\(\int \frac {(a+b \arcsin (c x))^2}{x^2 \sqrt {d+c d x} \sqrt {e-c e x}} \, dx\) [496]

Optimal result

Integrand size = 35, antiderivative size = 214 \[ \int \frac {(a+b \arcsin (c x))^2}{x^2 \sqrt {d+c d x} \sqrt {e-c e x}} \, dx=-\frac {i c \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{\sqrt {d+c d x} \sqrt {e-c e x}}-\frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{x \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {2 b c \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \log \left (1-e^{2 i \arcsin (c x)}\right )}{\sqrt {d+c d x} \sqrt {e-c e x}}-\frac {i b^2 c \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{\sqrt {d+c d x} \sqrt {e-c e x}} \] Output:

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

Mathematica [A] (verified)

Time = 2.32 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.88 \[ \int \frac {(a+b \arcsin (c x))^2}{x^2 \sqrt {d+c d x} \sqrt {e-c e x}} \, dx=\frac {b^2 \left (-1+c^2 x^2-i c x \sqrt {1-c^2 x^2}\right ) \arcsin (c x)^2+2 b \arcsin (c x) \left (-a+a c^2 x^2+b c x \sqrt {1-c^2 x^2} \log \left (1-e^{2 i \arcsin (c x)}\right )\right )+a \left (-a+a c^2 x^2+2 b c x \sqrt {1-c^2 x^2} \log (c x)\right )-i b^2 c x \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{x \sqrt {d+c d x} \sqrt {e-c e x}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.26 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.66, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5238, 5186, 5136, 3042, 25, 4200, 25, 2620, 2715, 2838}

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.

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

\(\Big \downarrow \) 5238

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

\(\Big \downarrow \) 5186

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

\(\Big \downarrow \) 5136

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 4200

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

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

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 5186

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 + 1)*((a + b 
*ArcSin[c*x])^n/(d*f*(m + 1))), x] - Simp[b*c*(n/(f*(m + 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*A 
rcSin[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[c^ 
2*d + e, 0] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[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. 538 vs. \(2 (210 ) = 420\).

Time = 2.43 (sec) , antiderivative size = 539, normalized size of antiderivative = 2.52

method	result	size
default	\(-\frac {a^{2} \sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}}{d e x}+b^{2} \left (-\frac {\sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, c x +c^{2} x^{2}-1\right ) \arcsin \left (c x \right )^{2}}{\left (c^{2} x^{2}-1\right ) x d e}+\frac {2 i \sqrt {-c^{2} x^{2}+1}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \left (i \arcsin \left (c x \right ) \ln \left (1-\sqrt {i c x +\sqrt {-c^{2} x^{2}+1}}\right )+i \arcsin \left (c x \right ) \ln \left (1+\sqrt {i c x +\sqrt {-c^{2} x^{2}+1}}\right )+i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right )^{2}+\operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (2, \sqrt {i c x +\sqrt {-c^{2} x^{2}+1}}\right )+2 \operatorname {polylog}\left (2, -\sqrt {i c x +\sqrt {-c^{2} x^{2}+1}}\right )\right ) c}{d e \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (\frac {2 i \sqrt {-c^{2} x^{2}+1}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \arcsin \left (c x \right ) c}{d e \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, c x +c^{2} x^{2}-1\right ) \arcsin \left (c x \right )}{\left (c^{2} x^{2}-1\right ) x d e}-\frac {\sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}-1\right ) c}{d e \left (c^{2} x^{2}-1\right )}\right )\)	\(539\)
parts	\(-\frac {a^{2} \sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}}{d e x}+b^{2} \left (-\frac {\sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, c x +c^{2} x^{2}-1\right ) \arcsin \left (c x \right )^{2}}{\left (c^{2} x^{2}-1\right ) x d e}+\frac {2 i \sqrt {-c^{2} x^{2}+1}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \left (i \arcsin \left (c x \right ) \ln \left (1-\sqrt {i c x +\sqrt {-c^{2} x^{2}+1}}\right )+i \arcsin \left (c x \right ) \ln \left (1+\sqrt {i c x +\sqrt {-c^{2} x^{2}+1}}\right )+i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+\arcsin \left (c x \right )^{2}+\operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (2, \sqrt {i c x +\sqrt {-c^{2} x^{2}+1}}\right )+2 \operatorname {polylog}\left (2, -\sqrt {i c x +\sqrt {-c^{2} x^{2}+1}}\right )\right ) c}{d e \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (\frac {2 i \sqrt {-c^{2} x^{2}+1}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \arcsin \left (c x \right ) c}{d e \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, c x +c^{2} x^{2}-1\right ) \arcsin \left (c x \right )}{\left (c^{2} x^{2}-1\right ) x d e}-\frac {\sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}-1\right ) c}{d e \left (c^{2} x^{2}-1\right )}\right )\)	\(539\)

Input:

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

Output:

-a^2*(d*(c*x+1))^(1/2)*(-e*(c*x-1))^(1/2)/d/e/x+b^2*(-(-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/(c^2*x 
^2-1)/x/d/e+2*I*(-c^2*x^2+1)^(1/2)*(d*(c*x+1))^(1/2)*(-e*(c*x-1))^(1/2)/d/ 
e/(c^2*x^2-1)*(I*arcsin(c*x)*ln(1-(I*c*x+(-c^2*x^2+1)^(1/2))^(1/2))+I*arcs 
in(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*pol 
ylog(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)+2*a*b*(2*I*(-c^2*x^2+1)^(1/2)*(d*(c*x+1))^(1/2)*(-e*(c*x 
-1))^(1/2)/d/e/(c^2*x^2-1)*arcsin(c*x)*c-(-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)/(c^2*x^2-1)/x/d/e-(-e 
*(c*x-1))^(1/2)*(d*(c*x+1))^(1/2)*(-c^2*x^2+1)^(1/2)/d/e/(c^2*x^2-1)*ln((I 
*c*x+(-c^2*x^2+1)^(1/2))^2-1)*c)
 

Fricas [F]

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

integrate((a+b*arcsin(c*x))^2/x^2/(c*d*x+d)^(1/2)/(-c*e*x+e)^(1/2),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)*sq 
rt(-c*e*x + e)/(c^2*d*e*x^4 - d*e*x^2), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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