\(\int \frac {(d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2}{x} \, dx\) [584]

Optimal result

Integrand size = 35, antiderivative size = 647 \[ \int \frac {(d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2}{x} \, dx=-\frac {22}{9} b^2 d e \sqrt {d+c d x} \sqrt {e-c e x}-\frac {2 a b c d e x \sqrt {d+c d x} \sqrt {e-c e x}}{\sqrt {1-c^2 x^2}}-\frac {2}{27} b^2 d e \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right )-\frac {2 b^2 c d e x \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x)}{\sqrt {1-c^2 x^2}}-\frac {2 b c d e x \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{3 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d e x^3 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{9 \sqrt {1-c^2 x^2}}+d e \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2+\frac {1}{3} d e \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-\frac {2 d e \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2 \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {2 i b d e \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {2 i b d e \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {2 b^2 d e \sqrt {d+c d x} \sqrt {e-c e x} \operatorname {PolyLog}\left (3,-e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {2 b^2 d e \sqrt {d+c d x} \sqrt {e-c e x} \operatorname {PolyLog}\left (3,e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}} \]

[Out]

-22/9*b^2*d*e*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)-2/27*b^2*d*e*(-c^2*x^2+1)*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)+d*e*
(a+b*arcsin(c*x))^2*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)+1/3*d*e*(-c^2*x^2+1)*(a+b*arcsin(c*x))^2*(c*d*x+d)^(1/2)*
(-c*e*x+e)^(1/2)-2*a*b*c*d*e*x*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)/(-c^2*x^2+1)^(1/2)-2*b^2*c*d*e*x*arcsin(c*x)*(
c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)/(-c^2*x^2+1)^(1/2)-2/3*b*c*d*e*x*(a+b*arcsin(c*x))*(c*d*x+d)^(1/2)*(-c*e*x+e)^
(1/2)/(-c^2*x^2+1)^(1/2)+2/9*b*c^3*d*e*x^3*(a+b*arcsin(c*x))*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)/(-c^2*x^2+1)^(1/
2)-2*d*e*(a+b*arcsin(c*x))^2*arctanh(I*c*x+(-c^2*x^2+1)^(1/2))*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)/(-c^2*x^2+1)^(
1/2)+2*I*b*d*e*(a+b*arcsin(c*x))*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)/(-c^2*x
^2+1)^(1/2)-2*I*b*d*e*(a+b*arcsin(c*x))*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)/(
-c^2*x^2+1)^(1/2)-2*b^2*d*e*polylog(3,-I*c*x-(-c^2*x^2+1)^(1/2))*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)/(-c^2*x^2+1)
^(1/2)+2*b^2*d*e*polylog(3,I*c*x+(-c^2*x^2+1)^(1/2))*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)/(-c^2*x^2+1)^(1/2)

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 647, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.371, Rules used = {4823, 4787, 4783, 4803, 4268, 2611, 2320, 6724, 4715, 267, 4739, 455, 45} \[ \int \frac {(d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2}{x} \, dx=-\frac {2 d e \sqrt {c d x+d} \sqrt {e-c e x} \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}+\frac {2 i b d e \sqrt {c d x+d} \sqrt {e-c e x} \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}-\frac {2 i b d e \sqrt {c d x+d} \sqrt {e-c e x} \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}-\frac {2 b c d e x \sqrt {c d x+d} \sqrt {e-c e x} (a+b \arcsin (c x))}{3 \sqrt {1-c^2 x^2}}+\frac {1}{3} d e \left (1-c^2 x^2\right ) \sqrt {c d x+d} \sqrt {e-c e x} (a+b \arcsin (c x))^2+\frac {2 b c^3 d e x^3 \sqrt {c d x+d} \sqrt {e-c e x} (a+b \arcsin (c x))}{9 \sqrt {1-c^2 x^2}}+d e \sqrt {c d x+d} \sqrt {e-c e x} (a+b \arcsin (c x))^2-\frac {2 a b c d e x \sqrt {c d x+d} \sqrt {e-c e x}}{\sqrt {1-c^2 x^2}}-\frac {2 b^2 d e \sqrt {c d x+d} \sqrt {e-c e x} \operatorname {PolyLog}\left (3,-e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {2 b^2 d e \sqrt {c d x+d} \sqrt {e-c e x} \operatorname {PolyLog}\left (3,e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {2 b^2 c d e x \arcsin (c x) \sqrt {c d x+d} \sqrt {e-c e x}}{\sqrt {1-c^2 x^2}}-\frac {2}{27} b^2 d e \left (1-c^2 x^2\right ) \sqrt {c d x+d} \sqrt {e-c e x}-\frac {22}{9} b^2 d e \sqrt {c d x+d} \sqrt {e-c e x} \]

