Integrand size = 27, antiderivative size = 1312 \[ \int x^m \left (d-c^2 d x^2\right )^3 (a+b \arccos (c x))^2 \, dx =\text {Too large to display} \] Output:
-12*b^2*c^4*d^3*x^(5+m)/(5+m)^3/(7+m)+48*d^3*x^(1+m)*(a+b*arccos(c*x))^2/( 5+m)/(7+m)/(m^2+4*m+3)+24*d^3*x^(1+m)*(-c^2*x^2+1)*(a+b*arccos(c*x))^2/(7+ m)/(m^2+8*m+15)+6*d^3*x^(1+m)*(-c^2*x^2+1)^2*(a+b*arccos(c*x))^2/(5+m)/(7+ m)+2*b^2*c^6*d^3*x^(7+m)/(7+m)^3-30*b*c*d^3*x^(2+m)*(a+b*arccos(c*x))*hype rgeom([1/2, 1+1/2*m],[2+1/2*m],c^2*x^2)/(5+m)/(7+m)^2/(m^2+5*m+6)+30*b^2*c ^2*d^3*x^(3+m)/(3+m)^2/(5+m)/(7+m)^2+36*b^2*c^2*d^3*x^(3+m)/(3+m)^2/(5+m)^ 2/(7+m)+12*b^2*c^2*d^3*x^(3+m)/(3+m)/(5+m)^2/(7+m)+48*b^2*c^2*d^3*x^(3+m)/ (3+m)^3/(5+m)/(7+m)-2*b*c*d^3*x^(2+m)*(-c^2*x^2+1)^(5/2)*(a+b*arccos(c*x)) /(7+m)^2-10*b*c*d^3*x^(2+m)*(-c^2*x^2+1)^(3/2)*(a+b*arccos(c*x))/(5+m)/(7+ m)^2-12*b*c*d^3*x^(2+m)*(-c^2*x^2+1)^(3/2)*(a+b*arccos(c*x))/(5+m)^2/(7+m) -36*b*c*d^3*x^(2+m)*(-c^2*x^2+1)^(1/2)*(a+b*arccos(c*x))/(3+m)/(5+m)^2/(7+ m)-48*b*c*d^3*x^(2+m)*(-c^2*x^2+1)^(1/2)*(a+b*arccos(c*x))/(3+m)^2/(5+m)/( 7+m)+2*b^2*c^2*d^3*x^(3+m)/(3+m)/(7+m)^2+10*b^2*c^2*d^3*x^(3+m)/(7+m)^2/(m ^2+8*m+15)-10*b^2*c^4*d^3*x^(5+m)/(5+m)^2/(7+m)^2-4*b^2*c^4*d^3*x^(5+m)/(5 +m)/(7+m)^2-30*b*c*d^3*x^(2+m)*(-c^2*x^2+1)^(1/2)*(a+b*arccos(c*x))/(7+m)^ 2/(m^2+8*m+15)+d^3*x^(1+m)*(-c^2*x^2+1)^3*(a+b*arccos(c*x))^2/(7+m)-36*b*c *d^3*x^(2+m)*(a+b*arccos(c*x))*hypergeom([1/2, 1+1/2*m],[2+1/2*m],c^2*x^2) /(5+m)^2/(7+m)/(m^2+5*m+6)-96*b*c*d^3*x^(2+m)*(a+b*arccos(c*x))*hypergeom( [1/2, 1+1/2*m],[2+1/2*m],c^2*x^2)/(5+m)/(7+m)/(m^3+6*m^2+11*m+6)+30*b^2*c^ 2*d^3*x^(3+m)*hypergeom([1, 3/2+1/2*m, 3/2+1/2*m],[2+1/2*m, 5/2+1/2*m],...
\[ \int x^m \left (d-c^2 d x^2\right )^3 (a+b \arccos (c x))^2 \, dx=\int x^m \left (d-c^2 d x^2\right )^3 (a+b \arccos (c x))^2 \, dx \] Input:
Integrate[x^m*(d - c^2*d*x^2)^3*(a + b*ArcCos[c*x])^2,x]
Output:
Integrate[x^m*(d - c^2*d*x^2)^3*(a + b*ArcCos[c*x])^2, x]
Time = 3.38 (sec) , antiderivative size = 1021, normalized size of antiderivative = 0.78, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {5203, 27, 5203, 244, 2009, 5203, 244, 2009, 5139, 5199, 15, 5221}
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.
