3.63 \(\int (f+g x)^2 (d-c^2 d x^2)^{3/2} (a+b \sin ^{-1}(c x))^2 \, dx\)

Optimal. Leaf size=1108 \[ \frac {b c^3 d g^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^6}{18 \sqrt {1-c^2 x^2}}+\frac {4 b c^3 d f g \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^5}{25 \sqrt {1-c^2 x^2}}+\frac {1}{108} b^2 c^2 d g^2 \sqrt {d-c^2 d x^2} x^5-\frac {7 b c d g^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^4}{48 \sqrt {1-c^2 x^2}}+\frac {1}{8} d g^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 x^3+\frac {1}{6} d g^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 x^3-\frac {8 b c d f g \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^3}{15 \sqrt {1-c^2 x^2}}-\frac {43 b^2 d g^2 \sqrt {d-c^2 d x^2} x^3}{1728}-\frac {3 b c d f^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^2}{8 \sqrt {1-c^2 x^2}}+\frac {b d g^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^2}{16 c \sqrt {1-c^2 x^2}}+\frac {3}{8} d f^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 x-\frac {d g^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 x}{16 c^2}+\frac {1}{4} d f^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 x+\frac {4 b d f g \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x}{5 c \sqrt {1-c^2 x^2}}-\frac {15}{64} b^2 d f^2 \sqrt {d-c^2 d x^2} x-\frac {7 b^2 d g^2 \sqrt {d-c^2 d x^2} x}{1152 c^2}-\frac {1}{32} b^2 d f^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} x+\frac {d f^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c \sqrt {1-c^2 x^2}}+\frac {d g^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{48 b c^3 \sqrt {1-c^2 x^2}}-\frac {2 d f g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2}+\frac {9 b^2 d f^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{64 c \sqrt {1-c^2 x^2}}+\frac {7 b^2 d g^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{1152 c^3 \sqrt {1-c^2 x^2}}+\frac {b d f^2 \left (1-c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac {4 b^2 d f g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{125 c^2}+\frac {32 b^2 d f g \sqrt {d-c^2 d x^2}}{75 c^2}+\frac {16 b^2 d f g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{225 c^2} \]

[Out]

