\(\int \frac {(a+b \arcsin (c x))^2}{(d+c d x)^{3/2} (e-c e x)^{5/2}} \, dx\) [94]

Optimal result

Integrand size = 32, antiderivative size = 742 \[ \int \frac {(a+b \arcsin (c x))^2}{(d+c d x)^{3/2} (e-c e x)^{5/2}} \, dx=\frac {b^2}{3 c d e^2 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {b^2 x}{3 d e^2 \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {b (a+b \arcsin (c x))}{3 c d e^2 \sqrt {d+c d x} \sqrt {e-c e x} \sqrt {1-c^2 x^2}}-\frac {b x (a+b \arcsin (c x))}{3 d e^2 \sqrt {d+c d x} \sqrt {e-c e x} \sqrt {1-c^2 x^2}}+\frac {2 x (a+b \arcsin (c x))^2}{3 d e^2 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {(a+b \arcsin (c x))^2}{3 c d e^2 \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right )}+\frac {x (a+b \arcsin (c x))^2}{3 d e^2 \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right )}-\frac {2 i \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{3 c d e^2 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {2 i b \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \arctan \left (e^{i \arcsin (c x)}\right )}{3 c d e^2 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {4 b \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \log \left (1+e^{2 i \arcsin (c x)}\right )}{3 c d e^2 \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{3 c d e^2 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{3 c d e^2 \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {2 i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )}{3 c d e^2 \sqrt {d+c d x} \sqrt {e-c e x}} \] Output:

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

Mathematica [A] (warning: unable to verify)

Time = 9.31 (sec) , antiderivative size = 764, normalized size of antiderivative = 1.03 \[ \int \frac {(a+b \arcsin (c x))^2}{(d+c d x)^{3/2} (e-c e x)^{5/2}} \, dx =\text {Too large to display} \] Input:

Integrate[(a + b*ArcSin[c*x])^2/((d + c*d*x)^(3/2)*(e - c*e*x)^(5/2)),x]
 

Output:

(Sqrt[-(e*(-1 + c*x))]*Sqrt[d*(1 + c*x)]*(a^2/(6*d^2*e^3*(-1 + c*x)^2) - ( 
5*a^2)/(12*d^2*e^3*(-1 + c*x)) - a^2/(4*d^2*e^3*(1 + c*x))))/c - (a*b*Sqrt 
[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[1 - c^2*x^2]*(2*ArcSin[c*x]*(2*c*x + Cos[ 
2*ArcSin[c*x]]) + Sqrt[1 - c^2*x^2]*(-1 + 5*Log[Cos[ArcSin[c*x]/2] - Sin[A 
rcSin[c*x]/2]] + 3*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]] - c*x*(5*L 
og[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] + 3*Log[Cos[ArcSin[c*x]/2] + S 
in[ArcSin[c*x]/2]]))))/(3*c*d*e^2*Sqrt[(-d - c*d*x)*(e - c*e*x)]*Sqrt[-(d* 
e*(1 - c^2*x^2))]*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])^3*(Cos[ArcSin[ 
c*x]/2] + Sin[ArcSin[c*x]/2])) - (b^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt 
[1 - c^2*x^2]*((9*I)*Pi*ArcSin[c*x] - ((-2 + ArcSin[c*x])*ArcSin[c*x])/(-1 
 + c*x) + (1 - 4*I)*ArcSin[c*x]^2 + 16*Pi*Log[1 + E^((-I)*ArcSin[c*x])] + 
3*(Pi + 2*ArcSin[c*x])*Log[1 - I*E^(I*ArcSin[c*x])] - 5*(Pi - 2*ArcSin[c*x 
])*Log[1 + I*E^(I*ArcSin[c*x])] - 16*Pi*Log[Cos[ArcSin[c*x]/2]] + 5*Pi*Log 
[-Cos[(Pi + 2*ArcSin[c*x])/4]] - 3*Pi*Log[Sin[(Pi + 2*ArcSin[c*x])/4]] - ( 
10*I)*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] - (6*I)*PolyLog[2, I*E^(I*ArcSin[ 
c*x])] + (2*ArcSin[c*x]^2*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] - Sin[Ar 
cSin[c*x]/2])^3 + ((4 + 5*ArcSin[c*x]^2)*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c 
*x]/2] - Sin[ArcSin[c*x]/2]) + (3*ArcSin[c*x]^2*Sin[ArcSin[c*x]/2])/(Cos[A 
rcSin[c*x]/2] + Sin[ArcSin[c*x]/2])))/(6*c*d*e^2*Sqrt[(-d - c*d*x)*(e - c* 
e*x)]*Sqrt[-(d*e*(1 - c^2*x^2))])
 

Rubi [A] (verified)

Time = 1.17 (sec) , antiderivative size = 386, normalized size of antiderivative = 0.52, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5178, 27, 5262, 2009}

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.

