3.544 \(\int \frac{\sqrt{e-c e x} (a+b \sin ^{-1}(c x))^2}{(d+c d x)^{3/2}} \, dx\)

Optimal. Leaf size=530 \[ \frac{4 i b^2 e^2 \left (1-c^2 x^2\right )^{3/2} \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac{4 i b^2 e^2 \left (1-c^2 x^2\right )^{3/2} \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac{2 i b^2 e^2 \left (1-c^2 x^2\right )^{3/2} \text{PolyLog}\left (2,-e^{2 i \sin ^{-1}(c x)}\right )}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac{e^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac{2 i e^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac{2 e^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}+\frac{2 e^2 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{(c d x+d)^{3/2} (e-c e x)^{3/2}}+\frac{4 b e^2 \left (1-c^2 x^2\right )^{3/2} \log \left (1+e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac{8 i b e^2 \left (1-c^2 x^2\right )^{3/2} \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{c (c d x+d)^{3/2} (e-c e x)^{3/2}} \]

[Out]

(-2*e^2*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) + (2*e^2*x*(1 - c^2*x^2)*
(a + b*ArcSin[c*x])^2)/((d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - ((2*I)*e^2*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*
x])^2)/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - (e^2*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^3)/(3*b*c*(d + c
*d*x)^(3/2)*(e - c*e*x)^(3/2)) - ((8*I)*b*e^2*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])
])/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) + (4*b*e^2*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])*Log[1 + E^((2*I)
*ArcSin[c*x])])/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) + ((4*I)*b^2*e^2*(1 - c^2*x^2)^(3/2)*PolyLog[2, (-I)*E
^(I*ArcSin[c*x])])/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - ((4*I)*b^2*e^2*(1 - c^2*x^2)^(3/2)*PolyLog[2, I*E
^(I*ArcSin[c*x])])/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - ((2*I)*b^2*e^2*(1 - c^2*x^2)^(3/2)*PolyLog[2, -E^
((2*I)*ArcSin[c*x])])/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2))

________________________________________________________________________________________

Rubi [A]  time = 0.947575, antiderivative size = 530, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 13, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.406, Rules used = {4673, 4775, 4763, 4651, 4675, 3719, 2190, 2279, 2391, 4677, 4657, 4181, 4641} \[ \frac{4 i b^2 e^2 \left (1-c^2 x^2\right )^{3/2} \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac{4 i b^2 e^2 \left (1-c^2 x^2\right )^{3/2} \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac{2 i b^2 e^2 \left (1-c^2 x^2\right )^{3/2} \text{PolyLog}\left (2,-e^{2 i \sin ^{-1}(c x)}\right )}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac{e^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac{2 i e^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac{2 e^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}+\frac{2 e^2 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{(c d x+d)^{3/2} (e-c e x)^{3/2}}+\frac{4 b e^2 \left (1-c^2 x^2\right )^{3/2} \log \left (1+e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac{8 i b e^2 \left (1-c^2 x^2\right )^{3/2} \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{c (c d x+d)^{3/2} (e-c e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*e^2*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) + (2*e^2*x*(1 - c^2*x^2)*
(a + b*ArcSin[c*x])^2)/((d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - ((2*I)*e^2*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*
x])^2)/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - (e^2*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^3)/(3*b*c*(d + c
*d*x)^(3/2)*(e - c*e*x)^(3/2)) - ((8*I)*b*e^2*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])
])/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) + (4*b*e^2*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])*Log[1 + E^((2*I)
*ArcSin[c*x])])/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) + ((4*I)*b^2*e^2*(1 - c^2*x^2)^(3/2)*PolyLog[2, (-I)*E
^(I*ArcSin[c*x])])/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - ((4*I)*b^2*e^2*(1 - c^2*x^2)^(3/2)*PolyLog[2, I*E
^(I*ArcSin[c*x])])/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - ((2*I)*b^2*e^2*(1 - c^2*x^2)^(3/2)*PolyLog[2, -E^
((2*I)*ArcSin[c*x])])/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2))

