\(\int x^m (d-c^2 d x^2)^3 (a+b \arcsin (c x))^2 \, dx\) [269]

Optimal result

Integrand size = 27, antiderivative size = 1312 \[ \int x^m \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2 \, dx =\text {Too large to display} \] Output:

96*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],c^2*x^2)/(3+m)^2/(5+m)/(7+m)/(m^2+3*m+2)-30*b*c*d^3*x^(2+m)*(a+b*arc 
sin(c*x))*hypergeom([1/2, 1+1/2*m],[2+1/2*m],c^2*x^2)/(5+m)/(7+m)^2/(m^2+5 
*m+6)-36*b*c*d^3*x^(2+m)*(a+b*arcsin(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*arcsin(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*c*d^3*x^(2+m)*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))/(7+m)^2/(m^2+8* 
m+15)+2*b^2*c^6*d^3*x^(7+m)/(7+m)^3+d^3*x^(1+m)*(-c^2*x^2+1)^3*(a+b*arcsin 
(c*x))^2/(7+m)+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*arcsin(c*x))/(7+m)^2+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)^2-12*b^2*c^4*d^3*x^(5+m)/(5+m)^3/(7+m 
)+48*d^3*x^(1+m)*(a+b*arcsin(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*arcsin(c*x))^2/(7+m)/(m^2+8*m+15)+6*d^3*x^(1+m)*(-c^2* 
x^2+1)^2*(a+b*arcsin(c*x))^2/(5+m)/(7+m)-36*b*c*d^3*x^(2+m)*(-c^2*x^2+1)^( 
1/2)*(a+b*arcsin(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*arcsin(c*x))/(3+m)^2/(5+m)/(7+m)-48*b*c*d^3*x^(2+m)*(a+b*arcsi 
n(c*x))*hypergeom([1/2, 1+1/2*m],[2+1/2*m],c^2*x^2)/(2+m)/(3+m)^2/(5+m)...
 

Mathematica [A] (warning: unable to verify)

Time = 0.58 (sec) , antiderivative size = 539, normalized size of antiderivative = 0.41 \[ \int x^m \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2 \, dx=d^3 x^{1+m} \left (\frac {(a+b \arcsin (c x))^2}{1+m}-\frac {3 c^2 x^2 (a+b \arcsin (c x))^2}{3+m}+\frac {3 c^4 x^4 (a+b \arcsin (c x))^2}{5+m}-\frac {c^6 x^6 (a+b \arcsin (c x))^2}{7+m}+\frac {2 b c x \left (-\left ((3+m) (a+b \arcsin (c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )\right )+b c x \, _3F_2\left (1,\frac {3}{2}+\frac {m}{2},\frac {3}{2}+\frac {m}{2};2+\frac {m}{2},\frac {5}{2}+\frac {m}{2};c^2 x^2\right )\right )}{(1+m) (2+m) (3+m)}-\frac {6 b c^3 x^3 \left (-\left ((5+m) (a+b \arcsin (c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {4+m}{2},\frac {6+m}{2},c^2 x^2\right )\right )+b c x \, _3F_2\left (1,\frac {5}{2}+\frac {m}{2},\frac {5}{2}+\frac {m}{2};3+\frac {m}{2},\frac {7}{2}+\frac {m}{2};c^2 x^2\right )\right )}{(3+m) (4+m) (5+m)}+\frac {6 b c^5 x^5 \left (-\left ((7+m) (a+b \arcsin (c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {6+m}{2},\frac {8+m}{2},c^2 x^2\right )\right )+b c x \, _3F_2\left (1,\frac {7}{2}+\frac {m}{2},\frac {7}{2}+\frac {m}{2};4+\frac {m}{2},\frac {9}{2}+\frac {m}{2};c^2 x^2\right )\right )}{(5+m) (6+m) (7+m)}-\frac {2 b c^7 x^7 \left (-\left ((9+m) (a+b \arcsin (c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},4+\frac {m}{2},5+\frac {m}{2},c^2 x^2\right )\right )+b c x \, _3F_2\left (1,\frac {9}{2}+\frac {m}{2},\frac {9}{2}+\frac {m}{2};5+\frac {m}{2},\frac {11}{2}+\frac {m}{2};c^2 x^2\right )\right )}{(7+m) (8+m) (9+m)}\right ) \] Input:

Integrate[x^m*(d - c^2*d*x^2)^3*(a + b*ArcSin[c*x])^2,x]
 

Output:

d^3*x^(1 + m)*((a + b*ArcSin[c*x])^2/(1 + m) - (3*c^2*x^2*(a + b*ArcSin[c* 
x])^2)/(3 + m) + (3*c^4*x^4*(a + b*ArcSin[c*x])^2)/(5 + m) - (c^6*x^6*(a + 
 b*ArcSin[c*x])^2)/(7 + m) + (2*b*c*x*(-((3 + m)*(a + b*ArcSin[c*x])*Hyper 
geometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2]) + b*c*x*HypergeometricPF 
Q[{1, 3/2 + m/2, 3/2 + m/2}, {2 + m/2, 5/2 + m/2}, c^2*x^2]))/((1 + m)*(2 
+ m)*(3 + m)) - (6*b*c^3*x^3*(-((5 + m)*(a + b*ArcSin[c*x])*Hypergeometric 
2F1[1/2, (4 + m)/2, (6 + m)/2, c^2*x^2]) + b*c*x*HypergeometricPFQ[{1, 5/2 
 + m/2, 5/2 + m/2}, {3 + m/2, 7/2 + m/2}, c^2*x^2]))/((3 + m)*(4 + m)*(5 + 
 m)) + (6*b*c^5*x^5*(-((7 + m)*(a + b*ArcSin[c*x])*Hypergeometric2F1[1/2, 
(6 + m)/2, (8 + m)/2, c^2*x^2]) + b*c*x*HypergeometricPFQ[{1, 7/2 + m/2, 7 
/2 + m/2}, {4 + m/2, 9/2 + m/2}, c^2*x^2]))/((5 + m)*(6 + m)*(7 + m)) - (2 
*b*c^7*x^7*(-((9 + m)*(a + b*ArcSin[c*x])*Hypergeometric2F1[1/2, 4 + m/2, 
5 + m/2, c^2*x^2]) + b*c*x*HypergeometricPFQ[{1, 9/2 + m/2, 9/2 + m/2}, {5 
 + m/2, 11/2 + m/2}, c^2*x^2]))/((7 + m)*(8 + m)*(9 + m)))
 

Rubi [A] (verified)

Time = 3.38 (sec) , antiderivative size = 1031, normalized size of antiderivative = 0.79, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {5202, 27, 5202, 244, 2009, 5202, 244, 2009, 5138, 5198, 15, 5220}

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.

\(\displaystyle \int x^m \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2 \, dx\)

\(\Big \downarrow \) 5202

\(\displaystyle -\frac {2 b c d^3 \int x^{m+1} \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))dx}{m+7}+\frac {6 d \int d^2 x^m \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2dx}{m+7}+\frac {d^3 \left (1-c^2 x^2\right )^3 x^{m+1} (a+b \arcsin (c x))^2}{m+7}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 b c d^3 \int x^{m+1} \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))dx}{m+7}+\frac {6 d^3 \int x^m \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2dx}{m+7}+\frac {d^3 \left (1-c^2 x^2\right )^3 x^{m+1} (a+b \arcsin (c x))^2}{m+7}\)

\(\Big \downarrow \) 5202

\(\displaystyle -\frac {2 b c d^3 \left (\frac {5 \int x^{m+1} \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))dx}{m+7}-\frac {b c \int x^{m+2} \left (1-c^2 x^2\right )^2dx}{m+7}+\frac {\left (1-c^2 x^2\right )^{5/2} x^{m+2} (a+b \arcsin (c x))}{m+7}\right )}{m+7}+\frac {6 d^3 \left (-\frac {2 b c \int x^{m+1} \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))dx}{m+5}+\frac {4 \int x^m \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2dx}{m+5}+\frac {\left (1-c^2 x^2\right )^2 x^{m+1} (a+b \arcsin (c x))^2}{m+5}\right )}{m+7}+\frac {d^3 \left (1-c^2 x^2\right )^3 x^{m+1} (a+b \arcsin (c x))^2}{m+7}\)

