Integrand size = 21, antiderivative size = 198 \[ \int x^2 \left (d+e x^2\right )^2 (a+b \arccos (c x)) \, dx=\frac {b \left (35 c^4 d^2+42 c^2 d e+15 e^2\right ) \sqrt {1-c^2 x^2}}{105 c^7}-\frac {b \left (35 c^4 d^2+84 c^2 d e+45 e^2\right ) \left (1-c^2 x^2\right )^{3/2}}{315 c^7}+\frac {b e \left (14 c^2 d+15 e\right ) \left (1-c^2 x^2\right )^{5/2}}{175 c^7}-\frac {b e^2 \left (1-c^2 x^2\right )^{7/2}}{49 c^7}+\frac {1}{3} d^2 x^3 (a+b \arccos (c x))+\frac {2}{5} d e x^5 (a+b \arccos (c x))+\frac {1}{7} e^2 x^7 (a+b \arccos (c x)) \] Output:
1/105*b*(35*c^4*d^2+42*c^2*d*e+15*e^2)*(-c^2*x^2+1)^(1/2)/c^7-1/315*b*(35* c^4*d^2+84*c^2*d*e+45*e^2)*(-c^2*x^2+1)^(3/2)/c^7+1/175*b*e*(14*c^2*d+15*e )*(-c^2*x^2+1)^(5/2)/c^7-1/49*b*e^2*(-c^2*x^2+1)^(7/2)/c^7+1/3*d^2*x^3*(a+ b*arccos(c*x))+2/5*d*e*x^5*(a+b*arccos(c*x))+1/7*e^2*x^7*(a+b*arccos(c*x))
Time = 0.12 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.80 \[ \int x^2 \left (d+e x^2\right )^2 (a+b \arccos (c x)) \, dx=\frac {105 a x^3 \left (35 d^2+42 d e x^2+15 e^2 x^4\right )-\frac {b \sqrt {1-c^2 x^2} \left (720 e^2+24 c^2 e \left (98 d+15 e x^2\right )+2 c^4 \left (1225 d^2+588 d e x^2+135 e^2 x^4\right )+c^6 \left (1225 d^2 x^2+882 d e x^4+225 e^2 x^6\right )\right )}{c^7}+105 b x^3 \left (35 d^2+42 d e x^2+15 e^2 x^4\right ) \arccos (c x)}{11025} \] Input:
Integrate[x^2*(d + e*x^2)^2*(a + b*ArcCos[c*x]),x]
Output:
(105*a*x^3*(35*d^2 + 42*d*e*x^2 + 15*e^2*x^4) - (b*Sqrt[1 - c^2*x^2]*(720* e^2 + 24*c^2*e*(98*d + 15*e*x^2) + 2*c^4*(1225*d^2 + 588*d*e*x^2 + 135*e^2 *x^4) + c^6*(1225*d^2*x^2 + 882*d*e*x^4 + 225*e^2*x^6)))/c^7 + 105*b*x^3*( 35*d^2 + 42*d*e*x^2 + 15*e^2*x^4)*ArcCos[c*x])/11025
Time = 0.49 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.01, 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.
