Integrand size = 27, antiderivative size = 27 \[ \int \frac {x^m (a+b \arccos (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=-\frac {b c x^{2+m} (a+b \arccos (c x))}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {b c (1-m) x^{2+m} (a+b \arccos (c x))}{6 d^3 \sqrt {1-c^2 x^2}}-\frac {b c (3-m) x^{2+m} (a+b \arccos (c x))}{4 d^3 \sqrt {1-c^2 x^2}}+\frac {x^{1+m} (a+b \arccos (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {(3-m) x^{1+m} (a+b \arccos (c x))^2}{8 d^3 \left (1-c^2 x^2\right )}+\frac {b c (1-m) (1+m) x^{2+m} (a+b \arccos (c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{6 d^3 (2+m)}+\frac {b c (3-m) (1+m) x^{2+m} (a+b \arccos (c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{4 d^3 (2+m)}+\frac {b^2 c^2 (1-m) x^{3+m} \operatorname {Hypergeometric2F1}\left (1,\frac {3+m}{2},\frac {5+m}{2},c^2 x^2\right )}{6 d^3 (3+m)}+\frac {b^2 c^2 (3-m) x^{3+m} \operatorname {Hypergeometric2F1}\left (1,\frac {3+m}{2},\frac {5+m}{2},c^2 x^2\right )}{4 d^3 (3+m)}+\frac {b^2 c^2 x^{3+m} \operatorname {Hypergeometric2F1}\left (2,\frac {3+m}{2},\frac {5+m}{2},c^2 x^2\right )}{6 d^3 (3+m)}-\frac {b^2 c^2 (1-m) (1+m) x^{3+m} \, _3F_2\left (1,\frac {3}{2}+\frac {m}{2},\frac {3}{2}+\frac {m}{2};2+\frac {m}{2},\frac {5}{2}+\frac {m}{2};c^2 x^2\right )}{6 d^3 \left (6+5 m+m^2\right )}-\frac {b^2 c^2 (3-m) (1+m) x^{3+m} \, _3F_2\left (1,\frac {3}{2}+\frac {m}{2},\frac {3}{2}+\frac {m}{2};2+\frac {m}{2},\frac {5}{2}+\frac {m}{2};c^2 x^2\right )}{4 d^3 \left (6+5 m+m^2\right )}+\frac {(1-m) (3-m) \text {Int}\left (\frac {x^m (a+b \arccos (c x))^2}{d-c^2 d x^2},x\right )}{8 d^2} \] Output:
-1/6*b*c*x^(2+m)*(a+b*arccos(c*x))/d^3/(-c^2*x^2+1)^(3/2)-1/6*b*c*(1-m)*x^ (2+m)*(a+b*arccos(c*x))/d^3/(-c^2*x^2+1)^(1/2)-1/4*b*c*(3-m)*x^(2+m)*(a+b* arccos(c*x))/d^3/(-c^2*x^2+1)^(1/2)+1/4*x^(1+m)*(a+b*arccos(c*x))^2/d^3/(- c^2*x^2+1)^2+1/8*(3-m)*x^(1+m)*(a+b*arccos(c*x))^2/d^3/(-c^2*x^2+1)+1/6*b* c*(1-m)*(1+m)*x^(2+m)*(a+b*arccos(c*x))*hypergeom([1/2, 1+1/2*m],[2+1/2*m] ,c^2*x^2)/d^3/(2+m)+1/4*b*c*(3-m)*(1+m)*x^(2+m)*(a+b*arccos(c*x))*hypergeo m([1/2, 1+1/2*m],[2+1/2*m],c^2*x^2)/d^3/(2+m)+1/6*b^2*c^2*(1-m)*x^(3+m)*hy pergeom([1, 3/2+1/2*m],[5/2+1/2*m],c^2*x^2)/d^3/(3+m)+1/4*b^2*c^2*(3-m)*x^ (3+m)*hypergeom([1, 3/2+1/2*m],[5/2+1/2*m],c^2*x^2)/d^3/(3+m)+1/6*b^2*c^2* x^(3+m)*hypergeom([2, 3/2+1/2*m],[5/2+1/2*m],c^2*x^2)/d^3/(3+m)-1/6*b^2*c^ 2*(1-m)*(1+m)*x^(3+m)*hypergeom([1, 3/2+1/2*m, 3/2+1/2*m],[2+1/2*m, 5/2+1/ 2*m],c^2*x^2)/d^3/(m^2+5*m+6)-1/4*b^2*c^2*(3-m)*(1+m)*x^(3+m)*hypergeom([1 , 3/2+1/2*m, 3/2+1/2*m],[2+1/2*m, 5/2+1/2*m],c^2*x^2)/d^3/(m^2+5*m+6)+1/8* (1-m)*(3-m)*Defer(Int)(x^m*(a+b*arccos(c*x))^2/(-c^2*d*x^2+d),x)/d^2
Not integrable
Time = 15.02 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {x^m (a+b \arccos (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=\int \frac {x^m (a+b \arccos (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx \] Input:
Integrate[(x^m*(a + b*ArcCos[c*x])^2)/(d - c^2*d*x^2)^3,x]
Output:
Integrate[(x^m*(a + b*ArcCos[c*x])^2)/(d - c^2*d*x^2)^3, x]
Not integrable
Time = 1.98 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 8, 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.
