3.573 \(\int \frac{(a+b \sin ^{-1}(c x))^2}{\sqrt{d+c d x} (e-c e x)^{5/2}} \, dx\)

Optimal. Leaf size=896 \[ \frac{c^2 d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2 x^3}{3 (c x d+d)^{5/2} (e-c e x)^{5/2}}-\frac{b c d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) x^2}{3 (c x d+d)^{5/2} (e-c e x)^{5/2}}+\frac{2 b^2 d^2 \left (1-c^2 x^2\right )^2 x}{3 (c x d+d)^{5/2} (e-c e x)^{5/2}}+\frac{2 d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2 x}{3 (c x d+d)^{5/2} (e-c e x)^{5/2}}+\frac{d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2 x}{3 (c x d+d)^{5/2} (e-c e x)^{5/2}}-\frac{2 b d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) x}{3 (c x d+d)^{5/2} (e-c e x)^{5/2}}+\frac{2 b^2 d^2 \left (1-c^2 x^2\right )^2}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}}-\frac{i d^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}}+\frac{2 d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}}-\frac{b^2 d^2 \left (1-c^2 x^2\right )^{5/2} \sin ^{-1}(c x)}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}}-\frac{b d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}}+\frac{4 i b d^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}}+\frac{2 b d^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}}-\frac{2 i b^2 d^2 \left (1-c^2 x^2\right )^{5/2} \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}}+\frac{2 i b^2 d^2 \left (1-c^2 x^2\right )^{5/2} \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}}-\frac{i b^2 d^2 \left (1-c^2 x^2\right )^{5/2} \text{PolyLog}\left (2,-e^{2 i \sin ^{-1}(c x)}\right )}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}} \]

[Out]

(2*b^2*d^2*(1 - c^2*x^2)^2)/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (2*b^2*d^2*x*(1 - c^2*x^2)^2)/(3*(d +
c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (b^2*d^2*(1 - c^2*x^2)^(5/2)*ArcSin[c*x])/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)
^(5/2)) - (b*d^2*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (2*b*d^2
*x*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(3*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (b*c*d^2*x^2*(1 - c^2*x^
2)^(3/2)*(a + b*ArcSin[c*x]))/(3*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (2*d^2*(1 - c^2*x^2)*(a + b*ArcSin[c*x
])^2)/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (d^2*x*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(3*(d + c*d*x)^(
5/2)*(e - c*e*x)^(5/2)) + (c^2*d^2*x^3*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(3*(d + c*d*x)^(5/2)*(e - c*e*x)^(
5/2)) + (2*d^2*x*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/(3*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - ((I/3)*d^2*(
1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^2)/(c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (((4*I)/3)*b*d^2*(1 - c^2*
x^2)^(5/2)*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])])/(c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (2*b*d^2*(
1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])*Log[1 + E^((2*I)*ArcSin[c*x])])/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2
)) - (((2*I)/3)*b^2*d^2*(1 - c^2*x^2)^(5/2)*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(c*(d + c*d*x)^(5/2)*(e - c*e*
x)^(5/2)) + (((2*I)/3)*b^2*d^2*(1 - c^2*x^2)^(5/2)*PolyLog[2, I*E^(I*ArcSin[c*x])])/(c*(d + c*d*x)^(5/2)*(e -
c*e*x)^(5/2)) - ((I/3)*b^2*d^2*(1 - c^2*x^2)^(5/2)*PolyLog[2, -E^((2*I)*ArcSin[c*x])])/(c*(d + c*d*x)^(5/2)*(e
- c*e*x)^(5/2))

