Integrand size = 29, antiderivative size = 29 \[ \int x^m \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2 \, dx=\frac {2 b^2 c^2 d x^{3+m} \sqrt {d-c^2 d x^2}}{(4+m)^3}-\frac {6 b c d x^{2+m} \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{(2+m)^2 (4+m) \sqrt {1-c^2 x^2}}-\frac {2 b c d x^{2+m} \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{\left (8+6 m+m^2\right ) \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d x^{4+m} \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{(4+m)^2 \sqrt {1-c^2 x^2}}+\frac {3 d x^{1+m} \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{8+6 m+m^2}+\frac {x^{1+m} \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2}{4+m}+\frac {6 b^2 c^2 d x^{3+m} \sqrt {d-c^2 d x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3+m}{2},\frac {5+m}{2},c^2 x^2\right )}{(2+m)^2 (3+m) (4+m) \sqrt {1-c^2 x^2}}+\frac {2 b^2 c^2 d (10+3 m) x^{3+m} \sqrt {d-c^2 d x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3+m}{2},\frac {5+m}{2},c^2 x^2\right )}{(2+m) (3+m) (4+m)^3 \sqrt {1-c^2 x^2}}+\frac {3 d^2 \text {Int}\left (\frac {x^m (a+b \arccos (c x))^2}{\sqrt {d-c^2 d x^2}},x\right )}{8+6 m+m^2} \] Output:
2*b^2*c^2*d*x^(3+m)*(-c^2*d*x^2+d)^(1/2)/(4+m)^3-6*b*c*d*x^(2+m)*(-c^2*d*x ^2+d)^(1/2)*(a+b*arccos(c*x))/(2+m)^2/(4+m)/(-c^2*x^2+1)^(1/2)-2*b*c*d*x^( 2+m)*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))/(m^2+6*m+8)/(-c^2*x^2+1)^(1/2) +2*b*c^3*d*x^(4+m)*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))/(4+m)^2/(-c^2*x^ 2+1)^(1/2)+3*d*x^(1+m)*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))^2/(m^2+6*m+8 )+x^(1+m)*(-c^2*d*x^2+d)^(3/2)*(a+b*arccos(c*x))^2/(4+m)+6*b^2*c^2*d*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)/(-c^2*x^2+1)^(1/2)+2*b^2*c^2*d*(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/(-c^2*x^2+1)^(1/2)+3*d^2*Defer(Int)(x^m*(a+b*arccos(c*x))^2/(-c^2*d *x^2+d)^(1/2),x)/(m^2+6*m+8)
Not integrable
Time = 0.71 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int x^m \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2 \, dx=\int x^m \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2 \, dx \] Input:
Integrate[x^m*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCos[c*x])^2,x]
Output:
Integrate[x^m*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCos[c*x])^2, x]
Not integrable
Time = 1.68 (sec) , antiderivative size = 29, 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.
| \(\displaystyle \int x^m \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2 \, dx\) |
| \(\Big \downarrow \) 5203 |
| \(\displaystyle \frac {2 b c d \sqrt {d-c^2 d x^2} \int x^{m+1} \left (1-c^2 x^2\right ) (a+b \arccos (c x))dx}{(m+4) \sqrt {1-c^2 x^2}}+\frac {3 d \int x^m \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2dx}{m+4}+\frac {x^{m+1} \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2}{m+4}\) |
| \(\Big \downarrow \) 5193 |
| \(\displaystyle \frac {2 b c d \sqrt {d-c^2 d x^2} \left (b c \int \frac {x^{m+2} \left (\frac {1}{m+2}-\frac {c^2 x^2}{m+4}\right )}{\sqrt {1-c^2 x^2}}dx-\frac {c^2 x^{m+4} (a+b \arccos (c x))}{m+4}+\frac {x^{m+2} (a+b \arccos (c x))}{m+2}\right )}{(m+4) \sqrt {1-c^2 x^2}}+\frac {3 d \int x^m \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2dx}{m+4}+\frac {x^{m+1} \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2}{m+4}\) |
| \(\Big \downarrow \) 363 |
