\(\int x^m (d-c^2 d x^2)^2 (a+b \arcsin (c x))^2 \, dx\) [277]

Optimal result

Integrand size = 27, antiderivative size = 756 \[ \int x^m \left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2 \, dx=\frac {6 b^2 c^2 d^2 x^{3+m}}{(3+m)^2 (5+m)^2}+\frac {2 b^2 c^2 d^2 x^{3+m}}{(3+m) (5+m)^2}+\frac {8 b^2 c^2 d^2 x^{3+m}}{(3+m)^3 (5+m)}-\frac {2 b^2 c^4 d^2 x^{5+m}}{(5+m)^3}-\frac {6 b c d^2 x^{2+m} \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{(3+m) (5+m)^2}-\frac {8 b c d^2 x^{2+m} \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{(3+m)^2 (5+m)}-\frac {2 b c d^2 x^{2+m} \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{(5+m)^2}+\frac {8 d^2 x^{1+m} (a+b \arcsin (c x))^2}{(5+m) \left (3+4 m+m^2\right )}+\frac {4 d^2 x^{1+m} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{15+8 m+m^2}+\frac {d^2 x^{1+m} \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{5+m}-\frac {8 b c d^2 x^{2+m} (a+b \arcsin (c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{(2+m) (3+m)^2 (5+m)}-\frac {6 b c d^2 x^{2+m} (a+b \arcsin (c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{(5+m)^2 \left (6+5 m+m^2\right )}-\frac {16 b c d^2 x^{2+m} (a+b \arcsin (c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{(5+m) \left (6+11 m+6 m^2+m^3\right )}+\frac {6 b^2 c^2 d^2 x^{3+m} \, _3F_2\left (1,\frac {3}{2}+\frac {m}{2},\frac {3}{2}+\frac {m}{2};2+\frac {m}{2},\frac {5}{2}+\frac {m}{2};c^2 x^2\right )}{(2+m) (3+m)^2 (5+m)^2}+\frac {8 b^2 c^2 d^2 x^{3+m} \, _3F_2\left (1,\frac {3}{2}+\frac {m}{2},\frac {3}{2}+\frac {m}{2};2+\frac {m}{2},\frac {5}{2}+\frac {m}{2};c^2 x^2\right )}{(2+m) (3+m)^3 (5+m)}+\frac {16 b^2 c^2 d^2 x^{3+m} \, _3F_2\left (1,\frac {3}{2}+\frac {m}{2},\frac {3}{2}+\frac {m}{2};2+\frac {m}{2},\frac {5}{2}+\frac {m}{2};c^2 x^2\right )}{(3+m)^2 (5+m) \left (2+3 m+m^2\right )} \]

[Out]

6*b^2*c^2*d^2*x^(3+m)/(3+m)^2/(5+m)^2+2*b^2*c^2*d^2*x^(3+m)/(3+m)/(5+m)^2+8*b^2*c^2*d^2*x^(3+m)/(3+m)^3/(5+m)-
2*b^2*c^4*d^2*x^(5+m)/(5+m)^3-2*b*c*d^2*x^(2+m)*(-c^2*x^2+1)^(3/2)*(a+b*arcsin(c*x))/(5+m)^2+8*d^2*x^(1+m)*(a+
b*arcsin(c*x))^2/(5+m)/(m^2+4*m+3)+4*d^2*x^(1+m)*(-c^2*x^2+1)*(a+b*arcsin(c*x))^2/(m^2+8*m+15)+d^2*x^(1+m)*(-c
^2*x^2+1)^2*(a+b*arcsin(c*x))^2/(5+m)-6*b*c*d^2*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/(m^2+5*m+6)-8*b*c*d^2*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/(m^2+7*m+10)-16*b*c*d^2*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)/(m^3
+6*m^2+11*m+6)+8*b^2*c^2*d^2*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)/(2+m)/(
3+m)^3/(5+m)+16*b^2*c^2*d^2*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)/(m^2+3*m+2)+6*b^2*c^2*d^2*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)/(2+m
)/(m^2+8*m+15)^2-6*b*c*d^2*x^(2+m)*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/(3+m)/(5+m)^2-8*b*c*d^2*x^(2+m)*(a+b*a
rcsin(c*x))*(-c^2*x^2+1)^(1/2)/(3+m)^2/(5+m)