________________________________________________________________________________________

Rubi [A]  time = 1.23074, antiderivative size = 896, normalized size of antiderivative = 1., number of steps used = 30, number of rules used = 18, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.562, Rules used = {4673, 4763, 4655, 4651, 4675, 3719, 2190, 2279, 2391, 4677, 191, 4657, 4181, 261, 4681, 4703, 288, 216} \[ \frac{c^2 d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2 x^3}{3 (c x d+d)^{5/2} (e-c e x)^{5/2}}-\frac{b c d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) x^2}{3 (c x d+d)^{5/2} (e-c e x)^{5/2}}+\frac{2 b^2 d^2 \left (1-c^2 x^2\right )^2 x}{3 (c x d+d)^{5/2} (e-c e x)^{5/2}}+\frac{2 d^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2 x}{3 (c x d+d)^{5/2} (e-c e x)^{5/2}}+\frac{d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2 x}{3 (c x d+d)^{5/2} (e-c e x)^{5/2}}-\frac{2 b d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) x}{3 (c x d+d)^{5/2} (e-c e x)^{5/2}}+\frac{2 b^2 d^2 \left (1-c^2 x^2\right )^2}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}}-\frac{i d^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}}+\frac{2 d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}}-\frac{b^2 d^2 \left (1-c^2 x^2\right )^{5/2} \sin ^{-1}(c x)}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}}-\frac{b d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}}+\frac{4 i b d^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}}+\frac{2 b d^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}}-\frac{2 i b^2 d^2 \left (1-c^2 x^2\right )^{5/2} \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}}+\frac{2 i b^2 d^2 \left (1-c^2 x^2\right )^{5/2} \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}}-\frac{i b^2 d^2 \left (1-c^2 x^2\right )^{5/2} \text{PolyLog}\left (2,-e^{2 i \sin ^{-1}(c x)}\right )}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*b^2*d^2*(1 - c^2*x^2)^2)/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (2*b^2*d^2*x*(1 - c^2*x^2)^2)/(3*(d +
c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (b^2*d^2*(1 - c^2*x^2)^(5/2)*ArcSin[c*x])/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)
^(5/2)) - (b*d^2*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (2*b*d^2
*x*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(3*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (b*c*d^2*x^2*(1 - c^2*x^
2)^(3/2)*(a + b*ArcSin[c*x]))/(3*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (2*d^2*(1 - c^2*x^2)*(a + b*ArcSin[c*x
])^2)/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (d^2*x*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(3*(d + c*d*x)^(
5/2)*(e - c*e*x)^(5/2)) + (c^2*d^2*x^3*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(3*(d + c*d*x)^(5/2)*(e - c*e*x)^(
5/2)) + (2*d^2*x*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/(3*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - ((I/3)*d^2*(
1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^2)/(c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (((4*I)/3)*b*d^2*(1 - c^2*
x^2)^(5/2)*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])])/(c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (2*b*d^2*(
1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])*Log[1 + E^((2*I)*ArcSin[c*x])])/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2
)) - (((2*I)/3)*b^2*d^2*(1 - c^2*x^2)^(5/2)*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(c*(d + c*d*x)^(5/2)*(e - c*e*
x)^(5/2)) + (((2*I)/3)*b^2*d^2*(1 - c^2*x^2)^(5/2)*PolyLog[2, I*E^(I*ArcSin[c*x])])/(c*(d + c*d*x)^(5/2)*(e -
c*e*x)^(5/2)) - ((I/3)*b^2*d^2*(1 - c^2*x^2)^(5/2)*PolyLog[2, -E^((2*I)*ArcSin[c*x])])/(c*(d + c*d*x)^(5/2)*(e
- c*e*x)^(5/2))