\(\displaystyle \int \frac {(a+b \arcsin (c x))^2}{(c d x+d)^{3/2} (e-c e x)^{5/2}} \, dx\)

\(\Big \downarrow \) 5178

\(\displaystyle \frac {\left (1-c^2 x^2\right )^{5/2} \int \frac {d (c x+1) (a+b \arcsin (c x))^2}{\left (1-c^2 x^2\right )^{5/2}}dx}{(c d x+d)^{5/2} (e-c e x)^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {d \left (1-c^2 x^2\right )^{5/2} \int \frac {(c x+1) (a+b \arcsin (c x))^2}{\left (1-c^2 x^2\right )^{5/2}}dx}{(c d x+d)^{5/2} (e-c e x)^{5/2}}\)

\(\Big \downarrow \) 5262

\(\displaystyle \frac {d \left (1-c^2 x^2\right )^{5/2} \int \left (\frac {c x (a+b \arcsin (c x))^2}{\left (1-c^2 x^2\right )^{5/2}}+\frac {(a+b \arcsin (c x))^2}{\left (1-c^2 x^2\right )^{5/2}}\right )dx}{(c d x+d)^{5/2} (e-c e x)^{5/2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {d \left (1-c^2 x^2\right )^{5/2} \left (\frac {2 i b \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{3 c}-\frac {b x (a+b \arcsin (c x))}{3 \left (1-c^2 x^2\right )}-\frac {b (a+b \arcsin (c x))}{3 c \left (1-c^2 x^2\right )}+\frac {2 x (a+b \arcsin (c x))^2}{3 \sqrt {1-c^2 x^2}}+\frac {x (a+b \arcsin (c x))^2}{3 \left (1-c^2 x^2\right )^{3/2}}+\frac {(a+b \arcsin (c x))^2}{3 c \left (1-c^2 x^2\right )^{3/2}}-\frac {2 i (a+b \arcsin (c x))^2}{3 c}+\frac {4 b \log \left (1+e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{3 c}-\frac {i b^2 \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{3 c}+\frac {i b^2 \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{3 c}-\frac {2 i b^2 \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )}{3 c}+\frac {b^2 x}{3 \sqrt {1-c^2 x^2}}+\frac {b^2}{3 c \sqrt {1-c^2 x^2}}\right )}{(c d x+d)^{5/2} (e-c e x)^{5/2}}\)

Input:

Int[(a + b*ArcSin[c*x])^2/((d + c*d*x)^(3/2)*(e - c*e*x)^(5/2)),x]
 

Output:

(d*(1 - c^2*x^2)^(5/2)*(b^2/(3*c*Sqrt[1 - c^2*x^2]) + (b^2*x)/(3*Sqrt[1 - 
c^2*x^2]) - (b*(a + b*ArcSin[c*x]))/(3*c*(1 - c^2*x^2)) - (b*x*(a + b*ArcS 
in[c*x]))/(3*(1 - c^2*x^2)) - (((2*I)/3)*(a + b*ArcSin[c*x])^2)/c + (a + b 
*ArcSin[c*x])^2/(3*c*(1 - c^2*x^2)^(3/2)) + (x*(a + b*ArcSin[c*x])^2)/(3*( 
1 - c^2*x^2)^(3/2)) + (2*x*(a + b*ArcSin[c*x])^2)/(3*Sqrt[1 - c^2*x^2]) + 
(((2*I)/3)*b*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])])/c + (4*b*(a + 
b*ArcSin[c*x])*Log[1 + E^((2*I)*ArcSin[c*x])])/(3*c) - ((I/3)*b^2*PolyLog[ 
2, (-I)*E^(I*ArcSin[c*x])])/c + ((I/3)*b^2*PolyLog[2, I*E^(I*ArcSin[c*x])] 
)/c - (((2*I)/3)*b^2*PolyLog[2, -E^((2*I)*ArcSin[c*x])])/c))/((d + c*d*x)^ 
(5/2)*(e - c*e*x)^(5/2))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

