Integrand size = 29, antiderivative size = 29 \[ \int x^m \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2 \, dx=\frac {10 b^2 c^2 d^2 x^{3+m} \sqrt {d-c^2 d x^2}}{(4+m)^3 (6+m)}+\frac {2 b^2 c^2 d^2 \left (52+15 m+m^2\right ) x^{3+m} \sqrt {d-c^2 d x^2}}{(4+m)^2 (6+m)^3}-\frac {2 b^2 c^4 d^2 x^{5+m} \sqrt {d-c^2 d x^2}}{(6+m)^3}-\frac {30 b c d^2 x^{2+m} \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{(2+m)^2 (4+m) (6+m) \sqrt {1-c^2 x^2}}-\frac {10 b c d^2 x^{2+m} \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{(6+m) \left (8+6 m+m^2\right ) \sqrt {1-c^2 x^2}}-\frac {2 b c d^2 x^{2+m} \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{\left (12+8 m+m^2\right ) \sqrt {1-c^2 x^2}}+\frac {10 b c^3 d^2 x^{4+m} \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{(4+m)^2 (6+m) \sqrt {1-c^2 x^2}}+\frac {4 b c^3 d^2 x^{4+m} \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{(4+m) (6+m) \sqrt {1-c^2 x^2}}-\frac {2 b c^5 d^2 x^{6+m} \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{(6+m)^2 \sqrt {1-c^2 x^2}}+\frac {15 d^2 x^{1+m} \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{(6+m) \left (8+6 m+m^2\right )}+\frac {5 d x^{1+m} \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2}{(4+m) (6+m)}+\frac {x^{1+m} \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2}{6+m}+\frac {30 b^2 c^2 d^2 x^{3+m} \sqrt {d-c^2 d x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3+m}{2},\frac {5+m}{2},c^2 x^2\right )}{(2+m)^2 (3+m) (4+m) (6+m) \sqrt {1-c^2 x^2}}+\frac {10 b^2 c^2 d^2 (10+3 m) x^{3+m} \sqrt {d-c^2 d x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3+m}{2},\frac {5+m}{2},c^2 x^2\right )}{(2+m) (3+m) (4+m)^3 (6+m) \sqrt {1-c^2 x^2}}+\frac {2 b^2 c^2 d^2 \left (264+130 m+15 m^2\right ) x^{3+m} \sqrt {d-c^2 d x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3+m}{2},\frac {5+m}{2},c^2 x^2\right )}{(2+m) (3+m) (4+m)^2 (6+m)^3 \sqrt {1-c^2 x^2}}+\frac {15 d^3 \text {Int}\left (\frac {x^m (a+b \arccos (c x))^2}{\sqrt {d-c^2 d x^2}},x\right )}{(6+m) \left (8+6 m+m^2\right )} \] Output:
10*b^2*c^2*d^2*x^(3+m)*(-c^2*d*x^2+d)^(1/2)/(4+m)^3/(6+m)+2*b^2*c^2*d^2*(m ^2+15*m+52)*x^(3+m)*(-c^2*d*x^2+d)^(1/2)/(4+m)^2/(6+m)^3-2*b^2*c^4*d^2*x^( 5+m)*(-c^2*d*x^2+d)^(1/2)/(6+m)^3-30*b*c*d^2*x^(2+m)*(-c^2*d*x^2+d)^(1/2)* (a+b*arccos(c*x))/(2+m)^2/(4+m)/(6+m)/(-c^2*x^2+1)^(1/2)-10*b*c*d^2*x^(2+m )*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))/(6+m)/(m^2+6*m+8)/(-c^2*x^2+1)^(1 /2)-2*b*c*d^2*x^(2+m)*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))/(m^2+8*m+12)/ (-c^2*x^2+1)^(1/2)+10*b*c^3*d^2*x^(4+m)*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c *x))/(4+m)^2/(6+m)/(-c^2*x^2+1)^(1/2)+4*b*c^3*d^2*x^(4+m)*(-c^2*d*x^2+d)^( 1/2)*(a+b*arccos(c*x))/(4+m)/(6+m)/(-c^2*x^2+1)^(1/2)-2*b*c^5*d^2*x^(6+m)* (-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))/(6+m)^2/(-c^2*x^2+1)^(1/2)+15*d^2*x ^(1+m)*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))^2/(6+m)/(m^2+6*m+8)+5*d*x^(1 +m)*(-c^2*d*x^2+d)^(3/2)*(a+b*arccos(c*x))^2/(4+m)/(6+m)+x^(1+m)*(-c^2*d*x ^2+d)^(5/2)*(a+b*arccos(c*x))^2/(6+m)+30*b^2*c^2*d^2*x^(3+m)*(-c^2*d*x^2+d )^(1/2)*hypergeom([1/2, 3/2+1/2*m],[5/2+1/2*m],c^2*x^2)/(2+m)^2/(3+m)/(4+m )/(6+m)/(-c^2*x^2+1)^(1/2)+10*b^2*c^2*d^2*(10+3*m)*x^(3+m)*(-c^2*d*x^2+d)^ (1/2)*hypergeom([1/2, 3/2+1/2*m],[5/2+1/2*m],c^2*x^2)/(2+m)/(3+m)/(4+m)^3/ (6+m)/(-c^2*x^2+1)^(1/2)+2*b^2*c^2*d^2*(15*m^2+130*m+264)*x^(3+m)*(-c^2*d* x^2+d)^(1/2)*hypergeom([1/2, 3/2+1/2*m],[5/2+1/2*m],c^2*x^2)/(2+m)/(3+m)/( 4+m)^2/(6+m)^3/(-c^2*x^2+1)^(1/2)+15*d^3*Defer(Int)(x^m*(a+b*arccos(c*x))^ 2/(-c^2*d*x^2+d)^(1/2),x)/(6+m)/(m^2+6*m+8)
