3.3.76 \(\int x^m (d-c^2 d x^2)^3 (a+b \text {ArcSin}(c x))^2 \, dx\) [276]

Optimal. Leaf size=1312 \[ \frac {2 b^2 c^2 d^3 x^{3+m}}{(3+m) (7+m)^2}+\frac {30 b^2 c^2 d^3 x^{3+m}}{(3+m)^2 (5+m) (7+m)^2}+\frac {36 b^2 c^2 d^3 x^{3+m}}{(3+m)^2 (5+m)^2 (7+m)}+\frac {12 b^2 c^2 d^3 x^{3+m}}{(3+m) (5+m)^2 (7+m)}+\frac {48 b^2 c^2 d^3 x^{3+m}}{(3+m)^3 (5+m) (7+m)}+\frac {10 b^2 c^2 d^3 x^{3+m}}{(7+m)^2 \left (15+8 m+m^2\right )}-\frac {10 b^2 c^4 d^3 x^{5+m}}{(5+m)^2 (7+m)^2}-\frac {4 b^2 c^4 d^3 x^{5+m}}{(5+m) (7+m)^2}-\frac {12 b^2 c^4 d^3 x^{5+m}}{(5+m)^3 (7+m)}+\frac {2 b^2 c^6 d^3 x^{7+m}}{(7+m)^3}-\frac {36 b c d^3 x^{2+m} \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{(3+m) (5+m)^2 (7+m)}-\frac {48 b c d^3 x^{2+m} \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{(3+m)^2 (5+m) (7+m)}-\frac {30 b c d^3 x^{2+m} \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{(7+m)^2 \left (15+8 m+m^2\right )}-\frac {10 b c d^3 x^{2+m} \left (1-c^2 x^2\right )^{3/2} (a+b \text {ArcSin}(c x))}{(5+m) (7+m)^2}-\frac {12 b c d^3 x^{2+m} \left (1-c^2 x^2\right )^{3/2} (a+b \text {ArcSin}(c x))}{(5+m)^2 (7+m)}-\frac {2 b c d^3 x^{2+m} \left (1-c^2 x^2\right )^{5/2} (a+b \text {ArcSin}(c x))}{(7+m)^2}+\frac {48 d^3 x^{1+m} (a+b \text {ArcSin}(c x))^2}{(5+m) (7+m) \left (3+4 m+m^2\right )}+\frac {24 d^3 x^{1+m} \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))^2}{(7+m) \left (15+8 m+m^2\right )}+\frac {6 d^3 x^{1+m} \left (1-c^2 x^2\right )^2 (a+b \text {ArcSin}(c x))^2}{(5+m) (7+m)}+\frac {d^3 x^{1+m} \left (1-c^2 x^2\right )^3 (a+b \text {ArcSin}(c x))^2}{7+m}-\frac {48 b c d^3 x^{2+m} (a+b \text {ArcSin}(c x)) \text {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{(2+m) (3+m)^2 (5+m) (7+m)}-\frac {30 b c d^3 x^{2+m} (a+b \text {ArcSin}(c x)) \text {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{(5+m) (7+m)^2 \left (6+5 m+m^2\right )}-\frac {36 b c d^3 x^{2+m} (a+b \text {ArcSin}(c x)) \text {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{(5+m)^2 (7+m) \left (6+5 m+m^2\right )}-\frac {96 b c d^3 x^{2+m} (a+b \text {ArcSin}(c x)) \text {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{(5+m) (7+m) \left (6+11 m+6 m^2+m^3\right )}+\frac {30 b^2 c^2 d^3 x^{3+m} \text {HypergeometricPFQ}\left (\left \{1,\frac {3}{2}+\frac {m}{2},\frac {3}{2}+\frac {m}{2}\right \},\left \{2+\frac {m}{2},\frac {5}{2}+\frac {m}{2}\right \},c^2 x^2\right )}{(2+m) (3+m)^2 (5+m) (7+m)^2}+\frac {36 b^2 c^2 d^3 x^{3+m} \text {HypergeometricPFQ}\left (\left \{1,\frac {3}{2}+\frac {m}{2},\frac {3}{2}+\frac {m}{2}\right \},\left \{2+\frac {m}{2},\frac {5}{2}+\frac {m}{2}\right \},c^2 x^2\right )}{(2+m) (3+m)^2 (5+m)^2 (7+m)}+\frac {48 b^2 c^2 d^3 x^{3+m} \text {HypergeometricPFQ}\left (\left \{1,\frac {3}{2}+\frac {m}{2},\frac {3}{2}+\frac {m}{2}\right \},\left \{2+\frac {m}{2},\frac {5}{2}+\frac {m}{2}\right \},c^2 x^2\right )}{(2+m) (3+m)^3 (5+m) (7+m)}+\frac {96 b^2 c^2 d^3 x^{3+m} \text {HypergeometricPFQ}\left (\left \{1,\frac {3}{2}+\frac {m}{2},\frac {3}{2}+\frac {m}{2}\right \},\left \{2+\frac {m}{2},\frac {5}{2}+\frac {m}{2}\right \},c^2 x^2\right )}{(3+m)^2 (5+m) (7+m) \left (2+3 m+m^2\right )} \]