[In]

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

[Out]

(-22*b^2*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/9 - (2*a*b*c*d*e*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/Sqrt[1 - c^2
*x^2] - (2*b^2*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(1 - c^2*x^2))/27 - (2*b^2*c*d*e*x*Sqrt[d + c*d*x]*Sqrt[e -
c*e*x]*ArcSin[c*x])/Sqrt[1 - c^2*x^2] - (2*b*c*d*e*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x]))/(3*
Sqrt[1 - c^2*x^2]) + (2*b*c^3*d*e*x^3*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x]))/(9*Sqrt[1 - c^2*x^2
]) + d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*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)/3 - (2*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x])^2*ArcTanh[E^(I*Arc
Sin[c*x])])/Sqrt[1 - c^2*x^2] + ((2*I)*b*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x])*PolyLog[2, -E
^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2] - ((2*I)*b*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x])*PolyLo
g[2, E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2] - (2*b^2*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*PolyLog[3, -E^(I*ArcSi
n[c*x])])/Sqrt[1 - c^2*x^2] + (2*b^2*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*PolyLog[3, E^(I*ArcSin[c*x])])/Sqrt[1
- c^2*x^2]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 267

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 455

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 4268

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

Rule 4715

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 4739

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

Rule 4783

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

Rule 4787

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 + 2*p + 1))), x] + (Dist[2*d*(p/(m + 2*p + 1)), Int[(
f*x)^m*(d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x])^n, x], x] - Dist[b*c*(n/(f*(m + 2*p + 1)))*Simp[(d + e*x^2)^p/(
1 - c^2*x^2)^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, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] &&  !LtQ[m, -1]

Rule 4803

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

Rule 4823

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)^Fr
acPart[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 6724