\(\displaystyle \int x^m \left (d-c^2 d x^2\right )^3 (a+b \arccos (c x))^2 \, dx\) |
\(\Big \downarrow \) 5203 |
\(\displaystyle \frac {2 b c d^3 \int x^{m+1} \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))dx}{m+7}+\frac {6 d \int d^2 x^m \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2dx}{m+7}+\frac {d^3 \left (1-c^2 x^2\right )^3 x^{m+1} (a+b \arccos (c x))^2}{m+7}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {2 b c d^3 \int x^{m+1} \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))dx}{m+7}+\frac {6 d^3 \int x^m \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2dx}{m+7}+\frac {d^3 \left (1-c^2 x^2\right )^3 x^{m+1} (a+b \arccos (c x))^2}{m+7}\) |
\(\Big \downarrow \) 5203 |
\(\displaystyle \frac {2 b c d^3 \left (\frac {5 \int x^{m+1} \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))dx}{m+7}+\frac {b c \int x^{m+2} \left (1-c^2 x^2\right )^2dx}{m+7}+\frac {\left (1-c^2 x^2\right )^{5/2} x^{m+2} (a+b \arccos (c x))}{m+7}\right )}{m+7}+\frac {6 d^3 \left (\frac {2 b c \int x^{m+1} \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))dx}{m+5}+\frac {4 \int x^m \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2dx}{m+5}+\frac {\left (1-c^2 x^2\right )^2 x^{m+1} (a+b \arccos (c x))^2}{m+5}\right )}{m+7}+\frac {d^3 \left (1-c^2 x^2\right )^3 x^{m+1} (a+b \arccos (c x))^2}{m+7}\) |
\(\Big \downarrow \) 244 |
\(\displaystyle \frac {6 d^3 \left (\frac {2 b c \int x^{m+1} \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))dx}{m+5}+\frac {4 \int x^m \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2dx}{m+5}+\frac {\left (1-c^2 x^2\right )^2 x^{m+1} (a+b \arccos (c x))^2}{m+5}\right )}{m+7}+\frac {2 b c d^3 \left (\frac {5 \int x^{m+1} \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))dx}{m+7}+\frac {b c \int \left (x^{m+2}-2 c^2 x^{m+4}+c^4 x^{m+6}\right )dx}{m+7}+\frac {\left (1-c^2 x^2\right )^{5/2} x^{m+2} (a+b \arccos (c x))}{m+7}\right )}{m+7}+\frac {d^3 \left (1-c^2 x^2\right )^3 x^{m+1} (a+b \arccos (c x))^2}{m+7}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {6 d^3 \left (\frac {2 b c \int x^{m+1} \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))dx}{m+5}+\frac {4 \int x^m \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2dx}{m+5}+\frac {\left (1-c^2 x^2\right )^2 x^{m+1} (a+b \arccos (c x))^2}{m+5}\right )}{m+7}+\frac {2 b c d^3 \left (\frac {5 \int x^{m+1} \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))dx}{m+7}+\frac {\left (1-c^2 x^2\right )^{5/2} x^{m+2} (a+b \arccos (c x))}{m+7}+\frac {b c \left (\frac {c^4 x^{m+7}}{m+7}-\frac {2 c^2 x^{m+5}}{m+5}+\frac {x^{m+3}}{m+3}\right )}{m+7}\right )}{m+7}+\frac {d^3 \left (1-c^2 x^2\right )^3 x^{m+1} (a+b \arccos (c x))^2}{m+7}\) |
\(\Big \downarrow \) 5203 |
\(\displaystyle \frac {6 d^3 \left (\frac {2 b c \left (\frac {3 \int x^{m+1} \sqrt {1-c^2 x^2} (a+b \arccos (c x))dx}{m+5}+\frac {b c \int x^{m+2} \left (1-c^2 x^2\right )dx}{m+5}+\frac {\left (1-c^2 x^2\right )^{3/2} x^{m+2} (a+b \arccos (c x))}{m+5}\right )}{m+5}+\frac {4 \left (\frac {2 b c \int x^{m+1} \sqrt {1-c^2 x^2} (a+b \arccos (c x))dx}{m+3}+\frac {2 \int x^m (a+b \arccos (c x))^2dx}{m+3}+\frac {\left (1-c^2 x^2\right ) x^{m+1} (a+b \arccos (c x))^2}{m+3}\right )}{m+5}+\frac {\left (1-c^2 x^2\right )^2 x^{m+1} (a+b \arccos (c x))^2}{m+5}\right )}{m+7}+\frac {2 b c d^3 \left (\frac {5 \left (\frac {3 \int x^{m+1} \sqrt {1-c^2 x^2} (a+b \arccos (c x))dx}{m+5}+\frac {b c \int x^{m+2} \left (1-c^2 x^2\right )dx}{m+5}+\frac {\left (1-c^2 x^2\right )^{3/2} x^{m+2} (a+b \arccos (c x))}{m+5}\right )}{m+7}+\frac {\left (1-c^2 x^2\right )^{5/2} x^{m+2} (a+b \arccos (c x))}{m+7}+\frac {b c \left (\frac {c^4 x^{m+7}}{m+7}-\frac {2 c^2 x^{m+5}}{m+5}+\frac {x^{m+3}}{m+3}\right )}{m+7}\right )}{m+7}+\frac {d^3 \left (1-c^2 x^2\right )^3 x^{m+1} (a+b \arccos (c x))^2}{m+7}\) |
\(\Big \downarrow \) 244 |
\(\displaystyle \frac {6 d^3 \left (\frac {2 b c \left (\frac {3 \int x^{m+1} \sqrt {1-c^2 x^2} (a+b \arccos (c x))dx}{m+5}+\frac {b c \int \left (x^{m+2}-c^2 x^{m+4}\right )dx}{m+5}+\frac {\left (1-c^2 x^2\right )^{3/2} x^{m+2} (a+b \arccos (c x))}{m+5}\right )}{m+5}+\frac {4 \left (\frac {2 b c \int x^{m+1} \sqrt {1-c^2 x^2} (a+b \arccos (c x))dx}{m+3}+\frac {2 \int x^m (a+b \arccos (c x))^2dx}{m+3}+\frac {\left (1-c^2 x^2\right ) x^{m+1} (a+b \arccos (c x))^2}{m+3}\right )}{m+5}+\frac {\left (1-c^2 x^2\right )^2 x^{m+1} (a+b \arccos (c x))^2}{m+5}\right )}{m+7}+\frac {2 b c d^3 \left (\frac {5 \left (\frac {3 \int x^{m+1} \sqrt {1-c^2 x^2} (a+b \arccos (c x))dx}{m+5}+\frac {b c \int \left (x^{m+2}-c^2 x^{m+4}\right )dx}{m+5}+\frac {\left (1-c^2 x^2\right )^{3/2} x^{m+2} (a+b \arccos (c x))}{m+5}\right )}{m+7}+\frac {\left (1-c^2 x^2\right )^{5/2} x^{m+2} (a+b \arccos (c x))}{m+7}+\frac {b c \left (\frac {c^4 x^{m+7}}{m+7}-\frac {2 c^2 x^{m+5}}{m+5}+\frac {x^{m+3}}{m+3}\right )}{m+7}\right )}{m+7}+\frac {d^3 \left (1-c^2 x^2\right )^3 x^{m+1} (a+b \arccos (c x))^2}{m+7}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {6 d^3 \left (\frac {2 b c \left (\frac {3 \int x^{m+1} \sqrt {1-c^2 x^2} (a+b \arccos (c x))dx}{m+5}+\frac {\left (1-c^2 x^2\right )^{3/2} x^{m+2} (a+b \arccos (c x))}{m+5}+\frac {b c \left (\frac {x^{m+3}}{m+3}-\frac {c^2 x^{m+5}}{m+5}\right )}{m+5}\right )}{m+5}+\frac {4 \left (\frac {2 b c \int x^{m+1} \sqrt {1-c^2 x^2} (a+b \arccos (c x))dx}{m+3}+\frac {2 \int x^m (a+b \arccos (c x))^2dx}{m+3}+\frac {\left (1-c^2 x^2\right ) x^{m+1} (a+b \arccos (c x))^2}{m+3}\right )}{m+5}+\frac {\left (1-c^2 x^2\right )^2 x^{m+1} (a+b \arccos (c x))^2}{m+5}\right )}{m+7}+\frac {2 b c d^3 \left (\frac {5 \left (\frac {3 \int x^{m+1} \sqrt {1-c^2 x^2} (a+b \arccos (c x))dx}{m+5}+\frac {\left (1-c^2 x^2\right )^{3/2} x^{m+2} (a+b \arccos (c x))}{m+5}+\frac {b c \left (\frac {x^{m+3}}{m+3}-\frac {c^2 x^{m+5}}{m+5}\right )}{m+5}\right )}{m+7}+\frac {\left (1-c^2 x^2\right )^{5/2} x^{m+2} (a+b \arccos (c x))}{m+7}+\frac {b c \left (\frac {c^4 x^{m+7}}{m+7}-\frac {2 c^2 x^{m+5}}{m+5}+\frac {x^{m+3}}{m+3}\right )}{m+7}\right )}{m+7}+\frac {d^3 \left (1-c^2 x^2\right )^3 x^{m+1} (a+b \arccos (c x))^2}{m+7}\) |
\(\Big \downarrow \) 5139 |