| \(\displaystyle \int x^2 \left (d+e x^2\right )^2 (a+b \arccos (c x)) \, dx\) |
| \(\Big \downarrow \) 5231 |
| \(\displaystyle b c \int \frac {x^3 \left (15 e^2 x^4+42 d e x^2+35 d^2\right )}{105 \sqrt {1-c^2 x^2}}dx+\frac {1}{3} d^2 x^3 (a+b \arccos (c x))+\frac {2}{5} d e x^5 (a+b \arccos (c x))+\frac {1}{7} e^2 x^7 (a+b \arccos (c x))\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{105} b c \int \frac {x^3 \left (15 e^2 x^4+42 d e x^2+35 d^2\right )}{\sqrt {1-c^2 x^2}}dx+\frac {1}{3} d^2 x^3 (a+b \arccos (c x))+\frac {2}{5} d e x^5 (a+b \arccos (c x))+\frac {1}{7} e^2 x^7 (a+b \arccos (c x))\) |
| \(\Big \downarrow \) 1578 |
| \(\displaystyle \frac {1}{210} b c \int \frac {x^2 \left (15 e^2 x^4+42 d e x^2+35 d^2\right )}{\sqrt {1-c^2 x^2}}dx^2+\frac {1}{3} d^2 x^3 (a+b \arccos (c x))+\frac {2}{5} d e x^5 (a+b \arccos (c x))+\frac {1}{7} e^2 x^7 (a+b \arccos (c x))\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \frac {1}{210} b c \int \left (-\frac {15 e^2 \left (1-c^2 x^2\right )^{5/2}}{c^6}+\frac {3 e \left (14 d c^2+15 e\right ) \left (1-c^2 x^2\right )^{3/2}}{c^6}+\frac {\left (-35 d^2 c^4-84 d e c^2-45 e^2\right ) \sqrt {1-c^2 x^2}}{c^6}+\frac {35 d^2 c^4+42 d e c^2+15 e^2}{c^6 \sqrt {1-c^2 x^2}}\right )dx^2+\frac {1}{3} d^2 x^3 (a+b \arccos (c x))+\frac {2}{5} d e x^5 (a+b \arccos (c x))+\frac {1}{7} e^2 x^7 (a+b \arccos (c x))\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{3} d^2 x^3 (a+b \arccos (c x))+\frac {2}{5} d e x^5 (a+b \arccos (c x))+\frac {1}{7} e^2 x^7 (a+b \arccos (c x))+\frac {1}{210} b c \left (-\frac {6 e \left (1-c^2 x^2\right )^{5/2} \left (14 c^2 d+15 e\right )}{5 c^8}+\frac {30 e^2 \left (1-c^2 x^2\right )^{7/2}}{7 c^8}+\frac {2 \left (1-c^2 x^2\right )^{3/2} \left (35 c^4 d^2+84 c^2 d e+45 e^2\right )}{3 c^8}-\frac {2 \sqrt {1-c^2 x^2} \left (35 c^4 d^2+42 c^2 d e+15 e^2\right )}{c^8}\right )\) |
Input:
Int[x^2*(d + e*x^2)^2*(a + b*ArcCos[c*x]),x]
Output:
(b*c*((-2*(35*c^4*d^2 + 42*c^2*d*e + 15*e^2)*Sqrt[1 - c^2*x^2])/c^8 + (2*( 35*c^4*d^2 + 84*c^2*d*e + 45*e^2)*(1 - c^2*x^2)^(3/2))/(3*c^8) - (6*e*(14* c^2*d + 15*e)*(1 - c^2*x^2)^(5/2))/(5*c^8) + (30*e^2*(1 - c^2*x^2)^(7/2))/ (7*c^8)))/210 + (d^2*x^3*(a + b*ArcCos[c*x]))/3 + (2*d*e*x^5*(a + b*ArcCos [c*x]))/5 + (e^2*x^7*(a + b*ArcCos[c*x]))/7
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 = 0.40 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.36
| method | result | size |
