3.80 \(\int \frac{(f+g x)^2 (a+b \sin ^{-1}(c x))^2}{(d-c^2 d x^2)^{5/2}} \, dx\)

Optimal. Leaf size=1025 \[ \frac{g^2 \left (a+b \sin ^{-1}(c x)\right )^2 x^3}{3 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}-\frac{b g^2 \left (a+b \sin ^{-1}(c x)\right ) x^2}{3 c d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}+\frac{2 f^2 \left (a+b \sin ^{-1}(c x)\right )^2 x}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{f^2 \left (a+b \sin ^{-1}(c x)\right )^2 x}{3 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}-\frac{2 b f g \left (a+b \sin ^{-1}(c x)\right ) x}{3 c d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}+\frac{b^2 f^2 x}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{b^2 g^2 x}{3 c^2 d^2 \sqrt{d-c^2 d x^2}}-\frac{2 i f^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c d^2 \sqrt{d-c^2 d x^2}}+\frac{i g^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}+\frac{2 f g \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}-\frac{b^2 g^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}-\frac{b f^2 \left (a+b \sin ^{-1}(c x)\right )}{3 c d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}+\frac{4 i b f g \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 c^2 d^2 \sqrt{d-c^2 d x^2}}+\frac{4 b f^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{3 c d^2 \sqrt{d-c^2 d x^2}}-\frac{2 b g^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}-\frac{2 i b^2 f g \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )}{3 c^2 d^2 \sqrt{d-c^2 d x^2}}+\frac{2 i b^2 f g \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )}{3 c^2 d^2 \sqrt{d-c^2 d x^2}}-\frac{2 i b^2 f^2 \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,-e^{2 i \sin ^{-1}(c x)}\right )}{3 c d^2 \sqrt{d-c^2 d x^2}}+\frac{i b^2 g^2 \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,-e^{2 i \sin ^{-1}(c x)}\right )}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}+\frac{2 b^2 f g}{3 c^2 d^2 \sqrt{d-c^2 d x^2}} \]

[Out]

(2*b^2*f*g)/(3*c^2*d^2*Sqrt[d - c^2*d*x^2]) + (b^2*f^2*x)/(3*d^2*Sqrt[d - c^2*d*x^2]) + (b^2*g^2*x)/(3*c^2*d^2
*Sqrt[d - c^2*d*x^2]) - (b^2*g^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(3*c^3*d^2*Sqrt[d - c^2*d*x^2]) - (b*f^2*(a +
b*ArcSin[c*x]))/(3*c*d^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]) - (2*b*f*g*x*(a + b*ArcSin[c*x]))/(3*c*d^2*Sqr
t[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]) - (b*g^2*x^2*(a + b*ArcSin[c*x]))/(3*c*d^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*
d*x^2]) + (2*f^2*x*(a + b*ArcSin[c*x])^2)/(3*d^2*Sqrt[d - c^2*d*x^2]) + (2*f*g*(a + b*ArcSin[c*x])^2)/(3*c^2*d
^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]) + (f^2*x*(a + b*ArcSin[c*x])^2)/(3*d^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]
) + (g^2*x^3*(a + b*ArcSin[c*x])^2)/(3*d^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]) - (((2*I)/3)*f^2*Sqrt[1 - c^2*x^
2]*(a + b*ArcSin[c*x])^2)/(c*d^2*Sqrt[d - c^2*d*x^2]) + ((I/3)*g^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(c
^3*d^2*Sqrt[d - c^2*d*x^2]) + (((4*I)/3)*b*f*g*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])]
)/(c^2*d^2*Sqrt[d - c^2*d*x^2]) + (4*b*f^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 + E^((2*I)*ArcSin[c*x])
])/(3*c*d^2*Sqrt[d - c^2*d*x^2]) - (2*b*g^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 + E^((2*I)*ArcSin[c*x]
)])/(3*c^3*d^2*Sqrt[d - c^2*d*x^2]) - (((2*I)/3)*b^2*f*g*Sqrt[1 - c^2*x^2]*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])
/(c^2*d^2*Sqrt[d - c^2*d*x^2]) + (((2*I)/3)*b^2*f*g*Sqrt[1 - c^2*x^2]*PolyLog[2, I*E^(I*ArcSin[c*x])])/(c^2*d^
2*Sqrt[d - c^2*d*x^2]) - (((2*I)/3)*b^2*f^2*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^((2*I)*ArcSin[c*x])])/(c*d^2*Sqrt[
d - c^2*d*x^2]) + ((I/3)*b^2*g^2*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^((2*I)*ArcSin[c*x])])/(c^3*d^2*Sqrt[d - c^2*d
*x^2])