| \(\displaystyle \int \frac {x^m (a+b \arccos (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 5209 |
| \(\displaystyle \frac {b c \int \frac {x^{m+1} (a+b \arccos (c x))}{\left (1-c^2 x^2\right )^{5/2}}dx}{2 d^3}+\frac {(3-m) \int \frac {x^m (a+b \arccos (c x))^2}{d^2 \left (1-c^2 x^2\right )^2}dx}{4 d}+\frac {x^{m+1} (a+b \arccos (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b c \int \frac {x^{m+1} (a+b \arccos (c x))}{\left (1-c^2 x^2\right )^{5/2}}dx}{2 d^3}+\frac {(3-m) \int \frac {x^m (a+b \arccos (c x))^2}{\left (1-c^2 x^2\right )^2}dx}{4 d^3}+\frac {x^{m+1} (a+b \arccos (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 5209 |
| \(\displaystyle \frac {b c \left (\frac {1}{3} (1-m) \int \frac {x^{m+1} (a+b \arccos (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx+\frac {1}{3} b c \int \frac {x^{m+2}}{\left (1-c^2 x^2\right )^2}dx+\frac {x^{m+2} (a+b \arccos (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}\right )}{2 d^3}+\frac {(3-m) \left (b c \int \frac {x^{m+1} (a+b \arccos (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx+\frac {1}{2} (1-m) \int \frac {x^m (a+b \arccos (c x))^2}{1-c^2 x^2}dx+\frac {x^{m+1} (a+b \arccos (c x))^2}{2 \left (1-c^2 x^2\right )}\right )}{4 d^3}+\frac {x^{m+1} (a+b \arccos (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 278 |
| \(\displaystyle \frac {b c \left (\frac {1}{3} (1-m) \int \frac {x^{m+1} (a+b \arccos (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx+\frac {x^{m+2} (a+b \arccos (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}+\frac {b c x^{m+3} \operatorname {Hypergeometric2F1}\left (2,\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{3 (m+3)}\right )}{2 d^3}+\frac {(3-m) \left (b c \int \frac {x^{m+1} (a+b \arccos (c x))}{\left (1-c^2 x^2\right )^{3/2}}dx+\frac {1}{2} (1-m) \int \frac {x^m (a+b \arccos (c x))^2}{1-c^2 x^2}dx+\frac {x^{m+1} (a+b \arccos (c x))^2}{2 \left (1-c^2 x^2\right )}\right )}{4 d^3}+\frac {x^{m+1} (a+b \arccos (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 5209 |
| \(\displaystyle \frac {b c \left (\frac {1}{3} (1-m) \left (-(m+1) \int \frac {x^{m+1} (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx+b c \int \frac {x^{m+2}}{1-c^2 x^2}dx+\frac {x^{m+2} (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}\right )+\frac {x^{m+2} (a+b \arccos (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}+\frac {b c x^{m+3} \operatorname {Hypergeometric2F1}\left (2,\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{3 (m+3)}\right )}{2 d^3}+\frac {(3-m) \left (b c \left (-(m+1) \int \frac {x^{m+1} (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx+b c \int \frac {x^{m+2}}{1-c^2 x^2}dx+\frac {x^{m+2} (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}\right )+\frac {1}{2} (1-m) \int \frac {x^m (a+b \arccos (c x))^2}{1-c^2 x^2}dx+\frac {x^{m+1} (a+b \arccos (c x))^2}{2 \left (1-c^2 x^2\right )}\right )}{4 d^3}+\frac {x^{m+1} (a+b \arccos (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 278 |
| \(\displaystyle \frac {b c \left (\frac {1}{3} (1-m) \left (-(m+1) \int \frac {x^{m+1} (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx+\frac {x^{m+2} (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}+\frac {b c x^{m+3} \operatorname {Hypergeometric2F1}\left (1,\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{m+3}\right )+\frac {x^{m+2} (a+b \arccos (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}+\frac {b c x^{m+3} \operatorname {Hypergeometric2F1}\left (2,\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{3 (m+3)}\right )}{2 d^3}+\frac {(3-m) \left (b c \left (-(m+1) \int \frac {x^{m+1} (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx+\frac {x^{m+2} (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}+\frac {b c x^{m+3} \operatorname {Hypergeometric2F1}\left (1,\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{m+3}\right )+\frac {1}{2} (1-m) \int \frac {x^m (a+b \arccos (c x))^2}{1-c^2 x^2}dx+\frac {x^{m+1} (a+b \arccos (c x))^2}{2 \left (1-c^2 x^2\right )}\right )}{4 d^3}+\frac {x^{m+1} (a+b \arccos (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 5221 |