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 756, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {4787, 4723, 4805, 4783, 30, 14} \[ \int x^m \left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2 \, dx=\frac {16 b^2 c^2 d^2 x^{m+3} \, _3F_2\left (1,\frac {m}{2}+\frac {3}{2},\frac {m}{2}+\frac {3}{2};\frac {m}{2}+2,\frac {m}{2}+\frac {5}{2};c^2 x^2\right )}{(m+3)^2 (m+5) \left (m^2+3 m+2\right )}+\frac {8 b^2 c^2 d^2 x^{m+3} \, _3F_2\left (1,\frac {m}{2}+\frac {3}{2},\frac {m}{2}+\frac {3}{2};\frac {m}{2}+2,\frac {m}{2}+\frac {5}{2};c^2 x^2\right )}{(m+2) (m+3)^3 (m+5)}+\frac {6 b^2 c^2 d^2 x^{m+3} \, _3F_2\left (1,\frac {m}{2}+\frac {3}{2},\frac {m}{2}+\frac {3}{2};\frac {m}{2}+2,\frac {m}{2}+\frac {5}{2};c^2 x^2\right )}{(m+2) (m+3)^2 (m+5)^2}-\frac {6 b c d^2 x^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},c^2 x^2\right ) (a+b \arcsin (c x))}{(m+5)^2 \left (m^2+5 m+6\right )}-\frac {16 b c d^2 x^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},c^2 x^2\right ) (a+b \arcsin (c x))}{(m+5) \left (m^3+6 m^2+11 m+6\right )}-\frac {8 b c d^2 x^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},c^2 x^2\right ) (a+b \arcsin (c x))}{(m+2) (m+3)^2 (m+5)}+\frac {4 d^2 \left (1-c^2 x^2\right ) x^{m+1} (a+b \arcsin (c x))^2}{m^2+8 m+15}+\frac {d^2 \left (1-c^2 x^2\right )^2 x^{m+1} (a+b \arcsin (c x))^2}{m+5}-\frac {2 b c d^2 \left (1-c^2 x^2\right )^{3/2} x^{m+2} (a+b \arcsin (c x))}{(m+5)^2}-\frac {8 b c d^2 \sqrt {1-c^2 x^2} x^{m+2} (a+b \arcsin (c x))}{(m+3)^2 (m+5)}-\frac {6 b c d^2 \sqrt {1-c^2 x^2} x^{m+2} (a+b \arcsin (c x))}{(m+3) (m+5)^2}+\frac {8 d^2 x^{m+1} (a+b \arcsin (c x))^2}{(m+5) \left (m^2+4 m+3\right )}-\frac {2 b^2 c^4 d^2 x^{m+5}}{(m+5)^3}+\frac {8 b^2 c^2 d^2 x^{m+3}}{(m+3)^3 (m+5)}+\frac {2 b^2 c^2 d^2 x^{m+3}}{(m+3) (m+5)^2}+\frac {6 b^2 c^2 d^2 x^{m+3}}{(m+3)^2 (m+5)^2} \]

[In]

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

[Out]