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

Rubi steps \begin{align*} \text {integral}& = \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} (a+b \arcsin (c x))^2}{x} \, dx}{\sqrt {1-c^2 x^2}} \\ & = \frac {1}{3} d e \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {\left (d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{x} \, dx}{\sqrt {1-c^2 x^2}}-\frac {\left (2 b c d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \left (1-c^2 x^2\right ) (a+b \arcsin (c x)) \, dx}{3 \sqrt {1-c^2 x^2}} \\ & = -\frac {2 b c d e x \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{3 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d e x^3 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{9 \sqrt {1-c^2 x^2}}+d e \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2+\frac {1}{3} d e \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {\left (d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {(a+b \arcsin (c x))^2}{x \sqrt {1-c^2 x^2}} \, dx}{\sqrt {1-c^2 x^2}}-\frac {\left (2 b c d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int (a+b \arcsin (c x)) \, dx}{\sqrt {1-c^2 x^2}}+\frac {\left (2 b^2 c^2 d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {x \left (1-\frac {c^2 x^2}{3}\right )}{\sqrt {1-c^2 x^2}} \, dx}{3 \sqrt {1-c^2 x^2}} \\ & = -\frac {2 a b c d e x \sqrt {d+c d x} \sqrt {e-c e x}}{\sqrt {1-c^2 x^2}}-\frac {2 b c d e x \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{3 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d e x^3 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{9 \sqrt {1-c^2 x^2}}+d e \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2+\frac {1}{3} d e \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {\left (d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \text {Subst}\left (\int (a+b x)^2 \csc (x) \, dx,x,\arcsin (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 ) \int \arcsin (c x) \, dx}{\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 ) \text {Subst}\left (\int \frac {1-\frac {c^2 x}{3}}{\sqrt {1-c^2 x}} \, dx,x,x^2\right )}{3 \sqrt {1-c^2 x^2}} \\ & = -\frac {2 a b c d e x \sqrt {d+c d x} \sqrt {e-c e x}}{\sqrt {1-c^2 x^2}}-\frac {2 b^2 c d e x \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x)}{\sqrt {1-c^2 x^2}}-\frac {2 b c d e x \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{3 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d e x^3 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{9 \sqrt {1-c^2 x^2}}+d e \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2+\frac {1}{3} d e \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-\frac {2 d e \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2 \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (2 b d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \text {Subst}\left (\int (a+b x) \log \left (1-e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{\sqrt {1-c^2 x^2}}+\frac {\left (2 b d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \text {Subst}\left (\int (a+b x) \log \left (1+e^{i x}\right ) \, dx,x,\arcsin (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 ) \text {Subst}\left (\int \left (\frac {2}{3 \sqrt {1-c^2 x}}+\frac {1}{3} \sqrt {1-c^2 x}\right ) \, dx,x,x^2\right )}{3 \sqrt {1-c^2 x^2}}+\frac {\left (2 b^2 c^2 d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {x}{\sqrt {1-c^2 x^2}} \, dx}{\sqrt {1-c^2 x^2}} \\ & = -\frac {22}{9} b^2 d e \sqrt {d+c d x} \sqrt {e-c e x}-\frac {2 a b c d e x \sqrt {d+c d x} \sqrt {e-c e x}}{\sqrt {1-c^2 x^2}}-\frac {2}{27} b^2 d e \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right )-\frac {2 b^2 c d e x \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x)}{\sqrt {1-c^2 x^2}}-\frac {2 b c d e x \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{3 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d e x^3 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{9 \sqrt {1-c^2 x^2}}+d e \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2+\frac {1}{3} d e \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-\frac {2 d e \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2 \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {2 i b d e \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {2 i b d e \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (2 i b^2 d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{\sqrt {1-c^2 x^2}}+\frac {\left (2 i b^2 d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{\sqrt {1-c^2 x^2}} \\ & = -\frac {22}{9} b^2 d e \sqrt {d+c d x} \sqrt {e-c e x}-\frac {2 a b c d e x \sqrt {d+c d x} \sqrt {e-c e x}}{\sqrt {1-c^2 x^2}}-\frac {2}{27} b^2 d e \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right )-\frac {2 b^2 c d e x \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x)}{\sqrt {1-c^2 x^2}}-\frac {2 b c d e x \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{3 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d e x^3 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{9 \sqrt {1-c^2 x^2}}+d e \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2+\frac {1}{3} d e \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-\frac {2 d e \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2 \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {2 i b d e \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {2 i b d e \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (2 b^2 d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {\left (2 b^2 d e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}} \\ & = -\frac {22}{9} b^2 d e \sqrt {d+c d x} \sqrt {e-c e x}-\frac {2 a b c d e x \sqrt {d+c d x} \sqrt {e-c e x}}{\sqrt {1-c^2 x^2}}-\frac {2}{27} b^2 d e \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right )-\frac {2 b^2 c d e x \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x)}{\sqrt {1-c^2 x^2}}-\frac {2 b c d e x \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{3 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d e x^3 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{9 \sqrt {1-c^2 x^2}}+d e \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2+\frac {1}{3} d e \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-\frac {2 d e \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2 \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {2 i b d e \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {2 i b d e \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {2 b^2 d e \sqrt {d+c d x} \sqrt {e-c e x} \operatorname {PolyLog}\left (3,-e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {2 b^2 d e \sqrt {d+c d x} \sqrt {e-c e x} \operatorname {PolyLog}\left (3,e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 5.08 (sec) , antiderivative size = 632, normalized size of antiderivative = 0.98 \[ \int \frac {(d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2}{x} \, dx=-\frac {1}{3} a^2 d e \sqrt {d+c d x} \sqrt {e-c e x} \left (-4+c^2 x^2\right )+\frac {2 a b d e \sqrt {d+c d x} \sqrt {e-c e x} \left (-3 c x+c^3 x^3+3 \left (1-c^2 x^2\right )^{3/2} \arcsin (c x)\right )}{9 \sqrt {1-c^2 x^2}}+a^2 d^{3/2} e^{3/2} \log (c x)-a^2 d^{3/2} e^{3/2} \log \left (d e+\sqrt {d} \sqrt {e} \sqrt {d+c d x} \sqrt {e-c e x}\right )-\frac {2 a b d e \sqrt {d+c d x} \sqrt {e-c e x} \left (c x-\sqrt {1-c^2 x^2} \arcsin (c x)-\arcsin (c x) \log \left (1-e^{i \arcsin (c x)}\right )+\arcsin (c x) \log \left (1+e^{i \arcsin (c x)}\right )-i \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )+i \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )\right )}{\sqrt {1-c^2 x^2}}-\frac {b^2 d e \sqrt {d+c d x} \sqrt {e-c e x} \left (2 \sqrt {1-c^2 x^2}+2 c x \arcsin (c x)-\sqrt {1-c^2 x^2} \arcsin (c x)^2-\arcsin (c x)^2 \log \left (1-e^{i \arcsin (c x)}\right )+\arcsin (c x)^2 \log \left (1+e^{i \arcsin (c x)}\right )-2 i \arcsin (c x) \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )+2 i \arcsin (c x) \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )+2 \operatorname {PolyLog}\left (3,-e^{i \arcsin (c x)}\right )-2 \operatorname {PolyLog}\left (3,e^{i \arcsin (c x)}\right )\right )}{\sqrt {1-c^2 x^2}}+\frac {b^2 d e \sqrt {d+c d x} \sqrt {e-c e x} \left (27 \sqrt {1-c^2 x^2} \left (-2+\arcsin (c x)^2\right )+\left (-2+9 \arcsin (c x)^2\right ) \cos (3 \arcsin (c x))-6 \arcsin (c x) (9 c x+\sin (3 \arcsin (c x)))\right )}{108 \sqrt {1-c^2 x^2}} \]

