\(\int x (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2 \, dx\) [488]

Optimal result

Integrand size = 33, antiderivative size = 338 \[ \int x (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2 \, dx=\frac {16 b^2 d e \sqrt {d+c d x} \sqrt {e-c e x}}{75 c^2}+\frac {8 b^2 d e \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right )}{225 c^2}+\frac {2 b^2 d e \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right )^2}{125 c^2}+\frac {2 b d e x \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{5 c \sqrt {1-c^2 x^2}}-\frac {4 b c d e x^3 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{15 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d e x^5 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{25 \sqrt {1-c^2 x^2}}-\frac {d e \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2}{5 c^2} \] Output:

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

Mathematica [A] (verified)

Time = 2.06 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.61 \[ \int x (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2 \, dx=-\frac {d e \sqrt {d+c d x} \sqrt {e-c e x} \left (225 a^2 \left (-1+c^2 x^2\right )^3+30 a b c x \sqrt {1-c^2 x^2} \left (15-10 c^2 x^2+3 c^4 x^4\right )+2 b^2 \left (149-187 c^2 x^2+47 c^4 x^4-9 c^6 x^6\right )+30 b \left (15 a \left (-1+c^2 x^2\right )^3+b c x \sqrt {1-c^2 x^2} \left (15-10 c^2 x^2+3 c^4 x^4\right )\right ) \arcsin (c x)+225 b^2 \left (-1+c^2 x^2\right )^3 \arcsin (c x)^2\right )}{1125 c^2 \left (-1+c^2 x^2\right )} \] Input:

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

Output:

-1/1125*(d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(225*a^2*(-1 + c^2*x^2)^3 + 3 
0*a*b*c*x*Sqrt[1 - c^2*x^2]*(15 - 10*c^2*x^2 + 3*c^4*x^4) + 2*b^2*(149 - 1 
87*c^2*x^2 + 47*c^4*x^4 - 9*c^6*x^6) + 30*b*(15*a*(-1 + c^2*x^2)^3 + b*c*x 
*Sqrt[1 - c^2*x^2]*(15 - 10*c^2*x^2 + 3*c^4*x^4))*ArcSin[c*x] + 225*b^2*(- 
1 + c^2*x^2)^3*ArcSin[c*x]^2))/(c^2*(-1 + c^2*x^2))
 

Rubi [A] (verified)

Time = 0.94 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.57, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {5238, 5182, 5154, 27, 1576, 1140, 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.

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

\(\Big \downarrow \) 5238

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

\(\Big \downarrow \) 5182

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

\(\Big \downarrow \) 5154

\(\displaystyle \frac {d e \sqrt {c d x+d} \sqrt {e-c e x} \left (\frac {2 b \left (-b c \int \frac {x \left (3 c^4 x^4-10 c^2 x^2+15\right )}{15 \sqrt {1-c^2 x^2}}dx+\frac {1}{5} c^4 x^5 (a+b \arcsin (c x))-\frac {2}{3} c^2 x^3 (a+b \arcsin (c x))+x (a+b \arcsin (c x))\right )}{5 c}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2}{5 c^2}\right )}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {d e \sqrt {c d x+d} \sqrt {e-c e x} \left (\frac {2 b \left (-\frac {1}{15} b c \int \frac {x \left (3 c^4 x^4-10 c^2 x^2+15\right )}{\sqrt {1-c^2 x^2}}dx+\frac {1}{5} c^4 x^5 (a+b \arcsin (c x))-\frac {2}{3} c^2 x^3 (a+b \arcsin (c x))+x (a+b \arcsin (c x))\right )}{5 c}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2}{5 c^2}\right )}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 1576

\(\displaystyle \frac {d e \sqrt {c d x+d} \sqrt {e-c e x} \left (\frac {2 b \left (-\frac {1}{30} b c \int \frac {3 c^4 x^4-10 c^2 x^2+15}{\sqrt {1-c^2 x^2}}dx^2+\frac {1}{5} c^4 x^5 (a+b \arcsin (c x))-\frac {2}{3} c^2 x^3 (a+b \arcsin (c x))+x (a+b \arcsin (c x))\right )}{5 c}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2}{5 c^2}\right )}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 1140