Not integrable
Time = 1.28 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int x^m \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2 \, dx=\int x^m \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2 \, dx \] Input:
Integrate[x^m*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCos[c*x])^2,x]
Output:
Integrate[x^m*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCos[c*x])^2, x]
Not integrable
Time = 3.22 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
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.
| \(\displaystyle \int x^m \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2 \, dx\) |
| \(\Big \downarrow \) 5203 |
| \(\displaystyle \frac {2 b c d^2 \sqrt {d-c^2 d x^2} \int x^{m+1} \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))dx}{(m+6) \sqrt {1-c^2 x^2}}+\frac {5 d \int x^m \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2dx}{m+6}+\frac {x^{m+1} \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2}{m+6}\) |
| \(\Big \downarrow \) 5193 |
| \(\displaystyle \frac {5 d \int x^m \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2dx}{m+6}+\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (b c \int \frac {x^{m+2} \left (\frac {c^4 x^4}{m+6}-\frac {2 c^2 x^2}{m+4}+\frac {1}{m+2}\right )}{\sqrt {1-c^2 x^2}}dx+\frac {c^4 x^{m+6} (a+b \arccos (c x))}{m+6}-\frac {2 c^2 x^{m+4} (a+b \arccos (c x))}{m+4}+\frac {x^{m+2} (a+b \arccos (c x))}{m+2}\right )}{(m+6) \sqrt {1-c^2 x^2}}+\frac {x^{m+1} \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2}{m+6}\) |
| \(\Big \downarrow \) 1590 |
| \(\displaystyle \frac {5 d \int x^m \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2dx}{m+6}+\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (b c \left (-\frac {\int -\frac {c^2 x^{m+2} \left (\frac {m+6}{m+2}-\frac {c^2 \left (m^2+15 m+52\right ) x^2}{(m+4) (m+6)}\right )}{\sqrt {1-c^2 x^2}}dx}{c^2 (m+6)}-\frac {c^2 \sqrt {1-c^2 x^2} x^{m+5}}{(m+6)^2}\right )+\frac {c^4 x^{m+6} (a+b \arccos (c x))}{m+6}-\frac {2 c^2 x^{m+4} (a+b \arccos (c x))}{m+4}+\frac {x^{m+2} (a+b \arccos (c x))}{m+2}\right )}{(m+6) \sqrt {1-c^2 x^2}}+\frac {x^{m+1} \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2}{m+6}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {5 d \int x^m \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2dx}{m+6}+\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (b c \left (\frac {\int \frac {c^2 x^{m+2} \left (\frac {m+6}{m+2}-\frac {c^2 \left (m^2+15 m+52\right ) x^2}{(m+4) (m+6)}\right )}{\sqrt {1-c^2 x^2}}dx}{c^2 (m+6)}-\frac {c^2 \sqrt {1-c^2 x^2} x^{m+5}}{(m+6)^2}\right )+\frac {c^4 x^{m+6} (a+b \arccos (c x))}{m+6}-\frac {2 c^2 x^{m+4} (a+b \arccos (c x))}{m+4}+\frac {x^{m+2} (a+b \arccos (c x))}{m+2}\right )}{(m+6) \sqrt {1-c^2 x^2}}+\frac {x^{m+1} \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2}{m+6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5 d \int x^m \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2dx}{m+6}+\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (b c \left (\frac {\int \frac {x^{m+2} \left (\frac {m+6}{m+2}-\frac {c^2 \left (m^2+15 m+52\right ) x^2}{(m+4) (m+6)}\right )}{\sqrt {1-c^2 x^2}}dx}{m+6}-\frac {c^2 \sqrt {1-c^2 x^2} x^{m+5}}{(m+6)^2}\right )+\frac {c^4 x^{m+6} (a+b \arccos (c x))}{m+6}-\frac {2 c^2 x^{m+4} (a+b \arccos (c x))}{m+4}+\frac {x^{m+2} (a+b \arccos (c x))}{m+2}\right )}{(m+6) \sqrt {1-c^2 x^2}}+\frac {x^{m+1} \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2}{m+6}\) |
| \(\Big \downarrow \) 363 |
| \(\displaystyle \frac {5 d \int x^m \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2dx}{m+6}+\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (b c \left (\frac {\frac {\left (15 m^2+130 m+264\right ) \int \frac {x^{m+2}}{\sqrt {1-c^2 x^2}}dx}{(m+2) (m+4)^2 (m+6)}+\frac {\left (m^2+15 m+52\right ) \sqrt {1-c^2 x^2} x^{m+3}}{(m+4)^2 (m+6)}}{m+6}-\frac {c^2 \sqrt {1-c^2 x^2} x^{m+5}}{(m+6)^2}\right )+\frac {c^4 x^{m+6} (a+b \arccos (c x))}{m+6}-\frac {2 c^2 x^{m+4} (a+b \arccos (c x))}{m+4}+\frac {x^{m+2} (a+b \arccos (c x))}{m+2}\right )}{(m+6) \sqrt {1-c^2 x^2}}+\frac {x^{m+1} \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2}{m+6}\) |
| \(\Big \downarrow \) 278 |
| \(\displaystyle \frac {5 d \int x^m \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2dx}{m+6}+\frac {x^{m+1} \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2}{m+6}+\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {c^4 x^{m+6} (a+b \arccos (c x))}{m+6}-\frac {2 c^2 x^{m+4} (a+b \arccos (c x))}{m+4}+\frac {x^{m+2} (a+b \arccos (c x))}{m+2}+b c \left (\frac {\frac {\left (15 m^2+130 m+264\right ) x^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{(m+2) (m+3) (m+4)^2 (m+6)}+\frac {\left (m^2+15 m+52\right ) \sqrt {1-c^2 x^2} x^{m+3}}{(m+4)^2 (m+6)}}{m+6}-\frac {c^2 \sqrt {1-c^2 x^2} x^{m+5}}{(m+6)^2}\right )\right )}{(m+6) \sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 5203 |
| \(\displaystyle \frac {5 d \left (\frac {2 b c d \sqrt {d-c^2 d x^2} \int x^{m+1} \left (1-c^2 x^2\right ) (a+b \arccos (c x))dx}{(m+4) \sqrt {1-c^2 x^2}}+\frac {3 d \int x^m \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2dx}{m+4}+\frac {x^{m+1} \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2}{m+4}\right )}{m+6}+\frac {x^{m+1} \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2}{m+6}+\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {c^4 x^{m+6} (a+b \arccos (c x))}{m+6}-\frac {2 c^2 x^{m+4} (a+b \arccos (c x))}{m+4}+\frac {x^{m+2} (a+b \arccos (c x))}{m+2}+b c \left (\frac {\frac {\left (15 m^2+130 m+264\right ) x^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{(m+2) (m+3) (m+4)^2 (m+6)}+\frac {\left (m^2+15 m+52\right ) \sqrt {1-c^2 x^2} x^{m+3}}{(m+4)^2 (m+6)}}{m+6}-\frac {c^2 \sqrt {1-c^2 x^2} x^{m+5}}{(m+6)^2}\right )\right )}{(m+6) \sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 5193 |