| parts | \(a \left (\frac {1}{7} e^{2} x^{7}+\frac {2}{5} d e \,x^{5}+\frac {1}{3} d^{2} x^{3}\right )+\frac {b \left (\frac {c^{3} \arccos \left (c x \right ) e^{2} x^{7}}{7}+\frac {2 c^{3} \arccos \left (c x \right ) d e \,x^{5}}{5}+\frac {\arccos \left (c x \right ) c^{3} x^{3} d^{2}}{3}+\frac {15 e^{2} \left (-\frac {c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{7}-\frac {6 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {8 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {16 \sqrt {-c^{2} x^{2}+1}}{35}\right )+35 d^{2} c^{4} \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )+42 d \,c^{2} e \left (-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-c^{2} x^{2}+1}}{15}\right )}{105 c^{4}}\right )}{c^{3}}\) | \(269\) |
| derivativedivides | \(\frac {\frac {a \left (\frac {1}{3} d^{2} c^{7} x^{3}+\frac {2}{5} d \,c^{7} e \,x^{5}+\frac {1}{7} e^{2} c^{7} x^{7}\right )}{c^{4}}+\frac {b \left (\frac {\arccos \left (c x \right ) d^{2} c^{7} x^{3}}{3}+\frac {2 \arccos \left (c x \right ) d \,c^{7} e \,x^{5}}{5}+\frac {\arccos \left (c x \right ) e^{2} c^{7} x^{7}}{7}+\frac {e^{2} \left (-\frac {c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{7}-\frac {6 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {8 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {16 \sqrt {-c^{2} x^{2}+1}}{35}\right )}{7}+\frac {d^{2} c^{4} \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )}{3}+\frac {2 d \,c^{2} e \left (-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-c^{2} x^{2}+1}}{15}\right )}{5}\right )}{c^{4}}}{c^{3}}\) | \(279\) |
| default | \(\frac {\frac {a \left (\frac {1}{3} d^{2} c^{7} x^{3}+\frac {2}{5} d \,c^{7} e \,x^{5}+\frac {1}{7} e^{2} c^{7} x^{7}\right )}{c^{4}}+\frac {b \left (\frac {\arccos \left (c x \right ) d^{2} c^{7} x^{3}}{3}+\frac {2 \arccos \left (c x \right ) d \,c^{7} e \,x^{5}}{5}+\frac {\arccos \left (c x \right ) e^{2} c^{7} x^{7}}{7}+\frac {e^{2} \left (-\frac {c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{7}-\frac {6 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {8 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {16 \sqrt {-c^{2} x^{2}+1}}{35}\right )}{7}+\frac {d^{2} c^{4} \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )}{3}+\frac {2 d \,c^{2} e \left (-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-c^{2} x^{2}+1}}{15}\right )}{5}\right )}{c^{4}}}{c^{3}}\) | \(279\) |
| orering | \(\frac {\left (2925 c^{8} e^{3} x^{10}+11727 c^{8} d \,e^{2} x^{8}+17199 c^{8} d^{2} e \,x^{6}+270 x^{8} e^{3} c^{6}+6125 c^{8} d^{3} x^{4}+1854 x^{6} e^{2} c^{6} d +7938 x^{4} e \,c^{6} d^{2}+540 x^{6} e^{3} c^{4}+2450 c^{6} d^{3} x^{2}+7236 x^{4} e^{2} c^{4} d -12348 x^{2} e \,c^{4} d^{2}+2160 x^{4} e^{3} c^{2}-4900 c^{4} d^{3}-13392 x^{2} e^{2} c^{2} d -4704 c^{2} d^{2} e -4320 x^{2} e^{3}-1440 d \,e^{2}\right ) \left (a +b \arccos \left (c x \right )\right )}{11025 \left (e \,x^{2}+d \right ) x \,c^{8}}-\frac {\left (225 x^{6} c^{6} e^{2}+882 x^{4} c^{6} d