32/75*b^2*d*f*g*(-c^2*d*x^2+d)^(1/2)/c^2-7/1152*b^2*d*g^2*x*(-c^2*d*x^2+d)^(1/2)/c^2+1/108*b^2*c^2*d*g^2*x^5*(
-c^2*d*x^2+d)^(1/2)-1/32*b^2*d*f^2*x*(-c^2*x^2+1)*(-c^2*d*x^2+d)^(1/2)-1/16*d*g^2*x*(a+b*arcsin(c*x))^2*(-c^2*
d*x^2+d)^(1/2)/c^2+1/4*d*f^2*x*(-c^2*x^2+1)*(a+b*arcsin(c*x))^2*(-c^2*d*x^2+d)^(1/2)+1/6*d*g^2*x^3*(-c^2*x^2+1
)*(a+b*arcsin(c*x))^2*(-c^2*d*x^2+d)^(1/2)-15/64*b^2*d*f^2*x*(-c^2*d*x^2+d)^(1/2)-43/1728*b^2*d*g^2*x^3*(-c^2*
d*x^2+d)^(1/2)+3/8*d*f^2*x*(a+b*arcsin(c*x))^2*(-c^2*d*x^2+d)^(1/2)+1/8*d*g^2*x^3*(a+b*arcsin(c*x))^2*(-c^2*d*
x^2+d)^(1/2)+4/5*b*d*f*g*x*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/c/(-c^2*x^2+1)^(1/2)-8/15*b*c*d*f*g*x^3*(a+b
*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+4/25*b*c^3*d*f*g*x^5*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1
/2)/(-c^2*x^2+1)^(1/2)-3/8*b*c*d*f^2*x^2*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+1/16*b*d*g^
2*x^2*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/c/(-c^2*x^2+1)^(1/2)-7/48*b*c*d*g^2*x^4*(a+b*arcsin(c*x))*(-c^2*d
*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+1/18*b*c^3*d*g^2*x^6*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2
)+16/225*b^2*d*f*g*(-c^2*x^2+1)*(-c^2*d*x^2+d)^(1/2)/c^2+4/125*b^2*d*f*g*(-c^2*x^2+1)^2*(-c^2*d*x^2+d)^(1/2)/c
^2+1/8*b*d*f^2*(-c^2*x^2+1)^(3/2)*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/c-2/5*d*f*g*(-c^2*x^2+1)^2*(a+b*arcsi
n(c*x))^2*(-c^2*d*x^2+d)^(1/2)/c^2+9/64*b^2*d*f^2*arcsin(c*x)*(-c^2*d*x^2+d)^(1/2)/c/(-c^2*x^2+1)^(1/2)+7/1152
*b^2*d*g^2*arcsin(c*x)*(-c^2*d*x^2+d)^(1/2)/c^3/(-c^2*x^2+1)^(1/2)+1/8*d*f^2*(a+b*arcsin(c*x))^3*(-c^2*d*x^2+d
)^(1/2)/b/c/(-c^2*x^2+1)^(1/2)+1/48*d*g^2*(a+b*arcsin(c*x))^3*(-c^2*d*x^2+d)^(1/2)/b/c^3/(-c^2*x^2+1)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 1.52, antiderivative size = 1108, normalized size of antiderivative = 1.00, number of steps used = 36, number of rules used = 21, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {4777, 4763, 4649, 4647, 4641, 4627, 321, 216, 4677, 195, 194, 4645, 12, 1247, 698, 4699, 4697, 4707, 14, 4687, 459} \[ \frac {b c^3 d g^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^6}{18 \sqrt {1-c^2 x^2}}+\frac {4 b c^3 d f g \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^5}{25 \sqrt {1-c^2 x^2}}+\frac {1}{108} b^2 c^2 d g^2 \sqrt {d-c^2 d x^2} x^5-\frac {7 b c d g^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^4}{48 \sqrt {1-c^2 x^2}}+\frac {1}{8} d g^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 x^3+\frac {1}{6} d g^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 x^3-\frac {8 b c d f g \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^3}{15 \sqrt {1-c^2 x^2}}-\frac {43 b^2 d g^2 \sqrt {d-c^2 d x^2} x^3}{1728}-\frac {3 b c d f^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^2}{8 \sqrt {1-c^2 x^2}}+\frac {b d g^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x^2}{16 c \sqrt {1-c^2 x^2}}+\frac {3}{8} d f^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 x-\frac {d g^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 x}{16 c^2}+\frac {1}{4} d f^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 x+\frac {4 b d f g \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) x}{5 c \sqrt {1-c^2 x^2}}-\frac {15}{64} b^2 d f^2 \sqrt {d-c^2 d x^2} x-\frac {7 b^2 d g^2 \sqrt {d-c^2 d x^2} x}{1152 c^2}-\frac {1}{32} b^2 d f^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} x+\frac {d f^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c \sqrt {1-c^2 x^2}}+\frac {d g^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{48 b c^3 \sqrt {1-c^2 x^2}}-\frac {2 d f g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2}+\frac {9 b^2 d f^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{64 c \sqrt {1-c^2 x^2}}+\frac {7 b^2 d g^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{1152 c^3 \sqrt {1-c^2 x^2}}+\frac {b d f^2 \left (1-c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac {4 b^2 d f g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{125 c^2}+\frac {32 b^2 d f g \sqrt {d-c^2 d x^2}}{75 c^2}+\frac {16 b^2 d f g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{225 c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(32*b^2*d*f*g*Sqrt[d - c^2*d*x^2])/(75*c^2) - (15*b^2*d*f^2*x*Sqrt[d - c^2*d*x^2])/64 - (7*b^2*d*g^2*x*Sqrt[d
- c^2*d*x^2])/(1152*c^2) - (43*b^2*d*g^2*x^3*Sqrt[d - c^2*d*x^2])/1728 + (b^2*c^2*d*g^2*x^5*Sqrt[d - c^2*d*x^2
])/108 + (16*b^2*d*f*g*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(225*c^2) - (b^2*d*f^2*x*(1 - c^2*x^2)*Sqrt[d - c^2*
d*x^2])/32 + (4*b^2*d*f*g*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2])/(125*c^2) + (9*b^2*d*f^2*Sqrt[d - c^2*d*x^2]*Ar
cSin[c*x])/(64*c*Sqrt[1 - c^2*x^2]) + (7*b^2*d*g^2*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(1152*c^3*Sqrt[1 - c^2*x^2
]) + (4*b*d*f*g*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(5*c*Sqrt[1 - c^2*x^2]) - (3*b*c*d*f^2*x^2*Sqrt[d -
c^2*d*x^2]*(a + b*ArcSin[c*x]))/(8*Sqrt[1 - c^2*x^2]) + (b*d*g^2*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))
/(16*c*Sqrt[1 - c^2*x^2]) - (8*b*c*d*f*g*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(15*Sqrt[1 - c^2*x^2]) -
(7*b*c*d*g^2*x^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(48*Sqrt[1 - c^2*x^2]) + (4*b*c^3*d*f*g*x^5*Sqrt[d
- c^2*d*x^2]*(a + b*ArcSin[c*x]))/(25*Sqrt[1 - c^2*x^2]) + (b*c^3*d*g^2*x^6*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[
c*x]))/(18*Sqrt[1 - c^2*x^2]) + (b*d*f^2*(1 - c^2*x^2)^(3/2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(8*c) +
(3*d*f^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/8 - (d*g^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/
(16*c^2) + (d*g^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/8 + (d*f^2*x*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2
]*(a + b*ArcSin[c*x])^2)/4 + (d*g^2*x^3*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/6 - (2*d*f*g*
(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(5*c^2) + (d*f^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[
c*x])^3)/(8*b*c*Sqrt[1 - c^2*x^2]) + (d*g^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/(48*b*c^3*Sqrt[1 - c^2*
x^2])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 194