| \(\displaystyle \frac {5 d \left (\frac {2 b c d \sqrt {d-c^2 d x^2} \left (b c \int \frac {x^{m+2} \left (\frac {1}{m+2}-\frac {c^2 x^2}{m+4}\right )}{\sqrt {1-c^2 x^2}}dx-\frac {c^2 x^{m+4} (a+b \arccos (c x))}{m+4}+\frac {x^{m+2} (a+b \arccos (c x))}{m+2}\right )}{(m+4) \sqrt {1-c^2 x^2}}+\frac {3 d \int x^m \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2dx}{m+4}+\frac {x^{m+1} \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2}{m+4}\right )}{m+6}+\frac {x^{m+1} \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2}{m+6}+\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {c^4 x^{m+6} (a+b \arccos (c x))}{m+6}-\frac {2 c^2 x^{m+4} (a+b \arccos (c x))}{m+4}+\frac {x^{m+2} (a+b \arccos (c x))}{m+2}+b c \left (\frac {\frac {\left (15 m^2+130 m+264\right ) x^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{(m+2) (m+3) (m+4)^2 (m+6)}+\frac {\left (m^2+15 m+52\right ) \sqrt {1-c^2 x^2} x^{m+3}}{(m+4)^2 (m+6)}}{m+6}-\frac {c^2 \sqrt {1-c^2 x^2} x^{m+5}}{(m+6)^2}\right )\right )}{(m+6) \sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 363 |
| \(\displaystyle \frac {5 d \left (\frac {2 b c d \sqrt {d-c^2 d x^2} \left (b c \left (\frac {(3 m+10) \int \frac {x^{m+2}}{\sqrt {1-c^2 x^2}}dx}{(m+2) (m+4)^2}+\frac {\sqrt {1-c^2 x^2} x^{m+3}}{(m+4)^2}\right )-\frac {c^2 x^{m+4} (a+b \arccos (c x))}{m+4}+\frac {x^{m+2} (a+b \arccos (c x))}{m+2}\right )}{(m+4) \sqrt {1-c^2 x^2}}+\frac {3 d \int x^m \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2dx}{m+4}+\frac {x^{m+1} \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2}{m+4}\right )}{m+6}+\frac {x^{m+1} \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2}{m+6}+\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {c^4 x^{m+6} (a+b \arccos (c x))}{m+6}-\frac {2 c^2 x^{m+4} (a+b \arccos (c x))}{m+4}+\frac {x^{m+2} (a+b \arccos (c x))}{m+2}+b c \left (\frac {\frac {\left (15 m^2+130 m+264\right ) x^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{(m+2) (m+3) (m+4)^2 (m+6)}+\frac {\left (m^2+15 m+52\right ) \sqrt {1-c^2 x^2} x^{m+3}}{(m+4)^2 (m+6)}}{m+6}-\frac {c^2 \sqrt {1-c^2 x^2} x^{m+5}}{(m+6)^2}\right )\right )}{(m+6) \sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 278 |
| \(\displaystyle \frac {5 d \left (\frac {3 d \int x^m \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2dx}{m+4}+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {c^2 x^{m+4} (a+b \arccos (c x))}{m+4}+\frac {x^{m+2} (a+b \arccos (c x))}{m+2}+b c \left (\frac {(3 m+10) x^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{(m+2) (m+3) (m+4)^2}+\frac {\sqrt {1-c^2 x^2} x^{m+3}}{(m+4)^2}\right )\right )}{(m+4) \sqrt {1-c^2 x^2}}+\frac {x^{m+1} \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2}{m+4}\right )}{m+6}+\frac {x^{m+1} \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2}{m+6}+\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {c^4 x^{m+6} (a+b \arccos (c x))}{m+6}-\frac {2 c^2 x^{m+4} (a+b \arccos (c x))}{m+4}+\frac {x^{m+2} (a+b \arccos (c x))}{m+2}+b c \left (\frac {\frac {\left (15 m^2+130 m+264\right ) x^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{(m+2) (m+3) (m+4)^2 (m+6)}+\frac {\left (m^2+15 m+52\right ) \sqrt {1-c^2 x^2} x^{m+3}}{(m+4)^2 (m+6)}}{m+6}-\frac {c^2 \sqrt {1-c^2 x^2} x^{m+5}}{(m+6)^2}\right )\right )}{(m+6) \sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 5203 |
