Integrand size = 27, antiderivative size = 310 \[ \int x^2 \left (d-c^2 d x^2\right )^2 (a+b \arccos (c x))^2 \, dx=-\frac {1636 b^2 d^2 x}{11025 c^2}-\frac {818 b^2 d^2 x^3}{33075}+\frac {136 b^2 c^2 d^2 x^5}{6125}-\frac {2}{343} b^2 c^4 d^2 x^7+\frac {32 b d^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{315 c^3}+\frac {16 b d^2 x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{315 c}+\frac {8 b d^2 \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))}{105 c^3}+\frac {2 b d^2 \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{175 c^3}-\frac {2 b d^2 \left (1-c^2 x^2\right )^{7/2} (a+b \arccos (c x))}{49 c^3}+\frac {8}{105} d^2 x^3 (a+b \arccos (c x))^2+\frac {4}{35} d^2 x^3 \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2+\frac {1}{7} d^2 x^3 \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2 \] Output:
-1636/11025*b^2*d^2*x/c^2-818/33075*b^2*d^2*x^3+136/6125*b^2*c^2*d^2*x^5-2 /343*b^2*c^4*d^2*x^7+32/315*b*d^2*(-c^2*x^2+1)^(1/2)*(a+b*arccos(c*x))/c^3 +16/315*b*d^2*x^2*(-c^2*x^2+1)^(1/2)*(a+b*arccos(c*x))/c+8/105*b*d^2*(-c^2 *x^2+1)^(3/2)*(a+b*arccos(c*x))/c^3+2/175*b*d^2*(-c^2*x^2+1)^(5/2)*(a+b*ar ccos(c*x))/c^3-2/49*b*d^2*(-c^2*x^2+1)^(7/2)*(a+b*arccos(c*x))/c^3+8/105*d ^2*x^3*(a+b*arccos(c*x))^2+4/35*d^2*x^3*(-c^2*x^2+1)*(a+b*arccos(c*x))^2+1 /7*d^2*x^3*(-c^2*x^2+1)^2*(a+b*arccos(c*x))^2
Time = 1.28 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.74 \[ \int x^2 \left (d-c^2 d x^2\right )^2 (a+b \arccos (c x))^2 \, dx=\frac {d^2 \left (11025 a^2 c^3 x^3 \left (35-42 c^2 x^2+15 c^4 x^4\right )-210 a b \sqrt {1-c^2 x^2} \left (818+409 c^2 x^2-612 c^4 x^4+225 c^6 x^6\right )-2 b^2 c x \left (85890+14315 c^2 x^2-12852 c^4 x^4+3375 c^6 x^6\right )-210 b \left (-105 a c^3 x^3 \left (35-42 c^2 x^2+15 c^4 x^4\right )+b \sqrt {1-c^2 x^2} \left (818+409 c^2 x^2-612 c^4 x^4+225 c^6 x^6\right )\right ) \arccos (c x)+11025 b^2 c^3 x^3 \left (35-42 c^2 x^2+15 c^4 x^4\right ) \arccos (c x)^2\right )}{1157625 c^3} \] Input:
Integrate[x^2*(d - c^2*d*x^2)^2*(a + b*ArcCos[c*x])^2,x]
Output:
(d^2*(11025*a^2*c^3*x^3*(35 - 42*c^2*x^2 + 15*c^4*x^4) - 210*a*b*Sqrt[1 - c^2*x^2]*(818 + 409*c^2*x^2 - 612*c^4*x^4 + 225*c^6*x^6) - 2*b^2*c*x*(8589 0 + 14315*c^2*x^2 - 12852*c^4*x^4 + 3375*c^6*x^6) - 210*b*(-105*a*c^3*x^3* (35 - 42*c^2*x^2 + 15*c^4*x^4) + b*Sqrt[1 - c^2*x^2]*(818 + 409*c^2*x^2 - 612*c^4*x^4 + 225*c^6*x^6))*ArcCos[c*x] + 11025*b^2*c^3*x^3*(35 - 42*c^2*x ^2 + 15*c^4*x^4)*ArcCos[c*x]^2))/(1157625*c^3)
Time = 1.87 (sec) , antiderivative size = 389, normalized size of antiderivative = 1.25, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {5203, 27, 5195, 27, 290, 2009, 5203, 5139, 5195, 27, 2009, 5211, 15, 5183, 24}
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.
