\(\int (d+c d x)^{5/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2 \, dx\) [75]

Optimal result
Mathematica [A] (verified)
Rubi [A] (verified)
Maple [C] (verified)
Fricas [F]
Sympy [F(-1)]
Maxima [F(-2)]
Giac [F]
Mupad [F(-1)]
Reduce [F]

Optimal result

Integrand size = 32, antiderivative size = 740 \[ \int (d+c d x)^{5/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2 \, dx=\frac {16 b^2 d^2 e \sqrt {d+c d x} \sqrt {e-c e x}}{75 c}-\frac {15}{64} b^2 d^2 e x \sqrt {d+c d x} \sqrt {e-c e x}+\frac {8 b^2 d^2 e \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right )}{225 c}-\frac {1}{32} b^2 d^2 e x \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right )+\frac {2 b^2 d^2 e \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right )^2}{125 c}+\frac {9 b^2 d^2 e \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x)}{64 c \sqrt {1-c^2 x^2}}+\frac {2 b d^2 e x \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{5 \sqrt {1-c^2 x^2}}-\frac {3 b c d^2 e x^2 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{8 \sqrt {1-c^2 x^2}}-\frac {4 b c^2 d^2 e x^3 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{15 \sqrt {1-c^2 x^2}}+\frac {2 b c^4 d^2 e x^5 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{25 \sqrt {1-c^2 x^2}}+\frac {b d^2 e \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{8 c}+\frac {3}{8} d^2 e x \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2+\frac {1}{4} d^2 e x \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-\frac {d^2 e \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{5 c}+\frac {d^2 e \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^3}{8 b c \sqrt {1-c^2 x^2}} \] Output:

16/75*b^2*d^2*e*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)/c-15/64*b^2*d^2*e*x*(c*d* 
x+d)^(1/2)*(-c*e*x+e)^(1/2)+8/225*b^2*d^2*e*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/ 
2)*(-c^2*x^2+1)/c-1/32*b^2*d^2*e*x*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)*(-c^2* 
x^2+1)+2/125*b^2*d^2*e*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)*(-c^2*x^2+1)^2/c+9 
/64*b^2*d^2*e*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)*arcsin(c*x)/c/(-c^2*x^2+1)^ 
(1/2)+2/5*b*d^2*e*x*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)*(a+b*arcsin(c*x))/(-c 
^2*x^2+1)^(1/2)-3/8*b*c*d^2*e*x^2*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)*(a+b*ar 
csin(c*x))/(-c^2*x^2+1)^(1/2)-4/15*b*c^2*d^2*e*x^3*(c*d*x+d)^(1/2)*(-c*e*x 
+e)^(1/2)*(a+b*arcsin(c*x))/(-c^2*x^2+1)^(1/2)+2/25*b*c^4*d^2*e*x^5*(c*d*x 
+d)^(1/2)*(-c*e*x+e)^(1/2)*(a+b*arcsin(c*x))/(-c^2*x^2+1)^(1/2)+1/8*b*d^2* 
e*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)*(-c^2*x^2+1)^(3/2)*(a+b*arcsin(c*x))/c+ 
3/8*d^2*e*x*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)*(a+b*arcsin(c*x))^2+1/4*d^2*e 
*x*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)*(-c^2*x^2+1)*(a+b*arcsin(c*x))^2-1/5*d 
^2*e*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)*(-c^2*x^2+1)^2*(a+b*arcsin(c*x))^2/c 
+1/8*d^2*e*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)*(a+b*arcsin(c*x))^3/b/c/(-c^2* 
x^2+1)^(1/2)
 

Mathematica [A] (verified)