Rule 4673

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_))^(p_)*((f_) + (g_.)*(x_))^(q_), x_Symbol] :> D
ist[((d + e*x)^q*(f + g*x)^q)/(1 - c^2*x^2)^q, Int[(d + e*x)^(p - q)*(1 - c^2*x^2)^q*(a + b*ArcSin[c*x])^n, x]
, x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^2 - e^2, 0] && HalfIntegerQ[p, q]
&& GeQ[p - q, 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{\left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{d+c d x} (e-c e x)^{5/2}} \, dx &=\frac{\left (1-c^2 x^2\right )^{5/2} \int \frac{(d+c d x)^2 \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{5/2}} \, dx}{(d+c d x)^{5/2} (e-c e x)^{5/2}}\\ &=\frac{\left (1-c^2 x^2\right )^{5/2} \int \left (\frac{d^2 \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{5/2}}+\frac{2 c d^2 x \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{5/2}}+\frac{c^2 d^2 x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{5/2}}\right ) \, dx}{(d+c d x)^{5/2} (e-c e x)^{5/2}}\\ &=\frac{\left (d^2 \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{5/2}} \, dx}{(d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{\left (2 c d^2 \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac{x \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{5/2}} \, dx}{(d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{\left (c^2 d^2 \left (1-c^2 x^2\right )^{5/2}\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+c d x)^{5/2} (e-c e x)^{5/2}}\\ &=\frac{2 d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{d^2 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{c^2 d^2 x^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{\left (2 d^2 \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{\left (4 b d^2 \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac{a+b \sin ^{-1}(c x)}{\left (1-c^2 x^2\right )^2} \, dx}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{\left (2 b c d^2 \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac{x \left (a+b \sin ^{-1}(c x)\right )}{\left (1-c^2 x^2\right )^2} \, dx}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{\left (2 b c^3 d^2 \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac{x^3 \left (a+b \sin ^{-1}(c x)\right )}{\left (1-c^2 x^2\right )^2} \, dx}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}\\ &=-\frac{b d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{2 b d^2 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{b c d^2 x^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{2 d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{d^2 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{c^2 d^2 x^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{2 d^2 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{\left (2 b d^2 \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac{a+b \sin ^{-1}(c x)}{1-c^2 x^2} \, dx}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{\left (b^2 d^2 \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac{1}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{\left (2 b c d^2 \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac{x \left (a+b \sin ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{\left (4 b c d^2 \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac{x \left (a+b \sin ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{\left (2 b^2 c d^2 \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac{x}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{\left (b^2 c^2 d^2 \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac{x^2}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}\\ &=\frac{2 b^2 d^2 \left (1-c^2 x^2\right )^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{2 b^2 d^2 x \left (1-c^2 x^2\right )^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{b d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{2 b d^2 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{b c d^2 x^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{2 d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{d^2 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{c^2 d^2 x^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{2 d^2 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{\left (b^2 d^2 \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{\left (2 b d^2 \left (1-c^2 x^2\right )^{5/2}\right ) \operatorname{Subst}\left (\int (a+b x) \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{\left (2 b d^2 \left (1-c^2 x^2\right )^{5/2}\right ) \operatorname{Subst}\left (\int (a+b x) \tan (x) \, dx,x,\sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{\left (4 b d^2 \left (1-c^2 x^2\right )^{5/2}\right ) \operatorname{Subst}\left (\int (a+b x) \tan (x) \, dx,x,\sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}\\ &=\frac{2 b^2 d^2 \left (1-c^2 x^2\right )^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{2 b^2 d^2 x \left (1-c^2 x^2\right )^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{b^2 d^2 \left (1-c^2 x^2\right )^{5/2} \sin ^{-1}(c x)}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{b d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{2 b d^2 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{b c d^2 x^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{2 d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{d^2 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{c^2 d^2 x^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{2 d^2 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{i d^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{4 i b d^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{\left (4 i b d^2 \left (1-c^2 x^2\right )^{5/2}\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+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{\left (8 i b d^2 \left (1-c^2 x^2\right )^{5/2}\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+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{\left (2 b^2 d^2 \left (1-c^2 x^2\right )^{5/2}\right ) \operatorname{Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{\left (2 b^2 d^2 \left (1-c^2 x^2\right )^{5/2}\right ) \operatorname{Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}\\ &=\frac{2 b^2 d^2 \left (1-c^2 x^2\right )^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{2 b^2 d^2 x \left (1-c^2 x^2\right )^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{b^2 d^2 \left (1-c^2 x^2\right )^{5/2} \sin ^{-1}(c x)}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{b d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{2 b d^2 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{b c d^2 x^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{2 d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{d^2 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{c^2 d^2 x^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{2 d^2 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{i d^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{4 i b d^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{2 b d^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{\left (2 i b^2 d^2 \left (1-c^2 x^2\right )^{5/2}\right ) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{\left (2 i b^2 d^2 \left (1-c^2 x^2\right )^{5/2}\right ) \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{\left (2 b^2 d^2 \left (1-c^2 x^2\right )^{5/2}\right ) \operatorname{Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{\left (4 b^2 d^2 \left (1-c^2 x^2\right )^{5/2}\right ) \operatorname{Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}\\ &=\frac{2 b^2 d^2 \left (1-c^2 x^2\right )^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{2 b^2 d^2 x \left (1-c^2 x^2\right )^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{b^2 d^2 \left (1-c^2 x^2\right )^{5/2} \sin ^{-1}(c x)}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{b d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{2 b d^2 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{b c d^2 x^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{2 d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{d^2 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{c^2 d^2 x^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{2 d^2 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{i d^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{4 i b d^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{2 b d^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{2 i b^2 d^2 \left (1-c^2 x^2\right )^{5/2} \text{Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{2 i b^2 d^2 \left (1-c^2 x^2\right )^{5/2} \text{Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{\left (i b^2 d^2 \left (1-c^2 x^2\right )^{5/2}\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{\left (2 i b^2 d^2 \left (1-c^2 x^2\right )^{5/2}\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}\\ &=\frac{2 b^2 d^2 \left (1-c^2 x^2\right )^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{2 b^2 d^2 x \left (1-c^2 x^2\right )^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{b^2 d^2 \left (1-c^2 x^2\right )^{5/2} \sin ^{-1}(c x)}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{b d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{2 b d^2 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{b c d^2 x^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{2 d^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{d^2 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{c^2 d^2 x^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{2 d^2 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{i d^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{4 i b d^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{2 b d^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{2 i b^2 d^2 \left (1-c^2 x^2\right )^{5/2} \text{Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}+\frac{2 i b^2 d^2 \left (1-c^2 x^2\right )^{5/2} \text{Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}-\frac{i b^2 d^2 \left (1-c^2 x^2\right )^{5/2} \text{Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )}{3 c (d+c d x)^{5/2} (e-c e x)^{5/2}}\\ \end{align*}

