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

Optimal. Leaf size=505 \[ -\frac{i b^2 c d e \sqrt{c d x+d} \sqrt{e-c e x} \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}+\frac{3 b c^3 d e x^2 \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt{1-c^2 x^2}}-\frac{c d e \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^3}{2 b \sqrt{1-c^2 x^2}}-\frac{i c d e \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{1-c^2 x^2}}+b c d e \sqrt{1-c^2 x^2} \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )-\frac{d e \left (1-c^2 x^2\right ) \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{x}+\frac{2 b c d e \sqrt{c d x+d} \sqrt{e-c e x} \log \left (1-e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}-\frac{3}{2} c^2 d e x \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{5 b^2 c d e \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)}{4 \sqrt{1-c^2 x^2}}+\frac{1}{4} b^2 c^2 d e x \sqrt{c d x+d} \sqrt{e-c e x} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.80743, antiderivative size = 505, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 14, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {4739, 4695, 4647, 4641, 4627, 321, 216, 4683, 4625, 3717, 2190, 2279, 2391, 195} \[ -\frac{i b^2 c d e \sqrt{c d x+d} \sqrt{e-c e x} \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt{1-c^2 x^2}}+\frac{3 b c^3 d e x^2 \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt{1-c^2 x^2}}-\frac{c d e \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^3}{2 b \sqrt{1-c^2 x^2}}-\frac{i c d e \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{1-c^2 x^2}}+b c d e \sqrt{1-c^2 x^2} \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )-\frac{d e \left (1-c^2 x^2\right ) \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{x}+\frac{2 b c d e \sqrt{c d x+d} \sqrt{e-c e x} \log \left (1-e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}-\frac{3}{2} c^2 d e x \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{5 b^2 c d e \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)}{4 \sqrt{1-c^2 x^2}}+\frac{1}{4} b^2 c^2 d e x \sqrt{c d x+d} \sqrt{e-c e x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4739

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((h_.)*(x_))^(m_.)*((d_) + (e_.)*(x_))^(p_)*((f_) + (g_.)*(x_))^(
q_), x_Symbol] :> Dist[((-((d^2*g)/e))^IntPart[q]*(d + e*x)^FracPart[q]*(f + g*x)^FracPart[q])/(1 - c^2*x^2)^F
racPart[q], Int[(h*x)^m*(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, h, m, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^2 - e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 0]

Rule 4695

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

Rule 4647

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

Rule 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 4627

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

Rule 321

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

Rule 216

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

Rule 4683

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

Rule 4625

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Subst[Int[(a + b*x)^n/Tan[x], x], x, ArcSin[c*
x]] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]

Rule 3717

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (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*k*Pi)*E^(2*I*(e + f*x)))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x)))
, x], x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && 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 195

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

Rubi steps

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

Mathematica [A]  time = 2.26623, size = 538, normalized size = 1.07 \[ \frac{-8 i b^2 c d e x \sqrt{c d x+d} \sqrt{e-c e x} \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )+12 a^2 c d^{3/2} e^{3/2} x \sqrt{1-c^2 x^2} \tan ^{-1}\left (\frac{c x \sqrt{c d x+d} \sqrt{e-c e x}}{\sqrt{d} \sqrt{e} \left (c^2 x^2-1\right )}\right )-4 a^2 c^2 d e x^2 \sqrt{1-c^2 x^2} \sqrt{c d x+d} \sqrt{e-c e x}-8 a^2 d e \sqrt{1-c^2 x^2} \sqrt{c d x+d} \sqrt{e-c e x}-2 b d e \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)^2 \left (6 a c x+4 b \sqrt{1-c^2 x^2}+4 i b c x+b c x \sin \left (2 \sin ^{-1}(c x)\right )\right )-2 b d e \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x) \left (8 a \sqrt{1-c^2 x^2}+2 a c x \sin \left (2 \sin ^{-1}(c x)\right )-8 b c x \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )+b c x \cos \left (2 \sin ^{-1}(c x)\right )\right )+16 a b c d e x \sqrt{c d x+d} \sqrt{e-c e x} \log (c x)-2 a b c d e x \sqrt{c d x+d} \sqrt{e-c e x} \cos \left (2 \sin ^{-1}(c x)\right )-4 b^2 c d e x \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)^3+b^2 c d e x \sqrt{c d x+d} \sqrt{e-c e x} \sin \left (2 \sin ^{-1}(c x)\right )}{8 x \sqrt{1-c^2 x^2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-8*a^2*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[1 - c^2*x^2] - 4*a^2*c^2*d*e*x^2*Sqrt[d + c*d*x]*Sqrt[e - c*e
*x]*Sqrt[1 - c^2*x^2] - 4*b^2*c*d*e*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]^3 + 12*a^2*c*d^(3/2)*e^(3/2)
*x*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/(Sqrt[d]*Sqrt[e]*(-1 + c^2*x^2))] - 2*a*b*c*
d*e*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Cos[2*ArcSin[c*x]] + 16*a*b*c*d*e*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Log[
c*x] - (8*I)*b^2*c*d*e*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*PolyLog[2, E^((2*I)*ArcSin[c*x])] + b^2*c*d*e*x*Sqrt[
d + c*d*x]*Sqrt[e - c*e*x]*Sin[2*ArcSin[c*x]] - 2*b*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]*(8*a*Sqrt[
1 - c^2*x^2] + b*c*x*Cos[2*ArcSin[c*x]] - 8*b*c*x*Log[1 - E^((2*I)*ArcSin[c*x])] + 2*a*c*x*Sin[2*ArcSin[c*x]])
- 2*b*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]^2*(6*a*c*x + (4*I)*b*c*x + 4*b*Sqrt[1 - c^2*x^2] + b*c*
x*Sin[2*ArcSin[c*x]]))/(8*x*Sqrt[1 - c^2*x^2])

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Maple [F]  time = 0.419, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\arcsin \left ( cx \right ) \right ) ^{2}}{{x}^{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)*(-c*e*x+e)^(3/2)*(a+b*arcsin(c*x))^2/x^2,x)

[Out]

int((c*d*x+d)^(3/2)*(-c*e*x+e)^(3/2)*(a+b*arcsin(c*x))^2/x^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((c*d*x+d)^(3/2)*(-c*e*x+e)^(3/2)*(a+b*arcsin(c*x))^2/x^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 (a^{2} c^{2} d e x^{2} - a^{2} d e +{\left (b^{2} c^{2} d e x^{2} - b^{2} d e\right )} \arcsin \left (c x\right )^{2} + 2 \,{\left (a b c^{2} d e x^{2} - a b d e\right )} \arcsin \left (c x\right )\right )} \sqrt{c d x + d} \sqrt{-c e x + e}}{x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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