12*e^2*(63*c^4*d^2 + 90*c^2*d*e + 35*e^2)*(1 - c^2*x^2)^(5/2))/(5*c^10) + (40*e^3*(9*c^2*d + 7 *e)*(1 - c^2*x^2)^(7/2))/(7*c^10) - (70*e^4*(1 - c^2*x^2)^(9/2))/(9*c^10)) )/630 + d^4*x*(a + b*ArcCos[c*x]) + (4*d^3*e*x^3*(a + b*ArcCos[c*x]))/3 + (6*d^2*e^2*x^5*(a + b*ArcCos[c*x]))/5 + (4*d*e^3*x^7*(a + b*ArcCos[c*x]))/ 7 + (e^4*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[(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_.) + ArcCos[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbo l] :> With[{u = IntHide[(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]] /; Fr eeQ[{a, b, c, d, e}, x] && NeQ[c^2*d + e, 0] && (IGtQ[p, 0] || ILtQ[p + 1/2 , 0])
Time = 0.20 (sec) , antiderivative size = 440, normalized size of antiderivative = 1.39
| method | result | size |
| parts | \(a \left (\frac {1}{9} e^{4} x^{9}+\frac {4}{7} d \,e^{3} x^{7}+\frac {6}{5} d^{2} e^{2} x^{5}+\frac {4}{3} d^{3} e \,x^{3}+d^{4} x \right )+\frac {b \left (\frac {c \arccos \left (c x \right ) e^{4} x^{9}}{9}+\frac {4 c \arccos \left (c x \right ) d \,e^{3} x^{7}}{7}+\frac {6 c \arccos \left (c x \right ) d^{2} e^{2} x^{5}}{5}+\frac {4 c \arccos \left (c x \right ) d^{3} e \,x^{3}}{3}+\arccos \left (c x \right ) d^{4} c x +\frac {35 e^{4} \left (-\frac {c^{8} x^{8} \sqrt {-c^{2} x^{2}+1}}{9}-\frac {8 c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{63}-\frac {16 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{105}-\frac {64 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{315}-\frac {128 \sqrt {-c^{2} x^{2}+1}}{315}\right )-315 d^{4} c^{8} \sqrt {-c^{2} x^{2}+1}+180 d \,c^{2} e^{3} \left (-\frac {c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{7}-\frac {6 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {8 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {16 \sqrt {-c^{2} x^{2}+1}}{35}\right )+378 d^{2} c^{4} e^{2} \left (-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-c^{2} x^{2}+1}}{15}\right )+420 d^{3} c^{6} e \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )}{315 c^{8}}\right )}{c}\) | \(440\) |
| derivativedivides | \(\frac {\frac {a \left (d^{4} c^{9} x +\frac {4}{3} d^{3} c^{9} e \,x^{3}+\frac {6}{5} d^{2} c^{9} e^{2} x^{5}+\frac {4}{7} d \,c^{9} e^{3} x^{7}+\frac {1}{9} e^{4} c^{9} x^{9}\right )}{c^{8}}+\frac {b \left (\arccos \left (c x \right ) d^{4} c^{9} x +\frac {4 \arccos \left (c x \right ) d^{3} c^{9} e \,x^{3}}{3}+\frac {6 \arccos \left (c x \right ) d^{2} c^{9} e^{2} x^{5}}{5}+\frac {4 \arccos \left (c x \right ) d \,c^{9} e^{3} x^{7}}{7}+\frac {\arccos \left (c x \right ) e^{4} c^{9} x^{9}}{9}+\frac {e^{4} \left (-\frac {c^{8} x^{8} \sqrt {-c^{2} x^{2}+1}}{9}-\frac {8 c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{63}-\frac {16 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{105}-\frac {64 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{315}-\frac {128 \sqrt {-c^{2} x^{2}+1}}{315}\right )}{9}-d^{4} c^{8} \sqrt {-c^{2} x^{2}+1}+\frac {4 d \,c^{2} e^{3} \left (-\frac {c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{7}-\frac {6 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {8 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {16 \sqrt {-c^{2} x^{2}+1}}{35}\right )}{7}+\frac {6 d^{2} c^{4} e^{2} \left (-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-c^{2} x^{2}+1}}{15}\right )}{5}+\frac {4 d^{3} c^{6} e \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )}{3}\right )}{c^{8}}}{c}\) | \(466\) |
| default | \(\frac {\frac {a \left (d^{4} c^{9} x +\frac {4}{3} d^{3} c^{9} e \,x^{3}+\frac {6}{5} d^{2} c^{9} e^{2} x^{5}+\frac {4}{7} d \,c^{9} e^{3} x^{7}+\frac {1}{9} e^{4} c^{9} x^{9}\right )}{c^{8}}+\frac {b \left (\arccos \left (c x \right ) d^{4} c^{9} x +\frac {4 \arccos \left (c x \right ) d^{3} c^{9} e \,x^{3}}{3}+\frac {6 \arccos \left (c x \right ) d^{2} c^{9} e^{2} x^{5}}{5}+\frac {4 \arccos \left (c x \right ) d \,c^{9} e^{3} x^{7}}{7}+\frac {\arccos \left (c x \right ) e^{4} c^{9} x^{9}}{9}+\frac {e^{4} \left (-\frac {c^{8} x^{8} \sqrt {-c^{2} x^{2}+1}}{9}-\frac {8 c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{63}-\frac {16 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{105}-\frac {64 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{315}-\frac {128 \sqrt {-c^{2} x^{2}+1}}{315}\right )}{9}-d^{4} c^{8} \sqrt {-c^{2} x^{2}+1}+\frac {4 d \,c^{2} e^{3} \left (-\frac {c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{7}-\frac {6 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {8 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {16 \sqrt {-c^{2} x^{2}+1}}{35}\right )}{7}+\frac {6 d^{2} c^{4} e^{2} \left (-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-c^{2} x^{2}+1}}{15}\right )}{5}+\frac {4 d^{3} c^{6} e \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )}{3}\right )}{c^{8}}}{c}\) | \(466\) |
| orering | \(\frac {x \left (20825 c^{10} e^{5} x^{10}+132525 c^{10} d \,e^{4} x^{8}+366282 c^{10} d^{2} e^{3} x^{6}+1400 c^{8} e^{5} x^{8}+604170 c^{10} d^{3} e^{2} x^{4}+12960 c^{8} d \,e^{4} x^{6}+1025325 c^{10} d^{4} e \,x^{2}+63504 c^{8} d^{2} e^{3} x^{4}+2240 c^{6} e^{5} x^{6}+99225 c^{10} d^{5}+352800 c^{8} d^{3} e^{2} x^{2}+25920 c^{6} d \,e^{4} x^{4}-793800 c^{8} d^{4} e +254016 c^{6} d^{2} e^{3} x^{2}+4480 c^{4} e^{5} x^{4}-705600 c^{6} d^{3} e^{2}+103680 