\(\int \frac {(a+b \arcsin (c x))^2}{x \sqrt {d+c d x} \sqrt {e-c e x}} \, dx\) [495]

Optimal result

Integrand size = 35, antiderivative size = 287 \[ \int \frac {(a+b \arcsin (c x))^2}{x \sqrt {d+c d x} \sqrt {e-c e x}} \, dx=-\frac {2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2 \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{\sqrt {d+c d x} \sqrt {e-c e x}}+\frac {2 i b \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{\sqrt {d+c d x} \sqrt {e-c e x}}-\frac {2 i b \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{\sqrt {d+c d x} \sqrt {e-c e x}}-\frac {2 b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (3,-e^{i \arcsin (c x)}\right )}{\sqrt {d+c d x} \sqrt {e-c e x}}+\frac {2 b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (3,e^{i \arcsin (c x)}\right )}{\sqrt {d+c d x} \sqrt {e-c e x}} \] Output:

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

Mathematica [A] (verified)

Time = 2.29 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.17 \[ \int \frac {(a+b \arcsin (c x))^2}{x \sqrt {d+c d x} \sqrt {e-c e x}} \, dx=\frac {a^2 \log (c x)}{\sqrt {d} \sqrt {e}}-\frac {a^2 \log \left (d e+\sqrt {d} \sqrt {e} \sqrt {d+c d x} \sqrt {e-c e x}\right )}{\sqrt {d} \sqrt {e}}+\frac {2 a b \sqrt {1-c^2 x^2} \left (\arcsin (c x) \left (\log \left (1-e^{i \arcsin (c x)}\right )-\log \left (1+e^{i \arcsin (c x)}\right )\right )+i \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-i \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )\right )}{\sqrt {d+c d x} \sqrt {e-c e x}}+\frac {b^2 \sqrt {1-c^2 x^2} \left (\arcsin (c x)^2 \log \left (1-e^{i \arcsin (c x)}\right )-\arcsin (c x)^2 \log \left (1+e^{i \arcsin (c x)}\right )+2 i \arcsin (c x) \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-2 i \arcsin (c x) \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )-2 \operatorname {PolyLog}\left (3,-e^{i \arcsin (c x)}\right )+2 \operatorname {PolyLog}\left (3,e^{i \arcsin (c x)}\right )\right )}{\sqrt {d+c d x} \sqrt {e-c e x}} \] Input:

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

Output:

(a^2*Log[c*x])/(Sqrt[d]*Sqrt[e]) - (a^2*Log[d*e + Sqrt[d]*Sqrt[e]*Sqrt[d + 
 c*d*x]*Sqrt[e - c*e*x]])/(Sqrt[d]*Sqrt[e]) + (2*a*b*Sqrt[1 - c^2*x^2]*(Ar 
cSin[c*x]*(Log[1 - E^(I*ArcSin[c*x])] - Log[1 + E^(I*ArcSin[c*x])]) + I*Po 
lyLog[2, -E^(I*ArcSin[c*x])] - I*PolyLog[2, E^(I*ArcSin[c*x])]))/(Sqrt[d + 
 c*d*x]*Sqrt[e - c*e*x]) + (b^2*Sqrt[1 - c^2*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)*ArcSi 
n[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*ArcSin[c*x])] + 2*PolyLog[3, E^(I*ArcSi 
n[c*x])]))/(Sqrt[d + c*d*x]*Sqrt[e - c*e*x])
 

Rubi [A] (verified)

Time = 1.07 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.52, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5238, 5218, 3042, 4671, 3011, 2720, 7143}

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.

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

\(\Big \downarrow \) 5238

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

\(\Big \downarrow \) 5218

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4671

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{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 3011

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] + Simp[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 3042

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

rule 4671

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] + (-Simp[d*(m/f)   Int[(c + 
d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Simp[d*(m/f)   Int[(c + d*x 
)^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IG 
tQ[m, 0]
 

rule 5218

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

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]
 

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> 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]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 600 vs. \(2 (295 ) = 590\).

Time = 3.08 (sec) , antiderivative size = 601, normalized size of antiderivative = 2.09