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

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 459

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(d*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(b*e*(m + n*(p + 1) + 1)), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m +
n*(p + 1) + 1)), Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0]
&& NeQ[m + n*(p + 1) + 1, 0]

Rule 698

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rule 1247

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[
Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]

Rule 4627

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 4641

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(a + b*ArcSin[c*x])^
(n + 1)/(b*c*Sqrt[d]*(n + 1)), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0] && NeQ[n,
-1]

Rule 4645

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(d + e*x^2)
^p, x]}, Dist[a + b*ArcSin[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]] /; F
reeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]

Rule 4647

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(x*Sqrt[d + e*x^2]*(
a + b*ArcSin[c*x])^n)/2, x] + (Dist[Sqrt[d + e*x^2]/(2*Sqrt[1 - c^2*x^2]), Int[(a + b*ArcSin[c*x])^n/Sqrt[1 -
c^2*x^2], x], x] - Dist[(b*c*n*Sqrt[d + e*x^2])/(2*Sqrt[1 - c^2*x^2]), Int[x*(a + b*ArcSin[c*x])^(n - 1), x],
x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0]

Rule 4649

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(x*(d + e*x^2)^p*(
a + b*ArcSin[c*x])^n)/(2*p + 1), x] + (Dist[(2*d*p)/(2*p + 1), Int[(d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x])^n,
x], x] - Dist[(b*c*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/((2*p + 1)*(1 - c^2*x^2)^FracPart[p]), Int[x*(1 - c
^2*x^2)^(p - 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && Gt
Q[n, 0] && GtQ[p, 0]

Rule 4677

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x^2)^
(p + 1)*(a + b*ArcSin[c*x])^n)/(2*e*(p + 1)), x] + Dist[(b*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*c*(p + 1
)*(1 - c^2*x^2)^FracPart[p]), Int[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x] /; FreeQ[{a, b,
c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]

Rule 4687

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u = I
ntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcSin[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 -
c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]

Rule 4697

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[Sqrt[d + e*x^2]/((m + 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] - Dist[(b*c*n*Sqrt[d + e*x^2])/(f*(m
+ 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] &&  !LtQ[m, -1] && (RationalQ[m] || EqQ[n, 1])

Rule 4699

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*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(
f*(m + 2*p + 1)*(1 - c^2*x^2)^FracPart[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]
&& (RationalQ[m] || EqQ[n, 1])

Rule 4707

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[
(f*(f*x)^(m - 1)*Sqrt[d + e*x^2]*(a + b*ArcSin[c*x])^n)/(e*m), x] + (Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)^(m
- 2)*(a + b*ArcSin[c*x])^n)/Sqrt[d + e*x^2], x], x] + Dist[(b*f*n*Sqrt[1 - c^2*x^2])/(c*m*Sqrt[d + e*x^2]), I
nt[(f*x)^(m - 1)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] &&
GtQ[n, 0] && GtQ[m, 1] && IntegerQ[m]

Rule 4763

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, (f + g*x)^m, x], x] /; FreeQ[{a, b, c, d, e, f, g},
x] && EqQ[c^2*d + e, 0] && IGtQ[m, 0] && IntegerQ[p + 1/2] && GtQ[d, 0] && IGtQ[n, 0] && (m == 1 || p > 0 ||
(n == 1 && p > -1) || (m == 2 && p < -2))