\(\Big \downarrow \) 244

\(\displaystyle \frac {6 d^3 \left (-\frac {2 b c \int x^{m+1} \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))dx}{m+5}+\frac {4 \int x^m \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2dx}{m+5}+\frac {\left (1-c^2 x^2\right )^2 x^{m+1} (a+b \arcsin (c x))^2}{m+5}\right )}{m+7}-\frac {2 b c d^3 \left (\frac {5 \int x^{m+1} \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))dx}{m+7}-\frac {b c \int \left (x^{m+2}-2 c^2 x^{m+4}+c^4 x^{m+6}\right )dx}{m+7}+\frac {\left (1-c^2 x^2\right )^{5/2} x^{m+2} (a+b \arcsin (c x))}{m+7}\right )}{m+7}+\frac {d^3 \left (1-c^2 x^2\right )^3 x^{m+1} (a+b \arcsin (c x))^2}{m+7}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {6 d^3 \left (-\frac {2 b c \int x^{m+1} \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))dx}{m+5}+\frac {4 \int x^m \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2dx}{m+5}+\frac {\left (1-c^2 x^2\right )^2 x^{m+1} (a+b \arcsin (c x))^2}{m+5}\right )}{m+7}-\frac {2 b c d^3 \left (\frac {5 \int x^{m+1} \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))dx}{m+7}+\frac {\left (1-c^2 x^2\right )^{5/2} x^{m+2} (a+b \arcsin (c x))}{m+7}-\frac {b c \left (\frac {c^4 x^{m+7}}{m+7}-\frac {2 c^2 x^{m+5}}{m+5}+\frac {x^{m+3}}{m+3}\right )}{m+7}\right )}{m+7}+\frac {d^3 \left (1-c^2 x^2\right )^3 x^{m+1} (a+b \arcsin (c x))^2}{m+7}\)

\(\Big \downarrow \) 5202

\(\displaystyle \frac {6 d^3 \left (-\frac {2 b c \left (\frac {3 \int x^{m+1} \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx}{m+5}-\frac {b c \int x^{m+2} \left (1-c^2 x^2\right )dx}{m+5}+\frac {\left (1-c^2 x^2\right )^{3/2} x^{m+2} (a+b \arcsin (c x))}{m+5}\right )}{m+5}+\frac {4 \left (-\frac {2 b c \int x^{m+1} \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx}{m+3}+\frac {2 \int x^m (a+b \arcsin (c x))^2dx}{m+3}+\frac {\left (1-c^2 x^2\right ) x^{m+1} (a+b \arcsin (c x))^2}{m+3}\right )}{m+5}+\frac {\left (1-c^2 x^2\right )^2 x^{m+1} (a+b \arcsin (c x))^2}{m+5}\right )}{m+7}-\frac {2 b c d^3 \left (\frac {5 \left (\frac {3 \int x^{m+1} \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx}{m+5}-\frac {b c \int x^{m+2} \left (1-c^2 x^2\right )dx}{m+5}+\frac {\left (1-c^2 x^2\right )^{3/2} x^{m+2} (a+b \arcsin (c x))}{m+5}\right )}{m+7}+\frac {\left (1-c^2 x^2\right )^{5/2} x^{m+2} (a+b \arcsin (c x))}{m+7}-\frac {b c \left (\frac {c^4 x^{m+7}}{m+7}-\frac {2 c^2 x^{m+5}}{m+5}+\frac {x^{m+3}}{m+3}\right )}{m+7}\right )}{m+7}+\frac {d^3 \left (1-c^2 x^2\right )^3 x^{m+1} (a+b \arcsin (c x))^2}{m+7}\)