| \(\displaystyle \frac {(3-m) \left (\frac {1}{2} (1-m) \int \frac {x^m (a+b \arccos (c x))^2}{1-c^2 x^2}dx+b c \left (-(m+1) \left (\frac {b c x^{m+3} \, _3F_2\left (1,\frac {m}{2}+\frac {3}{2},\frac {m}{2}+\frac {3}{2};\frac {m}{2}+2,\frac {m}{2}+\frac {5}{2};c^2 x^2\right )}{m^2+5 m+6}+\frac {x^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},c^2 x^2\right ) (a+b \arccos (c x))}{m+2}\right )+\frac {x^{m+2} (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}+\frac {b c x^{m+3} \operatorname {Hypergeometric2F1}\left (1,\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{m+3}\right )+\frac {x^{m+1} (a+b \arccos (c x))^2}{2 \left (1-c^2 x^2\right )}\right )}{4 d^3}+\frac {b c \left (\frac {1}{3} (1-m) \left (-(m+1) \left (\frac {b c x^{m+3} \, _3F_2\left (1,\frac {m}{2}+\frac {3}{2},\frac {m}{2}+\frac {3}{2};\frac {m}{2}+2,\frac {m}{2}+\frac {5}{2};c^2 x^2\right )}{m^2+5 m+6}+\frac {x^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},c^2 x^2\right ) (a+b \arccos (c x))}{m+2}\right )+\frac {x^{m+2} (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}+\frac {b c x^{m+3} \operatorname {Hypergeometric2F1}\left (1,\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{m+3}\right )+\frac {x^{m+2} (a+b \arccos (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}+\frac {b c x^{m+3} \operatorname {Hypergeometric2F1}\left (2,\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{3 (m+3)}\right )}{2 d^3}+\frac {x^{m+1} (a+b \arccos (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 5235 |
| \(\displaystyle \frac {(3-m) \left (\frac {1}{2} (1-m) \int \frac {x^m (a+b \arccos (c x))^2}{1-c^2 x^2}dx+b c \left (-(m+1) \left (\frac {b c x^{m+3} \, _3F_2\left (1,\frac {m}{2}+\frac {3}{2},\frac {m}{2}+\frac {3}{2};\frac {m}{2}+2,\frac {m}{2}+\frac {5}{2};c^2 x^2\right )}{m^2+5 m+6}+\frac {x^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},c^2 x^2\right ) (a+b \arccos (c x))}{m+2}\right )+\frac {x^{m+2} (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}+\frac {b c x^{m+3} \operatorname {Hypergeometric2F1}\left (1,\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{m+3}\right )+\frac {x^{m+1} (a+b \arccos (c x))^2}{2 \left (1-c^2 x^2\right )}\right )}{4 d^3}+\frac {b c \left (\frac {1}{3} (1-m) \left (-(m+1) \left (\frac {b c x^{m+3} \, _3F_2\left (1,\frac {m}{2}+\frac {3}{2},\frac {m}{2}+\frac {3}{2};\frac {m}{2}+2,\frac {m}{2}+\frac {5}{2};c^2 x^2\right )}{m^2+5 m+6}+\frac {x^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},c^2 x^2\right ) (a+b \arccos (c x))}{m+2}\right )+\frac {x^{m+2} (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}+\frac {b c x^{m+3} \operatorname {Hypergeometric2F1}\left (1,\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{m+3}\right )+\frac {x^{m+2} (a+b \arccos (c x))}{3 \left (1-c^2 x^2\right )^{3/2}}+\frac {b c x^{m+3} \operatorname {Hypergeometric2F1}\left (2,\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{3 (m+3)}\right )}{2 d^3}+\frac {x^{m+1} (a+b \arccos (c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}\) |
Input:
Int[(x^m*(a + b*ArcCos[c*x])^2)/(d - c^2*d*x^2)^3,x]
Output:
$Aborted
Not integrable
Time = 0.59 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00