________________________________________________________________________________________

Rubi [A]  time = 1.3072, antiderivative size = 1025, normalized size of antiderivative = 1., number of steps used = 30, number of rules used = 18, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.546, Rules used = {4777, 4763, 4655, 4651, 4675, 3719, 2190, 2279, 2391, 4677, 191, 4657, 4181, 261, 4681, 4703, 288, 216} \[ \frac{g^2 \left (a+b \sin ^{-1}(c x)\right )^2 x^3}{3 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}-\frac{b g^2 \left (a+b \sin ^{-1}(c x)\right ) x^2}{3 c d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}+\frac{2 f^2 \left (a+b \sin ^{-1}(c x)\right )^2 x}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{f^2 \left (a+b \sin ^{-1}(c x)\right )^2 x}{3 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}-\frac{2 b f g \left (a+b \sin ^{-1}(c x)\right ) x}{3 c d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}+\frac{b^2 f^2 x}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{b^2 g^2 x}{3 c^2 d^2 \sqrt{d-c^2 d x^2}}-\frac{2 i f^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c d^2 \sqrt{d-c^2 d x^2}}+\frac{i g^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}+\frac{2 f g \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}-\frac{b^2 g^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}-\frac{b f^2 \left (a+b \sin ^{-1}(c x)\right )}{3 c d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}+\frac{4 i b f g \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 c^2 d^2 \sqrt{d-c^2 d x^2}}+\frac{4 b f^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{3 c d^2 \sqrt{d-c^2 d x^2}}-\frac{2 b g^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}-\frac{2 i b^2 f g \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )}{3 c^2 d^2 \sqrt{d-c^2 d x^2}}+\frac{2 i b^2 f g \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )}{3 c^2 d^2 \sqrt{d-c^2 d x^2}}-\frac{2 i b^2 f^2 \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,-e^{2 i \sin ^{-1}(c x)}\right )}{3 c d^2 \sqrt{d-c^2 d x^2}}+\frac{i b^2 g^2 \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,-e^{2 i \sin ^{-1}(c x)}\right )}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}+\frac{2 b^2 f g}{3 c^2 d^2 \sqrt{d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

Int[((f + g*x)^2*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2)^(5/2),x]

[Out]

(2*b^2*f*g)/(3*c^2*d^2*Sqrt[d - c^2*d*x^2]) + (b^2*f^2*x)/(3*d^2*Sqrt[d - c^2*d*x^2]) + (b^2*g^2*x)/(3*c^2*d^2
*Sqrt[d - c^2*d*x^2]) - (b^2*g^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(3*c^3*d^2*Sqrt[d - c^2*d*x^2]) - (b*f^2*(a +
b*ArcSin[c*x]))/(3*c*d^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]) - (2*b*f*g*x*(a + b*ArcSin[c*x]))/(3*c*d^2*Sqr
t[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]) - (b*g^2*x^2*(a + b*ArcSin[c*x]))/(3*c*d^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*
d*x^2]) + (2*f^2*x*(a + b*ArcSin[c*x])^2)/(3*d^2*Sqrt[d - c^2*d*x^2]) + (2*f*g*(a + b*ArcSin[c*x])^2)/(3*c^2*d
^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]) + (f^2*x*(a + b*ArcSin[c*x])^2)/(3*d^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]
) + (g^2*x^3*(a + b*ArcSin[c*x])^2)/(3*d^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]) - (((2*I)/3)*f^2*Sqrt[1 - c^2*x^
2]*(a + b*ArcSin[c*x])^2)/(c*d^2*Sqrt[d - c^2*d*x^2]) + ((I/3)*g^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(c
^3*d^2*Sqrt[d - c^2*d*x^2]) + (((4*I)/3)*b*f*g*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])]
)/(c^2*d^2*Sqrt[d - c^2*d*x^2]) + (4*b*f^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 + E^((2*I)*ArcSin[c*x])
])/(3*c*d^2*Sqrt[d - c^2*d*x^2]) - (2*b*g^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 + E^((2*I)*ArcSin[c*x]
)])/(3*c^3*d^2*Sqrt[d - c^2*d*x^2]) - (((2*I)/3)*b^2*f*g*Sqrt[1 - c^2*x^2]*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])
/(c^2*d^2*Sqrt[d - c^2*d*x^2]) + (((2*I)/3)*b^2*f*g*Sqrt[1 - c^2*x^2]*PolyLog[2, I*E^(I*ArcSin[c*x])])/(c^2*d^
2*Sqrt[d - c^2*d*x^2]) - (((2*I)/3)*b^2*f^2*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^((2*I)*ArcSin[c*x])])/(c*d^2*Sqrt[
d - c^2*d*x^2]) + ((I/3)*b^2*g^2*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^((2*I)*ArcSin[c*x])])/(c^3*d^2*Sqrt[d - c^2*d
*x^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]

