\(\int \frac {(a+b \arcsin (c x))^2}{(d+c d x)^{3/2} (e-c e x)^{3/2}} \, dx\) [91]

Optimal result
Mathematica [B] (verified)
Rubi [A] (verified)
Maple [B] (verified)
Fricas [F]
Sympy [F]
Maxima [F]
Giac [F]
Mupad [F(-1)]
Reduce [F]

Optimal result

Integrand size = 32, antiderivative size = 231 \[ \int \frac {(a+b \arcsin (c x))^2}{(d+c d x)^{3/2} (e-c e x)^{3/2}} \, dx=\frac {x (a+b \arcsin (c x))^2}{d e \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {i \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{c d e \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {2 b \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \log \left (1+e^{2 i \arcsin (c x)}\right )}{c d e \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )}{c d e \sqrt {d+c d x} \sqrt {e-c e x}} \] Output:

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

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(550\) vs. \(2(231)=462\).

Time = 0.19 (sec) , antiderivative size = 550, normalized size of antiderivative = 2.38 \[ \int \frac {(a+b \arcsin (c x))^2}{(d+c d x)^{3/2} (e-c e x)^{3/2}} \, dx=\frac {a^2 c x+2 a b c x \arcsin (c x)+2 i b^2 \pi \sqrt {1-c^2 x^2} \arcsin (c x)+b^2 c x \arcsin (c x)^2-i b^2 \sqrt {1-c^2 x^2} \arcsin (c x)^2+4 b^2 \pi \sqrt {1-c^2 x^2} \log \left (1+e^{-i \arcsin (c x)}\right )+b^2 \pi \sqrt {1-c^2 x^2} \log \left (1-i e^{i \arcsin (c x)}\right )+2 b^2 \sqrt {1-c^2 x^2} \arcsin (c x) \log \left (1-i e^{i \arcsin (c x)}\right )-b^2 \pi \sqrt {1-c^2 x^2} \log \left (1+i e^{i \arcsin (c x)}\right )+2 b^2 \sqrt {1-c^2 x^2} \arcsin (c x) \log \left (1+i e^{i \arcsin (c x)}\right )-4 b^2 \pi \sqrt {1-c^2 x^2} \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )\right )+b^2 \pi \sqrt {1-c^2 x^2} \log \left (-\cos \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )+2 a b \sqrt {1-c^2 x^2} \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )-\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )+2 a b \sqrt {1-c^2 x^2} \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )-b^2 \pi \sqrt {1-c^2 x^2} \log \left (\sin \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )-2 i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-2 i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{c d e \sqrt {d+c d x} \sqrt {e-c e x}} \] Input:

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

Output:

(a^2*c*x + 2*a*b*c*x*ArcSin[c*x] + (2*I)*b^2*Pi*Sqrt[1 - c^2*x^2]*ArcSin[c 
*x] + b^2*c*x*ArcSin[c*x]^2 - I*b^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x]^2 + 4*b^ 
2*Pi*Sqrt[1 - c^2*x^2]*Log[1 + E^((-I)*ArcSin[c*x])] + b^2*Pi*Sqrt[1 - c^2 
*x^2]*Log[1 - I*E^(I*ArcSin[c*x])] + 2*b^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*L 
og[1 - I*E^(I*ArcSin[c*x])] - b^2*Pi*Sqrt[1 - c^2*x^2]*Log[1 + I*E^(I*ArcS 
in[c*x])] + 2*b^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x] 
)] - 4*b^2*Pi*Sqrt[1 - c^2*x^2]*Log[Cos[ArcSin[c*x]/2]] + b^2*Pi*Sqrt[1 - 
c^2*x^2]*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] + 2*a*b*Sqrt[1 - c^2*x^2]*Log[C 
os[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] + 2*a*b*Sqrt[1 - c^2*x^2]*Log[Cos[ 
ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]] - b^2*Pi*Sqrt[1 - c^2*x^2]*Log[Sin[(P 
i + 2*ArcSin[c*x])/4]] - (2*I)*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, (-I)*E^(I* 
ArcSin[c*x])] - (2*I)*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, I*E^(I*ArcSin[c*x]) 
])/(c*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])
 

Rubi [A] (verified)

Time = 0.97 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.61, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5178, 5160, 5180, 3042, 4202, 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.

