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

Optimal. Leaf size=713 \[ -\frac{8 i b^2 d^3 \left (1-c^2 x^2\right )^{3/2} \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}+\frac{8 i b^2 d^3 \left (1-c^2 x^2\right )^{3/2} \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac{4 i b^2 d^3 \left (1-c^2 x^2\right )^{3/2} \text{PolyLog}\left (2,-e^{2 i \sin ^{-1}(c x)}\right )}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac{2 a b d^3 x \left (1-c^2 x^2\right )^{3/2}}{(c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac{d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^3}{b c (c d x+d)^{3/2} (e-c e x)^{3/2}}+\frac{d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac{4 i d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}+\frac{4 d^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{(c d x+d)^{3/2} (e-c e x)^{3/2}}+\frac{4 d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}+\frac{8 b d^3 \left (1-c^2 x^2\right )^{3/2} \log \left (1+e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}+\frac{16 i b d^3 \left (1-c^2 x^2\right )^{3/2} \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac{2 b^2 d^3 \left (1-c^2 x^2\right )^2}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac{2 b^2 d^3 x \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x)}{(c d x+d)^{3/2} (e-c e x)^{3/2}} \]

[Out]

(-2*a*b*d^3*x*(1 - c^2*x^2)^(3/2))/((d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - (2*b^2*d^3*(1 - c^2*x^2)^2)/(c*(d +
c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - (2*b^2*d^3*x*(1 - c^2*x^2)^(3/2)*ArcSin[c*x])/((d + c*d*x)^(3/2)*(e - c*e*x
)^(3/2)) + (4*d^3*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) + (4*d^3*x*(1 -
c^2*x^2)*(a + b*ArcSin[c*x])^2)/((d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - ((4*I)*d^3*(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)) + (d^3*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/(c*(d +
c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - (d^3*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^3)/(b*c*(d + c*d*x)^(3/2)*(e -
c*e*x)^(3/2)) + ((16*I)*b*d^3*(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)) + (8*b*d^3*(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)) - ((8*I)*b^2*d^3*(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)) + ((8*I)*b^2*d^3*(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*d^3*(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 = 1.04947, antiderivative size = 713, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 15, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.469, Rules used = {4673, 4775, 4763, 4651, 4675, 3719, 2190, 2279, 2391, 4677, 4657, 4181, 4641, 4619, 261} \[ -\frac{8 i b^2 d^3 \left (1-c^2 x^2\right )^{3/2} \text{PolyLog}\left (2,-i e^{i \sin ^{-1}(c x)}\right )}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}+\frac{8 i b^2 d^3 \left (1-c^2 x^2\right )^{3/2} \text{PolyLog}\left (2,i e^{i \sin ^{-1}(c x)}\right )}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac{4 i b^2 d^3 \left (1-c^2 x^2\right )^{3/2} \text{PolyLog}\left (2,-e^{2 i \sin ^{-1}(c x)}\right )}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac{2 a b d^3 x \left (1-c^2 x^2\right )^{3/2}}{(c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac{d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^3}{b c (c d x+d)^{3/2} (e-c e x)^{3/2}}+\frac{d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac{4 i d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}+\frac{4 d^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{(c d x+d)^{3/2} (e-c e x)^{3/2}}+\frac{4 d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}+\frac{8 b d^3 \left (1-c^2 x^2\right )^{3/2} \log \left (1+e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}+\frac{16 i b d^3 \left (1-c^2 x^2\right )^{3/2} \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac{2 b^2 d^3 \left (1-c^2 x^2\right )^2}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac{2 b^2 d^3 x \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x)}{(c d x+d)^{3/2} (e-c e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a*b*d^3*x*(1 - c^2*x^2)^(3/2))/((d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - (2*b^2*d^3*(1 - c^2*x^2)^2)/(c*(d +
c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - (2*b^2*d^3*x*(1 - c^2*x^2)^(3/2)*ArcSin[c*x])/((d + c*d*x)^(3/2)*(e - c*e*x
)^(3/2)) + (4*d^3*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) + (4*d^3*x*(1 -
c^2*x^2)*(a + b*ArcSin[c*x])^2)/((d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - ((4*I)*d^3*(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)) + (d^3*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/(c*(d +
c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - (d^3*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^3)/(b*c*(d + c*d*x)^(3/2)*(e -
c*e*x)^(3/2)) + ((16*I)*b*d^3*(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)) + (8*b*d^3*(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)) - ((8*I)*b^2*d^3*(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)) + ((8*I)*b^2*d^3*(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*d^3*(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]