[Out]

-30*b*c*d^3*x^(2+m)*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/(7+m)^2/(m^2+8*m+15)-10*b*c*d^3*x^(2+m)*(-c^2*x^2+1)^
(3/2)*(a+b*arcsin(c*x))/(5+m)/(7+m)^2-12*b*c*d^3*x^(2+m)*(-c^2*x^2+1)^(3/2)*(a+b*arcsin(c*x))/(5+m)^2/(7+m)+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],c^2*x^2)/(3+m)^2/(7+m)^2/(m^2+7*
m+10)+48*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)^3/(7+m)/(
m^2+7*m+10)+2*b^2*c^6*d^3*x^(7+m)/(7+m)^3+30*b^2*c^2*d^3*x^(3+m)/(3+m)^2/(5+m)/(7+m)^2+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*arcsi
n(c*x))/(7+m)^2+36*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)/(m^2+
8*m+15)^2/(m^2+9*m+14)+2*b^2*c^2*d^3*x^(3+m)/(3+m)/(7+m)^2+36*b^2*c^2*d^3*x^(3+m)/(7+m)/(m^2+8*m+15)^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)-30*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)^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)-
48*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)/(3+m)^2/(7+m)/(m^2+7*m+10)-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)+
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)-48*b*c*d^3*x^(2+m)*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/(3+m)^2/(5+m)/(7+m)-36*b*c*d^3*x^(2+m)*(a+b
*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/(3+m)/(5+m)^2/(7+m)+d^3*x^(1+m)*(-c^2*x^2+1)^3*(a+b*arcsin(c*x))^2/(7+m)