e +1225 x^{2} c^{6} d^{2}+270 e^{2} x^{4} c^{4}+1176 c^{4} d e \,x^{2}+2450 c^{4} d^{2}+360 c^{2} e^{2} x^{2}+2352 c^{2} d e +720 e^{2}\right ) \left (c x -1\right ) \left (c x +1\right ) \left (2 x \left (e \,x^{2}+d \right )^{2} \left (a +b \arccos \left (c x \right )\right )+4 x^{3} \left (e \,x^{2}+d \right ) \left (a +b \arccos \left (c x \right )\right ) e -\frac {x^{2} \left (e \,x^{2}+d \right )^{2} b c}{\sqrt {-c^{2} x^{2}+1}}\right )}{11025 c^{8} x^{2} \left (e \,x^{2}+d \right )^{2}}\) | \(391\) |
Input:
int(x^2*(e*x^2+d)^2*(a+b*arccos(c*x)),x,method=_RETURNVERBOSE)
Output:
a*(1/7*e^2*x^7+2/5*d*e*x^5+1/3*d^2*x^3)+b/c^3*(1/7*c^3*arccos(c*x)*e^2*x^7 +2/5*c^3*arccos(c*x)*d*e*x^5+1/3*arccos(c*x)*c^3*x^3*d^2+1/105/c^4*(15*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))+35*d^2*c^4*(-1/3*c^2*x^2 *(-c^2*x^2+1)^(1/2)-2/3*(-c^2*x^2+1)^(1/2))+42*d*c^2*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 = 0.10 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.94 \[ \int x^2 \left (d+e x^2\right )^2 (a+b \arccos (c x)) \, dx=\frac {1575 \, a c^{7} e^{2} x^{7} + 4410 \, a c^{7} d e x^{5} + 3675 \, a c^{7} d^{2} x^{3} + 105 \, {\left (15 \, b c^{7} e^{2} x^{7} + 42 \, b c^{7} d e x^{5} + 35 \, b c^{7} d^{2} x^{3}\right )} \arccos \left (c x\right ) - {\left (225 \, b c^{6} e^{2} x^{6} + 2450 \, b c^{4} d^{2} + 2352 \, b c^{2} d e + 18 \, {\left (49 \, b c^{6} d e + 15 \, b c^{4} e^{2}\right )} x^{4} + 720 \, b e^{2} + {\left (1225 \, b c^{6} d^{2} + 1176 \, b c^{4} d e + 360 \, b c^{2} e^{2}\right )} x^{2}\right )} \sqrt {-c^{2} x^{2} + 1}}{11025 \, c^{7}} \] Input:
integrate(x^2*(e*x^2+d)^2*(a+b*arccos(c*x)),x, algorithm="fricas")
Output:
1/11025*(1575*a*c^7*e^2*x^7 + 4410*a*c^7*d*e*x^5 + 3675*a*c^7*d^2*x^3 + 10 5*(15*b*c^7*e^2*x^7 + 42*b*c^7*d*e*x^5 + 35*b*c^7*d^2*x^3)*arccos(c*x) - ( 225*b*c^6*e^2*x^6 + 2450*b*c^4*d^2 + 2352*b*c^2*d*e + 18*(49*b*c^6*d*e + 1 5*b*c^4*e^2)*x^4 + 720*b*e^2 + (1225*b*c^6*d^2 + 1176*b*c^4*d*e + 360*b*c^ 2*e^2)*x^2)*sqrt(-c^2*x^2 + 1))/c^7
Time = 0.73 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.71 \[ \int x^2 \left (d+e x^2\right )^2 (a+b \arccos (c x)) \, dx=\begin {cases} \frac {a d^{2} x^{3}}{3} + \frac {2 a d e x^{5}}{5} + \frac {a e^{2} x^{7}}{7} + \frac {b d^{2} x^{3} \operatorname {acos}{\left (c x \right )}}{3} + \frac {2 b d e x^{5} \operatorname {acos}{\left (c x \right )}}{5} + \frac {b e^{2} x^{7} \operatorname {acos}{\left (c x \right )}}{7} - \frac {b d^{2} x^{2} \sqrt {- c^{2} x^{2} + 1}}{9 c} - \frac {2 b d e x^{4} \sqrt {- c^{2} x^{2} + 1}}{25 c} - \frac {b e^{2} x^{6} \sqrt {- c^{2} x^{2} + 1}}{49 c} - \frac {2 b d^{2} \sqrt {- c^{2} x^{2} + 1}}{9 c^{3}} - \frac {8 b d e x^{2} \sqrt {- c^{2} x^{2} + 