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

\(\Big \downarrow \) 5178

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

\(\Big \downarrow \) 5160

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

\(\Big \downarrow \) 5180

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4202

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

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

Defintions of rubi rules used

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 4202
Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (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*( 
e + f*x))/(1 + E^(2*I*(e + f*x)))), x], x] /; FreeQ[{c, d, e, f}, x] && IGt 
Q[m, 0]
 

rule 5160
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2)^(3/2), x 
_Symbol] :> Simp[x*((a + b*ArcSin[c*x])^n/(d*Sqrt[d + e*x^2])), x] - Simp[b 
*c*(n/d)*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]   Int[x*((a + b*ArcSin[c*x 
])^(n - 1)/(1 - c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d 
 + e, 0] && GtQ[n, 0]
 

rule 5178
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_))^(p_)*((f_) 
 + (g_.)*(x_))^(q_), x_Symbol] :> Simp[(d + e*x)^q*((f + g*x)^q/(1 - c^2*x^ 
2)^q)   Int[(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, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^2 
- e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 0]
 

rule 5180
Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_))/((d_) + (e_.)*(x_)^2), 
x_Symbol] :> Simp[-e^(-1)   Subst[Int[(a + b*x)^n*Tan[x], x], x, ArcSin[c*x 
]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]
 
Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 704 vs. \(2 (227 ) = 454\).

Time = 0.01 (sec) , antiderivative size = 705, normalized size of antiderivative = 3.05

method result size
default \(a^{2} \left (-\frac {1}{c d e \sqrt {c d x +d}\, \sqrt {-c x e +e}}+\frac {\sqrt {c d x +d}}{c e \,d^{2} \sqrt {-c x e +e}}\right )-\frac {b^{2} \left (\arcsin \left (c x \right )^{2}+2 \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+2 \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-2 i \sqrt {-c^{2} x^{2}+1}\, \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) x c -2 i \sqrt {-c^{2} x^{2}+1}\, \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) x c -2 i \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) x^{2} c^{2}-2 i \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) x^{2} c^{2}+2 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) x c +2 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) x c -2 \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) x^{2} c^{2}-2 \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) x^{2} c^{2}+2 i \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+2 i \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )\right ) \left (-i \sqrt {-c^{2} x^{2}+1}+c x \right ) \sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}}{c \,d^{2} e^{2} \left (c^{2} x^{2}-1\right )}+\frac {2 a b \left (i \arcsin \left (c x \right ) x^{2} c^{2}-\ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right ) x^{2} c^{2}+\arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c x -i \arcsin \left (c x \right )+\ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )\right ) \sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-c^{2} x^{2}+1}}{c \,d^{2} e^{2} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}\) \(705\)
parts \(a^{2} \left (-\frac {1}{c d e \sqrt {c d x +d}\, \sqrt {-c x e +e}}+\frac {\sqrt {c d x +d}}{c e \,d^{2} \sqrt {-c x e +e}}\right )-\frac {b^{2} \left (\arcsin \left (c x \right )^{2}+2 \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+2 \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-2 i \sqrt {-c^{2} x^{2}+1}\, \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) x c -2 i \sqrt {-c^{2} x^{2}+1}\, \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) x c -2 i \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) x^{2} c^{2}-2 i \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) x^{2} c^{2}+2 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) x c +2 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) x c -2 \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) x^{2} c^{2}-2 \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) x^{2} c^{2}+2 i \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+2 i \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )\right ) \left (-i \sqrt {-c^{2} x^{2}+1}+c x \right ) \sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}}{c \,d^{2} e^{2} \left (c^{2} x^{2}-1\right )}+\frac {2 a b \left (i \arcsin \left (c x \right ) x^{2} c^{2}-\ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right ) x^{2} c^{2}+\arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c x -i \arcsin \left (c x \right )+\ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )\right ) \sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-c^{2} x^{2}+1}}{c \,d^{2} e^{2} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}\) \(705\)

Input:

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

Output:

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

Fricas [F]

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

integrate((a+b*arcsin(c*x))^2/(c*d*x+d)^(3/2)/(-c*e*x+e)^(3/2),x, algorith 
m="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^4 - 2*c^2*d^2*e^2*x^2 + d^2*e^2), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-b^2*integrate(arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2/((c^2*d*e*x^2 
- d*e)*sqrt(c*x + 1)*sqrt(-c*x + 1)), x)/(sqrt(d)*sqrt(e)) + 2*a*b*x*arcsi 
n(c*x)/(sqrt(-c^2*d*e*x^2 + d*e)*d*e) + a^2*x/(sqrt(-c^2*d*e*x^2 + d*e)*d* 
e) - a*b*sqrt(1/(d*e))*log(x^2 - 1/c^2)/(c*d*e)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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