| \(\displaystyle \int x^2 \left (d-c^2 d x^2\right )^2 (a+b \arccos (c x))^2 \, dx\) |
| \(\Big \downarrow \) 5203 |
| \(\displaystyle \frac {2}{7} b c d^2 \int x^3 \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))dx+\frac {4}{7} d \int d x^2 \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2dx+\frac {1}{7} d^2 x^3 \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {4}{7} d^2 \int x^2 \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2dx+\frac {2}{7} b c d^2 \int x^3 \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))dx+\frac {1}{7} d^2 x^3 \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2\) |
| \(\Big \downarrow \) 5195 |
| \(\displaystyle \frac {4}{7} d^2 \int x^2 \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2dx+\frac {2}{7} b c d^2 \left (b c \int -\frac {\left (1-c^2 x^2\right )^2 \left (5 c^2 x^2+2\right )}{35 c^4}dx+\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arccos (c x))}{7 c^4}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{5 c^4}\right )+\frac {1}{7} d^2 x^3 \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {4}{7} d^2 \int x^2 \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2dx+\frac {2}{7} b c d^2 \left (-\frac {b \int \left (1-c^2 x^2\right )^2 \left (5 c^2 x^2+2\right )dx}{35 c^3}+\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arccos (c x))}{7 c^4}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{5 c^4}\right )+\frac {1}{7} d^2 x^3 \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2\) |
| \(\Big \downarrow \) 290 |
| \(\displaystyle \frac {4}{7} d^2 \int x^2 \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2dx+\frac {2}{7} b c d^2 \left (-\frac {b \int \left (5 c^6 x^6-8 c^4 x^4+c^2 x^2+2\right )dx}{35 c^3}+\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arccos (c x))}{7 c^4}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{5 c^4}\right )+\frac {1}{7} d^2 x^3 \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {4}{7} d^2 \int x^2 \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2dx+\frac {1}{7} d^2 x^3 \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2+\frac {2}{7} b c d^2 \left (\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arccos (c x))}{7 c^4}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{5 c^4}-\frac {b \left (\frac {5 c^6 x^7}{7}-\frac {8 c^4 x^5}{5}+\frac {c^2 x^3}{3}+2 x\right )}{35 c^3}\right )\) |
| \(\Big \downarrow \) 5203 |
| \(\displaystyle \frac {4}{7} d^2 \left (\frac {2}{5} b c \int x^3 \sqrt {1-c^2 x^2} (a+b \arccos (c x))dx+\frac {2}{5} \int x^2 (a+b \arccos (c x))^2dx+\frac {1}{5} x^3 \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2\right )+\frac {1}{7} d^2 x^3 \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2+\frac {2}{7} b c d^2 \left (\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arccos (c x))}{7 c^4}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{5 c^4}-\frac {b \left (\frac {5 c^6 x^7}{7}-\frac {8 c^4 x^5}{5}+\frac {c^2 x^3}{3}+2 x\right )}{35 c^3}\right )\) |
| \(\Big \downarrow \) 5139 |
| \(\displaystyle \frac {4}{7} d^2 \left (\frac {2}{5} \left (\frac {2}{3} b c \int \frac {x^3 (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx+\frac {1}{3} x^3 (a+b \arccos (c x))^2\right )+\frac {2}{5} b c \int x^3 \sqrt {1-c^2 x^2} (a+b \arccos (c x))dx+\frac {1}{5} x^3 \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2\right )+\frac {1}{7} d^2 x^3 \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2+\frac {2}{7} b c d^2 \left (\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arccos (c x))}{7 c^4}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{5 c^4}-\frac {b \left (\frac {5 c^6 x^7}{7}-\frac {8 c^4 x^5}{5}+\frac {c^2 x^3}{3}+2 x\right )}{35 c^3}\right )\) |
| \(\Big \downarrow \) 5195 |
| \(\displaystyle \frac {4}{7} d^2 \left (\frac {2}{5} \left (\frac {2}{3} b c \int \frac {x^3 (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx+\frac {1}{3} x^3 (a+b \arccos (c x))^2\right )+\frac {2}{5} b c \left (b c \int -\frac {-3 c^4 x^4+c^2 x^2+2}{15 c^4}dx+\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{5 c^4}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))}{3 c^4}\right )+\frac {1}{5} x^3 \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2\right )+\frac {1}{7} d^2 x^3 \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2+\frac {2}{7} b c d^2 \left (\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arccos (c x))}{7 c^4}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{5 c^4}-\frac {b \left (\frac {5 c^6 x^7}{7}-\frac {8 c^4 x^5}{5}+\frac {c^2 x^3}{3}+2 x\right )}{35 c^3}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {4}{7} d^2 \left (\frac {2}{5} \left (\frac {2}{3} b c \int \frac {x^3 (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx+\frac {1}{3} x^3 (a+b \arccos (c x))^2\right )+\frac {2}{5} b c \left (-\frac {b \int \left (-3 c^4 x^4+c^2 