c^{4} d \,e^{4} x^{2}-508032 c^{4} d^{2} e^{3}+17920 c^{2} e^{5} x^{2}-207360 c^{2} d \,e^{4}-35840 e^{5}\right ) \left (a +b \arccos \left (c x \right )\right )}{99225 \left (e \,x^{2}+d \right ) c^{10}}-\frac {\left (1225 e^{4} x^{8} c^{8}+8100 c^{8} d \,e^{3} x^{6}+23814 c^{8} d^{2} e^{2} x^{4}+1400 c^{6} e^{4} x^{6}+44100 c^{8} d^{3} e \,x^{2}+9720 c^{6} d \,e^{3} x^{4}+99225 c^{8} d^{4}+31752 c^{6} d^{2} e^{2} x^{2}+1680 c^{4} e^{4} x^{4}+88200 c^{6} d^{3} e +12960 c^{4} d \,e^{3} x^{2}+63504 c^{4} d^{2} e^{2}+2240 c^{2} e^{4} x^{2}+25920 c^{2} d \,e^{3}+4480 e^{4}\right ) \left (c x -1\right ) \left (c x +1\right ) \left (8 \left (e \,x^{2}+d \right )^{3} \left (a +b \arccos \left (c x \right )\right ) e x -\frac {\left (e \,x^{2}+d \right )^{4} b c}{\sqrt {-c^{2} x^{2}+1}}\right )}{99225 c^{10} \left (e \,x^{2}+d \right )^{4}}\) | \(498\) |
Input:
int((e*x^2+d)^4*(a+b*arccos(c*x)),x,method=_RETURNVERBOSE)
Output:
a*(1/9*e^4*x^9+4/7*d*e^3*x^7+6/5*d^2*e^2*x^5+4/3*d^3*e*x^3+d^4*x)+b/c*(1/9 *c*arccos(c*x)*e^4*x^9+4/7*c*arccos(c*x)*d*e^3*x^7+6/5*c*arccos(c*x)*d^2*e ^2*x^5+4/3*c*arccos(c*x)*d^3*e*x^3+arccos(c*x)*d^4*c*x+1/315/c^8*(35*e^4*( -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))-315*d^4*c^8*(-c^2*x^2+1)^(1/2)+180*d*c^2*e^3*(-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))+378*d^2*c^4*e^2*(-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))+420*d^3*c^ 6*e*(-1/3*c^2*x^2*(-c^2*x^2+1)^(1/2)-2/3*(-c^2*x^2+1)^(1/2))))
Time = 0.15 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.02 \[ \int \left (d+e x^2\right )^4 (a+b \arccos (c x)) \, dx=\frac {11025 \, a c^{9} e^{4} x^{9} + 56700 \, a c^{9} d e^{3} x^{7} + 119070 \, a c^{9} d^{2} e^{2} x^{5} + 132300 \, a c^{9} d^{3} e x^{3} + 99225 \, a c^{9} d^{4} x + 315 \, {\left (35 \, b c^{9} e^{4} x^{9} + 180 \, b c^{9} d e^{3} x^{7} + 378 \, b c^{9} d^{2} e^{2} x^{5} + 420 \, b c^{9} d^{3} e x^{3} + 315 \, b c^{9} d^{4} x\right )} \arccos \left (c x\right ) - {\left (1225 \, b c^{8} e^{4} x^{8} + 99225 \, b c^{8} d^{4} + 88200 \, b c^{6} d^{3} e + 63504 \, b c^{4} d^{2} e^{2} + 25920 \, b c^{2} d e^{3} + 100 \, {\left (81 \, b c^{8} d e^{3} + 14 \, b c^{6} e^{4}\right )} x^{6} + 4480 \, b e^{4} + 6 \, {\left (3969 \, b c^{8} d^{2} e^{2} + 1620 \, b c^{6} d e^{3} + 280 \, b c^{4} e^{4}\right )} x^{4} + 4 \, {\left (11025 \, b c^{8} d^{3} e + 7938 \, b c^{6} d^{2} e^{2} + 3240 \, b c^{4} d e^{3} + 560 \, b c^{2} e^{4}\right )} x^{2}\right )} \sqrt {-c^{2} x^{2} + 1}}{99225 \, c^{9}} \] Input:
integrate((e*x^2+d)^4*(a+b*arccos(c*x)),x, algorithm="fricas")
Output:
1/99225*(11025*a*c^9*e^4*x^9 + 56700*a*c^9*d*e^3*x^7 + 119070*a*c^9*d^2*e^ 2*x^5 + 132300*a*c^9*d^3*e*x^3 + 99225*a*c^9*d^4*x + 315*(35*b*c^9*e^4*x^9 + 180*b*c^9*d*e^3*x^7 + 378*b*c^9*d^2*e^2*x^5 + 420*b*c^9*d^3*e*x^3 + 315 *b*c^9*d^4*x)*arccos(c*x) - (1225*b*c^8*e^4*x^8 + 99225*b*c^8*d^4 + 88200* b*c^6*d^3*e + 63504*b*c^4*d^2*e^2 + 25920*b*c^2*d*e^3 + 100*(81*b*c^8*d*e^ 3 + 14*b*c^6*e^4)*x^6 + 4480*b*e^4 + 6*(3969*b*c^8*d^2*e^2 + 1620*b*c^6*d* e^3 + 280*b*c^4*e^4)*x^4 + 4*(11025*b*c^8*d^3*e + 7938*b*c^6*d^2*e^2 + 324 0*b*c^4*d*e^3 + 560*b*c^2*e^4)*x^2)*sqrt(-c^2*x^2 + 1))/c^9
Time = 1.16 (sec) , antiderivative size = 598, normalized size of antiderivative = 1.89 \[ \int \left (d+e x^2\right )^4 (a+b \arccos (c x)) \, dx=\begin {cases} a d^{4} x + \frac {4 a d^{3} e x^{3}}{3} + \frac {6 a d^{2} e^{2} x^{5}}{5} + \frac {4 a d e^{3} x^{7}}{7} + \frac {a e^{4} x^{9}}{9} + b d^{4} x \operatorname {acos}{\left (c x \right )} + \frac {4 b d^{3} e x^{3} \operatorname {acos}{\left (c x \right )}}{3} + \frac {6 b d^{2} e^{2} x^{5} \operatorname {acos}{\left (c x \right )}}{5} + \frac {4 b d e^{3} x^{7} \operatorname {acos}{\left (c x \right )}}{7} + \frac {b e^{4} x^{9} \operatorname {acos}{\left (c x \right )}}{9} - \frac {b d^{4} \sqrt {- c^{2} x^{2} + 1}}{c} - \frac {4 b d^{3} e x^{2} \sqrt {- c^{2} x^{2} + 1}}{9 c} - \frac {6 b d^{2} e^{2} x^{4} \sqrt {- c^{2} x^{2} + 1}}{25 c} - \frac {4 b d e^{3} x^{6} \sqrt {- c^{2} x^{2} + 1}}{49 c} - \frac {b e^{4} x^{8} \sqrt {- c^{2} x^{2} + 1}}{81 c} - \frac {8 b d^{3} e \sqrt {- c^{2} x^{2} + 1}}{9 c^{3}} - \frac {8 b d^{2} e^{2} x^{2} \sqrt {- c^{2} x^{2} + 1}}{25 c^{3}} - \frac {24 b d e^{3} x^{4} \sqrt {- c^{2} x^{2} + 1}}{245 c^{3}} - \frac {8 b e^{4} x^{6} \sqrt {- c^{2} x^{2} + 1}}{567 c^{3}} - \frac {16 b d^{2} e^{2} \sqrt {- c^{2} x^{2} + 1}}{25 c^{5}} - \frac {32 b d e^{3} x^{2} \sqrt {- c^{2} x^{2} + 1}}{245 c^{5}} - \frac {16 b e^{4} x^{4} \sqrt {- c^{2} x^{2} + 1}}{945 c^{5}} - \frac {64 b d e^{3} \sqrt {- c^{2} x^{2} + 1}}{245 c^{7}} - \frac {64 b e^{4} x^{2} \sqrt {- c^{2} x^{2} + 1}}{2835 c^{7}} - \frac {128 b e^{4} \sqrt {- c^{2} x^{2} + 1}}{2835 c^{9}} & \text {for}\: c \neq 0 \\\left (a + \frac {\pi b}{2}\right ) \left (d^{4} x + \frac {4 d^{3} e x^{3}}{3} + \frac {6 d^{2} e^{2} x^{5}}{5} + \frac {4 d e^{3} x^{7}}{7} + \frac {e^{4} x^{9}}{9}\right ) & \text {otherwise} \end {cases} \] Input:
integrate((e*x**2+d)**4*(a+b*acos(c*x)),x)
Output:
Piecewise((a*d**4*x + 4*a*d**3*e*x**3/3 + 6*a*d**2*e**2*x**5/5 + 4*a*d*e** 3*x**7/7 + a*e**4*x**9/9 + b*d**4*x*acos(c*x) + 4*b*d**3*e*x**3*acos(c*x)/ 3 + 6*b*d**2*e**2*x**5*acos(c*x)/5 + 4*b*d*e**3*x**7*acos(c*x)/7 + b*e**4* x**9*acos(c*x)/9 - b*d**4*sqrt(-c**2*x**2 + 1)/c - 4*b*d**3*e*x**2*sqrt(-c **2*x**2 + 1)/(9*c) - 6*b*d**2*e**2*x**4*sqrt(-c**2*x**2 + 1)/(25*c) - 4*b *d*e**3*x**6*sqrt(-c**2*x**2 + 1)/(49*c) - b*e**4*x**8*sqrt(-c**2*x**2 + 1 )/(81*c) - 8*b*d**3*e*sqrt(-c**2*x**2 + 1)/(9*c**3) - 8*b*d**2*e**2*x**2*s qrt(-c**2*x**2 + 1)/(25*c**3) - 24*b*d*e**3*x**4*sqrt(-c**2*x**2 + 1)/(245 *c**3) - 8*b*e**4*x**6*sqrt(-c**2*x**2 + 1)/(567*c**3) - 16*b*d**2*e**2*sq rt(-c**2*x**2 + 1)/(25*c**5) - 32*b*d*e**3*x**2*sqrt(-c**2*x**2 + 1)/(245* c**5) - 16*b*e**4*x**4*sqrt(-c**2*x**2 + 1)/(945*c**5) - 64*b*d*e**3*sqrt( -c**2*x**2 + 1)/(245*c**7) - 64*b*e**4*x**2*sqrt(-c**2*x**2 + 1)/(2835*c** 7) - 128*b*e**4*sqrt(-c**2*x**2 + 1)/(2835*c**9), Ne(c, 0)), ((a + pi*b/2) *(d**4*x + 4*d**3*e*x**3/3 + 6*d**2*e**2*x**5/5 + 4*d*e**3*x**7/7 + e**4*x **9/9), True))
Time = 0.12 (sec) , antiderivative size = 430, normalized size of antiderivative = 1.36 \[ \int \left (d+e x^2\right )^4 (a+b \arccos (c x)) \, dx=\frac {1}{9} \, a e^{4} x^{9} + \frac {4}{7} \, a d e^{3} x^{7} + \frac {6}{5} \, a d^{2} e^{2} x^{5} + \frac {4}{3} \, a d^{3} e x^{3} + \frac {4}{9} \, {\left (3 \, x^{3} \arccos \left (c x\right ) - c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b d^{3} e + \frac {2}{25} \, {\left (15 \, x^{5} \arccos \left (c x\right ) - {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} b d^{2} e^{2} + \frac {4}{245} \, {\left (35 \, x^{7} \arccos \left (c x\right ) - {\left (\frac {5 \, \sqrt {-c^{2} x^{2} + 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {-c^{2} x^{2} + 1}}{c^{8}}\right )} c\right )} b d e^{3} + \frac {1}{2835} \, {\left (315 \, x^{9} \arccos \left (c x\right ) - {\left (\frac {35 \, \sqrt {-c^{2} x^{2} + 1} x^{8}}{c^{2}} + \frac {40 \, \sqrt {-c^{2} x^{2} + 1} x^{6}}{c^{4}} + \frac {48 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{6}} + \frac {64 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{8}} + \frac {128 \, \sqrt {-c^{2} x^{2} + 1}}{c^{10}}\right )} c\right )} b e^{4} + a d^{4} x + \frac {{\left (c x \arccos \left (c x\right ) - \sqrt {-c^{2} x^{2} + 1}\right )} b d^{4}}{c} \] Input:
integrate((e*x^2+d)^4*(a+b*arccos(c*x)),x, algorithm="maxima")
Output:
1/9*a*e^4*x^9 + 4/7*a*d*e^3*x^7 + 6/5*a*d^2*e^2*x^5 + 4/3*a*d^3*e*x^3 + 4/ 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*e + 2/25*(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*e^2 + 4/245*(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^3 + 1/2835*(315*x^9*arccos(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*sqr t(-c^2*x^2 + 1)*x^2/c^8 + 128*sqrt(-c^2*x^2 + 1)/c^10)*c)*b*e^4 + a*d^4*x + (c*x*arccos(c*x) - sqrt(-c^2*x^2 + 1))*b*d^4/c