\(\displaystyle \frac {6 d^3 \left (\frac {4 \left (\frac {2 \left (\frac {2 b c \int \frac {x^{m+1} (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx}{m+1}+\frac {x^{m+1} (a+b \arccos (c x))^2}{m+1}\right )}{m+3}+\frac {2 b c \int x^{m+1} \sqrt {1-c^2 x^2} (a+b \arccos (c x))dx}{m+3}+\frac {\left (1-c^2 x^2\right ) x^{m+1} (a+b \arccos (c x))^2}{m+3}\right )}{m+5}+\frac {2 b c \left (\frac {3 \int x^{m+1} \sqrt {1-c^2 x^2} (a+b \arccos (c x))dx}{m+5}+\frac {\left (1-c^2 x^2\right )^{3/2} x^{m+2} (a+b \arccos (c x))}{m+5}+\frac {b c \left (\frac {x^{m+3}}{m+3}-\frac {c^2 x^{m+5}}{m+5}\right )}{m+5}\right )}{m+5}+\frac {\left (1-c^2 x^2\right )^2 x^{m+1} (a+b \arccos (c x))^2}{m+5}\right )}{m+7}+\frac {2 b c d^3 \left (\frac {5 \left (\frac {3 \int x^{m+1} \sqrt {1-c^2 x^2} (a+b \arccos (c x))dx}{m+5}+\frac {\left (1-c^2 x^2\right )^{3/2} x^{m+2} (a+b \arccos (c x))}{m+5}+\frac {b c \left (\frac {x^{m+3}}{m+3}-\frac {c^2 x^{m+5}}{m+5}\right )}{m+5}\right )}{m+7}+\frac {\left (1-c^2 x^2\right )^{5/2} x^{m+2} (a+b \arccos (c x))}{m+7}+\frac {b c \left (\frac {c^4 x^{m+7}}{m+7}-\frac {2 c^2 x^{m+5}}{m+5}+\frac {x^{m+3}}{m+3}\right )}{m+7}\right )}{m+7}+\frac {d^3 \left (1-c^2 x^2\right )^3 x^{m+1} (a+b \arccos (c x))^2}{m+7}\) |
\(\Big \downarrow \) 5199 |
\(\displaystyle \frac {6 d^3 \left (\frac {4 \left (\frac {2 \left (\frac {2 b c \int \frac {x^{m+1} (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx}{m+1}+\frac {x^{m+1} (a+b \arccos (c x))^2}{m+1}\right )}{m+3}+\frac {2 b c \left (\frac {\int \frac {x^{m+1} (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx}{m+3}+\frac {b c \int x^{m+2}dx}{m+3}+\frac {\sqrt {1-c^2 x^2} x^{m+2} (a+b \arccos (c x))}{m+3}\right )}{m+3}+\frac {\left (1-c^2 x^2\right ) x^{m+1} (a+b \arccos (c x))^2}{m+3}\right )}{m+5}+\frac {2 b c \left (\frac {3 \left (\frac {\int \frac {x^{m+1} (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx}{m+3}+\frac {b c \int x^{m+2}dx}{m+3}+\frac {\sqrt {1-c^2 x^2} x^{m+2} (a+b \arccos (c x))}{m+3}\right )}{m+5}+\frac {\left (1-c^2 x^2\right )^{3/2} x^{m+2} (a+b \arccos (c x))}{m+5}+\frac {b c \left (\frac {x^{m+3}}{m+3}-\frac {c^2 x^{m+5}}{m+5}\right )}{m+5}\right )}{m+5}+\frac {\left (1-c^2 x^2\right )^2 x^{m+1} (a+b \arccos (c x))^2}{m+5}\right )}{m+7}+\frac {2 b c d^3 \left (\frac {5 \left (\frac {3 \left (\frac {\int \frac {x^{m+1} (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx}{m+3}+\frac {b c \int x^{m+2}dx}{m+3}+\frac {\sqrt {1-c^2 x^2} x^{m+2} (a+b \arccos (c x))}{m+3}\right )}{m+5}+\frac {\left (1-c^2 x^2\right )^{3/2} x^{m+2} (a+b \arccos (c x))}{m+5}+\frac {b c \left (\frac {x^{m+3}}{m+3}-\frac {c^2 x^{m+5}}{m+5}\right )}{m+5}\right )}{m+7}+\frac {\left (1-c^2 x^2\right )^{5/2} x^{m+2} (a+b \arccos (c x))}{m+7}+\frac {b c \left (\frac {c^4 x^{m+7}}{m+7}-\frac {2 c^2 x^{m+5}}{m+5}+\frac {x^{m+3}}{m+3}\right )}{m+7}\right )}{m+7}+\frac {d^3 \left (1-c^2 x^2\right )^3 x^{m+1} (a+b \arccos (c x))^2}{m+7}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle \frac {6 d^3 \left (\frac {4 \left (\frac {2 \left (\frac {2 b c \int \frac {x^{m+1} (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx}{m+1}+\frac {x^{m+1} (a+b \arccos (c x))^2}{m+1}\right )}{m+3}+\frac {2 b c \left (\frac {\int \frac {x^{m+1} (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx}{m+3}+\frac {\sqrt {1-c^2 x^2} x^{m+2} (a+b \arccos (c x))}{m+3}+\frac {b c x^{m+3}}{(m+3)^2}\right )}{m+3}+\frac {\left (1-c^2 x^2\right ) x^{m+1} (a+b \arccos (c x))^2}{m+3}\right )}{m+5}+\frac {2 b c \left (\frac {3 \left (\frac {\int \frac {x^{m+1} (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx}{m+3}+\frac {\sqrt {1-c^2 x^2} x^{m+2} (a+b \arccos (c x))}{m+3}+\frac {b c x^{m+3}}{(m+3)^2}\right )}{m+5}+\frac {\left (1-c^2 x^2\right )^{3/2} x^{m+2} (a+b \arccos (c x))}{m+5}+\frac {b c \left (\frac {x^{m+3}}{m+3}-\frac {c^2 x^{m+5}}{m+5}\right )}{m+5}\right )}{m+5}+\frac {\left (1-c^2 x^2\right )^2 x^{m+1} (a+b \arccos (c x))^2}{m+5}\right )}{m+7}+\frac {2 b c d^3 \left (\frac {5 \left (\frac {3 \left (\frac {\int \frac {x^{m+1} (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx}{m+3}+\frac {\sqrt {1-c^2 x^2} x^{m+2} (a+b \arccos (c x))}{m+3}+\frac {b c x^{m+3}}{(m+3)^2}\right )}{m+5}+\frac {\left (1-c^2 x^2\right )^{3/2} x^{m+2} (a+b \arccos (c x))}{m+5}+\frac {b c \left (\frac {x^{m+3}}{m+3}-\frac {c^2 x^{m+5}}{m+5}\right )}{m+5}\right )}{m+7}+\frac {\left (1-c^2 x^2\right )^{5/2} x^{m+2} (a+b \arccos (c x))}{m+7}+\frac {b c \left (\frac {c^4 x^{m+7}}{m+7}-\frac {2 c^2 x^{m+5}}{m+5}+\frac {x^{m+3}}{m+3}\right )}{m+7}\right )}{m+7}+\frac {d^3 \left (1-c^2 x^2\right )^3 x^{m+1} (a+b \arccos (c x))^2}{m+7}\) |
\(\Big \downarrow \) 5221 |
\(\displaystyle \frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arccos (c x))^2 x^{m+1}}{m+7}+\frac {6 d^3 \left (\frac {\left (1-c^2 x^2\right )^2 (a+b \arccos (c x))^2 x^{m+1}}{m+5}+\frac {4 \left (\frac {\left (1-c^2 x^2\right ) (a+b \arccos (c x))^2 x^{m+1}}{m+3}+\frac {2 \left (\frac {(a+b \arccos (c x))^2 x^{m+1}}{m+1}+\frac {2 b c \left (\frac {(a+b \arccos (c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},c^2 x^2\right ) x^{m+2}}{m+2}+\frac {b c \, _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 ) x^{m+3}}{m^2+5 m+6}\right )}{m+1}\right )}{m+3}+\frac {2 b c \left (\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x)) x^{m+2}}{m+3}+\frac {b c x^{m+3}}{(m+3)^2}+\frac {\frac {(a+b \arccos (c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},c^2 x^2\right ) x^{m+2}}{m+2}+\frac {b c \, _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 ) x^{m+3}}{m^2+5 m+6}}{m+3}\right )}{m+3}\right )}{m+5}+\frac {2 b c \left (\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x)) x^{m+2}}{m+5}+\frac {b c \left (\frac {x^{m+3}}{m+3}-\frac {c^2 x^{m+5}}{m+5}\right )}{m+5}+\frac {3 \left (\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x)) x^{m+2}}{m+3}+\frac {b c x^{m+3}}{(m+3)^2}+\frac {\frac {(a+b \arccos (c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},c^2 x^2\right ) x^{m+2}}{m+2}+\frac {b c \, _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 ) x^{m+3}}{m^2+5 m+6}}{m+3}\right )}{m+5}\right )}{m+5}\right )}{m+7}+\frac {2 b c d^3 \left (\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x)) x^{m+2}}{m+7}+\frac {b c \left (\frac {x^{m+3}}{m+3}-\frac {2 c^2 x^{m+5}}{m+5}+\frac {c^4 x^{m+7}}{m+7}\right )}{m+7}+\frac {5 \left (\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x)) x^{m+2}}{m+5}+\frac {b c \left (\frac {x^{m+3}}{m+3}-\frac {c^2 x^{m+5}}{m+5}\right )}{m+5}+\frac {3 \left (\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x)) x^{m+2}}{m+3}+\frac {b c x^{m+3}}{(m+3)^2}+\frac {\frac {(a+b \arccos (c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},c^2 x^2\right ) x^{m+2}}{m+2}+\frac {b c \, _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 ) x^{m+3}}{m^2+5 m+6}}{m+3}\right )}{m+5}\right )}{m+7}\right )}{m+7}\) |
Input:
Int[x^m*(d - c^2*d*x^2)^3*(a + b*ArcCos[c*x])^2,x]
Output:
(d^3*x^(1 + m)*(1 - c^2*x^2)^3*(a + b*ArcCos[c*x])^2)/(7 + m) + (6*d^3*((x ^(1 + m)*(1 - c^2*x^2)^2*(a + b*ArcCos[c*x])^2)/(5 + m) + (4*((x^(1 + m)*( 1 - c^2*x^2)*(a + b*ArcCos[c*x])^2)/(3 + m) + (2*((x^(1 + m)*(a + b*ArcCos [c*x])^2)/(1 + m) + (2*b*c*((x^(2 + m)*(a + b*ArcCos[c*x])*Hypergeometric2 F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/(2 + m) + (b*c*x^(3 + m)*Hypergeom etricPFQ[{1, 3/2 + m/2, 3/2 + m/2}, {2 + m/2, 5/2 + m/2}, c^2*x^2])/(6 + 5 *m + m^2)))/(1 + m)))/(3 + m) + (2*b*c*((b*c*x^(3 + m))/(3 + m)^2 + (x^(2 + m)*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x]))/(3 + m) + ((x^(2 + m)*(a + b*A rcCos[c*x])*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/(2 + m) + (b*c*x^(3 + m)*HypergeometricPFQ[{1, 3/2 + m/2, 3/2 + m/2}, {2 + m/2, 5 /2 + m/2}, c^2*x^2])/(6 + 5*m + m^2))/(3 + m)))/(3 + m)))/(5 + m) + (2*b*c *((b*c*(x^(3 + m)/(3 + m) - (c^2*x^(5 + m))/(5 + m)))/(5 + m) + (x^(2 + m) *(1 - c^2*x^2)^(3/2)*(a + b*ArcCos[c*x]))/(5 + m) + (3*((b*c*x^(3 + m))/(3 + m)^2 + (x^(2 + m)*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x]))/(3 + m) + ((x^ (2 + m)*(a + b*ArcCos[c*x])*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c ^2*x^2])/(2 + m) + (b*c*x^(3 + m)*HypergeometricPFQ[{1, 3/2 + m/2, 3/2 + m /2}, {2 + m/2, 5/2 + m/2}, c^2*x^2])/(6 + 5*m + m^2))/(3 + m)))/(5 + m)))/ (5 + m)))/(7 + m) + (2*b*c*d^3*((b*c*(x^(3 + m)/(3 + m) - (2*c^2*x^(5 + m) )/(5 + m) + (c^4*x^(7 + m))/(7 + m)))/(7 + m) + (x^(2 + m)*(1 - c^2*x^2)^( 5/2)*(a + b*ArcCos[c*x]))/(7 + m) + (5*((b*c*(x^(3 + m)/(3 + m) - (c^2*...
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[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 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_.)*((f_.)*(x_))^(m_)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*ArcC os[c*x])^n/(f*(m + 2))), x] + (Simp[(1/(m + 2))*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]] Int[(f*x)^m*((a + b*ArcCos[c*x])^n/Sqrt[1 - c^2*x^2]), x], x ] + Simp[b*c*(n/(f*(m + 2)))*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]] Int[ (f*x)^(m + 1)*(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] && (IGtQ[m, -2] || EqQ[n, 1])
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_.))*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_. )*(x_)^2], x_Symbol] :> Simp[((f*x)^(m + 1)/(f*(m + 1)))*(a + b*ArcCos[c*x] )*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2], x] + Simp[b*c*((f*x)^(m + 2)/(f^2*(m + 1)*(m + 2)))*S imp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*HypergeometricPFQ[{1, 1 + m/2, 1 + m /2}, {3/2 + m/2, 2 + m/2}, c^2*x^2], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && !IntegerQ[m]
\[\int x^{m} \left (-c^{2} d \,x^{2}+d \right )^{3} \left (a +b \arccos \left (c x \right )\right )^{2}d x\]
Input:
int(x^m*(-c^2*d*x^2+d)^3*(a+b*arccos(c*x))^2,x)
Output:
int(x^m*(-c^2*d*x^2+d)^3*(a+b*arccos(c*x))^2,x)
\[ \int x^m \left (d-c^2 d x^2\right )^3 (a+b \arccos (c x))^2 \, dx=\int { -{\left (c^{2} d x^{2} - d\right )}^{3} {\left (b \arccos \left (c x\right ) + a\right )}^{2} x^{m} \,d x } \] Input:
integrate(x^m*(-c^2*d*x^2+d)^3*(a+b*arccos(c*x))^2,x, algorithm="fricas")
Output:
integral(-(a^2*c^6*d^3*x^6 - 3*a^2*c^4*d^3*x^4 + 3*a^2*c^2*d^3*x^2 - a^2*d ^3 + (b^2*c^6*d^3*x^6 - 3*b^2*c^4*d^3*x^4 + 3*b^2*c^2*d^3*x^2 - b^2*d^3)*a rccos(c*x)^2 + 2*(a*b*c^6*d^3*x^6 - 3*a*b*c^4*d^3*x^4 + 3*a*b*c^2*d^3*x^2 - a*b*d^3)*arccos(c*x))*x^m, x)