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 4655

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[(x*(d + e*x^2)^(p
+ 1)*(a + b*ArcSin[c*x])^n)/(2*d*(p + 1)), x] + (Dist[(2*p + 3)/(2*d*(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] && GtQ[n, 0] && LtQ[p, -1] && NeQ[p, -3/2]

Rule 4651

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(x*(a + b*ArcSin[c
*x])^n)/(d*Sqrt[d + e*x^2]), x] - Dist[(b*c*n)/Sqrt[d], Int[(x*(a + b*ArcSin[c*x])^(n - 1))/(d + e*x^2), x], x
] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[d, 0]

Rule 4675

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

Rule 3719

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(I*(c + d*x)^(m + 1))/(d*(m + 1)), x
] - Dist[2*I, Int[((c + d*x)^m*E^(2*I*(e + f*x)))/(1 + E^(2*I*(e + f*x))), x], x] /; FreeQ[{c, d, e, f}, x] &&
IGtQ[m, 0]

Rule 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

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 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 4657

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

Rule 4181

Int[csc[(e_.) + Pi*(k_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c + d*x)^m*ArcTanh[E
^(I*k*Pi)*E^(I*(e + f*x))])/f, x] + (-Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Log[1 - E^(I*k*Pi)*E^(I*(e + f*x))],
x], x] + Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Log[1 + E^(I*k*Pi)*E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e,
f}, x] && IntegerQ[2*k] && IGtQ[m, 0]

Rule 261

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

Rule 4681

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 + 1)*(a + b*ArcSin[c*x])^n)/(d*f*(m + 1)), x] - Dist[(b*c*n*d^IntPart[p]*(d + e*x
^2)^FracPart[p])/(f*(m + 1)*(1 - c^2*x^2)^FracPart[p]), Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSi
n[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && EqQ[m + 2*p
+ 3, 0] && NeQ[m, -1]

Rule 4703

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

Rule 288

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*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

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]