Time = 3.84 (sec) , antiderivative size = 574, normalized size of antiderivative = 0.78 \[ \int (d+c d x)^{5/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2 \, dx=\frac {d^2 e \left (36000 b^2 \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x)^3-108000 a^2 \sqrt {d} \sqrt {e} \sqrt {1-c^2 x^2} \arctan \left (\frac {c x \sqrt {d+c d x} \sqrt {e-c e x}}{\sqrt {d} \sqrt {e} \left (-1+c^2 x^2\right )}\right )+1800 b \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x)^2 \left (-10 b \cos (3 \arcsin (c x))-2 b \cos (5 \arcsin (c x))+5 \left (12 a-4 b \sqrt {1-c^2 x^2}+8 b \sin (2 \arcsin (c x))+b \sin (4 \arcsin (c x))\right )\right )+\sqrt {d+c d x} \sqrt {e-c e x} \left (72000 a b \cos (2 \arcsin (c x))+4000 b^2 \cos (3 \arcsin (c x))+4500 a b \cos (4 \arcsin (c x))+288 b^2 \cos (5 \arcsin (c x))-15 \left (-4800 b^2 \sqrt {1-c^2 x^2}-512 a b c x \left (15-10 c^2 x^2+3 c^4 x^4\right )+480 a^2 \sqrt {1-c^2 x^2} \left (8-25 c x-16 c^2 x^2+10 c^3 x^3+8 c^4 x^4\right )+2400 b^2 \sin (2 \arcsin (c x))+75 b^2 \sin (4 \arcsin (c x))\right )\right )-60 b \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x) \left (-1200 b \cos (2 \arcsin (c x))-75 b \cos (4 \arcsin (c x))-4 \left (300 b c x-480 a \sqrt {1-c^2 x^2}+960 a c^2 x^2 \sqrt {1-c^2 x^2}-480 a c^4 x^4 \sqrt {1-c^2 x^2}+600 a \sin (2 \arcsin (c x))+50 b \sin (3 \arcsin (c x))+75 a \sin (4 \arcsin (c x))+6 b \sin (5 \arcsin (c x))\right )\right )\right )}{288000 c \sqrt {1-c^2 x^2}} \] Input:

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

Output:

(d^2*e*(36000*b^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]^3 - 108000*a 
^2*Sqrt[d]*Sqrt[e]*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[e - 
c*e*x])/(Sqrt[d]*Sqrt[e]*(-1 + c^2*x^2))] + 1800*b*Sqrt[d + c*d*x]*Sqrt[e 
- c*e*x]*ArcSin[c*x]^2*(-10*b*Cos[3*ArcSin[c*x]] - 2*b*Cos[5*ArcSin[c*x]] 
+ 5*(12*a - 4*b*Sqrt[1 - c^2*x^2] + 8*b*Sin[2*ArcSin[c*x]] + b*Sin[4*ArcSi 
n[c*x]])) + Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(72000*a*b*Cos[2*ArcSin[c*x]] 
+ 4000*b^2*Cos[3*ArcSin[c*x]] + 4500*a*b*Cos[4*ArcSin[c*x]] + 288*b^2*Cos[ 
5*ArcSin[c*x]] - 15*(-4800*b^2*Sqrt[1 - c^2*x^2] - 512*a*b*c*x*(15 - 10*c^ 
2*x^2 + 3*c^4*x^4) + 480*a^2*Sqrt[1 - c^2*x^2]*(8 - 25*c*x - 16*c^2*x^2 + 
10*c^3*x^3 + 8*c^4*x^4) + 2400*b^2*Sin[2*ArcSin[c*x]] + 75*b^2*Sin[4*ArcSi 
n[c*x]])) - 60*b*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]*(-1200*b*Cos[ 
2*ArcSin[c*x]] - 75*b*Cos[4*ArcSin[c*x]] - 4*(300*b*c*x - 480*a*Sqrt[1 - c 
^2*x^2] + 960*a*c^2*x^2*Sqrt[1 - c^2*x^2] - 480*a*c^4*x^4*Sqrt[1 - c^2*x^2 
] + 600*a*Sin[2*ArcSin[c*x]] + 50*b*Sin[3*ArcSin[c*x]] + 75*a*Sin[4*ArcSin 
[c*x]] + 6*b*Sin[5*ArcSin[c*x]]))))/(288000*c*Sqrt[1 - c^2*x^2])
 

Rubi [A] (verified)

Time = 1.06 (sec) , antiderivative size = 374, normalized size of antiderivative = 0.51, 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.

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

\(\Big \downarrow \) 5178

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5262

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {d (c d x+d)^{3/2} (e-c e x)^{3/2} \left (\frac {2}{25} b c^4 x^5 (a+b \arcsin (c x))-\frac {4}{15} b c^2 x^3 (a+b \arcsin (c x))+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2+\frac {3}{8} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2}{5 c}+\frac {b \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{8 c}-\frac {3}{8} b c x^2 (a+b \arcsin (c x))+\frac {2}{5} b x (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^3}{8 b c}+\frac {9 b^2 \arcsin (c x)}{64 c}-\frac {1}{32} b^2 x \left (1-c^2 x^2\right )^{3/2}-\frac {15}{64} b^2 x \sqrt {1-c^2 x^2}+\frac {2 b^2 \left (1-c^2 x^2\right )^{5/2}}{125 c}+\frac {8 b^2 \left (1-c^2 x^2\right )^{3/2}}{225 c}+\frac {16 b^2 \sqrt {1-c^2 x^2}}{75 c}\right )}{\left (1-c^2 x^2\right )^{3/2}}\)

Input:

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

Output:

(d*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)*((16*b^2*Sqrt[1 - c^2*x^2])/(75*c) 
- (15*b^2*x*Sqrt[1 - c^2*x^2])/64 + (8*b^2*(1 - c^2*x^2)^(3/2))/(225*c) - 
(b^2*x*(1 - c^2*x^2)^(3/2))/32 + (2*b^2*(1 - c^2*x^2)^(5/2))/(125*c) + (9* 
b^2*ArcSin[c*x])/(64*c) + (2*b*x*(a + b*ArcSin[c*x]))/5 - (3*b*c*x^2*(a + 
b*ArcSin[c*x]))/8 - (4*b*c^2*x^3*(a + b*ArcSin[c*x]))/15 + (2*b*c^4*x^5*(a 
 + b*ArcSin[c*x]))/25 + (b*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x]))/(8*c) + (3 
*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/8 + (x*(1 - c^2*x^2)^(3/2)*(a 
+ b*ArcSin[c*x])^2)/4 - ((1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^2)/(5*c) 
+ (a + b*ArcSin[c*x])^3/(8*b*c)))/(1 - c^2*x^2)^(3/2)
 

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 [C] (verified)

Result contains complex when optimal does not.

Time = 4.88 (sec) , antiderivative size = 2270, normalized size of antiderivative = 3.07

method result size
default \(\text {Expression too large to display}\) \(2270\)
parts \(\text {Expression too large to display}\) \(2270\)

Input:

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

Output:

-1/5*a^2/c/e*(c*d*x+d)^(5/2)*(-c*e*x+e)^(5/2)-1/4*a^2*d/c/e*(c*d*x+d)^(3/2 
)*(-c*e*x+e)^(5/2)-1/4*a^2*d^2/c/e*(c*d*x+d)^(1/2)*(-c*e*x+e)^(5/2)+1/8*a^ 
2*d^2/c*(-c*e*x+e)^(3/2)*(c*d*x+d)^(1/2)+3/8*a^2*d^2*e/c*(-c*e*x+e)^(1/2)* 
(c*d*x+d)^(1/2)+3/8*a^2*d^3*e^2*((c*d*x+d)*(-c*e*x+e))^(1/2)/(-c*e*x+e)^(1 
/2)/(c*d*x+d)^(1/2)/(c^2*d*e)^(1/2)*arctan((c^2*d*e)^(1/2)*x/(-c^2*d*e*x^2 
+d*e)^(1/2))+b^2*(-1/8*(-e*(c*x-1))^(1/2)*(d*(c*x+1))^(1/2)*(-c^2*x^2+1)^( 
1/2)/c/(c^2*x^2-1)*arcsin(c*x)^3*d^2*e-1/4000*(-e*(c*x-1))^(1/2)*(d*(c*x+1 
))^(1/2)*(16*c^6*x^6-28*c^4*x^4-16*I*(-c^2*x^2+1)^(1/2)*x^5*c^5+13*c^2*x^2 
+20*I*(-c^2*x^2+1)^(1/2)*x^3*c^3-5*I*(-c^2*x^2+1)^(1/2)*x*c-1)*(10*I*arcsi 
n(c*x)+25*arcsin(c*x)^2-2)*d^2*e/c/(c^2*x^2-1)-1/512*(-e*(c*x-1))^(1/2)*(d 
*(c*x+1))^(1/2)*(-8*I*(-c^2*x^2+1)^(1/2)*x^4*c^4+8*c^5*x^5+8*I*(-c^2*x^2+1 
)^(1/2)*x^2*c^2-12*c^3*x^3-I*(-c^2*x^2+1)^(1/2)+4*c*x)*(4*I*arcsin(c*x)+8* 
arcsin(c*x)^2-1)*d^2*e/c/(c^2*x^2-1)-1/16*(-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)^2-2-2*I*arcsin(c*x) 
)*d^2*e/c/(c^2*x^2-1)+1/16*(-e*(c*x-1))^(1/2)*(d*(c*x+1))^(1/2)*(2*I*(-c^2 
*x^2+1)^(1/2)*x^2*c^2+2*c^3*x^3-I*(-c^2*x^2+1)^(1/2)-2*c*x)*(2*arcsin(c*x) 
^2-1-2*I*arcsin(c*x))*d^2*e/c/(c^2*x^2-1)-1/18000*(-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)*(330*I*arcsin(c*x)+675*a 
rcsin(c*x)^2-134)*cos(4*arcsin(c*x))*d^2*e/c/(c^2*x^2-1)-1/9000*(-e*(c*x-1 
))^(1/2)*(d*(c*x+1))^(1/2)*(I*c^2*x^2-c*x*(-c^2*x^2+1)^(1/2)-I)*(210*I*...
 