Rule 4619

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

Rubi steps

\begin{align*} \int \frac{(d+c d x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{(e-c e x)^{3/2}} \, dx &=\frac{\left (1-c^2 x^2\right )^{3/2} \int \frac{(d+c d x)^3 \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{4 \left (d^3+c d^3 x\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{3/2}}-\frac{3 d^3 \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{1-c^2 x^2}}-\frac{c d^3 x \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 (4 \left (1-c^2 x^2\right )^{3/2}\right ) \int \frac{\left (d^3+c d^3 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 (3 d^3 \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{\left (c d^3 \left (1-c^2 x^2\right )^{3/2}\right ) \int \frac{x \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{d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac{d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^3}{b c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac{\left (4 \left (1-c^2 x^2\right )^{3/2}\right ) \int \left (\frac{d^3 \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{3/2}}+\frac{c d^3 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{\left (2 b d^3 \left (1-c^2 x^2\right )^{3/2}\right ) \int \left (a+b \sin ^{-1}(c x)\right ) \, dx}{(d+c d x)^{3/2} (e-c e x)^{3/2}}\\ &=-\frac{2 a b d^3 x \left (1-c^2 x^2\right )^{3/2}}{(d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac{d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac{d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^3}{b c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac{\left (4 d^3 \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 b^2 d^3 \left (1-c^2 x^2\right )^{3/2}\right ) \int \sin ^{-1}(c x) \, dx}{(d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac{\left (4 c d^3 \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 a b d^3 x \left (1-c^2 x^2\right )^{3/2}}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac{2 b^2 d^3 x \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x)}{(d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac{4 d^3 \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{4 d^3 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{d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac{d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^3}{b c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac{\left (8 b d^3 \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 (8 b c d^3 \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{\left (2 b^2 c d^3 \left (1-c^2 x^2\right )^{3/2}\right ) \int \frac{x}{\sqrt{1-c^2 x^2}} \, dx}{(d+c d x)^{3/2} (e-c e x)^{3/2}}\\ &=-\frac{2 a b d^3 x \left (1-c^2 x^2\right )^{3/2}}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac{2 b^2 d^3 \left (1-c^2 x^2\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac{2 b^2 d^3 x \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x)}{(d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac{4 d^3 \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{4 d^3 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{d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac{d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^3}{b c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac{\left (8 b d^3 \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 (8 b d^3 \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 a b d^3 x \left (1-c^2 x^2\right )^{3/2}}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac{2 b^2 d^3 \left (1-c^2 x^2\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac{2 b^2 d^3 x \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x)}{(d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac{4 d^3 \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{4 d^3 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{4 i d^3 \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{d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac{d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^3}{b c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac{16 i b d^3 \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 (16 i b d^3 \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 (8 b^2 d^3 \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 (8 b^2 d^3 \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 a b d^3 x \left (1-c^2 x^2\right )^{3/2}}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac{2 b^2 d^3 \left (1-c^2 x^2\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac{2 b^2 d^3 x \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x)}{(d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac{4 d^3 \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{4 d^3 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{4 i d^3 \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{d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac{d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^3}{b c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac{16 i b d^3 \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{8 b d^3 \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 (8 i b^2 d^3 \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 (8 i b^2 d^3 \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 (8 b^2 d^3 \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 a b d^3 x \left (1-c^2 x^2\right )^{3/2}}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac{2 b^2 d^3 \left (1-c^2 x^2\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac{2 b^2 d^3 x \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x)}{(d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac{4 d^3 \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{4 d^3 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{4 i d^3 \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{d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac{d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^3}{b c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac{16 i b d^3 \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{8 b d^3 \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{8 i b^2 d^3 \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{8 i b^2 d^3 \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 (4 i b^2 d^3 \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 a b d^3 x \left (1-c^2 x^2\right )^{3/2}}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac{2 b^2 d^3 \left (1-c^2 x^2\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac{2 b^2 d^3 x \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x)}{(d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac{4 d^3 \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{4 d^3 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{4 i d^3 \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{d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac{d^3 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^3}{b c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac{16 i b d^3 \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{8 b d^3 \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{8 i b^2 d^3 \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{8 i b^2 d^3 \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 d^3 \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 = 10.44, size = 1247, normalized size = 1.75 \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[-(e*(-1 + c*x))]*Sqrt[d*(1 + c*x)]*((a^2*d)/e^2 - (4*a^2*d)/(e^2*(-1 + c*x))))/c + (3*a^2*d^(3/2)*ArcTan
[(c*x*Sqrt[-(e*(-1 + c*x))]*Sqrt[d*(1 + c*x)])/(Sqrt[d]*Sqrt[e]*(-1 + c*x)*(1 + c*x))])/(c*e^(3/2)) - (a*b*d*(
1 + c*x)*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[-(d*e*(1 - c^2*x^2))]*(Cos[ArcSin[c*x]/2]*((-4 + ArcSin[c*x])*Ar
cSin[c*x] - 8*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]]) - (ArcSin[c*x]*(4 + ArcSin[c*x]) - 8*Log[Cos[ArcSi
n[c*x]/2] - Sin[ArcSin[c*x]/2]])*Sin[ArcSin[c*x]/2]))/(c*e^2*Sqrt[-((d + c*d*x)*(e - c*e*x))]*Sqrt[1 - c^2*x^2
]*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^2) + (2*a*b*d*(1 + c*x)*
Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[-(d*e*(1 - c^2*x^2))]*(Cos[ArcSin[c*x]/2]*(-(c*x) + 2*ArcSin[c*x] + Sqrt[
1 - c^2*x^2]*ArcSin[c*x] - ArcSin[c*x]^2 + 4*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]]) + (c*x + 2*ArcSin[c
*x] - Sqrt[1 - c^2*x^2]*ArcSin[c*x] + ArcSin[c*x]^2 - 4*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]])*Sin[ArcS
in[c*x]/2]))/(c*e^2*Sqrt[-((d + c*d*x)*(e - c*e*x))]*Sqrt[1 - c^2*x^2]*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2
])*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^2) - (b^2*d*(1 + c*x)*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[-(d*e*
(1 - c^2*x^2))]*((-18*I)*Pi*ArcSin[c*x] - (6 - 6*I)*ArcSin[c*x]^2 + ArcSin[c*x]^3 - 24*Pi*Log[1 + E^((-I)*ArcS
in[c*x])] + 12*(Pi - 2*ArcSin[c*x])*Log[1 + I*E^(I*ArcSin[c*x])] + 24*Pi*Log[Cos[ArcSin[c*x]/2]] - 12*Pi*Log[-
Cos[(Pi + 2*ArcSin[c*x])/4]] + (24*I)*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] - (12*ArcSin[c*x]^2*Sin[ArcSin[c*x]/2
])/(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])))/(3*c*e^2*Sqrt[-((d + c*d*x)*(e - c*e*x))]*Sqrt[1 - c^2*x^2]*(Co
s[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^2) - (b^2*d*(1 + c*x)*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[-(d*e*(1 - c
^2*x^2))]*(6 + (6*c*x*ArcSin[c*x])/Sqrt[1 - c^2*x^2] - 3*ArcSin[c*x]^2 - ((6 - 6*I)*ArcSin[c*x]^2)/Sqrt[1 - c^
2*x^2] + (2*ArcSin[c*x]^3)/Sqrt[1 - c^2*x^2] + (6*((-3*I)*Pi*ArcSin[c*x] - 4*Pi*Log[1 + E^((-I)*ArcSin[c*x])]
+ 2*(Pi - 2*ArcSin[c*x])*Log[1 + I*E^(I*ArcSin[c*x])] + 4*Pi*Log[Cos[ArcSin[c*x]/2]] - 2*Pi*Log[-Cos[(Pi + 2*A
rcSin[c*x])/4]] + (4*I)*PolyLog[2, (-I)*E^(I*ArcSin[c*x])]))/Sqrt[1 - c^2*x^2] - (12*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*e^2*Sqrt[-((d + c*d*x)*(e - c*e*
x))]*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^2)

________________________________________________________________________________________

Maple [F]  time = 0.202, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+b\arcsin \left ( cx \right ) \right ) ^{2} \left ( cdx+d \right ) ^{{\frac{3}{2}}} \left ( -cex+e \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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