Rule 4777

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :
> Dist[(d^IntPart[p]*(d + e*x^2)^FracPart[p])/(1 - c^2*x^2)^FracPart[p], Int[(f + g*x)^m*(1 - c^2*x^2)^p*(a +
b*ArcSin[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && IntegerQ
[p - 1/2] &&  !GtQ[d, 0]

Rubi steps

\begin {align*} \int (f+g x)^2 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \int (f+g x)^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{\sqrt {1-c^2 x^2}}\\ &=\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \int \left (f^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+2 f g x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+g^2 x^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2\right ) \, dx}{\sqrt {1-c^2 x^2}}\\ &=\frac {\left (d f^2 \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{\sqrt {1-c^2 x^2}}+\frac {\left (2 d f g \sqrt {d-c^2 d x^2}\right ) \int x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{\sqrt {1-c^2 x^2}}+\frac {\left (d g^2 \sqrt {d-c^2 d x^2}\right ) \int x^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{\sqrt {1-c^2 x^2}}\\ &=\frac {1}{4} d f^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{6} d g^2 x^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {2 d f g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2}+\frac {\left (3 d f^2 \sqrt {d-c^2 d x^2}\right ) \int \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{4 \sqrt {1-c^2 x^2}}-\frac {\left (b c d f^2 \sqrt {d-c^2 d x^2}\right ) \int x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right ) \, dx}{2 \sqrt {1-c^2 x^2}}+\frac {\left (4 b d f g \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{5 c \sqrt {1-c^2 x^2}}+\frac {\left (d g^2 \sqrt {d-c^2 d x^2}\right ) \int x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{2 \sqrt {1-c^2 x^2}}-\frac {\left (b c d g^2 \sqrt {d-c^2 d x^2}\right ) \int x^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right ) \, dx}{3 \sqrt {1-c^2 x^2}}\\ &=\frac {4 b d f g x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 c \sqrt {1-c^2 x^2}}-\frac {8 b c d f g x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt {1-c^2 x^2}}-\frac {b c d g^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{12 \sqrt {1-c^2 x^2}}+\frac {4 b c^3 d f g x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt {1-c^2 x^2}}+\frac {b c^3 d g^2 x^6 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 \sqrt {1-c^2 x^2}}+\frac {b d f^2 \left (1-c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac {3}{8} d f^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{8} d g^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} d f^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{6} d g^2 x^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {2 d f g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2}+\frac {\left (3 d f^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{8 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 d f^2 \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^{3/2} \, dx}{8 \sqrt {1-c^2 x^2}}-\frac {\left (3 b c d f^2 \sqrt {d-c^2 d x^2}\right ) \int x \left (a+b \sin ^{-1}(c x)\right ) \, dx}{4 \sqrt {1-c^2 x^2}}-\frac {\left (4 b^2 d f g \sqrt {d-c^2 d x^2}\right ) \int \frac {x \left (15-10 c^2 x^2+3 c^4 x^4\right )}{15 \sqrt {1-c^2 x^2}} \, dx}{5 \sqrt {1-c^2 x^2}}+\frac {\left (d g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{8 \sqrt {1-c^2 x^2}}-\frac {\left (b c d g^2 \sqrt {d-c^2 d x^2}\right ) \int x^3 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{4 \sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^2 d g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4 \left (3-2 c^2 x^2\right )}{12 \sqrt {1-c^2 x^2}} \, dx}{3 \sqrt {1-c^2 x^2}}\\ &=-\frac {1}{32} b^2 d f^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}+\frac {4 b d f g x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 c \sqrt {1-c^2 x^2}}-\frac {3 b c d f^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt {1-c^2 x^2}}-\frac {8 b c d f g x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt {1-c^2 x^2}}-\frac {7 b c d g^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 \sqrt {1-c^2 x^2}}+\frac {4 b c^3 d f g x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt {1-c^2 x^2}}+\frac {b c^3 d g^2 x^6 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 \sqrt {1-c^2 x^2}}+\frac {b d f^2 \left (1-c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac {3}{8} d f^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d g^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 c^2}+\frac {1}{8} d g^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} d f^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{6} d g^2 x^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {2 d f