\(\Big \downarrow \) 244

\(\displaystyle \frac {6 d^3 \left (-\frac {2 b c \left (\frac {3 \int x^{m+1} \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx}{m+5}-\frac {b c \int \left (x^{m+2}-c^2 x^{m+4}\right )dx}{m+5}+\frac {\left (1-c^2 x^2\right )^{3/2} x^{m+2} (a+b \arcsin (c x))}{m+5}\right )}{m+5}+\frac {4 \left (-\frac {2 b c \int x^{m+1} \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx}{m+3}+\frac {2 \int x^m (a+b \arcsin (c x))^2dx}{m+3}+\frac {\left (1-c^2 x^2\right ) x^{m+1} (a+b \arcsin (c x))^2}{m+3}\right )}{m+5}+\frac {\left (1-c^2 x^2\right )^2 x^{m+1} (a+b \arcsin (c x))^2}{m+5}\right )}{m+7}-\frac {2 b c d^3 \left (\frac {5 \left (\frac {3 \int x^{m+1} \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx}{m+5}-\frac {b c \int \left (x^{m+2}-c^2 x^{m+4}\right )dx}{m+5}+\frac {\left (1-c^2 x^2\right )^{3/2} x^{m+2} (a+b \arcsin (c x))}{m+5}\right )}{m+7}+\frac {\left (1-c^2 x^2\right )^{5/2} x^{m+2} (a+b \arcsin (c x))}{m+7}-\frac {b c \left (\frac {c^4 x^{m+7}}{m+7}-\frac {2 c^2 x^{m+5}}{m+5}+\frac {x^{m+3}}{m+3}\right )}{m+7}\right )}{m+7}+\frac {d^3 \left (1-c^2 x^2\right )^3 x^{m+1} (a+b \arcsin (c x))^2}{m+7}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {6 d^3 \left (-\frac {2 b c \left (\frac {3 \int x^{m+1} \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx}{m+5}+\frac {\left (1-c^2 x^2\right )^{3/2} x^{m+2} (a+b \arcsin (c x))}{m+5}-\frac {b c \left (\frac {x^{m+3}}{m+3}-\frac {c^2 x^{m+5}}{m+5}\right )}{m+5}\right )}{m+5}+\frac {4 \left (-\frac {2 b c \int x^{m+1} \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx}{m+3}+\frac {2 \int x^m (a+b \arcsin (c x))^2dx}{m+3}+\frac {\left (1-c^2 x^2\right ) x^{m+1} (a+b \arcsin (c x))^2}{m+3}\right )}{m+5}+\frac {\left (1-c^2 x^2\right )^2 x^{m+1} (a+b \arcsin (c x))^2}{m+5}\right )}{m+7}-\frac {2 b c d^3 \left (\frac {5 \left (\frac {3 \int x^{m+1} \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx}{m+5}+\frac {\left (1-c^2 x^2\right )^{3/2} x^{m+2} (a+b \arcsin (c x))}{m+5}-\frac {b c \left (\frac {x^{m+3}}{m+3}-\frac {c^2 x^{m+5}}{m+5}\right )}{m+5}\right )}{m+7}+\frac {\left (1-c^2 x^2\right )^{5/2} x^{m+2} (a+b \arcsin (c x))}{m+7}-\frac {b c \left (\frac {c^4 x^{m+7}}{m+7}-\frac {2 c^2 x^{m+5}}{m+5}+\frac {x^{m+3}}{m+3}\right )}{m+7}\right )}{m+7}+\frac {d^3 \left (1-c^2 x^2\right )^3 x^{m+1} (a+b \arcsin (c x))^2}{m+7}\)