Fricas [F]

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

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

Output:

integral(-(a^2*c^3*d^2*e*x^3 + a^2*c^2*d^2*e*x^2 - a^2*c*d^2*e*x - a^2*d^2 
*e + (b^2*c^3*d^2*e*x^3 + b^2*c^2*d^2*e*x^2 - b^2*c*d^2*e*x - b^2*d^2*e)*a 
rcsin(c*x)^2 + 2*(a*b*c^3*d^2*e*x^3 + a*b*c^2*d^2*e*x^2 - a*b*c*d^2*e*x - 
a*b*d^2*e)*arcsin(c*x))*sqrt(c*d*x + d)*sqrt(-c*e*x + e), x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

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

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (d+c d x)^{5/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2 \, dx=\frac {\sqrt {e}\, \sqrt {d}\, d^{2} e \left (-30 \mathit {asin} \left (\frac {\sqrt {-c x +1}}{\sqrt {2}}\right ) a^{2}-8 \sqrt {c x +1}\, \sqrt {-c x +1}\, a^{2} c^{4} x^{4}-10 \sqrt {c x +1}\, \sqrt {-c x +1}\, a^{2} c^{3} x^{3}+16 \sqrt {c x +1}\, \sqrt {-c x +1}\, a^{2} c^{2} x^{2}+25 \sqrt {c x +1}\, \sqrt {-c x +1}\, a^{2} c x -8 \sqrt {c x +1}\, \sqrt {-c x +1}\, a^{2}-80 \left (\int \sqrt {c x +1}\, \sqrt {-c x +1}\, \mathit {asin} \left (c x \right ) x^{3}d x \right ) a b \,c^{4}-80 \left (\int \sqrt {c x +1}\, \sqrt {-c x +1}\, \mathit {asin} \left (c x \right ) x^{2}d x \right ) a b \,c^{3}+80 \left (\int \sqrt {c x +1}\, \sqrt {-c x +1}\, \mathit {asin} \left (c x \right ) x d x \right ) a b \,c^{2}+80 \left (\int \sqrt {c x +1}\, \sqrt {-c x +1}\, \mathit {asin} \left (c x \right )d x \right ) a b c -40 \left (\int \sqrt {c x +1}\, \sqrt {-c x +1}\, \mathit {asin} \left (c x \right )^{2} x^{3}d x \right ) b^{2} c^{4}-40 \left (\int \sqrt {c x +1}\, \sqrt {-c x +1}\, \mathit {asin} \left (c x \right )^{2} x^{2}d x \right ) b^{2} c^{3}+40 \left (\int \sqrt {c x +1}\, \sqrt {-c x +1}\, \mathit {asin} \left (c x \right )^{2} x d x \right ) b^{2} c^{2}+40 \left (\int \sqrt {c x +1}\, \sqrt {-c x +1}\, \mathit {asin} \left (c x \right )^{2}d x \right ) b^{2} c \right )}{40 c} \] Input:

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

Output:

(sqrt(e)*sqrt(d)*d**2*e*( - 30*asin(sqrt( - c*x + 1)/sqrt(2))*a**2 - 8*sqr 
t(c*x + 1)*sqrt( - c*x + 1)*a**2*c**4*x**4 - 10*sqrt(c*x + 1)*sqrt( - c*x 
+ 1)*a**2*c**3*x**3 + 16*sqrt(c*x + 1)*sqrt( - c*x + 1)*a**2*c**2*x**2 + 2 
5*sqrt(c*x + 1)*sqrt( - c*x + 1)*a**2*c*x - 8*sqrt(c*x + 1)*sqrt( - c*x + 
1)*a**2 - 80*int(sqrt(c*x + 1)*sqrt( - c*x + 1)*asin(c*x)*x**3,x)*a*b*c**4 
 - 80*int(sqrt(c*x + 1)*sqrt( - c*x + 1)*asin(c*x)*x**2,x)*a*b*c**3 + 80*i 
nt(sqrt(c*x + 1)*sqrt( - c*x + 1)*asin(c*x)*x,x)*a*b*c**2 + 80*int(sqrt(c* 
x + 1)*sqrt( - c*x + 1)*asin(c*x),x)*a*b*c - 40*int(sqrt(c*x + 1)*sqrt( - 
c*x + 1)*asin(c*x)**2*x**3,x)*b**2*c**4 - 40*int(sqrt(c*x + 1)*sqrt( - c*x 
 + 1)*asin(c*x)**2*x**2,x)*b**2*c**3 + 40*int(sqrt(c*x + 1)*sqrt( - c*x + 
1)*asin(c*x)**2*x,x)*b**2*c**2 + 40*int(sqrt(c*x + 1)*sqrt( - c*x + 1)*asi 
n(c*x)**2,x)*b**2*c))/(40*c)