\(\int \frac {\sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2}{x^2} \, dx\) [486]

Optimal result

Integrand size = 35, antiderivative size = 257 \[ \int \frac {\sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2}{x^2} \, dx=-\frac {\sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2}{x}-\frac {i c \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}-\frac {c \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^3}{3 b \sqrt {1-c^2 x^2}}+\frac {2 b c \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x)) \log \left (1-e^{2 i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {i b^2 c \sqrt {d+c d x} \sqrt {e-c e x} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}} \] Output:

-(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)*(a+b*arcsin(c*x))^2/x-I*c*(c*d*x+d)^(1/2 
)*(-c*e*x+e)^(1/2)*(a+b*arcsin(c*x))^2/(-c^2*x^2+1)^(1/2)-1/3*c*(c*d*x+d)^ 
(1/2)*(-c*e*x+e)^(1/2)*(a+b*arcsin(c*x))^3/b/(-c^2*x^2+1)^(1/2)+2*b*c*(c*d 
*x+d)^(1/2)*(-c*e*x+e)^(1/2)*(a+b*arcsin(c*x))*ln(1-(I*c*x+(-c^2*x^2+1)^(1 
/2))^2)/(-c^2*x^2+1)^(1/2)-I*b^2*c*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)*polylo 
g(2,(I*c*x+(-c^2*x^2+1)^(1/2))^2)/(-c^2*x^2+1)^(1/2)
 

Mathematica [A] (verified)

Time = 1.15 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.46 \[ \int \frac {\sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2}{x^2} \, dx=\frac {-3 a^2 \sqrt {d+c d x} \sqrt {e-c e x} \sqrt {1-c^2 x^2}-3 i b \sqrt {d+c d x} \sqrt {e-c e x} \left (-i a c x+b c x-i b \sqrt {1-c^2 x^2}\right ) \arcsin (c x)^2-b^2 c x \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x)^3+3 a^2 c \sqrt {d} \sqrt {e} x \sqrt {1-c^2 x^2} \arctan \left (\frac {c x \sqrt {d+c d x} \sqrt {e-c e x}}{\sqrt {d} \sqrt {e} \left (-1+c^2 x^2\right )}\right )+6 b \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x) \left (-a \sqrt {1-c^2 x^2}+b c x \log \left (1-e^{2 i \arcsin (c x)}\right )\right )+6 a b c x \sqrt {d+c d x} \sqrt {e-c e x} \log (c x)-3 i b^2 c x \sqrt {d+c d x} \sqrt {e-c e x} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{3 x \sqrt {1-c^2 x^2}} \] Input:

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

Output:

(-3*a^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[1 - c^2*x^2] - (3*I)*b*Sqrt[d 
 + c*d*x]*Sqrt[e - c*e*x]*((-I)*a*c*x + b*c*x - I*b*Sqrt[1 - c^2*x^2])*Arc 
Sin[c*x]^2 - b^2*c*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]^3 + 3*a^2 
*c*Sqrt[d]*Sqrt[e]*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))] + 6*b*Sqrt[d + c*d*x]*Sqrt[e - 
 c*e*x]*ArcSin[c*x]*(-(a*Sqrt[1 - c^2*x^2]) + b*c*x*Log[1 - E^((2*I)*ArcSi 
n[c*x])]) + 6*a*b*c*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Log[c*x] - (3*I)*b^2 
*c*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*PolyLog[2, E^((2*I)*ArcSin[c*x])])/(3 
*x*Sqrt[1 - c^2*x^2])
 

Rubi [A] (verified)

Time = 1.49 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.62, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.314, Rules used = {5238, 5196, 5136, 3042, 25, 4200, 25, 2620, 2715, 2838, 5152}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\sqrt {c d x+d} \sqrt {e-c e x} (a+b \arcsin (c x))^2}{x^2} \, dx\)

\(\Big \downarrow \) 5238

\(\displaystyle \frac {\sqrt {c d x+d} \sqrt {e-c e x} \int \frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{x^2}dx}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 5196

