Integrand size = 20, antiderivative size = 569 \[ \int \left (d+e x^2\right )^3 (a+b \arccos (c x))^2 \, dx=-2 b^2 d^3 x-\frac {4 b^2 d^2 e x}{3 c^2}-\frac {16 b^2 d e^2 x}{25 c^4}-\frac {32 b^2 e^3 x}{245 c^6}-\frac {2}{9} b^2 d^2 e x^3-\frac {8 b^2 d e^2 x^3}{75 c^2}-\frac {16 b^2 e^3 x^3}{735 c^4}-\frac {6}{125} b^2 d e^2 x^5-\frac {12 b^2 e^3 x^5}{1225 c^2}-\frac {2}{343} b^2 e^3 x^7-\frac {2 b d^3 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{c}-\frac {4 b d^2 e \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{3 c^3}-\frac {16 b d e^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{25 c^5}-\frac {32 b e^3 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{245 c^7}-\frac {2 b d^2 e x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{3 c}-\frac {8 b d e^2 x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{25 c^3}-\frac {16 b e^3 x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{245 c^5}-\frac {6 b d e^2 x^4 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{25 c}-\frac {12 b e^3 x^4 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{245 c^3}-\frac {2 b e^3 x^6 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{49 c}+d^3 x (a+b \arccos (c x))^2+d^2 e x^3 (a+b \arccos (c x))^2+\frac {3}{5} d e^2 x^5 (a+b \arccos (c x))^2+\frac {1}{7} e^3 x^7 (a+b \arccos (c x))^2 \] Output:
-2*b^2*d^3*x-4/3*b^2*d^2*e*x/c^2-16/25*b^2*d*e^2*x/c^4-32/245*b^2*e^3*x/c^ 6-2/9*b^2*d^2*e*x^3-8/75*b^2*d*e^2*x^3/c^2-16/735*b^2*e^3*x^3/c^4-6/125*b^ 2*d*e^2*x^5-12/1225*b^2*e^3*x^5/c^2-2/343*b^2*e^3*x^7-2*b*d^3*(-c^2*x^2+1) ^(1/2)*(a+b*arccos(c*x))/c-4/3*b*d^2*e*(-c^2*x^2+1)^(1/2)*(a+b*arccos(c*x) )/c^3-16/25*b*d*e^2*(-c^2*x^2+1)^(1/2)*(a+b*arccos(c*x))/c^5-32/245*b*e^3* (-c^2*x^2+1)^(1/2)*(a+b*arccos(c*x))/c^7-2/3*b*d^2*e*x^2*(-c^2*x^2+1)^(1/2 )*(a+b*arccos(c*x))/c-8/25*b*d*e^2*x^2*(-c^2*x^2+1)^(1/2)*(a+b*arccos(c*x) )/c^3-16/245*b*e^3*x^2*(-c^2*x^2+1)^(1/2)*(a+b*arccos(c*x))/c^5-6/25*b*d*e ^2*x^4*(-c^2*x^2+1)^(1/2)*(a+b*arccos(c*x))/c-12/245*b*e^3*x^4*(-c^2*x^2+1 )^(1/2)*(a+b*arccos(c*x))/c^3-2/49*b*e^3*x^6*(-c^2*x^2+1)^(1/2)*(a+b*arcco s(c*x))/c+d^3*x*(a+b*arccos(c*x))^2+d^2*e*x^3*(a+b*arccos(c*x))^2+3/5*d*e^ 2*x^5*(a+b*arccos(c*x))^2+1/7*e^3*x^7*(a+b*arccos(c*x))^2
Time = 0.27 (sec) , antiderivative size = 445, normalized size of antiderivative = 0.78 \[ \int \left (d+e x^2\right )^3 (a+b \arccos (c x))^2 \, dx=\frac {11025 a^2 c^7 x \left (35 d^3+35 d^2 e x^2+21 d e^2 x^4+5 e^3 x^6\right )-210 a b \sqrt {1-c^2 x^2} \left (240 e^3+24 c^2 e^2 \left (49 d+5 e x^2\right )+2 c^4 e \left (1225 d^2+294 d e x^2+45 e^2 x^4\right )+c^6 \left (3675 d^3+1225 d^2 e x^2+441 d e^2 x^4+75 e^3 x^6\right )\right )-2 b^2 c x \left (25200 e^3+840 c^2 e^2 \left (147 d+5 e x^2\right )+210 c^4 e \left (1225 d^2+98 d e x^2+9 e^2 x^4\right )+c^6 \left (385875 d^3+42875 d^2 e x^2+9261 d e^2 x^4+1125 e^3 x^6\right )\right )-210 b \left (-105 a c^7 x \left (35 d^3+35 d^2 e x^2+21 d e^2 x^4+5 e^3 x^6\right )+b \sqrt {1-c^2 x^2} \left (240 e^3+24 c^2 e^2 \left (49 d+5 e x^2\right )+2 c^4 e \left (1225 d^2+294 d e x^2+45 e^2 x^4\right )+c^6 \left (3675 d^3+1225 d^2 e x^2+441 d e^2 x^4+75 e^3 x^6\right )\right )\right ) \arccos (c x)+11025 b^2 c^7 x \left (35 d^3+35 d^2 e x^2+21 d e^2 x^4+5 e^3 x^6\right ) \arccos (c x)^2}{385875 c^7} \] Input:
Integrate[(d + e*x^2)^3*(a + b*ArcCos[c*x])^2,x]
Output:
(11025*a^2*c^7*x*(35*d^3 + 35*d^2*e*x^2 + 21*d*e^2*x^4 + 5*e^3*x^6) - 210* a*b*Sqrt[1 - c^2*x^2]*(240*e^3 + 24*c^2*e^2*(49*d + 5*e*x^2) + 2*c^4*e*(12 25*d^2 + 294*d*e*x^2 + 45*e^2*x^4) + c^6*(3675*d^3 + 1225*d^2*e*x^2 + 441* d*e^2*x^4 + 75*e^3*x^6)) - 2*b^2*c*x*(25200*e^3 + 840*c^2*e^2*(147*d + 5*e *x^2) + 210*c^4*e*(1225*d^2 + 98*d*e*x^2 + 9*e^2*x^4) + c^6*(385875*d^3 + 42875*d^2*e*x^2 + 9261*d*e^2*x^4 + 1125*e^3*x^6)) - 210*b*(-105*a*c^7*x*(3 5*d^3 + 35*d^2*e*x^2 + 21*d*e^2*x^4 + 5*e^3*x^6) + b*Sqrt[1 - c^2*x^2]*(24 0*e^3 + 24*c^2*e^2*(49*d + 5*e*x^2) + 2*c^4*e*(1225*d^2 + 294*d*e*x^2 + 45 *e^2*x^4) + c^6*(3675*d^3 + 1225*d^2*e*x^2 + 441*d*e^2*x^4 + 75*e^3*x^6))) *ArcCos[c*x] + 11025*b^2*c^7*x*(35*d^3 + 35*d^2*e*x^2 + 21*d*e^2*x^4 + 5*e ^3*x^6)*ArcCos[c*x]^2)/(385875*c^7)
Time = 1.39 (sec) , antiderivative size = 569, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {5173, 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.
| \(\displaystyle \int \left (d+e x^2\right )^3 (a+b \arccos (c x))^2 \, dx\) |
| \(\Big \downarrow \) 5173 |
| \(\displaystyle \int \left (d^3 (a+b \arccos (c x))^2+3 d^2 e x^2 (a+b \arccos (c x))^2+3 d e^2 x^4 (a+b \arccos (c x))^2+e^3 x^6 (a+b \arccos (c x))^2\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 b d^3 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{c}-\frac {2 b d^2 e x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{3 c}-\frac {6 b d e^2 x^4 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{25 c}-\frac {2 b e^3 x^6 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{49 c}-\frac {32 b e^3 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{245 c^7}-\frac {16 b d e^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{25 c^5}-\frac {16 b e^3 x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{245 c^5}-\frac {4 b d^2 e \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{3 c^3}-\frac {8 b d e^2 x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{25 c^3}-\frac {12 b e^3 x^4 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{245 c^3}+d^3 x (a+b \arccos (c x))^2+d^2 e x^3 (a+b \arccos (c x))^2+\frac {3}{5} d e^2 x^5 (a+b \arccos (c x))^2+\frac {1}{7} e^3 x^7 (a+b \arccos (c x))^2-\frac {32 b^2 e^3 x}{245 c^6}-\frac {16 b^2 d e^2 x}{25 c^4}-\frac {16 b^2 e^3 x^3}{735 c^4}-\frac {4 b^2 d^2 e x}{3 c^2}-\frac {8 b^2 d e^2 x^3}{75 c^2}-\frac {12 b^2 e^3 x^5}{1225 c^2}-2 b^2 d^3 x-\frac {2}{9} b^2 d^2 e x^3-\frac {6}{125} b^2 d e^2 x^5-\frac {2}{343} b^2 e^3 x^7\) |
Input:
Int[(d + e*x^2)^3*(a + b*ArcCos[c*x])^2,x]
Output:
-2*b^2*d^3*x - (4*b^2*d^2*e*x)/(3*c^2) - (16*b^2*d*e^2*x)/(25*c^4) - (32*b ^2*e^3*x)/(245*c^6) - (2*b^2*d^2*e*x^3)/9 - (8*b^2*d*e^2*x^3)/(75*c^2) - ( 16*b^2*e^3*x^3)/(735*c^4) - (6*b^2*d*e^2*x^5)/125 - (12*b^2*e^3*x^5)/(1225 *c^2) - (2*b^2*e^3*x^7)/343 - (2*b*d^3*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x ]))/c - (4*b*d^2*e*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x]))/(3*c^3) - (16*b* d*e^2*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x]))/(25*c^5) - (32*b*e^3*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x]))/(245*c^7) - (2*b*d^2*e*x^2*Sqrt[1 - c^2*x^2 ]*(a + b*ArcCos[c*x]))/(3*c) - (8*b*d*e^2*x^2*Sqrt[1 - c^2*x^2]*(a + b*Arc Cos[c*x]))/(25*c^3) - (16*b*e^3*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x])) /(245*c^5) - (6*b*d*e^2*x^4*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x]))/(25*c) - (12*b*e^3*x^4*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x]))/(245*c^3) - (2*b*e^ 3*x^6*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x]))/(49*c) + d^3*x*(a + b*ArcCos[ c*x])^2 + d^2*e*x^3*(a + b*ArcCos[c*x])^2 + (3*d*e^2*x^5*(a + b*ArcCos[c*x ])^2)/5 + (e^3*x^7*(a + b*ArcCos[c*x])^2)/7