g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2}+\frac {d f^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c \sqrt {1-c^2 x^2}}-\frac {\left (3 b^2 d f^2 \sqrt {d-c^2 d x^2}\right ) \int \sqrt {1-c^2 x^2} \, dx}{32 \sqrt {1-c^2 x^2}}+\frac {\left (3 b^2 c^2 d f^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{8 \sqrt {1-c^2 x^2}}-\frac {\left (4 b^2 d f g \sqrt {d-c^2 d x^2}\right ) \int \frac {x \left (15-10 c^2 x^2+3 c^4 x^4\right )}{\sqrt {1-c^2 x^2}} \, dx}{75 \sqrt {1-c^2 x^2}}+\frac {\left (d g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{16 c^2 \sqrt {1-c^2 x^2}}+\frac {\left (b d g^2 \sqrt {d-c^2 d x^2}\right ) \int x \left (a+b \sin ^{-1}(c x)\right ) \, dx}{8 c \sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^2 d g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4 \left (3-2 c^2 x^2\right )}{\sqrt {1-c^2 x^2}} \, dx}{36 \sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^2 d g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4}{\sqrt {1-c^2 x^2}} \, dx}{16 \sqrt {1-c^2 x^2}}\\ &=-\frac {15}{64} b^2 d f^2 x \sqrt {d-c^2 d x^2}-\frac {1}{64} b^2 d g^2 x^3 \sqrt {d-c^2 d x^2}+\frac {1}{108} b^2 c^2 d g^2 x^5 \sqrt {d-c^2 d x^2}-\frac {1}{32} b^2 d f^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}+\frac {4 b d f g x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 c \sqrt {1-c^2 x^2}}-\frac {3 b c d f^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt {1-c^2 x^2}}+\frac {b d g^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 c \sqrt {1-c^2 x^2}}-\frac {8 b c d f g x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt {1-c^2 x^2}}-\frac {7 b c d g^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 \sqrt {1-c^2 x^2}}+\frac {4 b c^3 d f g x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt {1-c^2 x^2}}+\frac {b c^3 d g^2 x^6 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 \sqrt {1-c^2 x^2}}+\frac {b d f^2 \left (1-c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac {3}{8} d f^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d g^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 c^2}+\frac {1}{8} d g^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} d f^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{6} d g^2 x^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {2 d f g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2}+\frac {d f^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c \sqrt {1-c^2 x^2}}+\frac {d g^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{48 b c^3 \sqrt {1-c^2 x^2}}-\frac {\left (3 b^2 d f^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{64 \sqrt {1-c^2 x^2}}+\frac {\left (3 b^2 d f^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{16 \sqrt {1-c^2 x^2}}-\frac {\left (2 b^2 d f g \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {15-10 c^2 x+3 c^4 x^2}{\sqrt {1-c^2 x}} \, dx,x,x^2\right )}{75 \sqrt {1-c^2 x^2}}+\frac {\left (3 b^2 d g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{64 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 d g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{16 \sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^2 d g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4}{\sqrt {1-c^2 x^2}} \, dx}{27 \sqrt {1-c^2 x^2}}\\ &=-\frac {15}{64} b^2 d f^2 x \sqrt {d-c^2 d x^2}+\frac {b^2 d g^2 x \sqrt {d-c^2 d x^2}}{128 c^2}-\frac {43 b^2 d g^2 x^3 \sqrt {d-c^2 d x^2}}{1728}+\frac {1}{108} b^2 c^2 d g^2 x^5 \sqrt {d-c^2 d x^2}-\frac {1}{32} b^2 d f^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}+\frac {9 b^2 d f^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{64 c \sqrt {1-c^2 x^2}}+\frac {4 b d f g x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 c \sqrt {1-c^2 x^2}}-\frac {3 b c d f^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt {1-c^2 x^2}}+\frac {b d g^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 c \sqrt {1-c^2 x^2}}-\frac {8 b c d f g x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt {1-c^2 x^2}}-\frac {7 b c d g^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 \sqrt {1-c^2 x^2}}+\frac {4 b c^3 d f g x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt {1-c^2 x^2}}+\frac {b c^3 d g^2 x^6 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 \sqrt {1-c^2 x^2}}+\frac {b d f^2 \left (1-c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac {3}{8} d f^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d g^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 c^2}+\frac {1}{8} d g^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} d f^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{6} d g^2 x^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {2 d f g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2}+\frac {d