Rubi steps

\begin{align*} \int \frac{(f+g x)^2 \left (a+b \sin ^{-1}(c x)\right )^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx &=\frac{\sqrt{1-c^2 x^2} \int \frac{(f+g x)^2 \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{5/2}} \, dx}{d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{\sqrt{1-c^2 x^2} \int \left (\frac{f^2 \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{5/2}}+\frac{2 f g x \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{5/2}}+\frac{g^2 x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{5/2}}\right ) \, dx}{d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{\left (f^2 \sqrt{1-c^2 x^2}\right ) \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{5/2}} \, dx}{d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (2 f g \sqrt{1-c^2 x^2}\right ) \int \frac{x \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{5/2}} \, dx}{d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (g^2 \sqrt{1-c^2 x^2}\right ) \int \frac{x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{5/2}} \, dx}{d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{2 f g \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{f^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{g^2 x^3 \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{\left (2 f^2 \sqrt{1-c^2 x^2}\right ) \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (2 b c f^2 \sqrt{1-c^2 x^2}\right ) \int \frac{x \left (a+b \sin ^{-1}(c x)\right )}{\left (1-c^2 x^2\right )^2} \, dx}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (4 b f g \sqrt{1-c^2 x^2}\right ) \int \frac{a+b \sin ^{-1}(c x)}{\left (1-c^2 x^2\right )^2} \, dx}{3 c d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (2 b c g^2 \sqrt{1-c^2 x^2}\right ) \int \frac{x^3 \left (a+b \sin ^{-1}(c x)\right )}{\left (1-c^2 x^2\right )^2} \, dx}{3 d^2 \sqrt{d-c^2 d x^2}}\\ &=-\frac{b f^2 \left (a+b \sin ^{-1}(c x)\right )}{3 c d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}-\frac{2 b f g x \left (a+b \sin ^{-1}(c x)\right )}{3 c d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}-\frac{b g^2 x^2 \left (a+b \sin ^{-1}(c x)\right )}{3 c d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}+\frac{2 f^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{2 f g \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{f^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{g^2 x^3 \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{\left (b^2 f^2 \sqrt{1-c^2 x^2}\right ) \int \frac{1}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (4 b c f^2 \sqrt{1-c^2 x^2}\right ) \int \frac{x \left (a+b \sin ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (2 b^2 f g \sqrt{1-c^2 x^2}\right ) \int \frac{x}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (2 b f g \sqrt{1-c^2 x^2}\right ) \int \frac{a+b \sin ^{-1}(c x)}{1-c^2 x^2} \, dx}{3 c d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (b^2 g^2 \sqrt{1-c^2 x^2}\right ) \int \frac{x^2}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (2 b g^2 \sqrt{1-c^2 x^2}\right ) \int \frac{x \left (a+b \sin ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{3 c d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{2 b^2 f g}{3 c^2 d^2 \sqrt{d-c^2 d x^2}}+\frac{b^2 f^2 x}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{b^2 g^2 x}{3 c^2 d^2 \sqrt{d-c^2 d x^2}}-\frac{b f^2 \left (a+b \sin ^{-1}(c x)\right )}{3 c d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}-\frac{2 b f g x \left (a+b \sin ^{-1}(c x)\right )}{3 c d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}-\frac{b g^2 x^2 \left (a+b \sin ^{-1}(c x)\right )}{3 c d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}+\frac{2 f^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{2 f g \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{f^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{g^2 x^3 \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}-\frac{\left (4 b f^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \tan (x) \, dx,x,\sin ^{-1}(c x)\right )}{3 c d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (2 b f g \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{3 c^2 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (2 b g^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \tan (x) \, dx,x,\sin ^{-1}(c x)\right )}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (b^2 g^2 \sqrt{1-c^2 x^2}\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{3 c^2 d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{2 b^2 f g}{3 c^2 d^2 \sqrt{d-c^2 d x^2}}+\frac{b^2 f^2 x}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{b^2 g^2 x}{3 c^2 d^2 \sqrt{d-c^2 d x^2}}-\frac{b^2 g^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}-\frac{b f^2 \left (a+b \sin ^{-1}(c x)\right )}{3 c d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}-\frac{2 b f g x \left (a+b \sin ^{-1}(c x)\right )}{3 c d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}-\frac{b g^2 x^2 \left (a+b \sin ^{-1}(c x)\right )}{3 c d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}+\frac{2 f^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{2 f g \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{f^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{g^2 x^3 \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}-\frac{2 i f^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c d^2 \sqrt{d-c^2 d x^2}}+\frac{i g^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}+\frac{4 i b f g \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 c^2 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (8 i b f^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{2 i x} (a+b x)}{1+e^{2 i x}} \, dx,x,\sin ^{-1}(c x)\right )}{3 c d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (2 b^2 f g \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 c^2 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (2 b^2 f g \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 c^2 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (4 i b g^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{2 i x} (a+b x)}{1+e^{2 i x}} \, dx,x,\sin ^{-1}(c x)\right )}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{2 b^2 f g}{3 c^2 d^2 \sqrt{d-c^2 d x^2}}+\frac{b^2 f^2 x}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{b^2 g^2 x}{3 c^2 d^2 \sqrt{d-c^2 d x^2}}-\frac{b^2 g^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}-\frac{b f^2 \left (a+b \sin ^{-1}(c x)\right )}{3 c d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}-\frac{2 b f g x \left (a+b \sin ^{-1}(c x)\right )}{3 c d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}-\frac{b g^2 x^2 \left (a+b \sin ^{-1}(c x)\right )}{3 c d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}+\frac{2 f^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{2 f g \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{f^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{g^2 x^3 \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}-\frac{2 i f^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c d^2 \sqrt{d-c^2 d x^2}}+\frac{i g^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}+\frac{4 i b f g \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 c^2 d^2 \sqrt{d-c^2 d x^2}}+\frac{4 b f^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{3 c d^2 \sqrt{d-c^2 d x^2}}-\frac{2 b g^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (4 b^2 f^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 c d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (2 i b^2 f g \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{3 c^2 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (2 i b^2 f g \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{3 c^2 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (2 b^2 g^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{2 b^2 f g}{3 c^2 d^2 \sqrt{d-c^2 d x^2}}+\frac{b^2 f^2 x}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{b^2 g^2 x}{3 c^2 d^2 \sqrt{d-c^2 d x^2}}-\frac{b^2 g^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}-\frac{b f^2 \left (a+b \sin ^{-1}(c x)\right )}{3 c d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}-\frac{2 b f g x \left (a+b \sin ^{-1}(c x)\right )}{3 c d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}-\frac{b g^2 x^2 \left (a+b \sin ^{-1}(c x)\right )}{3 c d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}+\frac{2 f^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{2 f g \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{f^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{g^2 x^3 \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}-\frac{2 i f^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c d^2 \sqrt{d-c^2 d x^2}}+\frac{i g^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}+\frac{4 i b f g \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 c^2 d^2 \sqrt{d-c^2 d x^2}}+\frac{4 b f^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{3 c d^2 \sqrt{d-c^2 d x^2}}-\frac{2 b g^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}-\frac{2 i b^2 f g \sqrt{1-c^2 x^2} \text{Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{3 c^2 d^2 \sqrt{d-c^2 d x^2}}+\frac{2 i b^2 f g \sqrt{1-c^2 x^2} \text{Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{3 c^2 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (2 i b^2 f^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{3 c d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (i b^2 g^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{2 b^2 f g}{3 c^2 d^2 \sqrt{d-c^2 d x^2}}+\frac{b^2 f^2 x}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{b^2 g^2 x}{3 c^2 d^2 \sqrt{d-c^2 d x^2}}-\frac{b^2 g^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}-\frac{b f^2 \left (a+b \sin ^{-1}(c x)\right )}{3 c d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}-\frac{2 b f g x \left (a+b \sin ^{-1}(c x)\right )}{3 c d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}-\frac{b g^2 x^2 \left (a+b \sin ^{-1}(c x)\right )}{3 c d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}+\frac{2 f^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{2 f g \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{f^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{g^2 x^3 \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}-\frac{2 i f^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c d^2 \sqrt{d-c^2 d x^2}}+\frac{i g^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}+\frac{4 i b f g \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 c^2 d^2 \sqrt{d-c^2 d x^2}}+\frac{4 b f^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{3 c d^2 \sqrt{d-c^2 d x^2}}-\frac{2 b g^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}-\frac{2 i b^2 f g \sqrt{1-c^2 x^2} \text{Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{3 c^2 d^2 \sqrt{d-c^2 d x^2}}+\frac{2 i b^2 f g \sqrt{1-c^2 x^2} \text{Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{3 c^2 d^2 \sqrt{d-c^2 d x^2}}-\frac{2 i b^2 f^2 \sqrt{1-c^2 x^2} \text{Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )}{3 c d^2 \sqrt{d-c^2 d x^2}}+\frac{i b^2 g^2 \sqrt{1-c^2 x^2} \text{Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}\\ \end{align*}