| \(\displaystyle \frac {2 b c d \sqrt {d-c^2 d x^2} \left (b c \left (\frac {(3 m+10) \int \frac {x^{m+2}}{\sqrt {1-c^2 x^2}}dx}{(m+2) (m+4)^2}+\frac {\sqrt {1-c^2 x^2} x^{m+3}}{(m+4)^2}\right )-\frac {c^2 x^{m+4} (a+b \arccos (c x))}{m+4}+\frac {x^{m+2} (a+b \arccos (c x))}{m+2}\right )}{(m+4) \sqrt {1-c^2 x^2}}+\frac {3 d \int x^m \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2dx}{m+4}+\frac {x^{m+1} \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2}{m+4}\) |
| \(\Big \downarrow \) 278 |
| \(\displaystyle \frac {3 d \int x^m \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2dx}{m+4}+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {c^2 x^{m+4} (a+b \arccos (c x))}{m+4}+\frac {x^{m+2} (a+b \arccos (c x))}{m+2}+b c \left (\frac {(3 m+10) x^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{(m+2) (m+3) (m+4)^2}+\frac {\sqrt {1-c^2 x^2} x^{m+3}}{(m+4)^2}\right )\right )}{(m+4) \sqrt {1-c^2 x^2}}+\frac {x^{m+1} \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2}{m+4}\) |
| \(\Big \downarrow \) 5203 |
| \(\displaystyle \frac {3 d \left (\frac {2 b c \sqrt {d-c^2 d x^2} \int x^{m+1} (a+b \arccos (c x))dx}{(m+2) \sqrt {1-c^2 x^2}}+\frac {d \int \frac {x^m (a+b \arccos (c x))^2}{\sqrt {d-c^2 d x^2}}dx}{m+2}+\frac {x^{m+1} \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{m+2}\right )}{m+4}+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {c^2 x^{m+4} (a+b \arccos (c x))}{m+4}+\frac {x^{m+2} (a+b \arccos (c x))}{m+2}+b c \left (\frac {(3 m+10) x^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{(m+2) (m+3) (m+4)^2}+\frac {\sqrt {1-c^2 x^2} x^{m+3}}{(m+4)^2}\right )\right )}{(m+4) \sqrt {1-c^2 x^2}}+\frac {x^{m+1} \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2}{m+4}\) |
| \(\Big \downarrow \) 5139 |
| \(\displaystyle \frac {3 d \left (\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {b c \int \frac {x^{m+2}}{\sqrt {1-c^2 x^2}}dx}{m+2}+\frac {x^{m+2} (a+b \arccos (c x))}{m+2}\right )}{(m+2) \sqrt {1-c^2 x^2}}+\frac {d \int \frac {x^m (a+b \arccos (c x))^2}{\sqrt {d-c^2 d x^2}}dx}{m+2}+\frac {x^{m+1} \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{m+2}\right )}{m+4}+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {c^2 x^{m+4} (a+b \arccos (c x))}{m+4}+\frac {x^{m+2} (a+b \arccos (c x))}{m+2}+b c \left (\frac {(3 m+10) x^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{(m+2) (m+3) (m+4)^2}+\frac {\sqrt {1-c^2 x^2} x^{m+3}}{(m+4)^2}\right )\right )}{(m+4) \sqrt {1-c^2 x^2}}+\frac {x^{m+1} \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2}{m+4}\) |
| \(\Big \downarrow \) 278 |
| \(\displaystyle \frac {3 d \left (\frac {d \int \frac {x^m (a+b \arccos (c x))^2}{\sqrt {d-c^2 d x^2}}dx}{m+2}+\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {x^{m+2} (a+b \arccos (c x))}{m+2}+\frac {b c x^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{(m+2) (m+3)}\right )}{(m+2) \sqrt {1-c^2 x^2}}+\frac {x^{m+1} \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{m+2}\right )}{m+4}+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {c^2 x^{m+4} (a+b \arccos (c x))}{m+4}+\frac {x^{m+2} (a+b \arccos (c x))}{m+2}+b c \left (\frac {(3 m+10) x^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{(m+2) (m+3) (m+4)^2}+\frac {\sqrt {1-c^2 x^2} x^{m+3}}{(m+4)^2}\right )\right )}{(m+4) \sqrt {1-c^2 x^2}}+\frac {x^{m+1} \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2}{m+4}\) |
| \(\Big \downarrow \) 5235 |