\[ \int x^m \left (d-c^2 d x^2\right )^3 (a+b \arccos (c x))^2 \, dx=- d^{3} \left (\int \left (- a^{2} x^{m}\right )\, dx + \int \left (- b^{2} x^{m} \operatorname {acos}^{2}{\left (c x \right )}\right )\, dx + \int \left (- 2 a b x^{m} \operatorname {acos}{\left (c x \right )}\right )\, dx + \int 3 a^{2} c^{2} x^{2} x^{m}\, dx + \int \left (- 3 a^{2} c^{4} x^{4} x^{m}\right )\, dx + \int a^{2} c^{6} x^{6} x^{m}\, dx + \int 3 b^{2} c^{2} x^{2} x^{m} \operatorname {acos}^{2}{\left (c x \right )}\, dx + \int \left (- 3 b^{2} c^{4} x^{4} x^{m} \operatorname {acos}^{2}{\left (c x \right )}\right )\, dx + \int b^{2} c^{6} x^{6} x^{m} \operatorname {acos}^{2}{\left (c x \right )}\, dx + \int 6 a b c^{2} x^{2} x^{m} \operatorname {acos}{\left (c x \right )}\, dx + \int \left (- 6 a b c^{4} x^{4} x^{m} \operatorname {acos}{\left (c x \right )}\right )\, dx + \int 2 a b c^{6} x^{6} x^{m} \operatorname {acos}{\left (c x \right )}\, dx\right ) \] Input:
integrate(x**m*(-c**2*d*x**2+d)**3*(a+b*acos(c*x))**2,x)
Output:
-d**3*(Integral(-a**2*x**m, x) + Integral(-b**2*x**m*acos(c*x)**2, x) + In tegral(-2*a*b*x**m*acos(c*x), x) + Integral(3*a**2*c**2*x**2*x**m, x) + In tegral(-3*a**2*c**4*x**4*x**m, x) + Integral(a**2*c**6*x**6*x**m, x) + Int egral(3*b**2*c**2*x**2*x**m*acos(c*x)**2, x) + Integral(-3*b**2*c**4*x**4* x**m*acos(c*x)**2, x) + Integral(b**2*c**6*x**6*x**m*acos(c*x)**2, x) + In tegral(6*a*b*c**2*x**2*x**m*acos(c*x), x) + Integral(-6*a*b*c**4*x**4*x**m *acos(c*x), x) + Integral(2*a*b*c**6*x**6*x**m*acos(c*x), x))
\[ \int x^m \left (d-c^2 d x^2\right )^3 (a+b \arccos (c x))^2 \, dx=\int { -{\left (c^{2} d x^{2} - d\right )}^{3} {\left (b \arccos \left (c x\right ) + a\right )}^{2} x^{m} \,d x } \] Input:
integrate(x^m*(-c^2*d*x^2+d)^3*(a+b*arccos(c*x))^2,x, algorithm="maxima")
Output:
-a^2*c^6*d^3*x^(m + 7)/(m + 7) + 3*a^2*c^4*d^3*x^(m + 5)/(m + 5) - 3*a^2*c ^2*d^3*x^(m + 3)/(m + 3) + a^2*d^3*x^(m + 1)/(m + 1) - (((b^2*c^6*d^3*m^3 + 9*b^2*c^6*d^3*m^2 + 23*b^2*c^6*d^3*m + 15*b^2*c^6*d^3)*x^7 - 3*(b^2*c^4* d^3*m^3 + 11*b^2*c^4*d^3*m^2 + 31*b^2*c^4*d^3*m + 21*b^2*c^4*d^3)*x^5 + 3* (b^2*c^2*d^3*m^3 + 13*b^2*c^2*d^3*m^2 + 47*b^2*c^2*d^3*m + 35*b^2*c^2*d^3) *x^3 - (b^2*d^3*m^3 + 15*b^2*d^3*m^2 + 71*b^2*d^3*m + 105*b^2*d^3)*x)*x^m* arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)^2 - (m^4 + 16*m^3 + 86*m^2 + 17 6*m + 105)*integrate(-2*(((b^2*c^7*d^3*m^3 + 9*b^2*c^7*d^3*m^2 + 23*b^2*c^ 7*d^3*m + 15*b^2*c^7*d^3)*x^7 - 3*(b^2*c^5*d^3*m^3 + 11*b^2*c^5*d^3*m^2 + 31*b^2*c^5*d^3*m + 21*b^2*c^5*d^3)*x^5 + 3*(b^2*c^3*d^3*m^3 + 13*b^2*c^3*d ^3*m^2 + 47*b^2*c^3*d^3*m + 35*b^2*c^3*d^3)*x^3 - (b^2*c*d^3*m^3 + 15*b^2* c*d^3*m^2 + 71*b^2*c*d^3*m + 105*b^2*c*d^3)*x)*sqrt(c*x + 1)*sqrt(-c*x + 1 )*x^m*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x) - (a*b*d^3*m^4 + (a*b*c^8 *d^3*m^4 + 16*a*b*c^8*d^3*m^3 + 86*a*b*c^8*d^3*m^2 + 176*a*b*c^8*d^3*m + 1 05*a*b*c^8*d^3)*x^8 + 16*a*b*d^3*m^3 + 86*a*b*d^3*m^2 - 4*(a*b*c^6*d^3*m^4 + 16*a*b*c^6*d^3*m^3 + 86*a*b*c^6*d^3*m^2 + 176*a*b*c^6*d^3*m + 105*a*b*c ^6*d^3)*x^6 + 176*a*b*d^3*m + 105*a*b*d^3 + 6*(a*b*c^4*d^3*m^4 + 16*a*b*c^ 4*d^3*m^3 + 86*a*b*c^4*d^3*m^2 + 176*a*b*c^4*d^3*m + 105*a*b*c^4*d^3)*x^4 - 4*(a*b*c^2*d^3*m^4 + 16*a*b*c^2*d^3*m^3 + 86*a*b*c^2*d^3*m^2 + 176*a*b*c ^2*d^3*m + 105*a*b*c^2*d^3)*x^2)*x^m*arctan2(sqrt(c*x + 1)*sqrt(-c*x + ...