\(\displaystyle \frac {d e \sqrt {c d x+d} \sqrt {e-c e x} \left (\frac {2 b \left (-\frac {1}{30} b c \int \left (3 \left (1-c^2 x^2\right )^{3/2}+4 \sqrt {1-c^2 x^2}+\frac {8}{\sqrt {1-c^2 x^2}}\right )dx^2+\frac {1}{5} c^4 x^5 (a+b \arcsin (c x))-\frac {2}{3} c^2 x^3 (a+b \arcsin (c x))+x (a+b \arcsin (c x))\right )}{5 c}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2}{5 c^2}\right )}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {d e \sqrt {c d x+d} \sqrt {e-c e x} \left (\frac {2 b \left (\frac {1}{5} c^4 x^5 (a+b \arcsin (c x))-\frac {2}{3} c^2 x^3 (a+b \arcsin (c x))+x (a+b \arcsin (c x))-\frac {1}{30} b c \left (-\frac {6 \left (1-c^2 x^2\right )^{5/2}}{5 c^2}-\frac {8 \left (1-c^2 x^2\right )^{3/2}}{3 c^2}-\frac {16 \sqrt {1-c^2 x^2}}{c^2}\right )\right )}{5 c}-\frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2}{5 c^2}\right )}{\sqrt {1-c^2 x^2}}\)

Input:

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

Output:

(d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(-1/5*((1 - c^2*x^2)^(5/2)*(a + b*Arc 
Sin[c*x])^2)/c^2 + (2*b*(-1/30*(b*c*((-16*Sqrt[1 - c^2*x^2])/c^2 - (8*(1 - 
 c^2*x^2)^(3/2))/(3*c^2) - (6*(1 - c^2*x^2)^(5/2))/(5*c^2))) + x*(a + b*Ar 
cSin[c*x]) - (2*c^2*x^3*(a + b*ArcSin[c*x]))/3 + (c^4*x^5*(a + b*ArcSin[c* 
x]))/5))/(5*c)))/Sqrt[1 - c^2*x^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 1140

Int[((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x 
_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; 
FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 0]
 

rule 1576

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^( 
p_.), x_Symbol] :> Simp[1/2   Subst[Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x] 
, x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]
 

rule 2009

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

rule 5154

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbo 
l] :> With[{u = IntHide[(d + e*x^2)^p, x]}, Simp[(a + b*ArcSin[c*x])   u, x 
] - Simp[b*c   Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]] /; Fr 
eeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
 

rule 5182

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_ 
.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(2*e*(p + 
1))), x] + Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]   I 
nt[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x] /; FreeQ[{a, 
 b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]
 

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(678\) vs. \(2(290)=580\).

Time = 4.11 (sec) , antiderivative size = 679, normalized size of antiderivative = 2.01