\(\displaystyle \frac {\sqrt {c d x+d} \sqrt {e-c e x} \left (c^2 \left (-\int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx\right )+2 b c \int \frac {a+b \arcsin (c x)}{x}dx-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{x}\right )}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 5136

\(\displaystyle \frac {\sqrt {c d x+d} \sqrt {e-c e x} \left (c^2 \left (-\int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx\right )+2 b c \int \frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c x}d\arcsin (c x)-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{x}\right )}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sqrt {c d x+d} \sqrt {e-c e x} \left (c^2 \left (-\int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx\right )+2 b c \int -\left ((a+b \arcsin (c x)) \tan \left (\arcsin (c x)+\frac {\pi }{2}\right )\right )d\arcsin (c x)-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{x}\right )}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sqrt {c d x+d} \sqrt {e-c e x} \left (c^2 \left (-\int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx\right )-2 b c \int (a+b \arcsin (c x)) \tan \left (\arcsin (c x)+\frac {\pi }{2}\right )d\arcsin (c x)-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{x}\right )}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 4200

\(\displaystyle \frac {\sqrt {c d x+d} \sqrt {e-c e x} \left (c^2 \left (-\int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx\right )+2 b c \left (2 i \int -\frac {e^{2 i \arcsin (c x)} (a+b \arcsin (c x))}{1-e^{2 i \arcsin (c x)}}d\arcsin (c x)-\frac {i (a+b \arcsin (c x))^2}{2 b}\right )-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{x}\right )}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sqrt {c d x+d} \sqrt {e-c e x} \left (c^2 \left (-\int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx\right )+2 b c \left (-2 i \int \frac {e^{2 i \arcsin (c x)} (a+b \arcsin (c x))}{1-e^{2 i \arcsin (c x)}}d\arcsin (c x)-\frac {i (a+b \arcsin (c x))^2}{2 b}\right )-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{x}\right )}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 2620

\(\displaystyle \frac {\sqrt {c d x+d} \sqrt {e-c e x} \left (c^2 \left (-\int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx\right )+2 b c \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{2} i b \int \log \left (1-e^{2 i \arcsin (c x)}\right )d\arcsin (c x)\right )-\frac {i (a+b \arcsin (c x))^2}{2 b}\right )-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{x}\right )}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 2715

\(\displaystyle \frac {\sqrt {c d x+d} \sqrt {e-c e x} \left (c^2 \left (-\int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx\right )+2 b c \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \int e^{-2 i \arcsin (c x)} \log \left (1-e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}\right )-\frac {i (a+b \arcsin (c x))^2}{2 b}\right )-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{x}\right )}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {\sqrt {c d x+d} \sqrt {e-c e x} \left (c^2 \left (-\int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx\right )-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{x}+2 b c \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right )-\frac {i (a+b \arcsin (c x))^2}{2 b}\right )\right )}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 5152

\(\displaystyle \frac {\sqrt {c d x+d} \sqrt {e-c e x} \left (-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{x}+2 b c \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right )-\frac {i (a+b \arcsin (c x))^2}{2 b}\right )-\frac {c (a+b \arcsin (c x))^3}{3 b}\right )}{\sqrt {1-c^2 x^2}}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 2620

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/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si 
mp[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 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[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 2838

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 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4200

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol 
] :> Simp[I*((c + d*x)^(m + 1)/(d*(m + 1))), x] - Simp[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 5136

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

rule 5152

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

rule 5196

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*ArcS 
in[c*x])^n/(f*(m + 1))), x] + (-Simp[b*c*(n/(f*(m + 1)))*Simp[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] + Simp[(c^2/(f^2*(m + 1)))*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]]   Int 
[(f*x)^(m + 2)*((a + b*ArcSin[c*x])^n/Sqrt[1 - c^2*x^2]), x], x]) /; FreeQ[ 
{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[m, -1]
 

rule 5238

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((h_.)*(x_))^(m_.)*((d_) + (e_ 
.)*(x_))^(p_)*((f_) + (g_.)*(x_))^(q_), x_Symbol] :> Simp[((-d^2)*(g/e))^In 
tPart[q]*(d + e*x)^FracPart[q]*((f + g*x)^FracPart[q]/(1 - c^2*x^2)^FracPar 
t[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]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 686 vs. \(2 (245 ) = 490\).