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 5262

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_ 
) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^p*(a + 
 b*ArcSin[c*x])^n, (f + g*x)^m, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] & 
& EqQ[c^2*d + e, 0] && IGtQ[m, 0] && IntegerQ[p + 1/2] && GtQ[d, 0] && IGtQ 
[n, 0] && (m == 1 || p > 0 || (n == 1 && p > -1) || (m == 2 && p < -2))
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 3176 vs. \(2 (685 ) = 1370\).

Time = 3.37 (sec) , antiderivative size = 3177, normalized size of antiderivative = 4.28

method	result	size
default	\(\text {Expression too large to display}\)	\(3177\)
parts	\(\text {Expression too large to display}\)	\(3177\)

Input:

int((a+b*arcsin(c*x))^2/(c*d*x+d)^(3/2)/(-c*e*x+e)^(5/2),x,method=_RETURNV 
ERBOSE)
 

Output:

a^2*(-1/c/d/e/(c*d*x+d)^(1/2)/(-c*e*x+e)^(3/2)+2/d*(1/3/c/d/e/(-c*e*x+e)^( 
3/2)*(c*d*x+d)^(1/2)+1/3/c/d/e^2/(-c*e*x+e)^(1/2)*(c*d*x+d)^(1/2)))+2/3*I* 
b^2*(-e*(c*x-1))^(1/2)*(d*(c*x+1))^(1/2)/(4*c^4*x^4-9*c^3*x^3+c^2*x^2+9*c* 
x-5)/c/d^2/e^3*arcsin(c*x)*(-c^2*x^2+1)+6*I*b^2*(-e*(c*x-1))^(1/2)*(d*(c*x 
+1))^(1/2)/(4*c^4*x^4-9*c^3*x^3+c^2*x^2+9*c*x-5)/d^2/e^3*arcsin(c*x)^2*(-c 
^2*x^2+1)^(1/2)*x+2/3*I*b^2*(-e*(c*x-1))^(1/2)*(d*(c*x+1))^(1/2)/(4*c^4*x^ 
4-9*c^3*x^3+c^2*x^2+9*c*x-5)/d^2/e^3*arcsin(c*x)*(-c^2*x^2+1)*x+4/3*I*b^2* 
(-e*(c*x-1))^(1/2)*(d*(c*x+1))^(1/2)*(-c^2*x^2+1)^(1/2)/d^2/e^3/c/(c^2*x^2 
-1)*arcsin(c*x)^2+5/3*I*b^2*(-e*(c*x-1))^(1/2)*(d*(c*x+1))^(1/2)*(-c^2*x^2 
+1)^(1/2)/d^2/e^3/c/(c^2*x^2-1)*polylog(2,-I*(I*c*x+(-c^2*x^2+1)^(1/2)))+4 
/3*I*b^2*(-e*(c*x-1))^(1/2)*(d*(c*x+1))^(1/2)/(4*c^4*x^4-9*c^3*x^3+c^2*x^2 
+9*c*x-5)*c^4/d^2/e^3*arcsin(c*x)*x^5-8/3*I*b^2*(-e*(c*x-1))^(1/2)*(d*(c*x 
+1))^(1/2)/(4*c^4*x^4-9*c^3*x^3+c^2*x^2+9*c*x-5)*c^3/d^2/e^3*arcsin(c*x)*x 
^4-2/3*I*b^2*(-e*(c*x-1))^(1/2)*(d*(c*x+1))^(1/2)/(4*c^4*x^4-9*c^3*x^3+c^2 
*x^2+9*c*x-5)*c^2/d^2/e^3*arcsin(c*x)*x^3+10/3*I*b^2*(-e*(c*x-1))^(1/2)*(d 
*(c*x+1))^(1/2)/(4*c^4*x^4-9*c^3*x^3+c^2*x^2+9*c*x-5)*c/d^2/e^3*arcsin(c*x 
)*x^2-4/3*I*b^2*(-e*(c*x-1))^(1/2)*(d*(c*x+1))^(1/2)/(4*c^4*x^4-9*c^3*x^3+ 
c^2*x^2+9*c*x-5)*c/d^2/e^3*(-c^2*x^2+1)^(1/2)*x^2-10/3*I*b^2*(-e*(c*x-1))^ 
(1/2)*(d*(c*x+1))^(1/2)/(4*c^4*x^4-9*c^3*x^3+c^2*x^2+9*c*x-5)/c/d^2/e^3*ar 
csin(c*x)^2*(-c^2*x^2+1)^(1/2)-5/3*b^2*(-e*(c*x-1))^(1/2)*(d*(c*x+1))^(...
 