Mathematica [A]  time = 6.24611, size = 711, normalized size = 0.69 \[ \frac{\sqrt{1-c^2 x^2} \left (-\frac{(c f-g)^2 \left (-2 \left (-\tan \left (\frac{\pi }{4}-\frac{1}{2} \sin ^{-1}(c x)\right ) \left (a+b \sin ^{-1}(c x)\right )^2+i b \left (\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{b}-4 \left (i \log \left (1+e^{\frac{1}{2} i \left (\pi -2 \sin ^{-1}(c x)\right )}\right ) \left (a+b \sin ^{-1}(c x)\right )-b \text{PolyLog}\left (2,-e^{\frac{1}{2} i \left (\pi -2 \sin ^{-1}(c x)\right )}\right )\right )\right )\right )+2 b \sec ^2\left (\frac{\pi }{4}-\frac{1}{2} \sin ^{-1}(c x)\right ) \left (a+b \sin ^{-1}(c x)\right )+\tan \left (\frac{\pi }{4}-\frac{1}{2} \sin ^{-1}(c x)\right ) \sec ^2\left (\frac{\pi }{4}-\frac{1}{2} \sin ^{-1}(c x)\right ) \left (a+b \sin ^{-1}(c x)\right )^2+4 b^2 \tan \left (\frac{\pi }{4}-\frac{1}{2} \sin ^{-1}(c x)\right )\right )}{24 c^3}-\frac{(c f+g)^2 \left (2 \left (-\tan \left (\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right ) \left (a+b \sin ^{-1}(c x)\right )^2+i b \left (\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{b}+4 \left (b \text{PolyLog}\left (2,-e^{\frac{1}{2} i \left (2 \sin ^{-1}(c x)+\pi \right )}\right )+i \log \left (1+e^{\frac{1}{2} i \left (2 \sin ^{-1}(c x)+\pi \right )}\right ) \left (a+b \sin ^{-1}(c x)\right )\right )\right )\right )+2 b \sec ^2\left (\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right ) \left (a+b \sin ^{-1}(c x)\right )-\tan \left (\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right ) \sec ^2\left (\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right ) \left (a+b \sin ^{-1}(c x)\right )^2-4 b^2 \tan \left (\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right )\right )}{24 c^3}+\frac{\left (c^2 f^2-g^2\right ) \left (-\tan \left (\frac{\pi }{4}-\frac{1}{2} \sin ^{-1}(c x)\right ) \left (a+b \sin ^{-1}(c x)\right )^2+i b \left (\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{b}-4 \left (i \log \left (1+e^{\frac{1}{2} i \left (\pi -2 \sin ^{-1}(c x)\right )}\right ) \left (a+b \sin ^{-1}(c x)\right )-b \text{PolyLog}\left (2,-e^{\frac{1}{2} i \left (\pi -2 \sin ^{-1}(c x)\right )}\right )\right )\right )\right )}{4 c^3}-\frac{\left (c^2 f^2-g^2\right ) \left (-\tan \left (\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right ) \left (a+b \sin ^{-1}(c x)\right )^2+i b \left (\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{b}+4 \left (b \text{PolyLog}\left (2,-e^{\frac{1}{2} i \left (2 \sin ^{-1}(c x)+\pi \right )}\right )+i \log \left (1+e^{\frac{1}{2} i \left (2 \sin ^{-1}(c x)+\pi \right )}\right ) \left (a+b \sin ^{-1}(c x)\right )\right )\right )\right )}{4 c^3}\right )}{d^2 \sqrt{d-c^2 d x^2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[((f + g*x)^2*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2)^(5/2),x]