f^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c \sqrt {1-c^2 x^2}}+\frac {d g^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{48 b c^3 \sqrt {1-c^2 x^2}}-\frac {\left (2 b^2 d f g \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {8}{\sqrt {1-c^2 x}}+4 \sqrt {1-c^2 x}+3 \left (1-c^2 x\right )^{3/2}\right ) \, dx,x,x^2\right )}{75 \sqrt {1-c^2 x^2}}+\frac {\left (b^2 d g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{36 \sqrt {1-c^2 x^2}}+\frac {\left (3 b^2 d g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{128 c^2 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 d g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{32 c^2 \sqrt {1-c^2 x^2}}\\ &=\frac {32 b^2 d f g \sqrt {d-c^2 d x^2}}{75 c^2}-\frac {15}{64} b^2 d f^2 x \sqrt {d-c^2 d x^2}-\frac {7 b^2 d g^2 x \sqrt {d-c^2 d x^2}}{1152 c^2}-\frac {43 b^2 d g^2 x^3 \sqrt {d-c^2 d x^2}}{1728}+\frac {1}{108} b^2 c^2 d g^2 x^5 \sqrt {d-c^2 d x^2}+\frac {16 b^2 d f g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{225 c^2}-\frac {1}{32} b^2 d f^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}+\frac {4 b^2 d f g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{125 c^2}+\frac {9 b^2 d f^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{64 c \sqrt {1-c^2 x^2}}-\frac {b^2 d g^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{128 c^3 \sqrt {1-c^2 x^2}}+\frac {4 b d f g x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 c \sqrt {1-c^2 x^2}}-\frac {3 b c d f^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt {1-c^2 x^2}}+\frac {b d g^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 c \sqrt {1-c^2 x^2}}-\frac {8 b c d f g x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt {1-c^2 x^2}}-\frac {7 b c d g^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 \sqrt {1-c^2 x^2}}+\frac {4 b c^3 d f g x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt {1-c^2 x^2}}+\frac {b c^3 d g^2 x^6 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 \sqrt {1-c^2 x^2}}+\frac {b d f^2 \left (1-c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac {3}{8} d f^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d g^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 c^2}+\frac {1}{8} d g^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} d f^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{6} d g^2 x^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {2 d f g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2}+\frac {d f^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c \sqrt {1-c^2 x^2}}+\frac {d g^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{48 b c^3 \sqrt {1-c^2 x^2}}+\frac {\left (b^2 d g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{72 c^2 \sqrt {1-c^2 x^2}}\\ &=\frac {32 b^2 d f g \sqrt {d-c^2 d x^2}}{75 c^2}-\frac {15}{64} b^2 d f^2 x \sqrt {d-c^2 d x^2}-\frac {7 b^2 d g^2 x \sqrt {d-c^2 d x^2}}{1152 c^2}-\frac {43 b^2 d g^2 x^3 \sqrt {d-c^2 d x^2}}{1728}+\frac {1}{108} b^2 c^2 d g^2 x^5 \sqrt {d-c^2 d x^2}+\frac {16 b^2 d f g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{225 c^2}-\frac {1}{32} b^2 d f^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}+\frac {4 b^2 d f g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{125 c^2}+\frac {9 b^2 d f^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{64 c \sqrt {1-c^2 x^2}}+\frac {7 b^2 d g^2 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{1152 c^3 \sqrt {1-c^2 x^2}}+\frac {4 b d f g x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{5 c \sqrt {1-c^2 x^2}}-\frac {3 b c d f^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt {1-c^2 x^2}}+\frac {b d g^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 c \sqrt {1-c^2 x^2}}-\frac {8 b c d f g x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{15 \sqrt {1-c^2 x^2}}-\frac {7 b c d g^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 \sqrt {1-c^2 x^2}}+\frac {4 b c^3 d f g x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 \sqrt {1-c^2 x^2}}+\frac {b c^3 d g^2 x^6 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 \sqrt {1-c^2 x^2}}+\frac {b d f^2 \left (1-c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac {3}{8} d f^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d g^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 c^2}+\frac {1}{8} d g^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} d f^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{6} d g^2 x^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {2 d f g \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 c^2}+\frac {d f^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c \sqrt {1-c^2 x^2}}+\frac {d g^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{48 b c^3 \sqrt {1-c^2 x^2}}\\ \end {align*}