x^2+2\right )dx}{15 c^3}+\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{5 c^4}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))}{3 c^4}\right )+\frac {1}{5} x^3 \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2\right )+\frac {1}{7} d^2 x^3 \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2+\frac {2}{7} b c d^2 \left (\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arccos (c x))}{7 c^4}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{5 c^4}-\frac {b \left (\frac {5 c^6 x^7}{7}-\frac {8 c^4 x^5}{5}+\frac {c^2 x^3}{3}+2 x\right )}{35 c^3}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {4}{7} d^2 \left (\frac {2}{5} \left (\frac {2}{3} b c \int \frac {x^3 (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx+\frac {1}{3} x^3 (a+b \arccos (c x))^2\right )+\frac {1}{5} x^3 \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2+\frac {2}{5} b c \left (\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{5 c^4}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))}{3 c^4}-\frac {b \left (-\frac {3}{5} c^4 x^5+\frac {c^2 x^3}{3}+2 x\right )}{15 c^3}\right )\right )+\frac {1}{7} d^2 x^3 \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2+\frac {2}{7} b c d^2 \left (\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arccos (c x))}{7 c^4}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{5 c^4}-\frac {b \left (\frac {5 c^6 x^7}{7}-\frac {8 c^4 x^5}{5}+\frac {c^2 x^3}{3}+2 x\right )}{35 c^3}\right )\) |
| \(\Big \downarrow \) 5211 |
| \(\displaystyle \frac {4}{7} d^2 \left (\frac {2}{5} \left (\frac {2}{3} b c \left (\frac {2 \int \frac {x (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx}{3 c^2}-\frac {b \int x^2dx}{3 c}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{3 c^2}\right )+\frac {1}{3} x^3 (a+b \arccos (c x))^2\right )+\frac {1}{5} x^3 \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2+\frac {2}{5} b c \left (\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{5 c^4}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))}{3 c^4}-\frac {b \left (-\frac {3}{5} c^4 x^5+\frac {c^2 x^3}{3}+2 x\right )}{15 c^3}\right )\right )+\frac {1}{7} d^2 x^3 \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2+\frac {2}{7} b c d^2 \left (\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arccos (c x))}{7 c^4}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{5 c^4}-\frac {b \left (\frac {5 c^6 x^7}{7}-\frac {8 c^4 x^5}{5}+\frac {c^2 x^3}{3}+2 x\right )}{35 c^3}\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {4}{7} d^2 \left (\frac {2}{5} \left (\frac {2}{3} b c \left (\frac {2 \int \frac {x (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx}{3 c^2}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{3 c^2}-\frac {b x^3}{9 c}\right )+\frac {1}{3} x^3 (a+b \arccos (c x))^2\right )+\frac {1}{5} x^3 \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2+\frac {2}{5} b c \left (\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{5 c^4}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))}{3 c^4}-\frac {b \left (-\frac {3}{5} c^4 x^5+\frac {c^2 x^3}{3}+2 x\right )}{15 c^3}\right )\right )+\frac {1}{7} d^2 x^3 \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2+\frac {2}{7} b c d^2 \left (\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arccos (c x))}{7 c^4}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{5 c^4}-\frac {b \left (\frac {5 c^6 x^7}{7}-\frac {8 c^4 x^5}{5}+\frac {c^2 x^3}{3}+2 x\right )}{35 c^3}\right )\) |
| \(\Big \downarrow \) 5183 |
| \(\displaystyle \frac {4}{7} d^2 \left (\frac {2}{5} \left (\frac {2}{3} b c \left (\frac {2 \left (-\frac {b \int 1dx}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))}{c^2}\right )}{3 c^2}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{3 c^2}-\frac {b x^3}{9 c}\right )+\frac {1}{3} x^3 (a+b \arccos (c x))^2\right )+\frac {1}{5} x^3 \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2+\frac {2}{5} b c \left (\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{5 c^4}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))}{3 c^4}-\frac {b \left (-\frac {3}{5} c^4 x^5+\frac {c^2 x^3}{3}+2 x\right )}{15 c^3}\right )\right )+\frac {1}{7} d^2 x^3 \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2+\frac {2}{7} b c d^2 \left (\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arccos (c x))}{7 c^4}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{5 c^4}-\frac {b \left (\frac {5 c^6 x^7}{7}-\frac {8 c^4 x^5}{5}+\frac {c^2 x^3}{3}+2 x\right )}{35 c^3}\right )\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {1}{7} d^2 x^3 \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2+\frac {4}{7} d^2 \left (\frac {1}{5} x^3 \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2+\frac {2}{5} \left (\frac {2}{3} b c \left (-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{3 c^2}+\frac {2 \left (-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))}{c^2}-\frac {b x}{c}\right )}{3 c^2}-\frac {b x^3}{9 c}\right )+\frac {1}{3} x^3 (a+b \arccos (c x))^2\right )+\frac {2}{5} b c \left (\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{5 c^4}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))}{3 c^4}-\frac {b \left (-\frac {3}{5} c^4 x^5+\frac {c^2 x^3}{3}+2 x\right )}{15 c^3}\right )\right )+\frac {2}{7} b c d^2 \left (\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arccos (c x))}{7 c^4}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{5 c^4}-\frac {b \left (\frac {5 c^6 x^7}{7}-\frac {8 c^4 x^5}{5}+\frac {c^2 x^3}{3}+2 x\right )}{35 c^3}\right )\) |