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x _Symbol] :> Int[ExpandIntegrand[(a + b*ArcCos[c*x])^n, (d + e*x^2)^p, x], x ] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] && (G tQ[p, 0] || IGtQ[n, 0])
Time = 0.84 (sec) , antiderivative size = 702, normalized size of antiderivative = 1.23
| method | result | size |
| derivativedivides | \(\frac {\frac {a^{2} \left (d^{3} c^{7} x +d^{2} c^{7} e \,x^{3}+\frac {3}{5} d \,c^{7} e^{2} x^{5}+\frac {1}{7} e^{3} c^{7} x^{7}\right )}{c^{6}}+\frac {b^{2} \left (c^{6} d^{3} \left (\arccos \left (c x \right )^{2} c x -2 c x -2 \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )+\frac {c^{4} d^{2} e \left (9 \arccos \left (c x \right )^{2} c^{3} x^{3}-6 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) c^{2} x^{2}-2 c^{3} x^{3}-12 \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}-12 c x \right )}{9}+\frac {c^{2} d \,e^{2} \left (225 \arccos \left (c x \right )^{2} c^{5} x^{5}-90 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) c^{4} x^{4}-18 c^{5} x^{5}-120 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) c^{2} x^{2}-40 c^{3} x^{3}-240 \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}-240 c x \right )}{375}+\frac {e^{3} \left (3675 \arccos \left (c x \right )^{2} c^{7} x^{7}-1050 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) c^{6} x^{6}-150 c^{7} x^{7}-1260 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) c^{4} x^{4}-252 c^{5} x^{5}-1680 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) c^{2} x^{2}-560 c^{3} x^{3}-3360 \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}-3360 c x \right )}{25725}\right )}{c^{6}}+\frac {2 a b \left (\arccos \left (c x \right ) d^{3} c^{7} x +\arccos \left (c x \right ) d^{2} c^{7} e \,x^{3}+\frac {3 \arccos \left (c x \right ) d \,c^{7} e^{2} x^{5}}{5}+\frac {\arccos \left (c x \right ) e^{3} c^{7} x^{7}}{7}+\frac {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}-d^{3} c^{6} \sqrt {-c^{2} x^{2}+1}+\frac {3 d \,c^{2} 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}+d^{2} c^{4} e \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )\right )}{c^{6}}}{c}\) | \(702\) |
| default | \(\frac {\frac {a^{2} \left (d^{3} c^{7} x +d^{2} c^{7} e \,x^{3}+\frac {3}{5} d \,c^{7} e^{2} x^{5}+\frac {1}{7} e^{3} c^{7} x^{7}\right )}{c^{6}}+\frac {b^{2} \left (c^{6} d^{3} \left (\arccos \left (c x \right )^{2} c x -2 c x -2 \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )+\frac {c^{4} d^{2} e \left (9 \arccos \left (c x \right )^{2} c^{3} x^{3}-6 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) c^{2} x^{2}-2 c^{3} x^{3}-12 \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}-12 c x \right )}{9}+\frac {c^{2} d \,e^{2} \left (225 \arccos \left (c x \right )^{2} c^{5} x^{5}-90 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) c^{4} x^{4}-18 c^{5} x^{5}-120 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) c^{2} x^{2}-40 c^{3} x^{3}-240 \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}-240 c x \right )}{375}+\frac {e^{3} \left (3675 \arccos \left (c x \right )^{2} c^{7} x^{7}-1050 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) c^{6} x^{6}-150 c^{7} x^{7}-1260 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) c^{4} x^{4}-252 c^{5} x^{5}-1680 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) c^{2} x^{2}-560 c^{3} x^{3}-3360 \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}-3360 c x \right )}{25725}\right )}{c^{6}}+\frac {2 a b \left (\arccos \left (c x \right ) d^{3} c^{7} x +\arccos \left (c x \right ) d^{2} c^{7} e \,x^{3}+\frac {3 \arccos \left (c x \right ) d \,c^{7} e^{2} x^{5}}{5}+\frac {\arccos \left (c x \right ) e^{3} c^{7} x^{7}}{7}+\frac {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}-d^{3} c^{6} \sqrt {-c^{2} x^{2}+1}+\frac {3 d \,c^{2} 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}+d^{2} c^{4} e \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )\right )}{c^{6}}}{c}\) | \(702\) |
| parts | \(a^{2} \left (\frac {1}{7} e^{3} x^{7}+\frac {3}{5} d \,e^{2} x^{5}+d^{2} e \,x^{3}+d^{3} x \right )+\frac {b^{2} \left (55125 \arccos \left (c x \right )^{2} c^{7} x^{7} e^{3}+231525 \arccos \left (c x \right )^{2} c^{7} x^{5} d \,e^{2}+385875 \arccos \left (c x \right )^{2} c^{7} x^{3} d^{2} e +385875 \arccos \left (c x \right )^{2} c^{7} x \,d^{3}-15750 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) c^{6} x^{6} e^{3}-92610 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) c^{6} x^{4} d \,e^{2}-257250 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) c^{6} x^{2} d^{2} e -771750 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) c^{6} d^{3}-2250 e^{3} c^{7} x^{7}-18522 d \,c^{7} e^{2} x^{5}-85750 d^{2} c^{7} e \,x^{3}-771750 d^{3} c^{7} x -18900 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) c^{4} x^{4} e^{3}-123480 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) c^{4} x^{2} d \,e^{2}-514500 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) c^{4} d^{2} e -3780 c^{5} x^{5} e^{3}-41160 c^{5} x^{3} d \,e^{2}-514500 e \,c^{5} d^{2} x -25200 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) c^{2} x^{2} e^{3}-246960 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) c^{2} d \,e^{2}-8400 c^{3} x^{3} e^{3}-246960 e^{2} c^{3} d x -50400 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) e^{3}-50400 e^{3} c x \right )}{385875 c^{7}}+\frac {2 a b \left (\frac {c \arccos \left (c x \right ) e^{3} x^{7}}{7}+\frac {3 c \arccos \left (c x \right ) d \,e^{2} x^{5}}{5}+c \arccos \left (c x \right ) d^{2} e \,x^{3}+\arccos \left (c x \right ) d^{3} c x +\frac {5 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 )-35 d^{3} c^{6} \sqrt {-c^{2} x^{2}+1}+21 d \,c^{2} 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 )+35 d^{2} c^{4} e \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )}{35 c^{6}}\right )}{c}\) | \(751\) |
| orering | \(\frac {x \left (47625 c^{8} e^{5} x^{10}+328917 c^{8} d \,e^{4} x^{8}+1128666 c^{8} d^{2} e^{3} x^{6}+10080 c^{6} e^{5} x^{8}+5951050 c^{8} d^{3} e^{2} x^{4}+146016 c^{6} d \,e^{4} x^{6}-385875 c^{8} d^{4} e \,x^{2}+2711072 c^{6} d^{2} e^{3} x^{4}+30240 c^{4} e^{5} x^{6}+128625 c^{8} d^{5}-6860000 c^{6} d^{3} e^{2} x^{2}+1310400 c^{4} d \,e^{4} x^{4}-4445280 c^{4} d^{2} e^{3} x^{2}+268800 c^{2} e^{5} x^{4}-2042880 c^{2} d \,e^{4} x^{2}-403200 e^{5} x^{2}\right ) \left (a +b \arccos \left (c x \right )\right )^{2}}{128625 \left (e \,x^{2}+d \right )^{2} c^{8}}-\frac {\left (20250 c^{8} e^{4} x^{10}+154926 c^{8} d \,e^{3} x^{8}+637294 c^{8} d^{2} e^{2} x^{6}+8865 c^{6} e^{4} x^{8}+4716250 c^{8} d^{3} e \,x^{4}+130368 c^{6} d \,e^{3} x^{6}+2438730 c^{6} d^{2} e^{2} x^{4}+26670 c^{4} e^{4} x^{6}-5145000 c^{6} d^{3} e \,x^{2}+1172010 c^{4} d \,e^{3} x^{4}-385875 c^{6} d^{4}-3405990 c^{4} d^{2} e^{2} x^{2}+239400 c^{2} e^{4} x^{4}-257250 c^{4} d^{3} e -1617840 c^{2} d \,e^{3} x^{2}-123480 c^{2} d^{2} e^{2}-327600 e^{4} x^{2}-25200 d \,e^{3}\right ) \left (6 \left (e \,x^{2}+d \right )^{2} \left (a +b \arccos \left (c x \right )\right )^{2} e x -\frac {2 \left (e \,x^{2}+d \right )^{3} \left (a +b \arccos \left (c x \right )\right ) b c}{\sqrt {-c^{2} x^{2}+1}}\right )}{385875 \left (e \,x^{2}+d \right )^{4} c^{8}}+\frac {x \left (1125 e^{3} x^{6} c^{6}+9261 c^{6} d \,e^{2} x^{4}+42875 c^{6} d^{2} e \,x^{2}+1890 c^{4} e^{3} x^{4}+385875 c^{6} d^{3}+20580 c^{4} d \,e^{2} x^{2}+257250 c^{4} d^{2} e +4200 c^{2} e^{3} x^{2}+123480 c^{2} d \,e^{2}+25200 e^{3}\right ) \left (c x -1\right ) \left (c x +1\right ) \left (24 \left (e \,x^{2}+d \right ) \left (a +b \arccos \left (c x \right )\right )^{2} e^{2} x^{2}-\frac {24 \left (e \,x^{2}+d \right )^{2} \left (a +b \arccos \left (c x \right )\right ) e x b c}{\sqrt {-c^{2} x^{2}+1}}+6 \left (e \,x^{2}+d \right )^{2} \left (a +b \arccos \left (c x \right )\right )^{2} e +\frac {2 \left (e \,x^{2}+d \right )^{3} b^{2} c^{2}}{-c^{2} x^{2}+1}-\frac {2 \left (e \,x^{2}+d \right )^{3} \left (a +b \arccos \left (c x \right )\right ) b \,c^{3} x}{\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{385875 c^{8} \left (e \,x^{2}+d \right )^{3}}\) | \(764\) |
Input:
int((e*x^2+d)^3*(a+b*arccos(c*x))^2,x,method=_RETURNVERBOSE)
Output:
1/c*(a^2/c^6*(d^3*c^7*x+d^2*c^7*e*x^3+3/5*d*c^7*e^2*x^5+1/7*e^3*c^7*x^7)+b ^2/c^6*(c^6*d^3*(arccos(c*x)^2*c*x-2*c*x-2*arccos(c*x)*(-c^2*x^2+1)^(1/2)) +1/9*c^4*d^2*e*(9*arccos(c*x)^2*c^3*x^3-6*(-c^2*x^2+1)^(1/2)*arccos(c*x)*c ^2*x^2-2*c^3*x^3-12*arccos(c*x)*(-c^2*x^2+1)^(1/2)-12*c*x)+1/375*c^2*d*e^2 *(225*arccos(c*x)^2*c^5*x^5-90*(-c^2*x^2+1)^(1/2)*arccos(c*x)*c^4*x^4-18*c ^5*x^5-120*(-c^2*x^2+1)^(1/2)*arccos(c*x)*c^2*x^2-40*c^3*x^3-240*arccos(c* x)*(-c^2*x^2+1)^(1/2)-240*c*x)+1/25725*e^3*(3675*arccos(c*x)^2*c^7*x^7-105 0*(-c^2*x^2+1)^(1/2)*arccos(c*x)*c^6*x^6-150*c^7*x^7-1260*(-c^2*x^2+1)^(1/ 2)*arccos(c*x)*c^4*x^4-252*c^5*x^5-1680*(-c^2*x^2+1)^(1/2)*arccos(c*x)*c^2 *x^2-560*c^3*x^3-3360*arccos(c*x)*(-c^2*x^2+1)^(1/2)-3360*c*x))+2*a*b/c^6* (arccos(c*x)*d^3*c^7*x+arccos(c*x)*d^2*c^7*e*x^3+3/5*arccos(c*x)*d*c^7*e^2 *x^5+1/7*arccos(c*x)*e^3*c^7*x^7+1/7*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))-d^3*c^6*(-c^2*x^2+1)^(1/2)+3/5*d*c^2*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))+d ^2*c^4*e*(-1/3*c^2*x^2*(-c^2*x^2+1)^(1/2)-2/3*(-c^2*x^2+1)^(1/2))))
Time = 0.14 (sec) , antiderivative size = 555, normalized size of antiderivative = 0.98 \[ \int \left (d+e x^2\right )^3 (a+b \arccos (c x))^2 \, dx=\frac {1125 \, {\left (49 \, a^{2} - 2 \, b^{2}\right )} c^{7} e^{3} x^{7} + 189 \, {\left (49 \, {\left (25 \, a^{2} - 2 \, b^{2}\right )} c^{7} d e^{2} - 20 \, b^{2} c^{5} e^{3}\right )} x^{5} + 35 \, {\left (1225 \, {\left (9 \, a^{2} - 2 \, b^{2}\right )} c^{7} d^{2} e - 1176 \, b^{2} c^{5} d e^{2} - 240 \, b^{2} c^{3} e^{3}\right )} x^{3} + 11025 \, {\left (5 \, b^{2} c^{7} e^{3} x^{7} + 21 \, b^{2} c^{7} d e^{2} x^{5} + 35 \, b^{2} c^{7} d^{2} e x^{3} + 35 \, b^{2} c^{7} d^{3} x\right )} \arccos \left (c x\right )^{2} + 105 \, {\left (3675 \, {\left (a^{2} - 2 \, b^{2}\right )} c^{7} d^{3} - 4900 \, b^{2} c^{5} d^{2} e - 2352 \, b^{2} c^{3} d e^{2} - 480 \, b^{2} c e^{3}\right )} x + 22050 \, {\left (5 \, a b c^{7} e^{3} x^{7} + 21 \, a b c^{7} d e^{2} x^{5} + 35 \, a b c^{7} d^{2} e x^{3} + 35 \, a b c^{7} d^{3} x\right )} \arccos \left (c x\right ) - 210 \, {\left (75 \, a b c^{6} e^{3} x^{6} + 3675 \, a