Mathematica [A]  time = 6.60171, size = 388, normalized size = 0.43 \[ \frac{\sqrt{c d x+d} \sqrt{e-c e x} \left (\frac{b^2 \left (-8 i \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )+4 \tan \left (\frac{1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right )+\sin ^{-1}(c x) \left (-2 \sec ^2\left (\frac{1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right )+8 \log \left (1+i e^{i \sin ^{-1}(c x)}\right )\right )+\sin ^{-1}(c x)^2 \left (\tan \left (\frac{1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right ) \left (\sec ^2\left (\frac{1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right )+2\right )-2 i\right )\right )}{\sqrt{1-c^2 x^2}}-\frac{2 a^2 (c x-2)}{(c x-1)^2}+\frac{2 a b \left (2 \sin \left (\frac{1}{2} \sin ^{-1}(c x)\right ) \left (-\left (\sqrt{1-c^2 x^2}-1\right ) \sin ^{-1}(c x)-2 \left (\sqrt{1-c^2 x^2}+2\right ) \log \left (\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )-\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )+1\right )+\cos \left (\frac{3}{2} \sin ^{-1}(c x)\right ) \left (\sin ^{-1}(c x)-2 \log \left (\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )-\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )\right )+\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right ) \left (3 \sin ^{-1}(c x)+6 \log \left (\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )-\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )-2\right )\right )}{\sqrt{1-c^2 x^2} \left (\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )-\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )^3}\right )}{6 c d e^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*((-2*a^2*(-2 + c*x))/(-1 + c*x)^2 + (2*a*b*(Cos[(3*ArcSin[c*x])/2]*(ArcSin[c*
x] - 2*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]]) + Cos[ArcSin[c*x]/2]*(-2 + 3*ArcSin[c*x] + 6*Log[Cos[ArcS
in[c*x]/2] - Sin[ArcSin[c*x]/2]]) + 2*(1 - (-1 + Sqrt[1 - c^2*x^2])*ArcSin[c*x] - 2*(2 + Sqrt[1 - c^2*x^2])*Lo
g[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]])*Sin[ArcSin[c*x]/2]))/(Sqrt[1 - c^2*x^2]*(Cos[ArcSin[c*x]/2] - Sin[
ArcSin[c*x]/2])^3) + (b^2*((-8*I)*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] + ArcSin[c*x]*(8*Log[1 + I*E^(I*ArcSin[c*
x])] - 2*Sec[(Pi + 2*ArcSin[c*x])/4]^2) + 4*Tan[(Pi + 2*ArcSin[c*x])/4] + ArcSin[c*x]^2*(-2*I + (2 + Sec[(Pi +
2*ArcSin[c*x])/4]^2)*Tan[(Pi + 2*ArcSin[c*x])/4])))/Sqrt[1 - c^2*x^2]))/(6*c*d*e^3)

________________________________________________________________________________________

Maple [F]  time = 0.257, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+b\arcsin \left ( cx \right ) \right ) ^{2}{\frac{1}{\sqrt{cdx+d}}} \left ( -cex+e \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsin(c*x))^2/(c*d*x+d)^(1/2)/(-c*e*x+e)^(5/2),x)

[Out]

int((a+b*arcsin(c*x))^2/(c*d*x+d)^(1/2)/(-c*e*x+e)^(5/2),x)

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(c*x))^2/(c*d*x+d)^(1/2)/(-c*e*x+e)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(c*x))^2/(c*d*x+d)^(1/2)/(-c*e*x+e)^(5/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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((a+b*asin(c*x))**2/(c*d*x+d)**(1/2)/(-c*e*x+e)**(5/2),x)

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(c*x))^2/(c*d*x+d)^(1/2)/(-c*e*x+e)^(5/2),x, algorithm="giac")

[Out]

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