Fricas [F]

\[ \int \frac {(a+b \arcsin (c x))^2}{(d+c d x)^{3/2} (e-c e x)^{5/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (c d x + d\right )}^{\frac {3}{2}} {\left (-c e x + e\right )}^{\frac {5}{2}}} \,d x } \] Input:

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

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b \arcsin (c x))^2}{(d+c d x)^{3/2} (e-c e x)^{5/2}} \, dx=\text {Timed out} \] Input:

integrate((a+b*asin(c*x))**2/(c*d*x+d)**(3/2)/(-c*e*x+e)**(5/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(a+b \arcsin (c x))^2}{(d+c d x)^{3/2} (e-c e x)^{5/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (c d x + d\right )}^{\frac {3}{2}} {\left (-c e x + e\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((a+b*arcsin(c*x))^2/(c*d*x+d)^(3/2)/(-c*e*x+e)^(5/2),x, algorith 
m="maxima")
 

Output:

1/6*a*b*c*(2*sqrt(d)*sqrt(e)/(c^3*d^2*e^3*x - c^2*d^2*e^3) + 3*log(c*x + 1 
)/(c^2*d^(3/2)*e^(5/2)) + 5*log(c*x - 1)/(c^2*d^(3/2)*e^(5/2))) - 2/3*a*b* 
(1/(sqrt(-c^2*d*e*x^2 + d*e)*c^2*d*e^2*x - sqrt(-c^2*d*e*x^2 + d*e)*c*d*e^ 
2) - 2*x/(sqrt(-c^2*d*e*x^2 + d*e)*d*e^2))*arcsin(c*x) - 1/3*a^2*(1/(sqrt( 
-c^2*d*e*x^2 + d*e)*c^2*d*e^2*x - sqrt(-c^2*d*e*x^2 + d*e)*c*d*e^2) - 2*x/ 
(sqrt(-c^2*d*e*x^2 + d*e)*d*e^2)) + b^2*integrate(arctan2(c*x, sqrt(c*x + 
1)*sqrt(-c*x + 1))^2/((c^3*d*e^2*x^3 - c^2*d*e^2*x^2 - c*d*e^2*x + d*e^2)* 
sqrt(c*x + 1)*sqrt(-c*x + 1)), x)/(sqrt(d)*sqrt(e))
 

Giac [F]

\[ \int \frac {(a+b \arcsin (c x))^2}{(d+c d x)^{3/2} (e-c e x)^{5/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (c d x + d\right )}^{\frac {3}{2}} {\left (-c e x + e\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((a+b*arcsin(c*x))^2/(c*d*x+d)^(3/2)/(-c*e*x+e)^(5/2),x, algorith 
m="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \arcsin (c x))^2}{(d+c d x)^{3/2} (e-c e x)^{5/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{{\left (d+c\,d\,x\right )}^{3/2}\,{\left (e-c\,e\,x\right )}^{5/2}} \,d x \] Input:

int((a + b*asin(c*x))^2/((d + c*d*x)^(3/2)*(e - c*e*x)^(5/2)),x)
 

Output:

int((a + b*asin(c*x))^2/((d + c*d*x)^(3/2)*(e - c*e*x)^(5/2)), x)
 