\(\Big \downarrow \) 5138

\(\displaystyle \frac {6 d^3 \left (\frac {4 \left (\frac {2 \left (\frac {x^{m+1} (a+b \arcsin (c x))^2}{m+1}-\frac {2 b c \int \frac {x^{m+1} (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{m+1}\right )}{m+3}-\frac {2 b c \int x^{m+1} \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx}{m+3}+\frac {\left (1-c^2 x^2\right ) x^{m+1} (a+b \arcsin (c x))^2}{m+3}\right )}{m+5}-\frac {2 b c \left (\frac {3 \int x^{m+1} \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx}{m+5}+\frac {\left (1-c^2 x^2\right )^{3/2} x^{m+2} (a+b \arcsin (c x))}{m+5}-\frac {b c \left (\frac {x^{m+3}}{m+3}-\frac {c^2 x^{m+5}}{m+5}\right )}{m+5}\right )}{m+5}+\frac {\left (1-c^2 x^2\right )^2 x^{m+1} (a+b \arcsin (c x))^2}{m+5}\right )}{m+7}-\frac {2 b c d^3 \left (\frac {5 \left (\frac {3 \int x^{m+1} \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx}{m+5}+\frac {\left (1-c^2 x^2\right )^{3/2} x^{m+2} (a+b \arcsin (c x))}{m+5}-\frac {b c \left (\frac {x^{m+3}}{m+3}-\frac {c^2 x^{m+5}}{m+5}\right )}{m+5}\right )}{m+7}+\frac {\left (1-c^2 x^2\right )^{5/2} x^{m+2} (a+b \arcsin (c x))}{m+7}-\frac {b c \left (\frac {c^4 x^{m+7}}{m+7}-\frac {2 c^2 x^{m+5}}{m+5}+\frac {x^{m+3}}{m+3}\right )}{m+7}\right )}{m+7}+\frac {d^3 \left (1-c^2 x^2\right )^3 x^{m+1} (a+b \arcsin (c x))^2}{m+7}\)

\(\Big \downarrow \) 5198

\(\displaystyle \frac {6 d^3 \left (\frac {4 \left (\frac {2 \left (\frac {x^{m+1} (a+b \arcsin (c x))^2}{m+1}-\frac {2 b c \int \frac {x^{m+1} (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{m+1}\right )}{m+3}-\frac {2 b c \left (\frac {\int \frac {x^{m+1} (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{m+3}-\frac {b c \int x^{m+2}dx}{m+3}+\frac {\sqrt {1-c^2 x^2} x^{m+2} (a+b \arcsin (c x))}{m+3}\right )}{m+3}+\frac {\left (1-c^2 x^2\right ) x^{m+1} (a+b \arcsin (c x))^2}{m+3}\right )}{m+5}-\frac {2 b c \left (\frac {3 \left (\frac {\int \frac {x^{m+1} (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{m+3}-\frac {b c \int x^{m+2}dx}{m+3}+\frac {\sqrt {1-c^2 x^2} x^{m+2} (a+b \arcsin (c x))}{m+3}\right )}{m+5}+\frac {\left (1-c^2 x^2\right )^{3/2} x^{m+2} (a+b \arcsin (c x))}{m+5}-\frac {b c \left (\frac {x^{m+3}}{m+3}-\frac {c^2 x^{m+5}}{m+5}\right )}{m+5}\right )}{m+5}+\frac {\left (1-c^2 x^2\right )^2 x^{m+1} (a+b \arcsin (c x))^2}{m+5}\right )}{m+7}-\frac {2 b c d^3 \left (\frac {5 \left (\frac {3 \left (\frac {\int \frac {x^{m+1} (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{m+3}-\frac {b c \int x^{m+2}dx}{m+3}+\frac {\sqrt {1-c^2 x^2} x^{m+2} (a+b \arcsin (c x))}{m+3}\right )}{m+5}+\frac {\left (1-c^2 x^2\right )^{3/2} x^{m+2} (a+b \arcsin (c x))}{m+5}-\frac {b c \left (\frac {x^{m+3}}{m+3}-\frac {c^2 x^{m+5}}{m+5}\right )}{m+5}\right )}{m+7}+\frac {\left (1-c^2 x^2\right )^{5/2} x^{m+2} (a+b \arcsin (c x))}{m+7}-\frac {b c \left (\frac {c^4 x^{m+7}}{m+7}-\frac {2 c^2 x^{m+5}}{m+5}+\frac {x^{m+3}}{m+3}\right )}{m+7}\right )}{m+7}+\frac {d^3 \left (1-c^2 x^2\right )^3 x^{m+1} (a+b \arcsin (c x))^2}{m+7}\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {6 d^3 \left (\frac {4 \left (\frac {2 \left (\frac {x^{m+1} (a+b \arcsin (c x))^2}{m+1}-\frac {2 b c \int \frac {x^{m+1} (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{m+1}\right )}{m+3}-\frac {2 b c \left (\frac {\int \frac {x^{m+1} (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{m+3}+\frac {\sqrt {1-c^2 x^2} x^{m+2} (a+b \arcsin (c x))}{m+3}-\frac {b c x^{m+3}}{(m+3)^2}\right )}{m+3}+\frac {\left (1-c^2 x^2\right ) x^{m+1} (a+b \arcsin (c x))^2}{m+3}\right )}{m+5}-\frac {2 b c \left (\frac {3 \left (\frac {\int \frac {x^{m+1} (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{m+3}+\frac {\sqrt {1-c^2 x^2} x^{m+2} (a+b \arcsin (c x))}{m+3}-\frac {b c x^{m+3}}{(m+3)^2}\right )}{m+5}+\frac {\left (1-c^2 x^2\right )^{3/2} x^{m+2} (a+b \arcsin (c x))}{m+5}-\frac {b c \left (\frac {x^{m+3}}{m+3}-\frac {c^2 x^{m+5}}{m+5}\right )}{m+5}\right )}{m+5}+\frac {\left (1-c^2 x^2\right )^2 x^{m+1} (a+b \arcsin (c x))^2}{m+5}\right )}{m+7}-\frac {2 b c d^3 \left (\frac {5 \left (\frac {3 \left (\frac {\int \frac {x^{m+1} (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{m+3}+\frac {\sqrt {1-c^2 x^2} x^{m+2} (a+b \arcsin (c x))}{m+3}-\frac {b c x^{m+3}}{(m+3)^2}\right )}{m+5}+\frac {\left (1-c^2 x^2\right )^{3/2} x^{m+2} (a+b \arcsin (c x))}{m+5}-\frac {b c \left (\frac {x^{m+3}}{m+3}-\frac {c^2 x^{m+5}}{m+5}\right )}{m+5}\right )}{m+7}+\frac {\left (1-c^2 x^2\right )^{5/2} x^{m+2} (a+b \arcsin (c x))}{m+7}-\frac {b c \left (\frac {c^4 x^{m+7}}{m+7}-\frac {2 c^2 x^{m+5}}{m+5}+\frac {x^{m+3}}{m+3}\right )}{m+7}\right )}{m+7}+\frac {d^3 \left (1-c^2 x^2\right )^3 x^{m+1} (a+b \arcsin (c x))^2}{m+7}\)