| \(\displaystyle \frac {3 d \left (\frac {d \int \frac {x^m (a+b \arccos (c x))^2}{\sqrt {d-c^2 d x^2}}dx}{m+2}+\frac {2 b c \sqrt {d-c^2 d x^2} \left (\frac {x^{m+2} (a+b \arccos (c x))}{m+2}+\frac {b c x^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{(m+2) (m+3)}\right )}{(m+2) \sqrt {1-c^2 x^2}}+\frac {x^{m+1} \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{m+2}\right )}{m+4}+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (-\frac {c^2 x^{m+4} (a+b \arccos (c x))}{m+4}+\frac {x^{m+2} (a+b \arccos (c x))}{m+2}+b c \left (\frac {(3 m+10) x^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},c^2 x^2\right )}{(m+2) (m+3) (m+4)^2}+\frac {\sqrt {1-c^2 x^2} x^{m+3}}{(m+4)^2}\right )\right )}{(m+4) \sqrt {1-c^2 x^2}}+\frac {x^{m+1} \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2}{m+4}\) |
Input:
Int[x^m*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCos[c*x])^2,x]
Output:
$Aborted
Not integrable
Time = 3.96 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93
\[\int x^{m} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \arccos \left (c x \right )\right )^{2}d x\]
Input:
int(x^m*(-c^2*d*x^2+d)^(3/2)*(a+b*arccos(c*x))^2,x)
Output:
int(x^m*(-c^2*d*x^2+d)^(3/2)*(a+b*arccos(c*x))^2,x)
Not integrable
Time = 0.12 (sec) , antiderivative size = 85, normalized size of antiderivative = 2.93 \[ \int x^m \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2 \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \arccos \left (c x\right ) + a\right )}^{2} x^{m} \,d x } \] Input:
integrate(x^m*(-c^2*d*x^2+d)^(3/2)*(a+b*arccos(c*x))^2,x, algorithm="frica s")
Output:
integral(-(a^2*c^2*d*x^2 - a^2*d + (b^2*c^2*d*x^2 - b^2*d)*arccos(c*x)^2 + 2*(a*b*c^2*d*x^2 - a*b*d)*arccos(c*x))*sqrt(-c^2*d*x^2 + d)*x^m, x)
Timed out. \[ \int x^m \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2 \, dx=\text {Timed out} \] Input:
integrate(x**m*(-c**2*d*x**2+d)**(3/2)*(a+b*acos(c*x))**2,x)
Output:
Timed out
Not integrable
Time = 0.57 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int x^m \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2 \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \arccos \left (c x\right ) + a\right )}^{2} x^{m} \,d x } \] Input:
integrate(x^m*(-c^2*d*x^2+d)^(3/2)*(a+b*arccos(c*x))^2,x, algorithm="maxim a")
Output:
integrate((-c^2*d*x^2 + d)^(3/2)*(b*arccos(c*x) + a)^2*x^m, x)
Exception generated. \[ \int x^m \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2 \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^m*(-c^2*d*x^2+d)^(3/2)*(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 \[ \int x^m \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2 \, dx=\int x^m\,{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{3/2} \,d x \] Input:
int(x^m*(a + b*acos(c*x))^2*(d - c^2*d*x^2)^(3/2),x)
Output:
int(x^m*(a + b*acos(c*x))^2*(d - c^2*d*x^2)^(3/2), x)
Not integrable
Time = 0.29 (sec) , antiderivative size = 171, normalized size of antiderivative = 5.90 \[ \int x^m \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2 \, dx=\sqrt {d}\, d \left (-2 \left (\int x^{m} \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) x^{2}d x \right ) a b \,c^{2}+2 \left (\int x^{m} \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right )d x \right ) a b -\left (\int x^{m} \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right )^{2} x^{2}d x \right ) b^{2} c^{2}+\left (\int x^{m} \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right )^{2}d x \right ) b^{2}-\left (\int x^{m} \sqrt {-c^{2} x^{2}+1}\, x^{2}d x \right ) a^{2} c^{2}+\left (\int x^{m} \sqrt {-c^{2} x^{2}+1}d x \right ) a^{2}\right ) \] Input:
int(x^m*(-c^2*d*x^2+d)^(3/2)*(a+b*acos(c*x))^2,x)
Output:
sqrt(d)*d*( - 2*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**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**2,x)*a**2*c **2 + int(x**m*sqrt( - c**2*x**2 + 1),x)*a**2)