b c^{6} d^{3} + 2450 \, a b c^{4} d^{2} e + 1176 \, a b c^{2} d e^{2} + 240 \, a b e^{3} + 9 \, {\left (49 \, a b c^{6} d e^{2} + 10 \, a b c^{4} e^{3}\right )} x^{4} + {\left (1225 \, a b c^{6} d^{2} e + 588 \, a b c^{4} d e^{2} + 120 \, a b c^{2} e^{3}\right )} x^{2} + {\left (75 \, b^{2} c^{6} e^{3} x^{6} + 3675 \, b^{2} c^{6} d^{3} + 2450 \, b^{2} c^{4} d^{2} e + 1176 \, b^{2} c^{2} d e^{2} + 240 \, b^{2} e^{3} + 9 \, {\left (49 \, b^{2} c^{6} d e^{2} + 10 \, b^{2} c^{4} e^{3}\right )} x^{4} + {\left (1225 \, b^{2} c^{6} d^{2} e + 588 \, b^{2} c^{4} d e^{2} + 120 \, b^{2} c^{2} e^{3}\right )} x^{2}\right )} \arccos \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{385875 \, c^{7}} \] Input:
integrate((e*x^2+d)^3*(a+b*arccos(c*x))^2,x, algorithm="fricas")
Output:
1/385875*(1125*(49*a^2 - 2*b^2)*c^7*e^3*x^7 + 189*(49*(25*a^2 - 2*b^2)*c^7 *d*e^2 - 20*b^2*c^5*e^3)*x^5 + 35*(1225*(9*a^2 - 2*b^2)*c^7*d^2*e - 1176*b ^2*c^5*d*e^2 - 240*b^2*c^3*e^3)*x^3 + 11025*(5*b^2*c^7*e^3*x^7 + 21*b^2*c^ 7*d*e^2*x^5 + 35*b^2*c^7*d^2*e*x^3 + 35*b^2*c^7*d^3*x)*arccos(c*x)^2 + 105 *(3675*(a^2 - 2*b^2)*c^7*d^3 - 4900*b^2*c^5*d^2*e - 2352*b^2*c^3*d*e^2 - 4 80*b^2*c*e^3)*x + 22050*(5*a*b*c^7*e^3*x^7 + 21*a*b*c^7*d*e^2*x^5 + 35*a*b *c^7*d^2*e*x^3 + 35*a*b*c^7*d^3*x)*arccos(c*x) - 210*(75*a*b*c^6*e^3*x^6 + 3675*a*b*c^6*d^3 + 2450*a*b*c^4*d^2*e + 1176*a*b*c^2*d*e^2 + 240*a*b*e^3 + 9*(49*a*b*c^6*d*e^2 + 10*a*b*c^4*e^3)*x^4 + (1225*a*b*c^6*d^2*e + 588*a* b*c^4*d*e^2 + 120*a*b*c^2*e^3)*x^2 + (75*b^2*c^6*e^3*x^6 + 3675*b^2*c^6*d^ 3 + 2450*b^2*c^4*d^2*e + 1176*b^2*c^2*d*e^2 + 240*b^2*e^3 + 9*(49*b^2*c^6* d*e^2 + 10*b^2*c^4*e^3)*x^4 + (1225*b^2*c^6*d^2*e + 588*b^2*c^4*d*e^2 + 12 0*b^2*c^2*e^3)*x^2)*arccos(c*x))*sqrt(-c^2*x^2 + 1))/c^7
Time = 0.95 (sec) , antiderivative size = 994, normalized size of antiderivative = 1.75 \[ \int \left (d+e x^2\right )^3 (a+b \arccos (c x))^2 \, dx =\text {Too large to display} \] Input:
integrate((e*x**2+d)**3*(a+b*acos(c*x))**2,x)
Output:
Piecewise((a**2*d**3*x + a**2*d**2*e*x**3 + 3*a**2*d*e**2*x**5/5 + a**2*e* *3*x**7/7 + 2*a*b*d**3*x*acos(c*x) + 2*a*b*d**2*e*x**3*acos(c*x) + 6*a*b*d *e**2*x**5*acos(c*x)/5 + 2*a*b*e**3*x**7*acos(c*x)/7 - 2*a*b*d**3*sqrt(-c* *2*x**2 + 1)/c - 2*a*b*d**2*e*x**2*sqrt(-c**2*x**2 + 1)/(3*c) - 6*a*b*d*e* *2*x**4*sqrt(-c**2*x**2 + 1)/(25*c) - 2*a*b*e**3*x**6*sqrt(-c**2*x**2 + 1) /(49*c) - 4*a*b*d**2*e*sqrt(-c**2*x**2 + 1)/(3*c**3) - 8*a*b*d*e**2*x**2*s qrt(-c**2*x**2 + 1)/(25*c**3) - 12*a*b*e**3*x**4*sqrt(-c**2*x**2 + 1)/(245 *c**3) - 16*a*b*d*e**2*sqrt(-c**2*x**2 + 1)/(25*c**5) - 16*a*b*e**3*x**2*s qrt(-c**2*x**2 + 1)/(245*c**5) - 32*a*b*e**3*sqrt(-c**2*x**2 + 1)/(245*c** 7) + b**2*d**3*x*acos(c*x)**2 - 2*b**2*d**3*x + b**2*d**2*e*x**3*acos(c*x) **2 - 2*b**2*d**2*e*x**3/9 + 3*b**2*d*e**2*x**5*acos(c*x)**2/5 - 6*b**2*d* e**2*x**5/125 + b**2*e**3*x**7*acos(c*x)**2/7 - 2*b**2*e**3*x**7/343 - 2*b **2*d**3*sqrt(-c**2*x**2 + 1)*acos(c*x)/c - 2*b**2*d**2*e*x**2*sqrt(-c**2* x**2 + 1)*acos(c*x)/(3*c) - 6*b**2*d*e**2*x**4*sqrt(-c**2*x**2 + 1)*acos(c *x)/(25*c) - 2*b**2*e**3*x**6*sqrt(-c**2*x**2 + 1)*acos(c*x)/(49*c) - 4*b* *2*d**2*e*x/(3*c**2) - 8*b**2*d*e**2*x**3/(75*c**2) - 12*b**2*e**3*x**5/(1 225*c**2) - 4*b**2*d**2*e*sqrt(-c**2*x**2 + 1)*acos(c*x)/(3*c**3) - 8*b**2 *d*e**2*x**2*sqrt(-c**2*x**2 + 1)*acos(c*x)/(25*c**3) - 12*b**2*e**3*x**4* sqrt(-c**2*x**2 + 1)*acos(c*x)/(245*c**3) - 16*b**2*d*e**2*x/(25*c**4) - 1 6*b**2*e**3*x**3/(735*c**4) - 16*b**2*d*e**2*sqrt(-c**2*x**2 + 1)*acos(...