Input:
Int[x^2*(d - c^2*d*x^2)^2*(a + b*ArcCos[c*x])^2,x]
Output:
(d^2*x^3*(1 - c^2*x^2)^2*(a + b*ArcCos[c*x])^2)/7 + (2*b*c*d^2*(-1/35*(b*( 2*x + (c^2*x^3)/3 - (8*c^4*x^5)/5 + (5*c^6*x^7)/7))/c^3 - ((1 - c^2*x^2)^( 5/2)*(a + b*ArcCos[c*x]))/(5*c^4) + ((1 - c^2*x^2)^(7/2)*(a + b*ArcCos[c*x ]))/(7*c^4)))/7 + (4*d^2*((x^3*(1 - c^2*x^2)*(a + b*ArcCos[c*x])^2)/5 + (2 *b*c*(-1/15*(b*(2*x + (c^2*x^3)/3 - (3*c^4*x^5)/5))/c^3 - ((1 - c^2*x^2)^( 3/2)*(a + b*ArcCos[c*x]))/(3*c^4) + ((1 - c^2*x^2)^(5/2)*(a + b*ArcCos[c*x ]))/(5*c^4)))/5 + (2*((x^3*(a + b*ArcCos[c*x])^2)/3 + (2*b*c*(-1/9*(b*x^3) /c - (x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x]))/(3*c^2) + (2*(-((b*x)/c) - (Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x]))/c^2))/(3*c^2)))/3))/5))/7
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] :> I nt[ExpandIntegrand[(a + b*x^2)^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d }, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && IGtQ[q, 0]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcCos[c*x])^n/(d*(m + 1))), x] + Simp[b*c*(n /(d*(m + 1))) Int[(d*x)^(m + 1)*((a + b*ArcCos[c*x])^(n - 1)/Sqrt[1 - c^2 *x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_ .), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcCos[c*x])^n/(2*e*(p + 1))), x] - Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] I nt[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_) , x_Symbol] :> With[{u = IntHide[x^m*(d + e*x^2)^p, x]}, Simp[(a + b*ArcCos [c*x]) u, x] + Simp[b*c*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]] Int[Sim plifyIntegrand[u/Sqrt[d + e*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2] && NeQ[p, -2^(-1)] && (IGtQ[(m + 1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^p*((a + b*ArcC os[c*x])^n/(f*(m + 2*p + 1))), x] + (Simp[2*d*(p/(m + 2*p + 1)) Int[(f*x) ^m*(d + e*x^2)^(p - 1)*(a + b*ArcCos[c*x])^n, x], x] + Simp[b*c*(n/(f*(m + 2*p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] Int[(f*x)^(m + 1)*(1 - c^2 *x^2)^(p - 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] && !LtQ[m, -1]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcCos[c*x])^n/(e*(m + 2*p + 1))), x] + (Simp[f^2*((m - 1)/(c^2*(m + 2*p + 1))) Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcCos[c*x])^n, x], x] - S imp[b*f*(n/(c*(m + 2*p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] Int[(f* x)^(m - 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && IGtQ[m , 1] && NeQ[m + 2*p + 1, 0]
Time = 0.28 (sec) , antiderivative size = 399, normalized size of antiderivative = 1.29
| method | result | size |
| parts | \(d^{2} a^{2} \left (\frac {1}{7} c^{4} x^{7}-\frac {2}{5} c^{2} x^{5}+\frac {1}{3} x^{3}\right )+\frac {d^{2} b^{2} \left (\frac {\arccos \left (c x \right )^{2} \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{15}-\frac {2 \arccos \left (c x \right ) \left (c^{2} x^{2}-1\right )^{2} \sqrt {-c^{2} x^{2}+1}}{175}-\frac {2 \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{2625}+\frac {8 \arccos \left (c x \right ) \left (c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}}{315}+\frac {8 \left (c^{2} x^{2}-3\right ) c x}{945}-\frac {16 c x}{105}-\frac {16 \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{105}+\frac {\arccos \left (c x \right )^{2} \left (5 c^{6} x^{6}-21 c^{4} x^{4}+35 c^{2} x^{2}-35\right ) c x}{35}-\frac {2 \arccos \left (c x \right ) \left (c^{2} x^{2}-1\right )^{3} \sqrt {-c^{2} x^{2}+1}}{49}-\frac {2 \left (5 c^{6} x^{6}-21 c^{4} x^{4}+35 c^{2} x^{2}-35\right ) c x}{1715}\right )}{c^{3}}+\frac {2 d^{2} a b \left (\frac {\arccos \left (c x \right ) c^{7} x^{7}}{7}-\frac {2 \arccos \left (c x \right ) c^{5} x^{5}}{5}+\frac {c^{3} x^{3} \arccos \left (c x \right )}{3}-\frac {409 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{11025}-\frac {818 \sqrt {-c^{2} x^{2}+1}}{11025}+\frac {68 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{1225}-\frac {c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{49}\right )}{c^{3}}\) | \(399\) |