[In]

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

[Out]

-1/3*(a^2*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(-4 + c^2*x^2)) + (2*a*b*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(-3
*c*x + c^3*x^3 + 3*(1 - c^2*x^2)^(3/2)*ArcSin[c*x]))/(9*Sqrt[1 - c^2*x^2]) + a^2*d^(3/2)*e^(3/2)*Log[c*x] - a^
2*d^(3/2)*e^(3/2)*Log[d*e + Sqrt[d]*Sqrt[e]*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]] - (2*a*b*d*e*Sqrt[d + c*d*x]*Sqrt
[e - c*e*x]*(c*x - Sqrt[1 - c^2*x^2]*ArcSin[c*x] - ArcSin[c*x]*Log[1 - E^(I*ArcSin[c*x])] + ArcSin[c*x]*Log[1
+ E^(I*ArcSin[c*x])] - I*PolyLog[2, -E^(I*ArcSin[c*x])] + I*PolyLog[2, E^(I*ArcSin[c*x])]))/Sqrt[1 - c^2*x^2]
- (b^2*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(2*Sqrt[1 - c^2*x^2] + 2*c*x*ArcSin[c*x] - Sqrt[1 - c^2*x^2]*ArcSin
[c*x]^2 - ArcSin[c*x]^2*Log[1 - E^(I*ArcSin[c*x])] + ArcSin[c*x]^2*Log[1 + E^(I*ArcSin[c*x])] - (2*I)*ArcSin[c
*x]*PolyLog[2, -E^(I*ArcSin[c*x])] + (2*I)*ArcSin[c*x]*PolyLog[2, E^(I*ArcSin[c*x])] + 2*PolyLog[3, -E^(I*ArcS
in[c*x])] - 2*PolyLog[3, E^(I*ArcSin[c*x])]))/Sqrt[1 - c^2*x^2] + (b^2*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(27
*Sqrt[1 - c^2*x^2]*(-2 + ArcSin[c*x]^2) + (-2 + 9*ArcSin[c*x]^2)*Cos[3*ArcSin[c*x]] - 6*ArcSin[c*x]*(9*c*x + S
in[3*ArcSin[c*x]])))/(108*Sqrt[1 - c^2*x^2])

Maple [F]

\[\int \frac {\left (c d x +d \right )^{\frac {3}{2}} \left (-c e x +e \right )^{\frac {3}{2}} \left (a +b \arcsin \left (c x \right )\right )^{2}}{x}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {(d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2}{x} \, dx=\int { \frac {{\left (c d x + d\right )}^{\frac {3}{2}} {\left (-c e x + e\right )}^{\frac {3}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x} \,d x } \]

[In]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {(d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2}{x} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {(d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2}{x} \, dx=\text {Exception raised: ValueError} \]

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [F]

\[ \int \frac {(d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2}{x} \, dx=\int { \frac {{\left (c d x + d\right )}^{\frac {3}{2}} {\left (-c e x + e\right )}^{\frac {3}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2}{x} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\right )}^{3/2}\,{\left (e-c\,e\,x\right )}^{3/2}}{x} \,d x \]

[In]

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

[Out]

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