| \(\displaystyle \frac {5 d \left (\frac {3 d \left (\frac {2 b c \sqrt {d-c^2 d x^2} \int x^{m+1} (a+b \arccos (c x))dx}{(m+2) \sqrt {1-c^2 x^2}}+\frac {d \int \frac {x^m (a+b \arccos (c x))^2}{\sqrt {d-c^2 d x^2}}dx}{m+2}+\frac {x^{m+1} \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{m+2}\right )}{m+4}+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {c^2 x^{m+4} (a+b \arccos (c x))}{m+4}+\frac {x^{m+2} (a+b \arccos (c x))}{m+2}+b c \left (\frac {(3 m+10) x^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{(m+2) (m+3) (m+4)^2}+\frac {\sqrt {1-c^2 x^2} x^{m+3}}{(m+4)^2}\right )\right )}{(m+4) \sqrt {1-c^2 x^2}}+\frac {x^{m+1} \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2}{m+4}\right )}{m+6}+\frac {x^{m+1} \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2}{m+6}+\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {c^4 x^{m+6} (a+b \arccos (c x))}{m+6}-\frac {2 c^2 x^{m+4} (a+b \arccos (c x))}{m+4}+\frac {x^{m+2} (a+b \arccos (c x))}{m+2}+b c \left (\frac {\frac {\left (15 m^2+130 m+264\right ) x^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{(m+2) (m+3) (m+4)^2 (m+6)}+\frac {\left (m^2+15 m+52\right ) \sqrt {1-c^2 x^2} x^{m+3}}{(m+4)^2 (m+6)}}{m+6}-\frac {c^2 \sqrt {1-c^2 x^2} x^{m+5}}{(m+6)^2}\right )\right )}{(m+6) \sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 5139 |
| \(\displaystyle \frac {5 d \left (\frac {3 d \left (\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {b c \int \frac {x^{m+2}}{\sqrt {1-c^2 x^2}}dx}{m+2}+\frac {x^{m+2} (a+b \arccos (c x))}{m+2}\right )}{(m+2) \sqrt {1-c^2 x^2}}+\frac {d \int \frac {x^m (a+b \arccos (c x))^2}{\sqrt {d-c^2 d x^2}}dx}{m+2}+\frac {x^{m+1} \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{m+2}\right )}{m+4}+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {c^2 x^{m+4} (a+b \arccos (c x))}{m+4}+\frac {x^{m+2} (a+b \arccos (c x))}{m+2}+b c \left (\frac {(3 m+10) x^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{(m+2) (m+3) (m+4)^2}+\frac {\sqrt {1-c^2 x^2} x^{m+3}}{(m+4)^2}\right )\right )}{(m+4) \sqrt {1-c^2 x^2}}+\frac {x^{m+1} \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2}{m+4}\right )}{m+6}+\frac {x^{m+1} \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2}{m+6}+\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {c^4 x^{m+6} (a+b \arccos (c x))}{m+6}-\frac {2 c^2 x^{m+4} (a+b \arccos (c x))}{m+4}+\frac {x^{m+2} (a+b \arccos (c x))}{m+2}+b c \left (\frac {\frac {\left (15 m^2+130 m+264\right ) x^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{(m+2) (m+3) (m+4)^2 (m+6)}+\frac {\left (m^2+15 m+52\right ) \sqrt {1-c^2 x^2} x^{m+3}}{(m+4)^2 (m+6)}}{m+6}-\frac {c^2 \sqrt {1-c^2 x^2} x^{m+5}}{(m+6)^2}\right )\right )}{(m+6) \sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 278 |