________________________________________________________________________________________

Rubi [A]
time = 1.21, antiderivative size = 1312, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {4787, 4723, 4805, 4783, 30, 14, 276} \begin {gather*} \frac {d^3 \left (1-c^2 x^2\right )^3 (a+b \text {ArcSin}(c x))^2 x^{m+1}}{m+7}+\frac {6 d^3 \left (1-c^2 x^2\right )^2 (a+b \text {ArcSin}(c x))^2 x^{m+1}}{(m+5) (m+7)}+\frac {24 d^3 \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))^2 x^{m+1}}{(m+7) \left (m^2+8 m+15\right )}+\frac {48 d^3 (a+b \text {ArcSin}(c x))^2 x^{m+1}}{(m+5) (m+7) \left (m^2+4 m+3\right )}-\frac {2 b c d^3 \left (1-c^2 x^2\right )^{5/2} (a+b \text {ArcSin}(c x)) x^{m+2}}{(m+7)^2}-\frac {12 b c d^3 \left (1-c^2 x^2\right )^{3/2} (a+b \text {ArcSin}(c x)) x^{m+2}}{(m+5)^2 (m+7)}-\frac {10 b c d^3 \left (1-c^2 x^2\right )^{3/2} (a+b \text {ArcSin}(c x)) x^{m+2}}{(m+5) (m+7)^2}-\frac {48 b c d^3 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x)) x^{m+2}}{(m+3)^2 (m+5) (m+7)}-\frac {36 b c d^3 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x)) x^{m+2}}{(m+3) (m+5)^2 (m+7)}-\frac {30 b c d^3 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x)) x^{m+2}}{(m+7)^2 \left (m^2+8 m+15\right )}-\frac {48 b c d^3 (a+b \text {ArcSin}(c x)) \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};c^2 x^2\right ) x^{m+2}}{(m+2) (m+3)^2 (m+5) (m+7)}-\frac {36 b c d^3 (a+b \text {ArcSin}(c x)) \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};c^2 x^2\right ) x^{m+2}}{(m+5)^2 (m+7) \left (m^2+5 m+6\right )}-\frac {30 b c d^3 (a+b \text {ArcSin}(c x)) \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};c^2 x^2\right ) x^{m+2}}{(m+5) (m+7)^2 \left (m^2+5 m+6\right )}-\frac {96 b c d^3 (a+b \text {ArcSin}(c x)) \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};c^2 x^2\right ) x^{m+2}}{(m+5) (m+7) \left (m^3+6 m^2+11 m+6\right )}+\frac {48 b^2 c^2 d^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 ) x^{m+3}}{(m+2) (m+3)^3 (m+5) (m+7)}+\frac {36 b^2 c^2 d^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 ) x^{m+3}}{(m+2) (m+3)^2 (m+5)^2 (m+7)}+\frac {96 b^2 c^2 d^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 ) x^{m+3}}{(m+3)^2 (m+5) (m+7) \left (m^2+3 m+2\right )}+\frac {30 b^2 c^2 d^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 ) x^{m+3}}{(m+2) (m+3)^2 (m+5) (m+7)^2}+\frac {48 b^2 c^2 d^3 x^{m+3}}{(m+3)^3 (m+5) (m+7)}+\frac {12 b^2 c^2 d^3 x^{m+3}}{(m+3) (m+5)^2 (m+7)}+\frac {36 b^2 c^2 d^3 x^{m+3}}{(m+3)^2 (m+5)^2 (m+7)}+\frac {10 b^2 c^2 d^3 x^{m+3}}{(m+7)^2 \left (m^2+8 m+15\right )}+\frac {2 b^2 c^2 d^3 x^{m+3}}{(m+3) (m+7)^2}+\frac {30 b^2 c^2 d^3 x^{m+3}}{(m+3)^2 (m+5) (m+7)^2}-\frac {12 b^2 c^4 d^3 x^{m+5}}{(m+5)^3 (m+7)}-\frac {4 b^2 c^4 d^3 x^{m+5}}{(m+5) (m+7)^2}-\frac {10 b^2 c^4 d^3 x^{m+5}}{(m+5)^2 (m+7)^2}+\frac {2 b^2 c^6 d^3 x^{m+7}}{(m+7)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*b^2*c^2*d^3*x^(3 + m))/((3 + m)*(7 + m)^2) + (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)) + (10*b^2*c^2*d^3*x^(3 + m))/((7 + m)^2*(15 + 8*m +
m^2)) - (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)) + (2*b^2*c^6*d^3*x^(7 + m))/(7 + m)^3 - (36*b*c*d^3*x^(2 + m)*Sqrt
[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/((3 + m)*(5 + m)^2*(7 + m)) - (48*b*c*d^3*x^(2 + m)*Sqrt[1 - c^2*x^2]*(a +
b*ArcSin[c*x]))/((3 + m)^2*(5 + m)*(7 + m)) - (30*b*c*d^3*x^(2 + m)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/((7
+ m)^2*(15 + 8*m + m^2)) - (10*b*c*d^3*x^(2 + m)*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/((5 + m)*(7 + m)^2)
- (12*b*c*d^3*x^(2 + m)*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/((5 + m)^2*(7 + m)) - (2*b*c*d^3*x^(2 + m)*(
1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(7 + m)^2 + (48*d^3*x^(1 + m)*(a + b*ArcSin[c*x])^2)/((5 + m)*(7 + m)*
(3 + 4*m + m^2)) + (24*d^3*x^(1 + m)*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/((7 + m)*(15 + 8*m + m^2)) + (6*d^3*
x^(1 + m)*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/((5 + m)*(7 + m)) + (d^3*x^(1 + m)*(1 - c^2*x^2)^3*(a + b*Arc
Sin[c*x])^2)/(7 + m) - (48*b*c*d^3*x^(2 + m)*(a + b*ArcSin[c*x])*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2,
c^2*x^2])/((2 + m)*(3 + m)^2*(5 + m)*(7 + m)) - (30*b*c*d^3*x^(2 + m)*(a + b*ArcSin[c*x])*Hypergeometric2F1[1/
2, (2 + m)/2, (4 + m)/2, c^2*x^2])/((5 + m)*(7 + m)^2*(6 + 5*m + m^2)) - (36*b*c*d^3*x^(2 + m)*(a + b*ArcSin[c
*x])*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/((5 + m)^2*(7 + m)*(6 + 5*m + m^2)) - (96*b*c*d^3*
x^(2 + m)*(a + b*ArcSin[c*x])*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/((5 + m)*(7 + m)*(6 + 11*
m + 6*m^2 + m^3)) + (30*b^2*c^2*d^3*x^(3 + m)*HypergeometricPFQ[{1, 3/2 + m/2, 3/2 + m/2}, {2 + m/2, 5/2 + m/2
}, c^2*x^2])/((2 + m)*(3 + m)^2*(5 + m)*(7 + m)^2) + (36*b^2*c^2*d^3*x^(3 + m)*HypergeometricPFQ[{1, 3/2 + m/2
, 3/2 + m/2}, {2 + m/2, 5/2 + m/2}, c^2*x^2])/((2 + m)*(3 + m)^2*(5 + m)^2*(7 + m)) + (48*b^2*c^2*d^3*x^(3 + m
)*HypergeometricPFQ[{1, 3/2 + m/2, 3/2 + m/2}, {2 + m/2, 5/2 + m/2}, c^2*x^2])/((2 + m)*(3 + m)^3*(5 + m)*(7 +
m)) + (96*b^2*c^2*d^3*x^(3 + m)*HypergeometricPFQ[{1, 3/2 + m/2, 3/2 + m/2}, {2 + m/2, 5/2 + m/2}, c^2*x^2])/
((3 + m)^2*(5 + m)*(7 + m)*(2 + 3*m + m^2))