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 4775

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

Rubi steps

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

Mathematica [A]  time = 3.71217, size = 547, normalized size = 1.03 \[ \frac{\frac{b^2 \sqrt{c d x+d} \sqrt{e-c e x} \left (-24 i \left (\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )+\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right ) \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )+\sin ^{-1}(c x)^3 \left (-\left (\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )+\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )\right )-(6+6 i) \sin ^{-1}(c x)^2 \left (\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )+i \sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )+6 \sin ^{-1}(c x) \left (4 \log \left (1-i e^{i \sin ^{-1}(c x)}\right )+i \pi \right ) \left (\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )+\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )+12 \pi \left (\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )+\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right ) \left (2 \log \left (1+e^{-i \sin ^{-1}(c x)}\right )+\log \left (1-i e^{i \sin ^{-1}(c x)}\right )-\log \left (\sin \left (\frac{1}{4} \left (2 \sin ^{-1}(c x)+\pi \right )\right )\right )-2 \log \left (\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )\right )\right )}{\sqrt{1-c^2 x^2} \left (\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )+\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )}+3 a^2 \sqrt{d} \sqrt{e} \tan ^{-1}\left (\frac{c x \sqrt{c d x+d} \sqrt{e-c e x}}{\sqrt{d} \sqrt{e} \left (c^2 x^2-1\right )}\right )-\frac{6 a^2 \sqrt{c d x+d} \sqrt{e-c e x}}{c x+1}-\frac{3 a b \sqrt{c d x+d} \sqrt{e-c e x} \left (\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right ) \left (\sin ^{-1}(c x) \left (\sin ^{-1}(c x)+4\right )-8 \log \left (\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )+\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )\right )+\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right ) \left (\left (\sin ^{-1}(c x)-4\right ) \sin ^{-1}(c x)-8 \log \left (\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )+\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )\right )\right )}{\sqrt{1-c^2 x^2} \left (\sin \left (\frac{1}{2} \sin ^{-1}(c x)\right )+\cos \left (\frac{1}{2} \sin ^{-1}(c x)\right )\right )}}{3 c d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-6*a^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/(1 + c*x) + 3*a^2*Sqrt[d]*Sqrt[e]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[e
- c*e*x])/(Sqrt[d]*Sqrt[e]*(-1 + c^2*x^2))] - (3*a*b*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(Cos[ArcSin[c*x]/2]*(Arc
Sin[c*x]*(4 + ArcSin[c*x]) - 8*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]]) + ((-4 + ArcSin[c*x])*ArcSin[c*x]
- 8*Log[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])) + (b^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*((-6 - 6*I)*ArcSin[c*x]^2*(Cos[ArcSin[c*x]/2] +
I*Sin[ArcSin[c*x]/2]) - ArcSin[c*x]^3*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]) + 6*ArcSin[c*x]*(I*Pi + 4*Log
[1 - I*E^(I*ArcSin[c*x])])*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]) + 12*Pi*(2*Log[1 + E^((-I)*ArcSin[c*x])]
+ Log[1 - I*E^(I*ArcSin[c*x])] - 2*Log[Cos[ArcSin[c*x]/2]] - Log[Sin[(Pi + 2*ArcSin[c*x])/4]])*(Cos[ArcSin[c*x
]/2] + Sin[ArcSin[c*x]/2]) - (24*I)*PolyLog[2, I*E^(I*ArcSin[c*x])]*(Cos[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*c*d^2)

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Maple [F]  time = 0.214, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+b\arcsin \left ( cx \right ) \right ) ^{2}\sqrt{-cex+e} \left ( cdx+d \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsin(c*x))^2*(-c*e*x+e)^(1/2)/(c*d*x+d)^(3/2),x)

[Out]

int((a+b*arcsin(c*x))^2*(-c*e*x+e)^(1/2)/(c*d*x+d)^(3/2),x)

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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*e*x+e)^(1/2)/(c*d*x+d)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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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^{2} d^{2} x^{2} + 2 \, c d^{2} x + d^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(c*x))^2*(-c*e*x+e)^(1/2)/(c*d*x+d)^(3/2),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^2*d^2*x^2 + 2*c*d^2
*x + d^2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{- e \left (c x - 1\right )} \left (a + b \operatorname{asin}{\left (c x \right )}\right )^{2}}{\left (d \left (c x + 1\right )\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*asin(c*x))**2*(-c*e*x+e)**(1/2)/(c*d*x+d)**(3/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(c*x))^2*(-c*e*x+e)^(1/2)/(c*d*x+d)^(3/2),x, algorithm="giac")

[Out]

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