[Out]

(Sqrt[1 - c^2*x^2]*(((c^2*f^2 - g^2)*(I*b*((a + b*ArcSin[c*x])^2/b - 4*(I*(a + b*ArcSin[c*x])*Log[1 + E^((I/2)
*(Pi - 2*ArcSin[c*x]))] - b*PolyLog[2, -E^((I/2)*(Pi - 2*ArcSin[c*x]))])) - (a + b*ArcSin[c*x])^2*Tan[Pi/4 - A
rcSin[c*x]/2]))/(4*c^3) - ((c*f - g)^2*(2*b*(a + b*ArcSin[c*x])*Sec[Pi/4 - ArcSin[c*x]/2]^2 + 4*b^2*Tan[Pi/4 -
ArcSin[c*x]/2] + (a + b*ArcSin[c*x])^2*Sec[Pi/4 - ArcSin[c*x]/2]^2*Tan[Pi/4 - ArcSin[c*x]/2] - 2*(I*b*((a + b
*ArcSin[c*x])^2/b - 4*(I*(a + b*ArcSin[c*x])*Log[1 + E^((I/2)*(Pi - 2*ArcSin[c*x]))] - b*PolyLog[2, -E^((I/2)*
(Pi - 2*ArcSin[c*x]))])) - (a + b*ArcSin[c*x])^2*Tan[Pi/4 - ArcSin[c*x]/2])))/(24*c^3) - ((c^2*f^2 - g^2)*(I*b
*((a + b*ArcSin[c*x])^2/b + 4*(I*(a + b*ArcSin[c*x])*Log[1 + E^((I/2)*(Pi + 2*ArcSin[c*x]))] + b*PolyLog[2, -E
^((I/2)*(Pi + 2*ArcSin[c*x]))])) - (a + b*ArcSin[c*x])^2*Tan[Pi/4 + ArcSin[c*x]/2]))/(4*c^3) - ((c*f + g)^2*(2
*b*(a + b*ArcSin[c*x])*Sec[Pi/4 + ArcSin[c*x]/2]^2 - 4*b^2*Tan[Pi/4 + ArcSin[c*x]/2] - (a + b*ArcSin[c*x])^2*S
ec[Pi/4 + ArcSin[c*x]/2]^2*Tan[Pi/4 + ArcSin[c*x]/2] + 2*(I*b*((a + b*ArcSin[c*x])^2/b + 4*(I*(a + b*ArcSin[c*
x])*Log[1 + E^((I/2)*(Pi + 2*ArcSin[c*x]))] + b*PolyLog[2, -E^((I/2)*(Pi + 2*ArcSin[c*x]))])) - (a + b*ArcSin[
c*x])^2*Tan[Pi/4 + ArcSin[c*x]/2])))/(24*c^3)))/(d^2*Sqrt[d - c^2*d*x^2])