Reduce [F]

\[ \int \frac {(a+b \arcsin (c x))^2}{(d+c d x)^{3/2} (e-c e x)^{5/2}} \, dx=\frac {6 \sqrt {c x +1}\, \sqrt {-c x +1}\, \left (\int \frac {\mathit {asin} \left (c x \right )}{\sqrt {c x +1}\, \sqrt {-c x +1}\, c^{3} x^{3}-\sqrt {c x +1}\, \sqrt {-c x +1}\, c^{2} x^{2}-\sqrt {c x +1}\, \sqrt {-c x +1}\, c x +\sqrt {c x +1}\, \sqrt {-c x +1}}d x \right ) a b \,c^{2} x -6 \sqrt {c x +1}\, \sqrt {-c x +1}\, \left (\int \frac {\mathit {asin} \left (c x \right )}{\sqrt {c x +1}\, \sqrt {-c x +1}\, c^{3} x^{3}-\sqrt {c x +1}\, \sqrt {-c x +1}\, c^{2} x^{2}-\sqrt {c x +1}\, \sqrt {-c x +1}\, c x +\sqrt {c x +1}\, \sqrt {-c x +1}}d x \right ) a b c +3 \sqrt {c x +1}\, \sqrt {-c x +1}\, \left (\int \frac {\mathit {asin} \left (c x \right )^{2}}{\sqrt {c x +1}\, \sqrt {-c x +1}\, c^{3} x^{3}-\sqrt {c x +1}\, \sqrt {-c x +1}\, c^{2} x^{2}-\sqrt {c x +1}\, \sqrt {-c x +1}\, c x +\sqrt {c x +1}\, \sqrt {-c x +1}}d x \right ) b^{2} c^{2} x -3 \sqrt {c x +1}\, \sqrt {-c x +1}\, \left (\int \frac {\mathit {asin} \left (c x \right )^{2}}{\sqrt {c x +1}\, \sqrt {-c x +1}\, c^{3} x^{3}-\sqrt {c x +1}\, \sqrt {-c x +1}\, c^{2} x^{2}-\sqrt {c x +1}\, \sqrt {-c x +1}\, c x +\sqrt {c x +1}\, \sqrt {-c x +1}}d x \right ) b^{2} c +2 a^{2} c^{2} x^{2}-2 a^{2} c x -a^{2}}{3 \sqrt {e}\, \sqrt {d}\, \sqrt {c x +1}\, \sqrt {-c x +1}\, c d \,e^{2} \left (c x -1\right )} \] Input:

int((a+b*asin(c*x))^2/(c*d*x+d)^(3/2)/(-c*e*x+e)^(5/2),x)
                                                                                    
                                                                                    
 

Output:

(6*sqrt(c*x + 1)*sqrt( - c*x + 1)*int(asin(c*x)/(sqrt(c*x + 1)*sqrt( - c*x 
 + 1)*c**3*x**3 - sqrt(c*x + 1)*sqrt( - c*x + 1)*c**2*x**2 - sqrt(c*x + 1) 
*sqrt( - c*x + 1)*c*x + sqrt(c*x + 1)*sqrt( - c*x + 1)),x)*a*b*c**2*x - 6* 
sqrt(c*x + 1)*sqrt( - c*x + 1)*int(asin(c*x)/(sqrt(c*x + 1)*sqrt( - c*x + 
1)*c**3*x**3 - sqrt(c*x + 1)*sqrt( - c*x + 1)*c**2*x**2 - sqrt(c*x + 1)*sq 
rt( - c*x + 1)*c*x + sqrt(c*x + 1)*sqrt( - c*x + 1)),x)*a*b*c + 3*sqrt(c*x 
 + 1)*sqrt( - c*x + 1)*int(asin(c*x)**2/(sqrt(c*x + 1)*sqrt( - c*x + 1)*c* 
*3*x**3 - sqrt(c*x + 1)*sqrt( - c*x + 1)*c**2*x**2 - sqrt(c*x + 1)*sqrt( - 
 c*x + 1)*c*x + sqrt(c*x + 1)*sqrt( - c*x + 1)),x)*b**2*c**2*x - 3*sqrt(c* 
x + 1)*sqrt( - c*x + 1)*int(asin(c*x)**2/(sqrt(c*x + 1)*sqrt( - c*x + 1)*c 
**3*x**3 - sqrt(c*x + 1)*sqrt( - c*x + 1)*c**2*x**2 - sqrt(c*x + 1)*sqrt( 
- c*x + 1)*c*x + sqrt(c*x + 1)*sqrt( - c*x + 1)),x)*b**2*c + 2*a**2*c**2*x 
**2 - 2*a**2*c*x - a**2)/(3*sqrt(e)*sqrt(d)*sqrt(c*x + 1)*sqrt( - c*x + 1) 
*c*d*e**2*(c*x - 1))