(6*b^2*c^2*d^2*x^(3 + m))/((3 + m)^2*(5 + m)^2) + (2*b^2*c^2*d^2*x^(3 + m))/((3 + m)*(5 + m)^2) + (8*b^2*c^2*d
^2*x^(3 + m))/((3 + m)^3*(5 + m)) - (2*b^2*c^4*d^2*x^(5 + m))/(5 + m)^3 - (6*b*c*d^2*x^(2 + m)*Sqrt[1 - c^2*x^
2]*(a + b*ArcSin[c*x]))/((3 + m)*(5 + m)^2) - (8*b*c*d^2*x^(2 + m)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/((3
+ m)^2*(5 + m)) - (2*b*c*d^2*x^(2 + m)*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(5 + m)^2 + (8*d^2*x^(1 + m)*(
a + b*ArcSin[c*x])^2)/((5 + m)*(3 + 4*m + m^2)) + (4*d^2*x^(1 + m)*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(15 +
8*m + m^2) + (d^2*x^(1 + m)*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/(5 + m) - (8*b*c*d^2*x^(2 + m)*(a + b*ArcSi
n[c*x])*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/((2 + m)*(3 + m)^2*(5 + m)) - (6*b*c*d^2*x^(2 +
m)*(a + b*ArcSin[c*x])*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/((5 + m)^2*(6 + 5*m + m^2)) - (
16*b*c*d^2*x^(2 + m)*(a + b*ArcSin[c*x])*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/((5 + m)*(6 +
11*m + 6*m^2 + m^3)) + (6*b^2*c^2*d^2*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) + (8*b^2*c^2*d^2*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)) + (16*b^2*c^2*d^2*x^(3 + m)*HypergeometricP
FQ[{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)*(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 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*} \text {integral}& = \frac {d^2 x^{1+m} \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{5+m}+\frac {(4 d) \int x^m \left (d-c^2 d x^2\right ) (a+b \arcsin (c x))^2 \, dx}{5+m}-\frac {\left (2 b c d^2\right ) \int x^{1+m} \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x)) \, dx}{5+m} \\ & = -\frac {2 b c d^2 x^{2+m} \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{(5+m)^2}+\frac {4 d^2 x^{1+m} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{15+8 m+m^2}+\frac {d^2 x^{1+m} \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{5+m}-\frac {\left (6 b c d^2\right ) \int x^{1+m} \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \, dx}{(5+m)^2}+\frac {\left (2 b^2 c^2 d^2\right ) \int x^{2+m} \left (1-c^2 x^2\right ) \, dx}{(5+m)^2}+\frac {\left (8 d^2\right ) \int x^m (a+b \arcsin (c x))^2 \, dx}{15+8 m+m^2}-\frac {\left (8 b c d^2\right ) \int x^{1+m} \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \, dx}{15+8 m+m^2} \\ & = -\frac {6 b c d^2 x^{2+m} \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{(3+m) (5+m)^2}-\frac {8 b c d^2 x^{2+m} \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{(3+m)^2 (5+m)}-\frac {2 b c d^2 x^{2+m} \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{(5+m)^2}+\frac {8 d^2 x^{1+m} (a+b \arcsin (c x))^2}{15+23 m+9 m^2+m^3}+\frac {4 d^2 x^{1+m} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{15+8 m+m^2}+\frac {d^2 x^{1+m} \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{5+m}+\frac {\left (2 b^2 c^2 d^2\right ) \int \left (x^{2+m}-c^2 x^{4+m}\right ) \, dx}{(5+m)^2}-\frac {\left (6 b c d^2\right ) \int \frac {x^{1+m} (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}} \, dx}{(3+m) (5+m)^2}+\frac {\left (6 b^2 c^2 d^2\right ) \int x^{2+m} \, dx}{(3+m) (5+m)^2}-\frac {\left (8 b c d^2\right ) \int \frac {x^{1+m} (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}} \, dx}{(3+m)^2 (5+m)}+\frac {\left (8 b^2 c^2 d^2\right ) \int x^{2+m} \, dx}{(3+m)^2 (5+m)}-\frac {\left (16 b c d^2\right ) \int \frac {x^{1+m} (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}} \, dx}{15+23 m+9 m^2+m^3} \\ & = \frac {6 b^2 c^2 d^2 x^{3+m}}{(3+m)^2 (5+m)^2}+\frac {2 b^2 c^2 d^2 x^{3+m}}{(3+m) (5+m)^2}+\frac {8 b^2 c^2 d^2 x^{3+m}}{(3+m)^3 (5+m)}-\frac {2 b^2 c^4 d^2 x^{5+m}}{(5+m)^3}-\frac {6 b c d^2 x^{2+m} \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{(3+m) (5+m)^2}-\frac {8 b c d^2 x^{2+m} \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{(3+m)^2 (5+m)}-\frac {2 b c d^2 x^{2+m} \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{(5+m)^2}+\frac {8 d^2 x^{1+m} (a+b \arcsin (c x))^2}{15+23 m+9 m^2+m^3}+\frac {4 d^2 x^{1+m} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{15+8 m+m^2}+\frac {d^2 x^{1+m} \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{5+m}-\frac {6 b c d^2 x^{2+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 )}{(2+m) (3+m) (5+m)^2}-\frac {8 b c d^2 x^{2+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 )}{(2+m) (3+m)^2 (5+m)}-\frac {16 b c d^2 x^{2+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 )}{(2+m) \left (15+23 m+9 m^2+m^3\right )}+\frac {6 b^2 c^2 d^2 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 )}{(2+m) (3+m)^2 (5+m)^2}+\frac {8 b^2 c^2 d^2 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 )}{(2+m) (3+m)^3 (5+m)}+\frac {16 b^2 c^2 d^2 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 )}{(2+m) (3+m)^2 \left (5+6 m+m^2\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.54 (sec) , antiderivative size = 403, normalized size of antiderivative = 0.53 \[ \int x^m \left (d-c^2 d x^2\right )^2 (a+b \arcsin (c x))^2 \, dx=d^2 x^{1+m} \left (\frac {(a+b \arcsin (c x))^2}{1+m}-\frac {2 c^2 x^2 (a+b \arcsin (c x))^2}{3+m}+\frac {c^4 x^4 (a+b \arcsin (c x))^2}{5+m}+\frac {2 b c x \left (-\left ((3+m) (a+b \arcsin (c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )\right )+b c x \, _3F_2\left (1,\frac {3}{2}+\frac {m}{2},\frac {3}{2}+\frac {m}{2};2+\frac {m}{2},\frac {5}{2}+\frac {m}{2};c^2 x^2\right )\right )}{(1+m) (2+m) (3+m)}-\frac {4 b c^3 x^3 \left (-\left ((5+m) (a+b \arcsin (c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {4+m}{2},\frac {6+m}{2},c^2 x^2\right )\right )+b c x \, _3F_2\left (1,\frac {5}{2}+\frac {m}{2},\frac {5}{2}+\frac {m}{2};3+\frac {m}{2},\frac {7}{2}+\frac {m}{2};c^2 x^2\right )\right )}{(3+m) (4+m) (5+m)}+\frac {2 b c^5 x^5 \left (-\left ((7+m) (a+b \arcsin (c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {6+m}{2},\frac {8+m}{2},c^2 x^2\right )\right )+b c x \, _3F_2\left (1,\frac {7}{2}+\frac {m}{2},\frac {7}{2}+\frac {m}{2};4+\frac {m}{2},\frac {9}{2}+\frac {m}{2};c^2 x^2\right )\right )}{(5+m) (6+m) (7+m)}\right ) \]