1}}{75 c^{3}} - \frac {6 b e^{2} x^{4} \sqrt {- c^{2} x^{2} + 1}}{245 c^{3}} - \frac {16 b d e \sqrt {- c^{2} x^{2} + 1}}{75 c^{5}} - \frac {8 b e^{2} x^{2} \sqrt {- c^{2} x^{2} + 1}}{245 c^{5}} - \frac {16 b e^{2} \sqrt {- c^{2} x^{2} + 1}}{245 c^{7}} & \text {for}\: c \neq 0 \\\left (a + \frac {\pi b}{2}\right ) \left (\frac {d^{2} x^{3}}{3} + \frac {2 d e x^{5}}{5} + \frac {e^{2} x^{7}}{7}\right ) & \text {otherwise} \end {cases} \] Input:
integrate(x**2*(e*x**2+d)**2*(a+b*acos(c*x)),x)
Output:
Piecewise((a*d**2*x**3/3 + 2*a*d*e*x**5/5 + a*e**2*x**7/7 + b*d**2*x**3*ac os(c*x)/3 + 2*b*d*e*x**5*acos(c*x)/5 + b*e**2*x**7*acos(c*x)/7 - b*d**2*x* *2*sqrt(-c**2*x**2 + 1)/(9*c) - 2*b*d*e*x**4*sqrt(-c**2*x**2 + 1)/(25*c) - b*e**2*x**6*sqrt(-c**2*x**2 + 1)/(49*c) - 2*b*d**2*sqrt(-c**2*x**2 + 1)/( 9*c**3) - 8*b*d*e*x**2*sqrt(-c**2*x**2 + 1)/(75*c**3) - 6*b*e**2*x**4*sqrt (-c**2*x**2 + 1)/(245*c**3) - 16*b*d*e*sqrt(-c**2*x**2 + 1)/(75*c**5) - 8* b*e**2*x**2*sqrt(-c**2*x**2 + 1)/(245*c**5) - 16*b*e**2*sqrt(-c**2*x**2 + 1)/(245*c**7), Ne(c, 0)), ((a + pi*b/2)*(d**2*x**3/3 + 2*d*e*x**5/5 + e**2 *x**7/7), True))
Time = 0.14 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.29 \[ \int x^2 \left (d+e x^2\right )^2 (a+b \arccos (c x)) \, dx=\frac {1}{7} \, a e^{2} x^{7} + \frac {2}{5} \, a d e x^{5} + \frac {1}{3} \, a d^{2} x^{3} + \frac {1}{9} \, {\left (3 \, x^{3} \arccos \left (c x\right ) - c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b d^{2} + \frac {2}{75} \, {\left (15 \, x^{5} \arccos \left (c x\right ) - {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} b d e + \frac {1}{245} \, {\left (35 \, x^{7} \arccos \left (c x\right ) - {\left (\frac {5 \, \sqrt {-c^{2} x^{2} + 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {-c^{2} x^{2} + 1}}{c^{8}}\right )} c\right )} b e^{2} \] Input:
integrate(x^2*(e*x^2+d)^2*(a+b*arccos(c*x)),x, algorithm="maxima")
Output:
1/7*a*e^2*x^7 + 2/5*a*d*e*x^5 + 1/3*a*d^2*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^2 + 2/75*(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*e + 1/245*(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*e^2
Time = 0.13 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.36 \[ \int x^2 \left (d+e x^2\right )^2 (a+b \arccos (c x)) \, dx=\frac {1}{7} \, b e^{2} x^{7} \arccos \left (c x\right ) + \frac {1}{7} \, a e^{2} x^{7} + \frac {2}{5} \, b d e x^{5} \arccos \left (c x\right ) - \frac {\sqrt {-c^{2} x^{2} + 1} b e^{2} x^{6}}{49 \, c} + \frac {2}{5} \, a d e x^{5} + \frac {1}{3} \, b d^{2} x^{3} \arccos \left (c x\right ) - \frac {2 \, \sqrt {-c^{2} x^{2} + 1} b d e x^{4}}{25 \, c} + \frac {1}{3} \, a d^{2} x^{3} - \frac {\sqrt {-c^{2} x^{2} + 1} b d^{2} x^{2}}{9 \, c} - \frac {6 \, \sqrt {-c^{2} x^{2} + 1} b e^{2} x^{4}}{245 \, c^{3}} - \frac {8 \, \sqrt {-c^{2} x^{2} + 1} b d e x^{2}}{75 \, c^{3}} - \frac {2 \, \sqrt {-c^{2} x^{2} + 1} b d^{2}}{9 \, c^{3}} - \frac {8 \, \sqrt {-c^{2} x^{2} + 1} b e^{2} x^{2}}{245 \, c^{5}} - \frac {16 \, \sqrt {-c^{2} x^{2} + 1} b d e}{75 \, c^{5}} - \frac {16 \, \sqrt {-c^{2} x^{2} + 1} b e^{2}}{245 \, c^{7}} \] Input:
integrate(x^2*(e*x^2+d)^2*(a+b*arccos(c*x)),x, algorithm="giac")
Output:
1/7*b*e^2*x^7*arccos(c*x) + 1/7*a*e^2*x^7 + 2/5*b*d*e*x^5*arccos(c*x) - 1/ 49*sqrt(-c^2*x^2 + 1)*b*e^2*x^6/c + 2/5*a*d*e*x^5 + 1/3*b*d^2*x^3*arccos(c *x) - 2/25*sqrt(-c^2*x^2 + 1)*b*d*e*x^4/c + 1/3*a*d^2*x^3 - 1/9*sqrt(-c^2* x^2 + 1)*b*d^2*x^2/c - 6/245*sqrt(-c^2*x^2 + 1)*b*e^2*x^4/c^3 - 8/75*sqrt( -c^2*x^2 + 1)*b*d*e*x^2/c^3 - 2/9*sqrt(-c^2*x^2 + 1)*b*d^2/c^3 - 8/245*sqr t(-c^2*x^2 + 1)*b*e^2*x^2/c^5 - 16/75*sqrt(-c^2*x^2 + 1)*b*d*e/c^5 - 16/24 5*sqrt(-c^2*x^2 + 1)*b*e^2/c^7
Timed out. \[ \int x^2 \left (d+e x^2\right )^2 (a+b \arccos (c x)) \, dx=\int x^2\,\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^2 \,d x \] Input:
int(x^2*(a + b*acos(c*x))*(d + e*x^2)^2,x)
Output:
int(x^2*(a + b*acos(c*x))*(d + e*x^2)^2, x)
Time = 0.22 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.41 \[ \int x^2 \left (d+e x^2\right )^2 (a+b \arccos (c x)) \, dx=\frac {3675 \mathit {acos} \left (c x \right ) b \,c^{7} d^{2} x^{3}+4410 \mathit {acos} \left (c x \right ) b \,c^{7} d e \,x^{5}+1575 \mathit {acos} \left (c x \right ) b \,c^{7} e^{2} x^{7}-1225 \sqrt {-c^{2} x^{2}+1}\, b \,c^{6} d^{2} x^{2}-882 \sqrt {-c^{2} x^{2}+1}\, b \,c^{6} d e \,x^{4}-225 \sqrt {-c^{2} x^{2}+1}\, b \,c^{6} e^{2} x^{6}-2450 \sqrt {-c^{2} x^{2}+1}\, b \,c^{4} d^{2}-1176 \sqrt {-c^{2} x^{2}+1}\, b \,c^{4} d e \,x^{2}-270 \sqrt {-c^{2} x^{2}+1}\, b \,c^{4} e^{2} x^{4}-2352 \sqrt {-c^{2} x^{2}+1}\, b \,c^{2} d e -360 \sqrt {-c^{2} x^{2}+1}\, b \,c^{2} e^{2} x^{2}-720 \sqrt {-c^{2} x^{2}+1}\, b \,e^{2}+3675 a \,c^{7} d^{2} x^{3}+4410 a \,c^{7} d e \,x^{5}+1575 a \,c^{7} e^{2} x^{7}}{11025 c^{7}} \] Input:
int(x^2*(e*x^2+d)^2*(a+b*acos(c*x)),x)
Output:
(3675*acos(c*x)*b*c**7*d**2*x**3 + 4410*acos(c*x)*b*c**7*d*e*x**5 + 1575*a cos(c*x)*b*c**7*e**2*x**7 - 1225*sqrt( - c**2*x**2 + 1)*b*c**6*d**2*x**2 - 882*sqrt( - c**2*x**2 + 1)*b*c**6*d*e*x**4 - 225*sqrt( - c**2*x**2 + 1)*b *c**6*e**2*x**6 - 2450*sqrt( - c**2*x**2 + 1)*b*c**4*d**2 - 1176*sqrt( - c **2*x**2 + 1)*b*c**4*d*e*x**2 - 270*sqrt( - c**2*x**2 + 1)*b*c**4*e**2*x** 4 - 2352*sqrt( - c**2*x**2 + 1)*b*c**2*d*e - 360*sqrt( - c**2*x**2 + 1)*b* c**2*e**2*x**2 - 720*sqrt( - c**2*x**2 + 1)*b*e**2 + 3675*a*c**7*d**2*x**3 + 4410*a*c**7*d*e*x**5 + 1575*a*c**7*e**2*x**7)/(11025*c**7)