method	result	size
default	\(-\frac {a^{2} \sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \ln \left (\frac {2 \sqrt {d e}\, \sqrt {-d e \left (c^{2} x^{2}-1\right )}+2 d e}{x}\right )}{\sqrt {-d e \left (c^{2} x^{2}-1\right )}\, \sqrt {d e}}-\frac {b^{2} \sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (\arcsin \left (c x \right )^{2} \ln \left (1-\sqrt {i c x +\sqrt {-c^{2} x^{2}+1}}\right )+\arcsin \left (c x \right )^{2} \ln \left (1+\sqrt {i c x +\sqrt {-c^{2} x^{2}+1}}\right )-\arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-4 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, \sqrt {i c x +\sqrt {-c^{2} x^{2}+1}}\right )-4 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -\sqrt {i c x +\sqrt {-c^{2} x^{2}+1}}\right )-2 \operatorname {polylog}\left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+8 \operatorname {polylog}\left (3, \sqrt {i c x +\sqrt {-c^{2} x^{2}+1}}\right )+8 \operatorname {polylog}\left (3, -\sqrt {i c x +\sqrt {-c^{2} x^{2}+1}}\right )\right )}{d e \left (c^{2} x^{2}-1\right )}+\frac {2 i a b \sqrt {-c^{2} x^{2}+1}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \left (i \arcsin \left (c x \right ) \ln \left (1-\sqrt {i c x +\sqrt {-c^{2} x^{2}+1}}\right )+i \arcsin \left (c x \right ) \ln \left (1+\sqrt {i c x +\sqrt {-c^{2} x^{2}+1}}\right )-i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-\operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (2, \sqrt {i c x +\sqrt {-c^{2} x^{2}+1}}\right )+2 \operatorname {polylog}\left (2, -\sqrt {i c x +\sqrt {-c^{2} x^{2}+1}}\right )\right )}{d e \left (c^{2} x^{2}-1\right )}\)	\(601\)
parts	\(-\frac {a^{2} \sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \ln \left (\frac {2 \sqrt {d e}\, \sqrt {-d e \left (c^{2} x^{2}-1\right )}+2 d e}{x}\right )}{\sqrt {-d e \left (c^{2} x^{2}-1\right )}\, \sqrt {d e}}-\frac {b^{2} \sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (\arcsin \left (c x \right )^{2} \ln \left (1-\sqrt {i c x +\sqrt {-c^{2} x^{2}+1}}\right )+\arcsin \left (c x \right )^{2} \ln \left (1+\sqrt {i c x +\sqrt {-c^{2} x^{2}+1}}\right )-\arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-4 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, \sqrt {i c x +\sqrt {-c^{2} x^{2}+1}}\right )-4 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -\sqrt {i c x +\sqrt {-c^{2} x^{2}+1}}\right )-2 \operatorname {polylog}\left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+8 \operatorname {polylog}\left (3, \sqrt {i c x +\sqrt {-c^{2} x^{2}+1}}\right )+8 \operatorname {polylog}\left (3, -\sqrt {i c x +\sqrt {-c^{2} x^{2}+1}}\right )\right )}{d e \left (c^{2} x^{2}-1\right )}+\frac {2 i a b \sqrt {-c^{2} x^{2}+1}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \left (i \arcsin \left (c x \right ) \ln \left (1-\sqrt {i c x +\sqrt {-c^{2} x^{2}+1}}\right )+i \arcsin \left (c x \right ) \ln \left (1+\sqrt {i c x +\sqrt {-c^{2} x^{2}+1}}\right )-i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-\operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (2, \sqrt {i c x +\sqrt {-c^{2} x^{2}+1}}\right )+2 \operatorname {polylog}\left (2, -\sqrt {i c x +\sqrt {-c^{2} x^{2}+1}}\right )\right )}{d e \left (c^{2} x^{2}-1\right )}\)	\(601\)

Input:

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

Output:

-a^2*(d*(c*x+1))^(1/2)*(-e*(c*x-1))^(1/2)*ln(2*((d*e)^(1/2)*(-d*e*(c^2*x^2 
-1))^(1/2)+d*e)/x)/(-d*e*(c^2*x^2-1))^(1/2)/(d*e)^(1/2)-b^2*(-e*(c*x-1))^( 
1/2)*(d*(c*x+1))^(1/2)*(-c^2*x^2+1)^(1/2)*(arcsin(c*x)^2*ln(1-(I*c*x+(-c^2 
*x^2+1)^(1/2))^(1/2))+arcsin(c*x)^2*ln(1+(I*c*x+(-c^2*x^2+1)^(1/2))^(1/2)) 
-arcsin(c*x)^2*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))+2*I*arcsin(c*x)*polylog(2,-I 
*c*x-(-c^2*x^2+1)^(1/2))-4*I*arcsin(c*x)*polylog(2,(I*c*x+(-c^2*x^2+1)^(1/ 
2))^(1/2))-4*I*arcsin(c*x)*polylog(2,-(I*c*x+(-c^2*x^2+1)^(1/2))^(1/2))-2* 
polylog(3,-I*c*x-(-c^2*x^2+1)^(1/2))+8*polylog(3,(I*c*x+(-c^2*x^2+1)^(1/2) 
)^(1/2))+8*polylog(3,-(I*c*x+(-c^2*x^2+1)^(1/2))^(1/2)))/d/e/(c^2*x^2-1)+2 
*I*a*b*(-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))-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)))/d/e/(c^2*x^2-1)
 

Fricas [F]

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

integrate((a+b*arcsin(c*x))^2/x/(c*d*x+d)^(1/2)/(-c*e*x+e)^(1/2),x, algori 
thm="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^3 - d*e*x), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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