Time = 3.86 (sec) , antiderivative size = 687, normalized size of antiderivative = 2.67

method	result	size
default	\(\frac {a^{2} \left (-\arctan \left (\frac {\sqrt {c^{2} d e}\, x}{\sqrt {-d e \left (c^{2} x^{2}-1\right )}}\right ) x \,c^{2} d e -\sqrt {c^{2} d e}\, \sqrt {-d e \left (c^{2} x^{2}-1\right )}\right ) \sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}}{\sqrt {c^{2} d e}\, \sqrt {-d e \left (c^{2} x^{2}-1\right )}\, x}+b^{2} \left (\frac {\sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{3} c}{3 c^{2} x^{2}-3}-\frac {\sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, c x +c^{2} x^{2}-1\right ) \arcsin \left (c x \right )^{2}}{x \left (c^{2} x^{2}-1\right )}+\frac {2 i \sqrt {-c^{2} x^{2}+1}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \left (i \arcsin \left (c x \right ) \ln \left (1-\sqrt {i c x +\sqrt {-c^{2} x^{2}+1}}\right )+i \arcsin \left (c x \right ) \ln \left (1+\sqrt {i c x +\sqrt {-c^{2} x^{2}+1}}\right )+i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right )^{2}+\operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (2, \sqrt {i c x +\sqrt {-c^{2} x^{2}+1}}\right )+2 \operatorname {polylog}\left (2, -\sqrt {i c x +\sqrt {-c^{2} x^{2}+1}}\right )\right ) c}{c^{2} x^{2}-1}\right )+2 a b \left (\frac {\sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{2} c}{2 c^{2} x^{2}-2}+\frac {2 i \sqrt {-c^{2} x^{2}+1}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \arcsin \left (c x \right ) c}{c^{2} x^{2}-1}-\frac {\sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, c x +c^{2} x^{2}-1\right ) \arcsin \left (c x \right )}{x \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}-1\right ) c}{c^{2} x^{2}-1}\right )\)	\(687\)
parts	\(\frac {a^{2} \left (-\arctan \left (\frac {\sqrt {c^{2} d e}\, x}{\sqrt {-d e \left (c^{2} x^{2}-1\right )}}\right ) x \,c^{2} d e -\sqrt {c^{2} d e}\, \sqrt {-d e \left (c^{2} x^{2}-1\right )}\right ) \sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}}{\sqrt {c^{2} d e}\, \sqrt {-d e \left (c^{2} x^{2}-1\right )}\, x}+b^{2} \left (\frac {\sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{3} c}{3 c^{2} x^{2}-3}-\frac {\sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, c x +c^{2} x^{2}-1\right ) \arcsin \left (c x \right )^{2}}{x \left (c^{2} x^{2}-1\right )}+\frac {2 i \sqrt {-c^{2} x^{2}+1}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \left (i \arcsin \left (c x \right ) \ln \left (1-\sqrt {i c x +\sqrt {-c^{2} x^{2}+1}}\right )+i \arcsin \left (c x \right ) \ln \left (1+\sqrt {i c x +\sqrt {-c^{2} x^{2}+1}}\right )+i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right )^{2}+\operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (2, \sqrt {i c x +\sqrt {-c^{2} x^{2}+1}}\right )+2 \operatorname {polylog}\left (2, -\sqrt {i c x +\sqrt {-c^{2} x^{2}+1}}\right )\right ) c}{c^{2} x^{2}-1}\right )+2 a b \left (\frac {\sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{2} c}{2 c^{2} x^{2}-2}+\frac {2 i \sqrt {-c^{2} x^{2}+1}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \arcsin \left (c x \right ) c}{c^{2} x^{2}-1}-\frac {\sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, c x +c^{2} x^{2}-1\right ) \arcsin \left (c x \right )}{x \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}-1\right ) c}{c^{2} x^{2}-1}\right )\)	\(687\)