| derivativedivides | \(\frac {d^{2} a^{2} \left (\frac {1}{7} c^{7} x^{7}-\frac {2}{5} c^{5} x^{5}+\frac {1}{3} c^{3} x^{3}\right )+d^{2} b^{2} \left (\frac {\arccos \left (c x \right )^{2} \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{15}-\frac {2 \arccos \left (c x \right ) \left (c^{2} x^{2}-1\right )^{2} \sqrt {-c^{2} x^{2}+1}}{175}-\frac {2 \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{2625}+\frac {8 \arccos \left (c x \right ) \left (c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}}{315}+\frac {8 \left (c^{2} x^{2}-3\right ) c x}{945}-\frac {16 c x}{105}-\frac {16 \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{105}+\frac {\arccos \left (c x \right )^{2} \left (5 c^{6} x^{6}-21 c^{4} x^{4}+35 c^{2} x^{2}-35\right ) c x}{35}-\frac {2 \arccos \left (c x \right ) \left (c^{2} x^{2}-1\right )^{3} \sqrt {-c^{2} x^{2}+1}}{49}-\frac {2 \left (5 c^{6} x^{6}-21 c^{4} x^{4}+35 c^{2} x^{2}-35\right ) c x}{1715}\right )+2 d^{2} a b \left (\frac {\arccos \left (c x \right ) c^{7} x^{7}}{7}-\frac {2 \arccos \left (c x \right ) c^{5} x^{5}}{5}+\frac {c^{3} x^{3} \arccos \left (c x \right )}{3}-\frac {409 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{11025}-\frac {818 \sqrt {-c^{2} x^{2}+1}}{11025}+\frac {68 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{1225}-\frac {c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{49}\right )}{c^{3}}\) | \(400\) |
| default | \(\frac {d^{2} a^{2} \left (\frac {1}{7} c^{7} x^{7}-\frac {2}{5} c^{5} x^{5}+\frac {1}{3} c^{3} x^{3}\right )+d^{2} b^{2} \left (\frac {\arccos \left (c x \right )^{2} \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{15}-\frac {2 \arccos \left (c x \right ) \left (c^{2} x^{2}-1\right )^{2} \sqrt {-c^{2} x^{2}+1}}{175}-\frac {2 \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{2625}+\frac {8 \arccos \left (c x \right ) \left (c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}}{315}+\frac {8 \left (c^{2} x^{2}-3\right ) c x}{945}-\frac {16 c x}{105}-\frac {16 \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{105}+\frac {\arccos \left (c x \right )^{2} \left (5 c^{6} x^{6}-21 c^{4} x^{4}+35 c^{2} x^{2}-35\right ) c x}{35}-\frac {2 \arccos \left (c x \right ) \left (c^{2} x^{2}-1\right )^{3} \sqrt {-c^{2} x^{2}+1}}{49}-\frac {2 \left (5 c^{6} x^{6}-21 c^{4} x^{4}+35 c^{2} x^{2}-35\right ) c x}{1715}\right )+2 d^{2} a b \left (\frac {\arccos \left (c x \right ) c^{7} x^{7}}{7}-\frac {2 \arccos \left (c x \right ) c^{5} x^{5}}{5}+\frac {c^{3} x^{3} \arccos \left (c x \right )}{3}-\frac {409 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{11025}-\frac {818 \sqrt {-c^{2} x^{2}+1}}{11025}+\frac {68 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{1225}-\frac {c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{49}\right )}{c^{3}}\) | \(400\) |
| orering | \(\frac {\left (428625 c^{10} x^{10}-1739907 c^{8} x^{8}+2486259 c^{6} x^{6}+2357383 c^{4} x^{4}-2404920 c^{2} x^{2}+687120\right ) \left (-c^{2} d \,x^{2}+d \right )^{2} \left (a +b \arccos \left (c x \right )\right )^{2}}{1157625 x \,c^{4} \left (c x -1\right ) \left (c x +1\right ) \left (c^{2} x^{2}-1\right )^{2}}-\frac {\left (60750 c^{8} x^{8}-242757 c^{6} x^{6}+316078 c^{4} x^{4}+930475 c^{2} x^{2}-429450\right ) \left (2 x \left (-c^{2} d \,x^{2}+d \right )^{2} \left (a +b \arccos \left (c x \right )\right )^{2}-4 x^{3} \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \arccos \left (c x \right )\right )^{2} d \,c^{2}-\frac {2 x^{2} \left (-c^{2} d \,x^{2}+d \right )^{2} \left (a +b \arccos \left (c x \right )\right ) b c}{\sqrt {-c^{2} x^{2}+1}}\right )}{1157625 x^{2} c^{4} \left (c x -1\right ) \left (c x +1\right ) \left (c^{2} x^{2}-1\right )}+\frac {\left (3375 c^{6} x^{6}-12852 c^{4} x^{4}+14315 c^{2} x^{2}+85890\right ) \left (2 \left (-c^{2} d \,x^{2}+d \right )^{2} \left (a +b \arccos \left (c x \right )\right )^{2}-20 x^{2} \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \arccos \left (c x \right )\right )^{2} d \,c^{2}-\frac {8 x \left (-c^{2} d \,x^{2}+d \right )^{2} \left (a +b \arccos \left (c x \right )\right ) b c}{\sqrt {-c^{2} x^{2}+1}}+8 x^{4} d^{2} c^{4} \left (a +b \arccos \left (c x \right )\right )^{2}+\frac {16 x^{3} \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \arccos \left (c x \right )\right ) d \,c^{3} b}{\sqrt {-c^{2} x^{2}+1}}+\frac {2 x^{2} \left (-c^{2} d \,x^{2}+d \right )^{2} b^{2} c^{2}}{-c^{2} x^{2}+1}-\frac {2 x^{3} \left (-c^{2} d \,x^{2}+d \right )^{2} \left (a +b \arccos \left (c x \right )\right ) b \,c^{3}}{\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{1157625 x \,c^{4} \left (c x -1\right ) \left (c x +1\right )}\) | \(546\) |
Input:
int(x^2*(-c^2*d*x^2+d)^2*(a+b*arccos(c*x))^2,x,method=_RETURNVERBOSE)
Output:
d^2*a^2*(1/7*c^4*x^7-2/5*c^2*x^5+1/3*x^3)+d^2*b^2/c^3*(1/15*arccos(c*x)^2* (3*c^4*x^4-10*c^2*x^2+15)*c*x-2/175*arccos(c*x)*(c^2*x^2-1)^2*(-c^2*x^2+1) ^(1/2)-2/2625*(3*c^4*x^4-10*c^2*x^2+15)*c*x+8/315*arccos(c*x)*(c^2*x^2-1)* (-c^2*x^2+1)^(1/2)+8/945*(c^2*x^2-3)*c*x-16/105*c*x-16/105*arccos(c*x)*(-c ^2*x^2+1)^(1/2)+1/35*arccos(c*x)^2*(5*c^6*x^6-21*c^4*x^4+35*c^2*x^2-35)*c* x-2/49*arccos(c*x)*(c^2*x^2-1)^3*(-c^2*x^2+1)^(1/2)-2/1715*(5*c^6*x^6-21*c ^4*x^4+35*c^2*x^2-35)*c*x)+2*d^2*a*b/c^3*(1/7*arccos(c*x)*c^7*x^7-2/5*arcc os(c*x)*c^5*x^5+1/3*c^3*x^3*arccos(c*x)-409/11025*c^2*x^2*(-c^2*x^2+1)^(1/ 2)-818/11025*(-c^2*x^2+1)^(1/2)+68/1225*c^4*x^4*(-c^2*x^2+1)^(1/2)-1/49*c^ 6*x^6*(-c^2*x^2+1)^(1/2))
Time = 0.11 (sec) , antiderivative size = 296, normalized size of antiderivative = 0.95 \[ \int x^2 \left (d-c^2 d x^2\right )^2 (a+b \arccos (c x))^2 \, dx=\frac {3375 \, {\left (49 \, a^{2} - 2 \, b^{2}\right )} c^{7} d^{2} x^{7} - 378 \, {\left (1225 \, a^{2} - 68 \, b^{2}\right )} c^{5} d^{2} x^{5} + 35 \, {\left (11025 \, a^{2} - 818 \, b^{2}\right )} c^{3} d^{2} x^{3} - 171780 \, b^{2} c d^{2} x + 11025 \, {\left (15 \, b^{2} c^{7} d^{2} x^{7} - 42 \, b^{2} c^{5} d^{2} x^{5} + 35 \, b^{2} c^{3} d^{2} x^{3}\right )} \arccos \left (c x\right )^{2} + 22050 \, {\left (15 \, a b c^{7} d^{2} x^{7} - 42 \, a b c^{5} d^{2} x^{5} + 35 \, a b c^{3} d^{2} x^{3}\right )} \arccos \left (c x\right ) - 210 \, {\left (225 \, a b c^{6} d^{2} x^{6} - 612 \, a b c^{4} d^{2} x^{4} + 409 \, a b c^{2} d^{2} x^{2} + 818 \, a b d^{2} + {\left (225 \, b^{2} c^{6} d^{2} x^{6} - 612 \, b^{2} c^{4} d^{2} x^{4} + 409 \, b^{2} c^{2} d^{2} x^{2} + 818 \, b^{2} d^{2}\right )} \arccos \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{1157625 \, c^{3}} \] Input:
integrate(x^2*(-c^2*d*x^2+d)^2*(a+b*arccos(c*x))^2,x, algorithm="fricas")
Output:
1/1157625*(3375*(49*a^2 - 2*b^2)*c^7*d^2*x^7 - 378*(1225*a^2 - 68*b^2)*c^5 *d^2*x^5 + 35*(11025*a^2 - 818*b^2)*c^3*d^2*x^3 - 171780*b^2*c*d^2*x + 110 25*(15*b^2*c^7*d^2*x^7 - 42*b^2*c^5*d^2*x^5 + 35*b^2*c^3*d^2*x^3)*arccos(c *x)^2 + 22050*(15*a*b*c^7*d^2*x^7 - 42*a*b*c^5*d^2*x^5 + 35*a*b*c^3*d^2*x^ 3)*arccos(c*x) - 210*(225*a*b*c^6*d^2*x^6 - 612*a*b*c^4*d^2*x^4 + 409*a*b* c^2*d^2*x^2 + 818*a*b*d^2 + (225*b^2*c^6*d^2*x^6 - 612*b^2*c^4*d^2*x^4 + 4 09*b^2*c^2*d^2*x^2 + 818*b^2*d^2)*arccos(c*x))*sqrt(-c^2*x^2 + 1))/c^3
Time = 0.89 (sec) , antiderivative size = 488, normalized size of antiderivative = 1.57 \[ \int x^2 \left (d-c^2 d x^2\right )^2 (a+b \arccos (c x))^2 \, dx=\begin {cases} \frac {a^{2} c^{4} d^{2} x^{7}}{7} - \frac {2 a^{2} c^{2} d^{2} x^{5}}{5} + \frac {a^{2} d^{2} x^{3}}{3} + \frac {2 a b c^{4} d^{2} x^{7} \operatorname {acos}{\left (c x \right )}}{7} - \frac {2 a b c^{3} d^{2} x^{6} \sqrt {- c^{2} x^{2} + 1}}{49} - \frac {4 a b c^{2} d^{2} x^{5} \operatorname {acos}{\left (c x \right )}}{5} + \frac {136 a b c d^{2} x^{4} \sqrt {- c^{2} x^{2} + 1}}{1225} + \frac {2 a b d^{2} x^{3} \operatorname {acos}{\left (c x \right )}}{3} - \frac {818 a b d^{2} x^{2} \sqrt {- c^{2} x^{2} + 1}}{11025 c} - \frac {1636 a b d^{2} \sqrt {- c^{2} x^{2} + 1}}{11025 c^{3}} + \frac {b^{2} c^{4} d^{2} x^{7} \operatorname {acos}^{2}{\left (c x \right )}}{7} - \frac {2 b^{2} c^{4} d^{2} x^{7}}{343} - \frac {2 b^{2} c^{3} d^{2} x^{6} \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}{\left (c x \right )}}{49} - \frac {2 b^{2} c^{2} d^{2} x^{5} \operatorname {acos}^{2}{\left (c x \right )}}{5} + \frac {136 b^{2} c^{2} d^{2} x^{5}}{6125} + \frac {136 b^{2} c d^{2} x^{4} \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}{\left (c x \right )}}{1225} + \frac {b^{2} d^{2} x^{3} \operatorname {acos}^{2}{\left (c x \right )}}{3} - \frac {818 b^{2} d^{2} x^{3}}{33075} - \frac {818 b^{2} d^{2} x^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}{\left (c x \right )}}{11025 c} - \frac {1636 b^{2} d^{2} x}{11025 c^{2}} - \frac {1636 b^{2} d^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}{\left (c x \right )}}{11025 c^{3}} & \text {for}\: c \neq 0 \\\frac {d^{2} x^{3} \left (a + \frac {\pi b}{2}\right )^{2}}{3} & \text {otherwise} \end {cases} \] Input:
integrate(x**2*(-c**2*d*x**2+d)**2*(a+b*acos(c*x))**2,x)
Output:
Piecewise((a**2*c**4*d**2*x**7/7 - 2*a**2*c**2*d**2*x**5/5 + a**2*d**2*x** 3/3 + 2*a*b*c**4*d**2*x**7*acos(c*x)/7 - 2*a*b*c**3*d**2*x**6*sqrt(-c**2*x **2 + 1)/49 - 4*a*b*c**2*d**2*x**5*acos(c*x)/5 + 136*a*b*c*d**2*x**4*sqrt( -c**2*x**2 + 1)/1225 + 2*a*b*d**2*x**3*acos(c*x)/3 - 818*a*b*d**2*x**2*sqr t(-c**2*x**2 + 1)/(11025*c) - 1636*a*b*d**2*sqrt(-c**2*x**2 + 1)/(11025*c* *3) + b**2*c**4*d**2*x**7*acos(c*x)**2/7 - 2*b**2*c**4*d**2*x**7/343 - 2*b **2*c**3*d**2*x**6*sqrt(-c**2*x**2 + 1)*acos(c*x)/49 - 2*b**2*c**2*d**2*x* *5*acos(c*x)**2/5 + 136*b**2*c**2*d**2*x**5/6125 + 136*b**2*c*d**2*x**4*sq rt(-c**2*x**2 + 1)*acos(c*x)/1225 + b**2*d**2*x**3*acos(c*x)**2/3 - 818*b* *2*d**2*x**3/33075 - 818*b**2*d**2*x**2*sqrt(-c**2*x**2 + 1)*acos(c*x)/(11 025*c) - 1636*b**2*d**2*x/(11025*c**2) - 1636*b**2*d**2*sqrt(-c**2*x**2 + 1)*acos(c*x)/(11025*c**3), Ne(c, 0)), (d**2*x**3*(a + pi*b/2)**2/3, True))
Leaf count of result is larger than twice the leaf count of optimal. 634 vs. \(2 (274) = 548\).