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 30

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

Rule 276

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 4723

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcSi
n[c*x])^n/(d*(m + 1))), x] - Dist[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 4783

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*ArcSin[c*x])^n/(f*(m + 2))), x] + (Dist[(1/(m + 2))*Simp[Sqrt[d + e*x^2]/S
qrt[1 - c^2*x^2]], Int[(f*x)^m*((a + b*ArcSin[c*x])^n/Sqrt[1 - c^2*x^2]), x], x] - Dist[b*c*(n/(f*(m + 2)))*Si
mp[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 4787

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*ArcSin[c*x])^n/(f*(m + 2*p + 1))), x] + (Dist[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] - Dist[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 4805

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)))*Simp[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]

Rubi steps

\begin {align*} \int x^m \left (d-c^2 d x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\int x^m \left (d-c^2 d x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx\\ \end {align*}

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Mathematica [F]
time = 3.12, size = 0, normalized size = 0.00 \begin {gather*} \int x^m \left (d-c^2 d x^2\right )^3 (a+b \text {ArcSin}(c x))^2 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [F]
time = 9.44, size = 0, normalized size = 0.00 \[\int x^{m} \left (-c^{2} d \,x^{2}+d \right )^{3} \left (a +b \arcsin \left (c x \right )\right )^{2}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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 + 176*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 + 105*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*x + 1)))/(m^4 + 16*m^3 - (c^2*m^4 + 16*c^2*m^3 + 86*c^2*m^2 +
176*c^2*m + 105*c^2)*x^2 + 86*m^2 + 176*m + 105), x))/(m^4 + 16*m^3 + 86*m^2 + 176*m + 105)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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)*arcsin(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)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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