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Mathematica [A]  time = 1.11, size = 616, normalized size = 0.56 \[ \frac {d \sqrt {d-c^2 d x^2} \left (9000 a^3 \left (6 c^2 f^2+g^2\right )+15 b \sin ^{-1}(c x) \left (1800 a^2 \left (6 c^2 f^2+g^2\right )-240 a b c \sqrt {1-c^2 x^2} \left (30 c^2 f^2 x \left (2 c^2 x^2-5\right )+96 f g \left (c^2 x^2-1\right )^2+5 g^2 x \left (8 c^4 x^4-14 c^2 x^2+3\right )\right )+b^2 \left (16 c^6 x^4 \left (225 f^2+288 f g x+100 g^2 x^2\right )-120 c^4 x^2 \left (150 f^2+128 f g x+35 g^2 x^2\right )+90 c^2 \left (85 f^2+256 f g x+20 g^2 x^2\right )+175 g^2\right )\right )-1800 a^2 b c \sqrt {1-c^2 x^2} \left (30 c^2 f^2 x \left (2 c^2 x^2-5\right )+96 f g \left (c^2 x^2-1\right )^2+5 g^2 x \left (8 c^4 x^4-14 c^2 x^2+3\right )\right )+120 a b^2 c^2 x \left (450 c^2 f^2 x \left (c^2 x^2-5\right )+192 f g \left (3 c^4 x^4-10 c^2 x^2+15\right )+25 g^2 x \left (8 c^4 x^4-21 c^2 x^2+9\right )\right )+1800 b^2 \sin ^{-1}(c x)^2 \left (15 a \left (6 c^2 f^2+g^2\right )-b c \sqrt {1-c^2 x^2} \left (30 c^2 f^2 x \left (2 c^2 x^2-5\right )+96 f g \left (c^2 x^2-1\right )^2+5 g^2 x \left (8 c^4 x^4-14 c^2 x^2+3\right )\right )\right )+9000 b^3 \left (6 c^2 f^2+g^2\right ) \sin ^{-1}(c x)^3+b^3 c \sqrt {1-c^2 x^2} \left (6750 c^2 f^2 x \left (2 c^2 x^2-17\right )+1536 f g \left (9 c^4 x^4-38 c^2 x^2+149\right )+125 g^2 x \left (32 c^4 x^4-86 c^2 x^2-21\right )\right )\right )}{432000 b c^3 \sqrt {1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d*Sqrt[d - c^2*d*x^2]*(9000*a^3*(6*c^2*f^2 + g^2) + 120*a*b^2*c^2*x*(450*c^2*f^2*x*(-5 + c^2*x^2) + 192*f*g*(
15 - 10*c^2*x^2 + 3*c^4*x^4) + 25*g^2*x*(9 - 21*c^2*x^2 + 8*c^4*x^4)) - 1800*a^2*b*c*Sqrt[1 - c^2*x^2]*(96*f*g
*(-1 + c^2*x^2)^2 + 30*c^2*f^2*x*(-5 + 2*c^2*x^2) + 5*g^2*x*(3 - 14*c^2*x^2 + 8*c^4*x^4)) + b^3*c*Sqrt[1 - c^2
*x^2]*(6750*c^2*f^2*x*(-17 + 2*c^2*x^2) + 1536*f*g*(149 - 38*c^2*x^2 + 9*c^4*x^4) + 125*g^2*x*(-21 - 86*c^2*x^
2 + 32*c^4*x^4)) + 15*b*(1800*a^2*(6*c^2*f^2 + g^2) + b^2*(175*g^2 + 90*c^2*(85*f^2 + 256*f*g*x + 20*g^2*x^2)
- 120*c^4*x^2*(150*f^2 + 128*f*g*x + 35*g^2*x^2) + 16*c^6*x^4*(225*f^2 + 288*f*g*x + 100*g^2*x^2)) - 240*a*b*c
*Sqrt[1 - c^2*x^2]*(96*f*g*(-1 + c^2*x^2)^2 + 30*c^2*f^2*x*(-5 + 2*c^2*x^2) + 5*g^2*x*(3 - 14*c^2*x^2 + 8*c^4*
x^4)))*ArcSin[c*x] + 1800*b^2*(15*a*(6*c^2*f^2 + g^2) - b*c*Sqrt[1 - c^2*x^2]*(96*f*g*(-1 + c^2*x^2)^2 + 30*c^
2*f^2*x*(-5 + 2*c^2*x^2) + 5*g^2*x*(3 - 14*c^2*x^2 + 8*c^4*x^4)))*ArcSin[c*x]^2 + 9000*b^3*(6*c^2*f^2 + g^2)*A
rcSin[c*x]^3))/(432000*b*c^3*Sqrt[1 - c^2*x^2])