Time = 0.17 (sec) , antiderivative size = 700, normalized size of antiderivative = 1.23 \[ \int \left (d+e x^2\right )^3 (a+b \arccos (c x))^2 \, dx =\text {Too large to display} \] Input:
integrate((e*x^2+d)^3*(a+b*arccos(c*x))^2,x, algorithm="maxima")
Output:
1/7*b^2*e^3*x^7*arccos(c*x)^2 + 1/7*a^2*e^3*x^7 + 3/5*b^2*d*e^2*x^5*arccos (c*x)^2 + 3/5*a^2*d*e^2*x^5 + b^2*d^2*e*x^3*arccos(c*x)^2 + a^2*d^2*e*x^3 + b^2*d^3*x*arccos(c*x)^2 + 2/3*(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))*a*b*d^2*e - 2/9*(3*c*(sqrt(-c^2*x^2 + 1)*x^2/c^2 + 2*sqrt(-c^2*x^2 + 1)/c^4)*arccos(c*x) + (c^2*x^3 + 6*x)/c^2 )*b^2*d^2*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)*a*b*d*e^2 - 2/3 75*(15*(3*sqrt(-c^2*x^2 + 1)*x^4/c^2 + 4*sqrt(-c^2*x^2 + 1)*x^2/c^4 + 8*sq rt(-c^2*x^2 + 1)/c^6)*c*arccos(c*x) + (9*c^4*x^5 + 20*c^2*x^3 + 120*x)/c^4 )*b^2*d*e^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)*a*b*e^3 - 2/25725*(105*(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*arccos(c*x) + (75*c^6*x^7 + 126*c^4*x^5 + 280*c^2*x^3 + 16 80*x)/c^6)*b^2*e^3 - 2*b^2*d^3*(x + sqrt(-c^2*x^2 + 1)*arccos(c*x)/c) + a^ 2*d^3*x + 2*(c*x*arccos(c*x) - sqrt(-c^2*x^2 + 1))*a*b*d^3/c
Time = 0.17 (sec) , antiderivative size = 826, normalized size of antiderivative = 1.45 \[ \int \left (d+e x^2\right )^3 (a+b \arccos (c x))^2 \, dx =\text {Too large to display} \] Input:
integrate((e*x^2+d)^3*(a+b*arccos(c*x))^2,x, algorithm="giac")
Output:
1/7*b^2*e^3*x^7*arccos(c*x)^2 + 2/7*a*b*e^3*x^7*arccos(c*x) + 1/7*a^2*e^3* x^7 - 2/343*b^2*e^3*x^7 + 3/5*b^2*d*e^2*x^5*arccos(c*x)^2 - 2/49*sqrt(-c^2 *x^2 + 1)*b^2*e^3*x^6*arccos(c*x)/c + 6/5*a*b*d*e^2*x^5*arccos(c*x) - 2/49 *sqrt(-c^2*x^2 + 1)*a*b*e^3*x^6/c + 3/5*a^2*d*e^2*x^5 - 6/125*b^2*d*e^2*x^ 5 + b^2*d^2*e*x^3*arccos(c*x)^2 - 6/25*sqrt(-c^2*x^2 + 1)*b^2*d*e^2*x^4*ar ccos(c*x)/c + 2*a*b*d^2*e*x^3*arccos(c*x) - 6/25*sqrt(-c^2*x^2 + 1)*a*b*d* e^2*x^4/c + a^2*d^2*e*x^3 - 2/9*b^2*d^2*e*x^3 - 12/1225*b^2*e^3*x^5/c^2 + b^2*d^3*x*arccos(c*x)^2 - 2/3*sqrt(-c^2*x^2 + 1)*b^2*d^2*e*x^2*arccos(c*x) /c - 12/245*sqrt(-c^2*x^2 + 1)*b^2*e^3*x^4*arccos(c*x)/c^3 + 2*a*b*d^3*x*a rccos(c*x) - 2/3*sqrt(-c^2*x^2 + 1)*a*b*d^2*e*x^2/c - 12/245*sqrt(-c^2*x^2 + 1)*a*b*e^3*x^4/c^3 + a^2*d^3*x - 2*b^2*d^3*x - 8/75*b^2*d*e^2*x^3/c^2 - 2*sqrt(-c^2*x^2 + 1)*b^2*d^3*arccos(c*x)/c - 8/25*sqrt(-c^2*x^2 + 1)*b^2* d*e^2*x^2*arccos(c*x)/c^3 - 2*sqrt(-c^2*x^2 + 1)*a*b*d^3/c - 8/25*sqrt(-c^ 2*x^2 + 1)*a*b*d*e^2*x^2/c^3 - 4/3*b^2*d^2*e*x/c^2 - 16/735*b^2*e^3*x^3/c^ 4 - 4/3*sqrt(-c^2*x^2 + 1)*b^2*d^2*e*arccos(c*x)/c^3 - 16/245*sqrt(-c^2*x^ 2 + 1)*b^2*e^3*x^2*arccos(c*x)/c^5 - 4/3*sqrt(-c^2*x^2 + 1)*a*b*d^2*e/c^3 - 16/245*sqrt(-c^2*x^2 + 1)*a*b*e^3*x^2/c^5 - 16/25*b^2*d*e^2*x/c^4 - 16/2 5*sqrt(-c^2*x^2 + 1)*b^2*d*e^2*arccos(c*x)/c^5 - 16/25*sqrt(-c^2*x^2 + 1)* a*b*d*e^2/c^5 - 32/245*b^2*e^3*x/c^6 - 32/245*sqrt(-c^2*x^2 + 1)*b^2*e^3*a rccos(c*x)/c^7 - 32/245*sqrt(-c^2*x^2 + 1)*a*b*e^3/c^7