[In]

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

[Out]

d^2*x^(1 + m)*((a + b*ArcSin[c*x])^2/(1 + m) - (2*c^2*x^2*(a + b*ArcSin[c*x])^2)/(3 + m) + (c^4*x^4*(a + b*Arc
Sin[c*x])^2)/(5 + m) + (2*b*c*x*(-((3 + m)*(a + b*ArcSin[c*x])*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^
2*x^2]) + b*c*x*HypergeometricPFQ[{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)) - (4*b*c^3*x^3*(-((5 + m)*(a + b*ArcSin[c*x])*Hypergeometric2F1[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
)) + (2*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)))

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral((a^2*c^4*d^2*x^4 - 2*a^2*c^2*d^2*x^2 + a^2*d^2 + (b^2*c^4*d^2*x^4 - 2*b^2*c^2*d^2*x^2 + b^2*d^2)*arcs
in(c*x)^2 + 2*(a*b*c^4*d^2*x^4 - 2*a*b*c^2*d^2*x^2 + a*b*d^2)*arcsin(c*x))*x^m, x)

Sympy [F]

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

[In]

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

[Out]

d**2*(Integral(a**2*x**m, x) + Integral(b**2*x**m*asin(c*x)**2, x) + Integral(2*a*b*x**m*asin(c*x), x) + Integ
ral(-2*a**2*c**2*x**2*x**m, x) + Integral(a**2*c**4*x**4*x**m, x) + Integral(-2*b**2*c**2*x**2*x**m*asin(c*x)*
*2, x) + Integral(b**2*c**4*x**4*x**m*asin(c*x)**2, x) + Integral(-4*a*b*c**2*x**2*x**m*asin(c*x), x) + Integr
al(2*a*b*c**4*x**4*x**m*asin(c*x), x))

Maxima [F]

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

[In]

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

[Out]

a^2*c^4*d^2*x^(m + 5)/(m + 5) - 2*a^2*c^2*d^2*x^(m + 3)/(m + 3) + a^2*d^2*x^(m + 1)/(m + 1) + (((b^2*c^4*d^2*m
^2 + 4*b^2*c^4*d^2*m + 3*b^2*c^4*d^2)*x^5 - 2*(b^2*c^2*d^2*m^2 + 6*b^2*c^2*d^2*m + 5*b^2*c^2*d^2)*x^3 + (b^2*d
^2*m^2 + 8*b^2*d^2*m + 15*b^2*d^2)*x)*x^m*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + (m^3 + 9*m^2 + 23*m +
15)*integrate(-2*(((b^2*c^5*d^2*m^2 + 4*b^2*c^5*d^2*m + 3*b^2*c^5*d^2)*x^5 - 2*(b^2*c^3*d^2*m^2 + 6*b^2*c^3*d
^2*m + 5*b^2*c^3*d^2)*x^3 + (b^2*c*d^2*m^2 + 8*b^2*c*d^2*m + 15*b^2*c*d^2)*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^2*m^3 - (a*b*c^6*d^2*m^3 + 9*a*b*c^6*d^2*m^2 + 23*a*b*c^6
*d^2*m + 15*a*b*c^6*d^2)*x^6 + 9*a*b*d^2*m^2 + 23*a*b*d^2*m + 3*(a*b*c^4*d^2*m^3 + 9*a*b*c^4*d^2*m^2 + 23*a*b*
c^4*d^2*m + 15*a*b*c^4*d^2)*x^4 + 15*a*b*d^2 - 3*(a*b*c^2*d^2*m^3 + 9*a*b*c^2*d^2*m^2 + 23*a*b*c^2*d^2*m + 15*
a*b*c^2*d^2)*x^2)*x^m*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))/(m^3 - (c^2*m^3 + 9*c^2*m^2 + 23*c^2*m + 15*
c^2)*x^2 + 9*m^2 + 23*m + 15), x))/(m^3 + 9*m^2 + 23*m + 15)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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