\(\int \frac {(a+b \arcsin (c x))^2}{x (d+c d x)^{3/2} (e-c e x)^{3/2}} \, dx\) [500]

Optimal result

Integrand size = 35, antiderivative size = 548 \[ \int \frac {(a+b \arcsin (c x))^2}{x (d+c d x)^{3/2} (e-c e x)^{3/2}} \, dx=\frac {(a+b \arcsin (c x))^2}{d e \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {4 i b \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \arctan \left (e^{i \arcsin (c x)}\right )}{d e \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2 \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{d e \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {2 i b \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{d e \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {2 i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{d e \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {2 i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{d e \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {2 i b \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{d e \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {2 b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (3,-e^{i \arcsin (c x)}\right )}{d e \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {2 b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (3,e^{i \arcsin (c x)}\right )}{d e \sqrt {d+c d x} \sqrt {e-c e x}} \] Output:

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

Mathematica [A] (verified)

Time = 5.61 (sec) , antiderivative size = 877, normalized size of antiderivative = 1.60 \[ \int \frac {(a+b \arcsin (c x))^2}{x (d+c d x)^{3/2} (e-c e x)^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(-((a^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/(-1 + c^2*x^2)) + a^2*Sqrt[d]*Sqr 
t[e]*Log[c*x] - a^2*Sqrt[d]*Sqrt[e]*Log[d*e + Sqrt[d]*Sqrt[e]*Sqrt[d + c*d 
*x]*Sqrt[e - c*e*x]] + (2*a*b*d*e*(ArcSin[c*x] + Sqrt[1 - c^2*x^2]*ArcSin[ 
c*x]*Log[1 - E^(I*ArcSin[c*x])] - Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 + E^ 
(I*ArcSin[c*x])] + Sqrt[1 - c^2*x^2]*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c 
*x]/2]] - Sqrt[1 - c^2*x^2]*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]] + 
 I*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^(I*ArcSin[c*x])] - I*Sqrt[1 - c^2*x^2]* 
PolyLog[2, E^(I*ArcSin[c*x])]))/(Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) + (b^2*d 
*e*(I*Pi*Sqrt[1 - c^2*x^2]*ArcSin[c*x] + ArcSin[c*x]^2 + Sqrt[1 - c^2*x^2] 
*ArcSin[c*x]^2*Log[1 - E^(I*ArcSin[c*x])] - Pi*Sqrt[1 - c^2*x^2]*Log[1 - I 
*E^(I*ArcSin[c*x])] - 2*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 - I*E^(I*ArcSi 
n[c*x])] - Pi*Sqrt[1 - c^2*x^2]*Log[1 + I*E^(I*ArcSin[c*x])] + 2*Sqrt[1 - 
c^2*x^2]*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])] - Sqrt[1 - c^2*x^2]*ArcS 
in[c*x]^2*Log[1 + E^(I*ArcSin[c*x])] + Pi*Sqrt[1 - c^2*x^2]*Log[-Cos[(Pi + 
 2*ArcSin[c*x])/4]] + Pi*Sqrt[1 - c^2*x^2]*Log[Sin[(Pi + 2*ArcSin[c*x])/4] 
] + (2*I)*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*PolyLog[2, -E^(I*ArcSin[c*x])] - ( 
2*I)*Sqrt[1 - c^2*x^2]*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] + (2*I)*Sqrt[1 - 
 c^2*x^2]*PolyLog[2, I*E^(I*ArcSin[c*x])] - (2*I)*Sqrt[1 - c^2*x^2]*ArcSin 
[c*x]*PolyLog[2, E^(I*ArcSin[c*x])] - 2*Sqrt[1 - c^2*x^2]*PolyLog[3, -E^(I 
*ArcSin[c*x])] + 2*Sqrt[1 - c^2*x^2]*PolyLog[3, E^(I*ArcSin[c*x])]))/(S...
 

Rubi [A] (verified)

Time = 2.42 (sec) , antiderivative size = 250, normalized size of antiderivative = 0.46, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.371, Rules used = {5238, 5208, 5164, 3042, 4669, 2715, 2838, 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.

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

\(\Big \downarrow \) 5238

\(\displaystyle \frac {\sqrt {1-c^2 x^2} \int \frac {(a+b \arcsin (c x))^2}{x \left (1-c^2 x^2\right )^{3/2}}dx}{d e \sqrt {c d x+d} \sqrt {e-c e x}}\)

\(\Big \downarrow \) 5208

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

\(\Big \downarrow \) 5164

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4669

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 5218

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4671

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

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

Defintions of rubi rules used

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 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 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 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 4669