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fricas [F]  time = 0.75, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (a^{2} c^{2} d g^{2} x^{4} + 2 \, a^{2} c^{2} d f g x^{3} - 2 \, a^{2} d f g x - a^{2} d f^{2} + {\left (a^{2} c^{2} d f^{2} - a^{2} d g^{2}\right )} x^{2} + {\left (b^{2} c^{2} d g^{2} x^{4} + 2 \, b^{2} c^{2} d f g x^{3} - 2 \, b^{2} d f g x - b^{2} d f^{2} + {\left (b^{2} c^{2} d f^{2} - b^{2} d g^{2}\right )} x^{2}\right )} \arcsin \left (c x\right )^{2} + 2 \, {\left (a b c^{2} d g^{2} x^{4} + 2 \, a b c^{2} d f g x^{3} - 2 \, a b d f g x - a b d f^{2} + {\left (a b c^{2} d f^{2} - a b d g^{2}\right )} x^{2}\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(a^2*c^2*d*g^2*x^4 + 2*a^2*c^2*d*f*g*x^3 - 2*a^2*d*f*g*x - a^2*d*f^2 + (a^2*c^2*d*f^2 - a^2*d*g^2)*x
^2 + (b^2*c^2*d*g^2*x^4 + 2*b^2*c^2*d*f*g*x^3 - 2*b^2*d*f*g*x - b^2*d*f^2 + (b^2*c^2*d*f^2 - b^2*d*g^2)*x^2)*a
rcsin(c*x)^2 + 2*(a*b*c^2*d*g^2*x^4 + 2*a*b*c^2*d*f*g*x^3 - 2*a*b*d*f*g*x - a*b*d*f^2 + (a*b*c^2*d*f^2 - a*b*d
*g^2)*x^2)*arcsin(c*x))*sqrt(-c^2*d*x^2 + d), x)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> An error occurred running a Giac command:INPUT:sage2OUTPUT:sym2poly/r2sym(co
nst gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [C]  time = 1.21, size = 9278, normalized size = 8.37 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{8} \, {\left (2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x + 3 \, \sqrt {-c^{2} d x^{2} + d} d x + \frac {3 \, d^{\frac {3}{2}} \arcsin \left (c x\right )}{c}\right )} a^{2} f^{2} + \frac {1}{48} \, a^{2} g^{2} {\left (\frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x}{c^{2}} - \frac {8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x}{c^{2} d} + \frac {3 \, \sqrt {-c^{2} d x^{2} + d} d x}{c^{2}} + \frac {3 \, d^{\frac {3}{2}} \arcsin \left (c x\right )}{c^{3}}\right )} - \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} a^{2} f g}{5 \, c^{2} d} + \sqrt {d} \int -{\left ({\left (b^{2} c^{2} d g^{2} x^{4} + 2 \, b^{2} c^{2} d f g x^{3} - 2 \, b^{2} d f g x - b^{2} d f^{2} + {\left (b^{2} c^{2} d f^{2} - b^{2} d g^{2}\right )} x^{2}\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} + 2 \, {\left (a b c^{2} d g^{2} x^{4} + 2 \, a b c^{2} d f g x^{3} - 2 \, a b d f g x - a b d f^{2} + {\left (a b c^{2} d f^{2} - a b d g^{2}\right )} x^{2}\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )\right )} \sqrt {c x + 1} \sqrt {-c x + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(2*(-c^2*d*x^2 + d)^(3/2)*x + 3*sqrt(-c^2*d*x^2 + d)*d*x + 3*d^(3/2)*arcsin(c*x)/c)*a^2*f^2 + 1/48*a^2*g^2
*(2*(-c^2*d*x^2 + d)^(3/2)*x/c^2 - 8*(-c^2*d*x^2 + d)^(5/2)*x/(c^2*d) + 3*sqrt(-c^2*d*x^2 + d)*d*x/c^2 + 3*d^(
3/2)*arcsin(c*x)/c^3) - 2/5*(-c^2*d*x^2 + d)^(5/2)*a^2*f*g/(c^2*d) + sqrt(d)*integrate(-((b^2*c^2*d*g^2*x^4 +
2*b^2*c^2*d*f*g*x^3 - 2*b^2*d*f*g*x - b^2*d*f^2 + (b^2*c^2*d*f^2 - b^2*d*g^2)*x^2)*arctan2(c*x, sqrt(c*x + 1)*
sqrt(-c*x + 1))^2 + 2*(a*b*c^2*d*g^2*x^4 + 2*a*b*c^2*d*f*g*x^3 - 2*a*b*d*f*g*x - a*b*d*f^2 + (a*b*c^2*d*f^2 -
a*b*d*g^2)*x^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))*sqrt(c*x + 1)*sqrt(-c*x + 1), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (f+g\,x\right )}^2\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{3/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2} \left (f + g x\right )^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)**2*(-c**2*d*x**2+d)**(3/2)*(a+b*asin(c*x))**2,x)

[Out]

Integral((-d*(c*x - 1)*(c*x + 1))**(3/2)*(a + b*asin(c*x))**2*(f + g*x)**2, x)

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