Input:

int((c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)*(a+b*arcsin(c*x))^2/x^2,x,method=_RET 
URNVERBOSE)
 

Output:

a^2*(-arctan((c^2*d*e)^(1/2)*x/(-d*e*(c^2*x^2-1))^(1/2))*x*c^2*d*e-(c^2*d* 
e)^(1/2)*(-d*e*(c^2*x^2-1))^(1/2))*(d*(c*x+1))^(1/2)*(-e*(c*x-1))^(1/2)/(c 
^2*d*e)^(1/2)/(-d*e*(c^2*x^2-1))^(1/2)/x+b^2*(1/3*(-e*(c*x-1))^(1/2)*(d*(c 
*x+1))^(1/2)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)^3*c/(c^2*x^2-1)-(-e*(c*x-1))^( 
1/2)*(d*(c*x+1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*c*x+c^2*x^2-1)*arcsin(c*x)^2/ 
x/(c^2*x^2-1)+2*I*(-c^2*x^2+1)^(1/2)*(d*(c*x+1))^(1/2)*(-e*(c*x-1))^(1/2)* 
(I*arcsin(c*x)*ln(1-(I*c*x+(-c^2*x^2+1)^(1/2))^(1/2))+I*arcsin(c*x)*ln(1+( 
I*c*x+(-c^2*x^2+1)^(1/2))^(1/2))+I*arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1/ 
2))+arcsin(c*x)^2+polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))+2*polylog(2,(I*c*x+ 
(-c^2*x^2+1)^(1/2))^(1/2))+2*polylog(2,-(I*c*x+(-c^2*x^2+1)^(1/2))^(1/2))) 
*c/(c^2*x^2-1))+2*a*b*(1/2*(-e*(c*x-1))^(1/2)*(d*(c*x+1))^(1/2)*(-c^2*x^2+ 
1)^(1/2)*arcsin(c*x)^2*c/(c^2*x^2-1)+2*I*(-c^2*x^2+1)^(1/2)*(d*(c*x+1))^(1 
/2)*(-e*(c*x-1))^(1/2)*arcsin(c*x)*c/(c^2*x^2-1)-(-e*(c*x-1))^(1/2)*(d*(c* 
x+1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*c*x+c^2*x^2-1)*arcsin(c*x)/x/(c^2*x^2-1) 
-(-e*(c*x-1))^(1/2)*(d*(c*x+1))^(1/2)*(-c^2*x^2+1)^(1/2)*ln((I*c*x+(-c^2*x 
^2+1)^(1/2))^2-1)*c/(c^2*x^2-1))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

\[ \int \frac {\sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2}{x^2} \, dx=\int \frac {\sqrt {d \left (c x + 1\right )} \sqrt {- e \left (c x - 1\right )} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{x^{2}}\, dx \] Input:

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

Output:

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

Maxima [F(-2)]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2}{x^2} \, dx=\frac {\sqrt {e}\, \sqrt {d}\, \left (2 \mathit {asin} \left (\frac {\sqrt {-c x +1}}{\sqrt {2}}\right ) a^{2} c x -\sqrt {c x +1}\, \sqrt {-c x +1}\, a^{2}+2 \left (\int \frac {\sqrt {c x +1}\, \sqrt {-c x +1}\, \mathit {asin} \left (c x \right )}{x^{2}}d x \right ) a b x +\left (\int \frac {\sqrt {c x +1}\, \sqrt {-c x +1}\, \mathit {asin} \left (c x \right )^{2}}{x^{2}}d x \right ) b^{2} x \right )}{x} \] Input:

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

Output:

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