\[\int \frac {x^{m} \left (a +b \arccos \left (c x \right )\right )^{2}}{\left (-c^{2} d \,x^{2}+d \right )^{3}}d x\]
Input:
int(x^m*(a+b*arccos(c*x))^2/(-c^2*d*x^2+d)^3,x)
Output:
int(x^m*(a+b*arccos(c*x))^2/(-c^2*d*x^2+d)^3,x)
Not integrable
Time = 0.10 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.56 \[ \int \frac {x^m (a+b \arccos (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {{\left (b \arccos \left (c x\right ) + a\right )}^{2} x^{m}}{{\left (c^{2} d x^{2} - d\right )}^{3}} \,d x } \] Input:
integrate(x^m*(a+b*arccos(c*x))^2/(-c^2*d*x^2+d)^3,x, algorithm="fricas")
Output:
integral(-(b^2*arccos(c*x)^2 + 2*a*b*arccos(c*x) + a^2)*x^m/(c^6*d^3*x^6 - 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 - d^3), x)
Not integrable
Time = 98.59 (sec) , antiderivative size = 119, normalized size of antiderivative = 4.41 \[ \int \frac {x^m (a+b \arccos (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=- \frac {\int \frac {a^{2} x^{m}}{c^{6} x^{6} - 3 c^{4} x^{4} + 3 c^{2} x^{2} - 1}\, dx + \int \frac {b^{2} x^{m} \operatorname {acos}^{2}{\left (c x \right )}}{c^{6} x^{6} - 3 c^{4} x^{4} + 3 c^{2} x^{2} - 1}\, dx + \int \frac {2 a b x^{m} \operatorname {acos}{\left (c x \right )}}{c^{6} x^{6} - 3 c^{4} x^{4} + 3 c^{2} x^{2} - 1}\, dx}{d^{3}} \] Input:
integrate(x**m*(a+b*acos(c*x))**2/(-c**2*d*x**2+d)**3,x)
Output:
-(Integral(a**2*x**m/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1), x) + Int egral(b**2*x**m*acos(c*x)**2/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1), x) + Integral(2*a*b*x**m*acos(c*x)/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1), x))/d**3
Not integrable
Time = 0.60 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.19 \[ \int \frac {x^m (a+b \arccos (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {{\left (b \arccos \left (c x\right ) + a\right )}^{2} x^{m}}{{\left (c^{2} d x^{2} - d\right )}^{3}} \,d x } \] Input:
integrate(x^m*(a+b*arccos(c*x))^2/(-c^2*d*x^2+d)^3,x, algorithm="maxima")
Output:
-integrate((b*arccos(c*x) + a)^2*x^m/(c^2*d*x^2 - d)^3, x)
Exception generated. \[ \int \frac {x^m (a+b \arccos (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(x^m*(a+b*arccos(c*x))^2/(-c^2*d*x^2+d)^3,x, algorithm="giac")
Output:
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve cteur & l) Error: Bad Argument Value
Not integrable
Time = 0.36 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {x^m (a+b \arccos (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=\int \frac {x^m\,{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^2}{{\left (d-c^2\,d\,x^2\right )}^3} \,d x \] Input:
int((x^m*(a + b*acos(c*x))^2)/(d - c^2*d*x^2)^3,x)
Output:
int((x^m*(a + b*acos(c*x))^2)/(d - c^2*d*x^2)^3, x)
Not integrable
Time = 0.22 (sec) , antiderivative size = 128, normalized size of antiderivative = 4.74 \[ \int \frac {x^m (a+b \arccos (c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=\frac {-\left (\int \frac {x^{m}}{c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1}d x \right ) a^{2}-2 \left (\int \frac {x^{m} \mathit {acos} \left (c x \right )}{c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1}d x \right ) a b -\left (\int \frac {x^{m} \mathit {acos} \left (c x \right )^{2}}{c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1}d x \right ) b^{2}}{d^{3}} \] Input:
int(x^m*(a+b*acos(c*x))^2/(-c^2*d*x^2+d)^3,x)
Output:
( - int(x**m/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1),x)*a**2 - 2*int(( x**m*acos(c*x))/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1),x)*a*b - int(( x**m*acos(c*x)**2)/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1),x)*b**2)/d* *3