\(\Big \downarrow \) 5220

\(\displaystyle \frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))^2 x^{m+1}}{m+7}+\frac {6 d^3 \left (\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2 x^{m+1}}{m+5}+\frac {4 \left (\frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2 x^{m+1}}{m+3}+\frac {2 \left (\frac {x^{m+1} (a+b \arcsin (c x))^2}{m+1}-\frac {2 b c \left (\frac {x^{m+2} (a+b \arcsin (c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},c^2 x^2\right )}{m+2}-\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}\right )}{m+1}\right )}{m+3}-\frac {2 b c \left (\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x)) x^{m+2}}{m+3}-\frac {b c x^{m+3}}{(m+3)^2}+\frac {\frac {x^{m+2} (a+b \arcsin (c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},c^2 x^2\right )}{m+2}-\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}}{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 \arcsin (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 \arcsin (c x)) x^{m+2}}{m+3}-\frac {b c x^{m+3}}{(m+3)^2}+\frac {\frac {x^{m+2} (a+b \arcsin (c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},c^2 x^2\right )}{m+2}-\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}}{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 \arcsin (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 \arcsin (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 \arcsin (c x)) x^{m+2}}{m+3}-\frac {b c x^{m+3}}{(m+3)^2}+\frac {\frac {x^{m+2} (a+b \arcsin (c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},c^2 x^2\right )}{m+2}-\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}}{m+3}\right )}{m+5}\right )}{m+7}\right )}{m+7}\)

Input:

Int[x^m*(d - c^2*d*x^2)^3*(a + b*ArcSin[c*x])^2,x]
 

Output:

(d^3*x^(1 + m)*(1 - c^2*x^2)^3*(a + b*ArcSin[c*x])^2)/(7 + m) + (6*d^3*((x 
^(1 + m)*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/(5 + m) + (4*((x^(1 + m)*( 
1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(3 + m) + (2*((x^(1 + m)*(a + b*ArcSin 
[c*x])^2)/(1 + m) - (2*b*c*((x^(2 + m)*(a + b*ArcSin[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*ArcSin[c*x]))/(3 + m) + ((x^(2 + m)*(a + 
b*ArcSin[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*ArcSin[c*x]))/(5 + m) + (3*(-((b*c*x^(3 
+ m))/(3 + m)^2) + (x^(2 + m)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(3 + 
m) + ((x^(2 + m)*(a + b*ArcSin[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*ArcSin[c*x]))/(7 + m) + (5*(-((b*c*(x^(3 + m)/(...
 

Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 244

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]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 5138

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] 
:> Simp[(d*x)^(m + 1)*((a + b*ArcSin[c*x])^n/(d*(m + 1))), x] - Simp[b*c*(n 
/(d*(m + 1)))   Int[(d*x)^(m + 1)*((a + b*ArcSin[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]
 

rule 5198

Int[((a_.) + ArcSin[(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*ArcS 
in[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*ArcSin[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*ArcSin[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])
 

rule 5202

Int[((a_.) + ArcSin[(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*ArcS 
in[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*ArcSin[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*ArcSin[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]
 

rule 5220

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_. 
)*(x_)^2], x_Symbol] :> Simp[((f*x)^(m + 1)/(f*(m + 1)))*Simp[Sqrt[1 - c^2* 
x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSin[c*x])*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]
 

Maple [F]

\[\int x^{m} \left (-c^{2} d \,x^{2}+d \right )^{3} \left (a +b \arcsin \left (c x \right )\right )^{2}d x\]

Input:

int(x^m*(-c^2*d*x^2+d)^3*(a+b*arcsin(c*x))^2,x)
 

Output:

int(x^m*(-c^2*d*x^2+d)^3*(a+b*arcsin(c*x))^2,x)
 

Fricas [F]

\[ \int x^m \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2 \, dx=\int { -{\left (c^{2} d x^{2} - d\right )}^{3} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{m} \,d x } \] Input:

integrate(x^m*(-c^2*d*x^2+d)^3*(a+b*arcsin(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 
rcsin(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)*arcsin(c*x))*x^m, x)
 

Sympy [F]

\[ \int x^m \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2 \, dx=- d^{3} \left (\int \left (- a^{2} x^{m}\right )\, dx + \int \left (- b^{2} x^{m} \operatorname {asin}^{2}{\left (c x \right )}\right )\, dx + \int \left (- 2 a b x^{m} \operatorname {asin}{\left (c x \right )}\right )\, dx + \int 3 a^{2} c^{2} x^{2} x^{m}\, dx + \int \left (- 3 a^{2} c^{4} x^{4} x^{m}\right )\, dx + \int a^{2} c^{6} x^{6} x^{m}\, dx + \int 3 b^{2} c^{2} x^{2} x^{m} \operatorname {asin}^{2}{\left (c x \right )}\, dx + \int \left (- 3 b^{2} c^{4} x^{4} x^{m} \operatorname {asin}^{2}{\left (c x \right )}\right )\, dx + \int b^{2} c^{6} x^{6} x^{m} \operatorname {asin}^{2}{\left (c x \right )}\, dx + \int 6 a b c^{2} x^{2} x^{m} \operatorname {asin}{\left (c x \right )}\, dx + \int \left (- 6 a b c^{4} x^{4} x^{m} \operatorname {asin}{\left (c x \right )}\right )\, dx + \int 2 a b c^{6} x^{6} x^{m} \operatorname {asin}{\left (c x \right )}\, dx\right ) \] Input:

integrate(x**m*(-c**2*d*x**2+d)**3*(a+b*asin(c*x))**2,x)
 

Output:

-d**3*(Integral(-a**2*x**m, x) + Integral(-b**2*x**m*asin(c*x)**2, x) + In 
tegral(-2*a*b*x**m*asin(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*asin(c*x)**2, x) + Integral(-3*b**2*c**4*x**4* 
x**m*asin(c*x)**2, x) + Integral(b**2*c**6*x**6*x**m*asin(c*x)**2, x) + In 
tegral(6*a*b*c**2*x**2*x**m*asin(c*x), x) + Integral(-6*a*b*c**4*x**4*x**m 
*asin(c*x), x) + Integral(2*a*b*c**6*x**6*x**m*asin(c*x), x))
 

Maxima [F]

\[ \int x^m \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2 \, dx=\int { -{\left (c^{2} d x^{2} - d\right )}^{3} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{m} \,d x } \] Input:

integrate(x^m*(-c^2*d*x^2+d)^3*(a+b*arcsin(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(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^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(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + (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(c*x, sqrt(c*x + 1)*sqrt(-c...
                                                                                    
                                                                                    
 

Giac [F]

\[ \int x^m \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2 \, dx=\int { -{\left (c^{2} d x^{2} - d\right )}^{3} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{m} \,d x } \] Input:

integrate(x^m*(-c^2*d*x^2+d)^3*(a+b*arcsin(c*x))^2,x, algorithm="giac")
 

Output:

integrate(-(c^2*d*x^2 - d)^3*(b*arcsin(c*x) + a)^2*x^m, x)
 

Mupad [F(-1)]

Timed out. \[ \int x^m \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2 \, dx=\int x^m\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^3 \,d x \] Input:

int(x^m*(a + b*asin(c*x))^2*(d - c^2*d*x^2)^3,x)
 

Output:

int(x^m*(a + b*asin(c*x))^2*(d - c^2*d*x^2)^3, x)
 

Reduce [F]

\[ \int x^m \left (d-c^2 d x^2\right )^3 (a+b \arcsin (c x))^2 \, dx =\text {Too large to display} \] Input:

int(x^m*(-c^2*d*x^2+d)^3*(a+b*asin(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*asin(c*x)*x** 
6,x)*a*b*c**6*m**4 - 32*int(x**m*asin(c*x)*x**6,x)*a*b*c**6*m**3 - 172*int 
(x**m*asin(c*x)*x**6,x)*a*b*c**6*m**2 - 352*int(x**m*asin(c*x)*x**6,x)*a*b 
*c**6*m - 210*int(x**m*asin(c*x)*x**6,x)*a*b*c**6 + 6*int(x**m*asin(c*x)*x 
**4,x)*a*b*c**4*m**4 + 96*int(x**m*asin(c*x)*x**4,x)*a*b*c**4*m**3 + 516*i 
nt(x**m*asin(c*x)*x**4,x)*a*b*c**4*m**2 + 1056*int(x**m*asin(c*x)*x**4,x)* 
a*b*c**4*m + 630*int(x**m*asin(c*x)*x**4,x)*a*b*c**4 - 6*int(x**m*asin(c*x 
)*x**2,x)*a*b*c**2*m**4 - 96*int(x**m*asin(c*x)*x**2,x)*a*b*c**2*m**3 - 51 
6*int(x**m*asin(c*x)*x**2,x)*a*b*c**2*m**2 - 1056*int(x**m*asin(c*x)*x**2, 
x)*a*b*c**2*m - 630*int(x**m*asin(c*x)*x**2,x)*a*b*c**2 + 2*int(x**m*asin( 
c*x),x)*a*b*m**4 + 32*int(x**m*asin(c*x),x)*a*b*m**3 + 172*int(x**m*asin(c 
*x),x)*a*b*m**2 + 352*int(x**m*asin(c*x),x)*a*b*m + 210*int(x**m*asin(c*x) 
,x)*a*b - int(x**m*asin(c*x)**2*x**6,x)*b**2*c**6*m**4 - 16*int(x**m*asin( 
c*x)**2*x**6,x)*b**2*c**6*m**3 - 86*int(x**m*asin(c*x)**2*x**6,x)*b**2*c** 
6*m**2 - 176*int(x**m*asin(c*x)**2*x**6,x)*b**2*c**6*m - 105*int(x**m*a...