Time = 0.14 (sec) , antiderivative size = 634, normalized size of antiderivative = 2.05 \[ \int x^2 \left (d-c^2 d x^2\right )^2 (a+b \arccos (c x))^2 \, dx=\frac {1}{7} \, b^{2} c^{4} d^{2} x^{7} \arccos \left (c x\right )^{2} + \frac {1}{7} \, a^{2} c^{4} d^{2} x^{7} - \frac {2}{5} \, b^{2} c^{2} d^{2} x^{5} \arccos \left (c x\right )^{2} - \frac {2}{5} \, a^{2} c^{2} d^{2} x^{5} + \frac {2}{245} \, {\left (35 \, x^{7} \arccos \left (c x\right ) - {\left (\frac {5 \, \sqrt {-c^{2} x^{2} + 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {-c^{2} x^{2} + 1}}{c^{8}}\right )} c\right )} a b c^{4} d^{2} - \frac {2}{25725} \, {\left (105 \, {\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 \arccos \left (c x\right ) + \frac {75 \, c^{6} x^{7} + 126 \, c^{4} x^{5} + 280 \, c^{2} x^{3} + 1680 \, x}{c^{6}}\right )} b^{2} c^{4} d^{2} + \frac {1}{3} \, b^{2} d^{2} x^{3} \arccos \left (c x\right )^{2} - \frac {4}{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 )} a b c^{2} d^{2} + \frac {4}{1125} \, {\left (15 \, {\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 \arccos \left (c x\right ) + \frac {9 \, c^{4} x^{5} + 20 \, c^{2} x^{3} + 120 \, x}{c^{4}}\right )} b^{2} c^{2} d^{2} + \frac {1}{3} \, a^{2} d^{2} x^{3} + \frac {2}{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 )} a b d^{2} - \frac {2}{27} \, {\left (3 \, c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )} \arccos \left (c x\right ) + \frac {c^{2} x^{3} + 6 \, x}{c^{2}}\right )} b^{2} d^{2} \] Input:
integrate(x^2*(-c^2*d*x^2+d)^2*(a+b*arccos(c*x))^2,x, algorithm="maxima")
Output:
1/7*b^2*c^4*d^2*x^7*arccos(c*x)^2 + 1/7*a^2*c^4*d^2*x^7 - 2/5*b^2*c^2*d^2* x^5*arccos(c*x)^2 - 2/5*a^2*c^2*d^2*x^5 + 2/245*(35*x^7*arccos(c*x) - (5*s qrt(-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*c^4*d^2 - 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 + 1680*x)/c^6)*b^2*c^4*d^2 + 1/3*b^2*d^2*x^3*a rccos(c*x)^2 - 4/75*(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)*a*b*c^2*d^2 + 4/1125*(15*(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*arccos(c*x) + (9*c^4*x^5 + 20*c^2*x^3 + 120*x) /c^4)*b^2*c^2*d^2 + 1/3*a^2*d^2*x^3 + 2/9*(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))*a*b*d^2 - 2/27*(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
Time = 0.17 (sec) , antiderivative size = 406, normalized size of antiderivative = 1.31 \[ \int x^2 \left (d-c^2 d x^2\right )^2 (a+b \arccos (c x))^2 \, dx=\frac {1}{7} \, b^{2} c^{4} d^{2} x^{7} \arccos \left (c x\right )^{2} + \frac {2}{7} \, a b c^{4} d^{2} x^{7} \arccos \left (c x\right ) + \frac {1}{7} \, a^{2} c^{4} d^{2} x^{7} - \frac {2}{343} \, b^{2} c^{4} d^{2} x^{7} - \frac {2}{49} \, \sqrt {-c^{2} x^{2} + 1} b^{2} c^{3} d^{2} x^{6} \arccos \left (c x\right ) - \frac {2}{49} \, \sqrt {-c^{2} x^{2} + 1} a b c^{3} d^{2} x^{6} - \frac {2}{5} \, b^{2} c^{2} d^{2} x^{5} \arccos \left (c x\right )^{2} - \frac {4}{5} \, a b c^{2} d^{2} x^{5} \arccos \left (c x\right ) - \frac {2}{5} \, a^{2} c^{2} d^{2} x^{5} + \frac {136}{6125} \, b^{2} c^{2} d^{2} x^{5} + \frac {136}{1225} \, \sqrt {-c^{2} x^{2} + 1} b^{2} c d^{2} x^{4} \arccos \left (c x\right ) + \frac {136}{1225} \, \sqrt {-c^{2} x^{2} + 1} a b c d^{2} x^{4} + \frac {1}{3} \, b^{2} d^{2} x^{3} \arccos \left (c x\right )^{2} + \frac {2}{3} \, a b d^{2} x^{3} \arccos \left (c x\right ) + \frac {1}{3} \, a^{2} d^{2} x^{3} - \frac {818}{33075} \, b^{2} d^{2} x^{3} - \frac {818 \, \sqrt {-c^{2} x^{2} + 1} b^{2} d^{2} x^{2} \arccos \left (c x\right )}{11025 \, c} - \frac {818 \, \sqrt {-c^{2} x^{2} + 1} a b d^{2} x^{2}}{11025 \, c} - \frac {1636 \, b^{2} d^{2} x}{11025 \, c^{2}} - \frac {1636 \, \sqrt {-c^{2} x^{2} + 1} b^{2} d^{2} \arccos \left (c x\right )}{11025 \, c^{3}} - \frac {1636 \, \sqrt {-c^{2} x^{2} + 1} a b d^{2}}{11025 \, c^{3}} \] Input:
integrate(x^2*(-c^2*d*x^2+d)^2*(a+b*arccos(c*x))^2,x, algorithm="giac")
Output:
1/7*b^2*c^4*d^2*x^7*arccos(c*x)^2 + 2/7*a*b*c^4*d^2*x^7*arccos(c*x) + 1/7* a^2*c^4*d^2*x^7 - 2/343*b^2*c^4*d^2*x^7 - 2/49*sqrt(-c^2*x^2 + 1)*b^2*c^3* d^2*x^6*arccos(c*x) - 2/49*sqrt(-c^2*x^2 + 1)*a*b*c^3*d^2*x^6 - 2/5*b^2*c^ 2*d^2*x^5*arccos(c*x)^2 - 4/5*a*b*c^2*d^2*x^5*arccos(c*x) - 2/5*a^2*c^2*d^ 2*x^5 + 136/6125*b^2*c^2*d^2*x^5 + 136/1225*sqrt(-c^2*x^2 + 1)*b^2*c*d^2*x ^4*arccos(c*x) + 136/1225*sqrt(-c^2*x^2 + 1)*a*b*c*d^2*x^4 + 1/3*b^2*d^2*x ^3*arccos(c*x)^2 + 2/3*a*b*d^2*x^3*arccos(c*x) + 1/3*a^2*d^2*x^3 - 818/330 75*b^2*d^2*x^3 - 818/11025*sqrt(-c^2*x^2 + 1)*b^2*d^2*x^2*arccos(c*x)/c - 818/11025*sqrt(-c^2*x^2 + 1)*a*b*d^2*x^2/c - 1636/11025*b^2*d^2*x/c^2 - 16 36/11025*sqrt(-c^2*x^2 + 1)*b^2*d^2*arccos(c*x)/c^3 - 1636/11025*sqrt(-c^2 *x^2 + 1)*a*b*d^2/c^3
Timed out. \[ \int x^2 \left (d-c^2 d x^2\right )^2 (a+b \arccos (c x))^2 \, dx=\int x^2\,{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^2 \,d x \] Input:
int(x^2*(a + b*acos(c*x))^2*(d - c^2*d*x^2)^2,x)
Output:
int(x^2*(a + b*acos(c*x))^2*(d - c^2*d*x^2)^2, x)
\[ \int x^2 \left (d-c^2 d x^2\right )^2 (a+b \arccos (c x))^2 \, dx=\frac {d^{2} \left (3150 \mathit {acos} \left (c x \right ) a b \,c^{7} x^{7}-8820 \mathit {acos} \left (c x \right ) a b \,c^{5} x^{5}+7350 \mathit {acos} \left (c x \right ) a b \,c^{3} x^{3}-450 \sqrt {-c^{2} x^{2}+1}\, a b \,c^{6} x^{6}+1224 \sqrt {-c^{2} x^{2}+1}\, a b \,c^{4} x^{4}-818 \sqrt {-c^{2} x^{2}+1}\, a b \,c^{2} x^{2}-1636 \sqrt {-c^{2} x^{2}+1}\, a b +11025 \left (\int \mathit {acos} \left (c x \right )^{2} x^{6}d x \right ) b^{2} c^{7}-22050 \left (\int \mathit {acos} \left (c x \right )^{2} x^{4}d x \right ) b^{2} c^{5}+11025 \left (\int \mathit {acos} \left (c x \right )^{2} x^{2}d x \right ) b^{2} c^{3}+1575 a^{2} c^{7} x^{7}-4410 a^{2} c^{5} x^{5}+3675 a^{2} c^{3} x^{3}\right )}{11025 c^{3}} \] Input:
int(x^2*(-c^2*d*x^2+d)^2*(a+b*acos(c*x))^2,x)
Output:
(d**2*(3150*acos(c*x)*a*b*c**7*x**7 - 8820*acos(c*x)*a*b*c**5*x**5 + 7350* acos(c*x)*a*b*c**3*x**3 - 450*sqrt( - c**2*x**2 + 1)*a*b*c**6*x**6 + 1224* sqrt( - c**2*x**2 + 1)*a*b*c**4*x**4 - 818*sqrt( - c**2*x**2 + 1)*a*b*c**2 *x**2 - 1636*sqrt( - c**2*x**2 + 1)*a*b + 11025*int(acos(c*x)**2*x**6,x)*b **2*c**7 - 22050*int(acos(c*x)**2*x**4,x)*b**2*c**5 + 11025*int(acos(c*x)* *2*x**2,x)*b**2*c**3 + 1575*a**2*c**7*x**7 - 4410*a**2*c**5*x**5 + 3675*a* *2*c**3*x**3))/(11025*c**3)