\[ \int x^m \left (d-c^2 d x^2\right )^3 (a+b \arccos (c x))^2 \, dx=\int { -{\left (c^{2} d x^{2} - d\right )}^{3} {\left (b \arccos \left (c x\right ) + a\right )}^{2} x^{m} \,d x } \] Input:
integrate(x^m*(-c^2*d*x^2+d)^3*(a+b*arccos(c*x))^2,x, algorithm="giac")
Output:
integrate(-(c^2*d*x^2 - d)^3*(b*arccos(c*x) + a)^2*x^m, x)
Timed out. \[ \int x^m \left (d-c^2 d x^2\right )^3 (a+b \arccos (c x))^2 \, dx=\int x^m\,{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^3 \,d x \] 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)
\[ \int x^m \left (d-c^2 d x^2\right )^3 (a+b \arccos (c x))^2 \, dx =\text {Too large to display} \] Input:
int(x^m*(-c^2*d*x^2+d)^3*(a+b*acos(c*x))^2,x)
Output:
(d**3*( - x**m*a**2*c**6*m**3*x**7 - 9*x**m*a**2*c**6*m**2*x**7 - 23*x**m* a**2*c**6*m*x**7 - 15*x**m*a**2*c**6*x**7 + 3*x**m*a**2*c**4*m**3*x**5 + 3 3*x**m*a**2*c**4*m**2*x**5 + 93*x**m*a**2*c**4*m*x**5 + 63*x**m*a**2*c**4* x**5 - 3*x**m*a**2*c**2*m**3*x**3 - 39*x**m*a**2*c**2*m**2*x**3 - 141*x**m *a**2*c**2*m*x**3 - 105*x**m*a**2*c**2*x**3 + x**m*a**2*m**3*x + 15*x**m*a **2*m**2*x + 71*x**m*a**2*m*x + 105*x**m*a**2*x - 2*int(x**m*acos(c*x)*x** 6,x)*a*b*c**6*m**4 - 32*int(x**m*acos(c*x)*x**6,x)*a*b*c**6*m**3 - 172*int (x**m*acos(c*x)*x**6,x)*a*b*c**6*m**2 - 352*int(x**m*acos(c*x)*x**6,x)*a*b *c**6*m - 210*int(x**m*acos(c*x)*x**6,x)*a*b*c**6 + 6*int(x**m*acos(c*x)*x **4,x)*a*b*c**4*m**4 + 96*int(x**m*acos(c*x)*x**4,x)*a*b*c**4*m**3 + 516*i nt(x**m*acos(c*x)*x**4,x)*a*b*c**4*m**2 + 1056*int(x**m*acos(c*x)*x**4,x)* a*b*c**4*m + 630*int(x**m*acos(c*x)*x**4,x)*a*b*c**4 - 6*int(x**m*acos(c*x )*x**2,x)*a*b*c**2*m**4 - 96*int(x**m*acos(c*x)*x**2,x)*a*b*c**2*m**3 - 51 6*int(x**m*acos(c*x)*x**2,x)*a*b*c**2*m**2 - 1056*int(x**m*acos(c*x)*x**2, x)*a*b*c**2*m - 630*int(x**m*acos(c*x)*x**2,x)*a*b*c**2 + 2*int(x**m*acos( c*x),x)*a*b*m**4 + 32*int(x**m*acos(c*x),x)*a*b*m**3 + 172*int(x**m*acos(c *x),x)*a*b*m**2 + 352*int(x**m*acos(c*x),x)*a*b*m + 210*int(x**m*acos(c*x) ,x)*a*b - int(x**m*acos(c*x)**2*x**6,x)*b**2*c**6*m**4 - 16*int(x**m*acos( c*x)**2*x**6,x)*b**2*c**6*m**3 - 86*int(x**m*acos(c*x)**2*x**6,x)*b**2*c** 6*m**2 - 176*int(x**m*acos(c*x)**2*x**6,x)*b**2*c**6*m - 105*int(x**m*a...