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Maple [B]  time = 0.454, size = 9710, normalized size = 9.5 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x+f)^2*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(5/2),x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{3} \, a b c f^{2}{\left (\frac{1}{c^{4} d^{\frac{5}{2}} x^{2} - c^{2} d^{\frac{5}{2}}} + \frac{2 \, \log \left (c x + 1\right )}{c^{2} d^{\frac{5}{2}}} + \frac{2 \, \log \left (c x - 1\right )}{c^{2} d^{\frac{5}{2}}}\right )} + \frac{2}{3} \, a b f^{2}{\left (\frac{2 \, x}{\sqrt{-c^{2} d x^{2} + d} d^{2}} + \frac{x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}} d}\right )} \arcsin \left (c x\right ) + \frac{1}{3} \, a^{2} f^{2}{\left (\frac{2 \, x}{\sqrt{-c^{2} d x^{2} + d} d^{2}} + \frac{x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}} d}\right )} - \frac{1}{3} \, a^{2} g^{2}{\left (\frac{x}{\sqrt{-c^{2} d x^{2} + d} c^{2} d^{2}} - \frac{x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}} c^{2} d}\right )} + \sqrt{d} \int \frac{{\left (b^{2} g^{2} x^{2} + 2 \, b^{2} f g x + b^{2} f^{2}\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )^{2} + 2 \,{\left (a b g^{2} x^{2} + 2 \, a b f g x\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )}{{\left (c^{4} d^{3} x^{4} - 2 \, c^{2} d^{3} x^{2} + d^{3}\right )} \sqrt{c x + 1} \sqrt{-c x + 1}}\,{d x} + \frac{2 \, a^{2} f g}{3 \,{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}} c^{2} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^2*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (a^{2} g^{2} x^{2} + 2 \, a^{2} f g x + a^{2} f^{2} +{\left (b^{2} g^{2} x^{2} + 2 \, b^{2} f g x + b^{2} f^{2}\right )} \arcsin \left (c x\right )^{2} + 2 \,{\left (a b g^{2} x^{2} + 2 \, a b f g x + a b f^{2}\right )} \arcsin \left (c x\right )\right )} \sqrt{-c^{2} d x^{2} + d}}{c^{6} d^{3} x^{6} - 3 \, c^{4} d^{3} x^{4} + 3 \, c^{2} d^{3} x^{2} - d^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^2*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(5/2),x, algorithm="fricas")

[Out]

integral(-(a^2*g^2*x^2 + 2*a^2*f*g*x + a^2*f^2 + (b^2*g^2*x^2 + 2*b^2*f*g*x + b^2*f^2)*arcsin(c*x)^2 + 2*(a*b*
g^2*x^2 + 2*a*b*f*g*x + a*b*f^2)*arcsin(c*x))*sqrt(-c^2*d*x^2 + d)/(c^6*d^3*x^6 - 3*c^4*d^3*x^4 + 3*c^2*d^3*x^
2 - d^3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)**2*(a+b*asin(c*x))**2/(-c**2*d*x**2+d)**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (g x + f\right )}^{2}{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^2*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(5/2),x, algorithm="giac")

[Out]

integrate((g*x + f)^2*(b*arcsin(c*x) + a)^2/(-c^2*d*x^2 + d)^(5/2), x)