Timed out. \[ \int \left (d+e x^2\right )^3 (a+b \arccos (c x))^2 \, dx=\int {\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^2\,{\left (e\,x^2+d\right )}^3 \,d x \] Input:
int((a + b*acos(c*x))^2*(d + e*x^2)^3,x)
Output:
int((a + b*acos(c*x))^2*(d + e*x^2)^3, x)
\[ \int \left (d+e x^2\right )^3 (a+b \arccos (c x))^2 \, dx=\frac {3675 \mathit {acos} \left (c x \right )^{2} b^{2} c^{7} d^{3} x -7350 \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) b^{2} c^{6} d^{3}+7350 \mathit {acos} \left (c x \right ) a b \,c^{7} d^{3} x +7350 \mathit {acos} \left (c x \right ) a b \,c^{7} d^{2} e \,x^{3}+4410 \mathit {acos} \left (c x \right ) a b \,c^{7} d \,e^{2} x^{5}+1050 \mathit {acos} \left (c x \right ) a b \,c^{7} e^{3} x^{7}-7350 \sqrt {-c^{2} x^{2}+1}\, a b \,c^{6} d^{3}-2450 \sqrt {-c^{2} x^{2}+1}\, a b \,c^{6} d^{2} e \,x^{2}-882 \sqrt {-c^{2} x^{2}+1}\, a b \,c^{6} d \,e^{2} x^{4}-150 \sqrt {-c^{2} x^{2}+1}\, a b \,c^{6} e^{3} x^{6}-4900 \sqrt {-c^{2} x^{2}+1}\, a b \,c^{4} d^{2} e -1176 \sqrt {-c^{2} x^{2}+1}\, a b \,c^{4} d \,e^{2} x^{2}-180 \sqrt {-c^{2} x^{2}+1}\, a b \,c^{4} e^{3} x^{4}-2352 \sqrt {-c^{2} x^{2}+1}\, a b \,c^{2} d \,e^{2}-240 \sqrt {-c^{2} x^{2}+1}\, a b \,c^{2} e^{3} x^{2}-480 \sqrt {-c^{2} x^{2}+1}\, a b \,e^{3}+3675 \left (\int \mathit {acos} \left (c x \right )^{2} x^{6}d x \right ) b^{2} c^{7} e^{3}+11025 \left (\int \mathit {acos} \left (c x \right )^{2} x^{4}d x \right ) b^{2} c^{7} d \,e^{2}+11025 \left (\int \mathit {acos} \left (c x \right )^{2} x^{2}d x \right ) b^{2} c^{7} d^{2} e +3675 a^{2} c^{7} d^{3} x +3675 a^{2} c^{7} d^{2} e \,x^{3}+2205 a^{2} c^{7} d \,e^{2} x^{5}+525 a^{2} c^{7} e^{3} x^{7}-7350 b^{2} c^{7} d^{3} x}{3675 c^{7}} \] Input:
int((e*x^2+d)^3*(a+b*acos(c*x))^2,x)
Output:
(3675*acos(c*x)**2*b**2*c**7*d**3*x - 7350*sqrt( - c**2*x**2 + 1)*acos(c*x )*b**2*c**6*d**3 + 7350*acos(c*x)*a*b*c**7*d**3*x + 7350*acos(c*x)*a*b*c** 7*d**2*e*x**3 + 4410*acos(c*x)*a*b*c**7*d*e**2*x**5 + 1050*acos(c*x)*a*b*c **7*e**3*x**7 - 7350*sqrt( - c**2*x**2 + 1)*a*b*c**6*d**3 - 2450*sqrt( - c **2*x**2 + 1)*a*b*c**6*d**2*e*x**2 - 882*sqrt( - c**2*x**2 + 1)*a*b*c**6*d *e**2*x**4 - 150*sqrt( - c**2*x**2 + 1)*a*b*c**6*e**3*x**6 - 4900*sqrt( - c**2*x**2 + 1)*a*b*c**4*d**2*e - 1176*sqrt( - c**2*x**2 + 1)*a*b*c**4*d*e* *2*x**2 - 180*sqrt( - c**2*x**2 + 1)*a*b*c**4*e**3*x**4 - 2352*sqrt( - c** 2*x**2 + 1)*a*b*c**2*d*e**2 - 240*sqrt( - c**2*x**2 + 1)*a*b*c**2*e**3*x** 2 - 480*sqrt( - c**2*x**2 + 1)*a*b*e**3 + 3675*int(acos(c*x)**2*x**6,x)*b* *2*c**7*e**3 + 11025*int(acos(c*x)**2*x**4,x)*b**2*c**7*d*e**2 + 11025*int (acos(c*x)**2*x**2,x)*b**2*c**7*d**2*e + 3675*a**2*c**7*d**3*x + 3675*a**2 *c**7*d**2*e*x**3 + 2205*a**2*c**7*d*e**2*x**5 + 525*a**2*c**7*e**3*x**7 - 7350*b**2*c**7*d**3*x)/(3675*c**7)