Time = 0.14 (sec) , antiderivative size = 475, normalized size of antiderivative = 1.50 \[ \int \left (d+e x^2\right )^4 (a+b \arccos (c x)) \, dx=\frac {1}{9} \, b e^{4} x^{9} \arccos \left (c x\right ) + \frac {1}{9} \, a e^{4} x^{9} + \frac {4}{7} \, b d e^{3} x^{7} \arccos \left (c x\right ) - \frac {\sqrt {-c^{2} x^{2} + 1} b e^{4} x^{8}}{81 \, c} + \frac {4}{7} \, a d e^{3} x^{7} + \frac {6}{5} \, b d^{2} e^{2} x^{5} \arccos \left (c x\right ) - \frac {4 \, \sqrt {-c^{2} x^{2} + 1} b d e^{3} x^{6}}{49 \, c} + \frac {6}{5} \, a d^{2} e^{2} x^{5} + \frac {4}{3} \, b d^{3} e x^{3} \arccos \left (c x\right ) - \frac {6 \, \sqrt {-c^{2} x^{2} + 1} b d^{2} e^{2} x^{4}}{25 \, c} - \frac {8 \, \sqrt {-c^{2} x^{2} + 1} b e^{4} x^{6}}{567 \, c^{3}} + \frac {4}{3} \, a d^{3} e x^{3} + b d^{4} x \arccos \left (c x\right ) - \frac {4 \, \sqrt {-c^{2} x^{2} + 1} b d^{3} e x^{2}}{9 \, c} - \frac {24 \, \sqrt {-c^{2} x^{2} + 1} b d e^{3} x^{4}}{245 \, c^{3}} + a d^{4} x - \frac {\sqrt {-c^{2} x^{2} + 1} b d^{4}}{c} - \frac {8 \, \sqrt {-c^{2} x^{2} + 1} b d^{2} e^{2} x^{2}}{25 \, c^{3}} - \frac {16 \, \sqrt {-c^{2} x^{2} + 1} b e^{4} x^{4}}{945 \, c^{5}} - \frac {8 \, \sqrt {-c^{2} x^{2} + 1} b d^{3} e}{9 \, c^{3}} - \frac {32 \, \sqrt {-c^{2} x^{2} + 1} b d e^{3} x^{2}}{245 \, c^{5}} - \frac {16 \, \sqrt {-c^{2} x^{2} + 1} b d^{2} e^{2}}{25 \, c^{5}} - \frac {64 \, \sqrt {-c^{2} x^{2} + 1} b e^{4} x^{2}}{2835 \, c^{7}} - \frac {64 \, \sqrt {-c^{2} x^{2} + 1} b d e^{3}}{245 \, c^{7}} - \frac {128 \, \sqrt {-c^{2} x^{2} + 1} b e^{4}}{2835 \, c^{9}} \] Input:
integrate((e*x^2+d)^4*(a+b*arccos(c*x)),x, algorithm="giac")
Output:
1/9*b*e^4*x^9*arccos(c*x) + 1/9*a*e^4*x^9 + 4/7*b*d*e^3*x^7*arccos(c*x) - 1/81*sqrt(-c^2*x^2 + 1)*b*e^4*x^8/c + 4/7*a*d*e^3*x^7 + 6/5*b*d^2*e^2*x^5* arccos(c*x) - 4/49*sqrt(-c^2*x^2 + 1)*b*d*e^3*x^6/c + 6/5*a*d^2*e^2*x^5 + 4/3*b*d^3*e*x^3*arccos(c*x) - 6/25*sqrt(-c^2*x^2 + 1)*b*d^2*e^2*x^4/c - 8/ 567*sqrt(-c^2*x^2 + 1)*b*e^4*x^6/c^3 + 4/3*a*d^3*e*x^3 + b*d^4*x*arccos(c* x) - 4/9*sqrt(-c^2*x^2 + 1)*b*d^3*e*x^2/c - 24/245*sqrt(-c^2*x^2 + 1)*b*d* e^3*x^4/c^3 + a*d^4*x - sqrt(-c^2*x^2 + 1)*b*d^4/c - 8/25*sqrt(-c^2*x^2 + 1)*b*d^2*e^2*x^2/c^3 - 16/945*sqrt(-c^2*x^2 + 1)*b*e^4*x^4/c^5 - 8/9*sqrt( -c^2*x^2 + 1)*b*d^3*e/c^3 - 32/245*sqrt(-c^2*x^2 + 1)*b*d*e^3*x^2/c^5 - 16 /25*sqrt(-c^2*x^2 + 1)*b*d^2*e^2/c^5 - 64/2835*sqrt(-c^2*x^2 + 1)*b*e^4*x^ 2/c^7 - 64/245*sqrt(-c^2*x^2 + 1)*b*d*e^3/c^7 - 128/2835*sqrt(-c^2*x^2 + 1 )*b*e^4/c^9
Timed out. \[ \int \left (d+e x^2\right )^4 (a+b \arccos (c x)) \, dx=\int \left (a+b\,\mathrm {acos}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^4 \,d x \] Input:
int((a + b*acos(c*x))*(d + e*x^2)^4,x)
Output:
int((a + b*acos(c*x))*(d + e*x^2)^4, x)
Time = 0.24 (sec) , antiderivative size = 494, normalized size of antiderivative = 1.56 \[ \int \left (d+e x^2\right )^4 (a+b \arccos (c x)) \, dx=\frac {-99225 \sqrt {-c^{2} x^{2}+1}\, b \,c^{8} d^{4}+99225 a \,c^{9} d^{4} x +11025 a \,c^{9} e^{4} x^{9}+99225 \mathit {acos} \left (c x \right ) b \,c^{9} d^{4} x +132300 \mathit {acos} \left (c x \right ) b \,c^{9} d^{3} e \,x^{3}+119070 \mathit {acos} \left (c x \right ) b \,c^{9} d^{2} e^{2} x^{5}+56700 \mathit {acos} \left (c x \right ) b \,c^{9} d \,e^{3} x^{7}-44100 \sqrt {-c^{2} x^{2}+1}\, b \,c^{8} d^{3} e \,x^{2}-23814 \sqrt {-c^{2} x^{2}+1}\, b \,c^{8} d^{2} e^{2} x^{4}-8100 \sqrt {-c^{2} x^{2}+1}\, b \,c^{8} d \,e^{3} x^{6}-31752 \sqrt {-c^{2} x^{2}+1}\, b \,c^{6} d^{2} e^{2} x^{2}-9720 \sqrt {-c^{2} x^{2}+1}\, b \,c^{6} d \,e^{3} x^{4}-4480 \sqrt {-c^{2} x^{2}+1}\, b \,e^{4}+11025 \mathit {acos} \left (c x \right ) b \,c^{9} e^{4} x^{9}-1225 \sqrt {-c^{2} x^{2}+1}\, b \,c^{8} e^{4} x^{8}-88200 \sqrt {-c^{2} x^{2}+1}\, b \,c^{6} d^{3} e -1400 \sqrt {-c^{2} x^{2}+1}\, b \,c^{6} e^{4} x^{6}-63504 \sqrt {-c^{2} x^{2}+1}\, b \,c^{4} d^{2} e^{2}-1680 \sqrt {-c^{2} x^{2}+1}\, b \,c^{4} e^{4} x^{4}-25920 \sqrt {-c^{2} x^{2}+1}\, b \,c^{2} d \,e^{3}-2240 \sqrt {-c^{2} x^{2}+1}\, b \,c^{2} e^{4} x^{2}+132300 a \,c^{9} d^{3} e \,x^{3}+119070 a \,c^{9} d^{2} e^{2} x^{5}+56700 a \,c^{9} d \,e^{3} x^{7}-12960 \sqrt {-c^{2} x^{2}+1}\, b \,c^{4} d \,e^{3} x^{2}}{99225 c^{9}} \] Input:
int((e*x^2+d)^4*(a+b*acos(c*x)),x)
Output:
(99225*acos(c*x)*b*c**9*d**4*x + 132300*acos(c*x)*b*c**9*d**3*e*x**3 + 119 070*acos(c*x)*b*c**9*d**2*e**2*x**5 + 56700*acos(c*x)*b*c**9*d*e**3*x**7 + 11025*acos(c*x)*b*c**9*e**4*x**9 - 99225*sqrt( - c**2*x**2 + 1)*b*c**8*d* *4 - 44100*sqrt( - c**2*x**2 + 1)*b*c**8*d**3*e*x**2 - 23814*sqrt( - c**2* x**2 + 1)*b*c**8*d**2*e**2*x**4 - 8100*sqrt( - c**2*x**2 + 1)*b*c**8*d*e** 3*x**6 - 1225*sqrt( - c**2*x**2 + 1)*b*c**8*e**4*x**8 - 88200*sqrt( - c**2 *x**2 + 1)*b*c**6*d**3*e - 31752*sqrt( - c**2*x**2 + 1)*b*c**6*d**2*e**2*x **2 - 9720*sqrt( - c**2*x**2 + 1)*b*c**6*d*e**3*x**4 - 1400*sqrt( - c**2*x **2 + 1)*b*c**6*e**4*x**6 - 63504*sqrt( - c**2*x**2 + 1)*b*c**4*d**2*e**2 - 12960*sqrt( - c**2*x**2 + 1)*b*c**4*d*e**3*x**2 - 1680*sqrt( - c**2*x**2 + 1)*b*c**4*e**4*x**4 - 25920*sqrt( - c**2*x**2 + 1)*b*c**2*d*e**3 - 2240 *sqrt( - c**2*x**2 + 1)*b*c**2*e**4*x**2 - 4480*sqrt( - c**2*x**2 + 1)*b*e **4 + 99225*a*c**9*d**4*x + 132300*a*c**9*d**3*e*x**3 + 119070*a*c**9*d**2 *e**2*x**5 + 56700*a*c**9*d*e**3*x**7 + 11025*a*c**9*e**4*x**9)/(99225*c** 9)