| \(\displaystyle \frac {5 d \left (\frac {3 d \left (\frac {d \int \frac {x^m (a+b \arccos (c x))^2}{\sqrt {d-c^2 d x^2}}dx}{m+2}+\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {x^{m+2} (a+b \arccos (c x))}{m+2}+\frac {b c x^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{(m+2) (m+3)}\right )}{(m+2) \sqrt {1-c^2 x^2}}+\frac {x^{m+1} \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{m+2}\right )}{m+4}+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {c^2 x^{m+4} (a+b \arccos (c x))}{m+4}+\frac {x^{m+2} (a+b \arccos (c x))}{m+2}+b c \left (\frac {(3 m+10) x^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{(m+2) (m+3) (m+4)^2}+\frac {\sqrt {1-c^2 x^2} x^{m+3}}{(m+4)^2}\right )\right )}{(m+4) \sqrt {1-c^2 x^2}}+\frac {x^{m+1} \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2}{m+4}\right )}{m+6}+\frac {x^{m+1} \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2}{m+6}+\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {c^4 x^{m+6} (a+b \arccos (c x))}{m+6}-\frac {2 c^2 x^{m+4} (a+b \arccos (c x))}{m+4}+\frac {x^{m+2} (a+b \arccos (c x))}{m+2}+b c \left (\frac {\frac {\left (15 m^2+130 m+264\right ) x^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{(m+2) (m+3) (m+4)^2 (m+6)}+\frac {\left (m^2+15 m+52\right ) \sqrt {1-c^2 x^2} x^{m+3}}{(m+4)^2 (m+6)}}{m+6}-\frac {c^2 \sqrt {1-c^2 x^2} x^{m+5}}{(m+6)^2}\right )\right )}{(m+6) \sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 5235 |
| \(\displaystyle \frac {5 d \left (\frac {3 d \left (\frac {d \int \frac {x^m (a+b \arccos (c x))^2}{\sqrt {d-c^2 d x^2}}dx}{m+2}+\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {x^{m+2} (a+b \arccos (c x))}{m+2}+\frac {b c x^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{(m+2) (m+3)}\right )}{(m+2) \sqrt {1-c^2 x^2}}+\frac {x^{m+1} \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{m+2}\right )}{m+4}+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {c^2 x^{m+4} (a+b \arccos (c x))}{m+4}+\frac {x^{m+2} (a+b \arccos (c x))}{m+2}+b c \left (\frac {(3 m+10) x^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{(m+2) (m+3) (m+4)^2}+\frac {\sqrt {1-c^2 x^2} x^{m+3}}{(m+4)^2}\right )\right )}{(m+4) \sqrt {1-c^2 x^2}}+\frac {x^{m+1} \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2}{m+4}\right )}{m+6}+\frac {x^{m+1} \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2}{m+6}+\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (\frac {c^4 x^{m+6} (a+b \arccos (c x))}{m+6}-\frac {2 c^2 x^{m+4} (a+b \arccos (c x))}{m+4}+\frac {x^{m+2} (a+b \arccos (c x))}{m+2}+b c \left (\frac {\frac {\left (15 m^2+130 m+264\right ) x^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{(m+2) (m+3) (m+4)^2 (m+6)}+\frac {\left (m^2+15 m+52\right ) \sqrt {1-c^2 x^2} x^{m+3}}{(m+4)^2 (m+6)}}{m+6}-\frac {c^2 \sqrt {1-c^2 x^2} x^{m+5}}{(m+6)^2}\right )\right )}{(m+6) \sqrt {1-c^2 x^2}}\) |
Input:
Int[x^m*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCos[c*x])^2,x]
Output:
$Aborted
Not integrable
Time = 9.34 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93
\[\int x^{m} \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \arccos \left (c x \right )\right )^{2}d x\]
Input:
int(x^m*(-c^2*d*x^2+d)^(5/2)*(a+b*arccos(c*x))^2,x)
Output:
int(x^m*(-c^2*d*x^2+d)^(5/2)*(a+b*arccos(c*x))^2,x)
Not integrable
Time = 0.13 (sec) , antiderivative size = 134, normalized size of antiderivative = 4.62 \[ \int x^m \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2 \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \arccos \left (c x\right ) + a\right )}^{2} x^{m} \,d x } \] Input:
integrate(x^m*(-c^2*d*x^2+d)^(5/2)*(a+b*arccos(c*x))^2,x, algorithm="frica s")
Output:
integral((a^2*c^4*d^2*x^4 - 2*a^2*c^2*d^2*x^2 + a^2*d^2 + (b^2*c^4*d^2*x^4 - 2*b^2*c^2*d^2*x^2 + b^2*d^2)*arccos(c*x)^2 + 2*(a*b*c^4*d^2*x^4 - 2*a*b *c^2*d^2*x^2 + a*b*d^2)*arccos(c*x))*sqrt(-c^2*d*x^2 + d)*x^m, x)
Timed out. \[ \int x^m \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2 \, dx=\text {Timed out} \] Input:
integrate(x**m*(-c**2*d*x**2+d)**(5/2)*(a+b*acos(c*x))**2,x)
Output:
Timed out
Not integrable
Time = 0.76 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int x^m \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2 \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \arccos \left (c x\right ) + a\right )}^{2} x^{m} \,d x } \] Input:
integrate(x^m*(-c^2*d*x^2+d)^(5/2)*(a+b*arccos(c*x))^2,x, algorithm="maxim a")
Output:
integrate((-c^2*d*x^2 + d)^(5/2)*(b*arccos(c*x) + a)^2*x^m, x)
Exception generated. \[ \int x^m \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2 \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^m*(-c^2*d*x^2+d)^(5/2)*(a+b*arccos(c*x))^2,x, algorithm="giac" )
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Not integrable
Time = 0.44 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int x^m \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2 \, dx=\int x^m\,{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \] Input:
int(x^m*(a + b*acos(c*x))^2*(d - c^2*d*x^2)^(5/2),x)
Output:
int(x^m*(a + b*acos(c*x))^2*(d - c^2*d*x^2)^(5/2), x)
Not integrable
Time = 0.31 (sec) , antiderivative size = 264, normalized size of antiderivative = 9.10 \[ \int x^m \left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))^2 \, dx=\sqrt {d}\, d^{2} \left (2 \left (\int x^{m} \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) x^{4}d x \right ) a b \,c^{4}-4 \left (\int x^{m} \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) x^{2}d x \right ) a b \,c^{2}+2 \left (\int x^{m} \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right )d x \right ) a b +\left (\int x^{m} \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right )^{2} x^{4}d x \right ) b^{2} c^{4}-2 \left (\int x^{m} \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right )^{2} x^{2}d x \right ) b^{2} c^{2}+\left (\int x^{m} \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right )^{2}d x \right ) b^{2}+\left (\int x^{m} \sqrt {-c^{2} x^{2}+1}\, x^{4}d x \right ) a^{2} c^{4}-2 \left (\int x^{m} \sqrt {-c^{2} x^{2}+1}\, x^{2}d x \right ) a^{2} c^{2}+\left (\int x^{m} \sqrt {-c^{2} x^{2}+1}d x \right ) a^{2}\right ) \] Input:
int(x^m*(-c^2*d*x^2+d)^(5/2)*(a+b*acos(c*x))^2,x)
Output:
sqrt(d)*d**2*(2*int(x**m*sqrt( - c**2*x**2 + 1)*acos(c*x)*x**4,x)*a*b*c**4 - 4*int(x**m*sqrt( - c**2*x**2 + 1)*acos(c*x)*x**2,x)*a*b*c**2 + 2*int(x* *m*sqrt( - c**2*x**2 + 1)*acos(c*x),x)*a*b + int(x**m*sqrt( - c**2*x**2 + 1)*acos(c*x)**2*x**4,x)*b**2*c**4 - 2*int(x**m*sqrt( - c**2*x**2 + 1)*acos (c*x)**2*x**2,x)*b**2*c**2 + int(x**m*sqrt( - c**2*x**2 + 1)*acos(c*x)**2, x)*b**2 + int(x**m*sqrt( - c**2*x**2 + 1)*x**4,x)*a**2*c**4 - 2*int(x**m*s qrt( - c**2*x**2 + 1)*x**2,x)*a**2*c**2 + int(x**m*sqrt( - c**2*x**2 + 1), x)*a**2)