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

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 5164

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2), x_Symbo 
l] :> Simp[1/(c*d)   Subst[Int[(a + b*x)^n*Sec[x], x], x, ArcSin[c*x]], x] 
/; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]
 

rule 5208

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/(2*d*f*(p + 1))), x] + (Simp[(m + 2*p + 3)/(2*d*(p + 1)) 
   Int[(f*x)^m*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^n, x], x] + Simp[b*c 
*(n/(2*f*(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] && LtQ[p, -1] &&  !G 
tQ[m, 1] && (IntegerQ[m] || IntegerQ[p] || EqQ[n, 1])
 

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 [A] (warning: unable to verify)

Time = 4.00 (sec) , antiderivative size = 1083, normalized size of antiderivative = 1.98

method	result	size
default	\(\text {Expression too large to display}\)	\(1083\)
parts	\(\text {Expression too large to display}\)	\(1083\)

Input:

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

Output:

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

Fricas [F]

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

integrate((a+b*arcsin(c*x))^2/x/(c*d*x+d)^(3/2)/(-c*e*x+e)^(3/2),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)*sqr 
t(-c*e*x + e)/(c^4*d^2*e^2*x^5 - 2*c^2*d^2*e^2*x^3 + d^2*e^2*x), x)
 

Sympy [F]

\[ \int \frac {(a+b \arcsin (c x))^2}{x (d+c d x)^{3/2} (e-c e x)^{3/2}} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{x \left (d \left (c x + 1\right )\right )^{\frac {3}{2}} \left (- e \left (c x - 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:

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

Output:

Integral((a + b*asin(c*x))**2/(x*(d*(c*x + 1))**(3/2)*(-e*(c*x - 1))**(3/2 
)), x)
 

Maxima [F(-2)]

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

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

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(a+b \arcsin (c x))^2}{x (d+c d x)^{3/2} (e-c e x)^{3/2}} \, dx=\frac {-2 \sqrt {c x +1}\, \sqrt {-c x +1}\, \left (\int \frac {\mathit {asin} \left (c x \right )}{\sqrt {c x +1}\, \sqrt {-c x +1}\, c^{2} x^{3}-\sqrt {c x +1}\, \sqrt {-c x +1}\, x}d x \right ) a b -\sqrt {c x +1}\, \sqrt {-c x +1}\, \left (\int \frac {\mathit {asin} \left (c x \right )^{2}}{\sqrt {c x +1}\, \sqrt {-c x +1}\, c^{2} x^{3}-\sqrt {c x +1}\, \sqrt {-c x +1}\, x}d x \right ) b^{2}-\sqrt {c x +1}\, \sqrt {-c x +1}\, \mathrm {log}\left (-\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-c x +1}}{\sqrt {2}}\right )}{2}\right )-1\right ) a^{2}+\sqrt {c x +1}\, \sqrt {-c x +1}\, \mathrm {log}\left (-\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-c x +1}}{\sqrt {2}}\right )}{2}\right )+1\right ) a^{2}-\sqrt {c x +1}\, \sqrt {-c x +1}\, \mathrm {log}\left (\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-c x +1}}{\sqrt {2}}\right )}{2}\right )-1\right ) a^{2}+\sqrt {c x +1}\, \sqrt {-c x +1}\, \mathrm {log}\left (\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-c x +1}}{\sqrt {2}}\right )}{2}\right )+1\right ) a^{2}+a^{2}}{\sqrt {e}\, \sqrt {d}\, \sqrt {c x +1}\, \sqrt {-c x +1}\, d e} \] Input:

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

Output:

( - 2*sqrt(c*x + 1)*sqrt( - c*x + 1)*int(asin(c*x)/(sqrt(c*x + 1)*sqrt( - 
c*x + 1)*c**2*x**3 - sqrt(c*x + 1)*sqrt( - c*x + 1)*x),x)*a*b - sqrt(c*x + 
 1)*sqrt( - c*x + 1)*int(asin(c*x)**2/(sqrt(c*x + 1)*sqrt( - c*x + 1)*c**2 
*x**3 - sqrt(c*x + 1)*sqrt( - c*x + 1)*x),x)*b**2 - sqrt(c*x + 1)*sqrt( - 
c*x + 1)*log( - sqrt(2) + tan(asin(sqrt( - c*x + 1)/sqrt(2))/2) - 1)*a**2 
+ sqrt(c*x + 1)*sqrt( - c*x + 1)*log( - sqrt(2) + tan(asin(sqrt( - c*x + 1 
)/sqrt(2))/2) + 1)*a**2 - sqrt(c*x + 1)*sqrt( - c*x + 1)*log(sqrt(2) + tan 
(asin(sqrt( - c*x + 1)/sqrt(2))/2) - 1)*a**2 + sqrt(c*x + 1)*sqrt( - c*x + 
 1)*log(sqrt(2) + tan(asin(sqrt( - c*x + 1)/sqrt(2))/2) + 1)*a**2 + a**2)/ 
(sqrt(e)*sqrt(d)*sqrt(c*x + 1)*sqrt( - c*x + 1)*d*e)