method	result	size
orering	\(\frac {\left (549 c^{8} x^{8}-1982 c^{6} x^{6}+4355 c^{4} x^{4}-1420 c^{2} x^{2}+298\right ) \left (c d x +d \right )^{\frac {3}{2}} \left (-c x e +e \right )^{\frac {3}{2}} \left (a +b \arcsin \left (c x \right )\right )^{2}}{1125 c^{4} x^{2} \left (c x -1\right )^{2} \left (c x +1\right )^{2}}-\frac {2 \left (54 c^{6} x^{6}-217 c^{4} x^{4}+672 c^{2} x^{2}-149\right ) \left (\left (c d x +d \right )^{\frac {3}{2}} \left (-c x e +e \right )^{\frac {3}{2}} \left (a +b \arcsin \left (c x \right )\right )^{2}+\frac {3 x \sqrt {c d x +d}\, \left (-c x e +e \right )^{\frac {3}{2}} \left (a +b \arcsin \left (c x \right )\right )^{2} c d}{2}-\frac {3 x \left (c d x +d \right )^{\frac {3}{2}} \sqrt {-c x e +e}\, \left (a +b \arcsin \left (c x \right )\right )^{2} c e}{2}+\frac {2 x \left (c d x +d \right )^{\frac {3}{2}} \left (-c x e +e \right )^{\frac {3}{2}} \left (a +b \arcsin \left (c x \right )\right ) b c}{\sqrt {-c^{2} x^{2}+1}}\right )}{1125 c^{4} \left (c x -1\right ) x^{2} \left (c x +1\right )}+\frac {\left (9 c^{4} x^{4}-38 c^{2} x^{2}+149\right ) \left (3 \sqrt {c d x +d}\, \left (-c x e +e \right )^{\frac {3}{2}} \left (a +b \arcsin \left (c x \right )\right )^{2} c d -3 \left (c d x +d \right )^{\frac {3}{2}} \sqrt {-c x e +e}\, \left (a +b \arcsin \left (c x \right )\right )^{2} c e +\frac {4 \left (c d x +d \right )^{\frac {3}{2}} \left (-c x e +e \right )^{\frac {3}{2}} \left (a +b \arcsin \left (c x \right )\right ) b c}{\sqrt {-c^{2} x^{2}+1}}+\frac {3 x \left (-c x e +e \right )^{\frac {3}{2}} \left (a +b \arcsin \left (c x \right )\right )^{2} c^{2} d^{2}}{4 \sqrt {c d x +d}}-\frac {9 c^{2} d e x \sqrt {c d x +d}\, \sqrt {-c x e +e}\, \left (a +b \arcsin \left (c x \right )\right )^{2}}{2}+\frac {6 x \sqrt {c d x +d}\, \left (-c x e +e \right )^{\frac {3}{2}} \left (a +b \arcsin \left (c x \right )\right ) c^{2} d b}{\sqrt {-c^{2} x^{2}+1}}+\frac {3 x \left (c d x +d \right )^{\frac {3}{2}} \left (a +b \arcsin \left (c x \right )\right )^{2} c^{2} e^{2}}{4 \sqrt {-c x e +e}}-\frac {6 x \left (c d x +d \right )^{\frac {3}{2}} \sqrt {-c x e +e}\, \left (a +b \arcsin \left (c x \right )\right ) c^{2} e b}{\sqrt {-c^{2} x^{2}+1}}+\frac {2 x \left (c d x +d \right )^{\frac {3}{2}} \left (-c x e +e \right )^{\frac {3}{2}} b^{2} c^{2}}{-c^{2} x^{2}+1}+\frac {2 x^{2} \left (c d x +d \right )^{\frac {3}{2}} \left (-c x e +e \right )^{\frac {3}{2}} \left (a +b \arcsin \left (c x \right )\right ) b \,c^{3}}{\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{1125 c^{4} x}\)	\(679\)
default	\(\text {Expression too large to display}\)	\(1142\)
parts	\(\text {Expression too large to display}\)	\(1142\)

Input:

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

Output:

1/1125*(549*c^8*x^8-1982*c^6*x^6+4355*c^4*x^4-1420*c^2*x^2+298)/c^4/x^2/(c 
*x-1)^2/(c*x+1)^2*(c*d*x+d)^(3/2)*(-c*e*x+e)^(3/2)*(a+b*arcsin(c*x))^2-2/1 
125*(54*c^6*x^6-217*c^4*x^4+672*c^2*x^2-149)/c^4/(c*x-1)/x^2/(c*x+1)*((c*d 
*x+d)^(3/2)*(-c*e*x+e)^(3/2)*(a+b*arcsin(c*x))^2+3/2*x*(c*d*x+d)^(1/2)*(-c 
*e*x+e)^(3/2)*(a+b*arcsin(c*x))^2*c*d-3/2*x*(c*d*x+d)^(3/2)*(-c*e*x+e)^(1/ 
2)*(a+b*arcsin(c*x))^2*c*e+2*x*(c*d*x+d)^(3/2)*(-c*e*x+e)^(3/2)*(a+b*arcsi 
n(c*x))*b*c/(-c^2*x^2+1)^(1/2))+1/1125*(9*c^4*x^4-38*c^2*x^2+149)/c^4/x*(3 
*(c*d*x+d)^(1/2)*(-c*e*x+e)^(3/2)*(a+b*arcsin(c*x))^2*c*d-3*(c*d*x+d)^(3/2 
)*(-c*e*x+e)^(1/2)*(a+b*arcsin(c*x))^2*c*e+4*(c*d*x+d)^(3/2)*(-c*e*x+e)^(3 
/2)*(a+b*arcsin(c*x))*b*c/(-c^2*x^2+1)^(1/2)+3/4*x/(c*d*x+d)^(1/2)*(-c*e*x 
+e)^(3/2)*(a+b*arcsin(c*x))^2*c^2*d^2-9/2*c^2*d*e*x*(c*d*x+d)^(1/2)*(-c*e* 
x+e)^(1/2)*(a+b*arcsin(c*x))^2+6*x*(c*d*x+d)^(1/2)*(-c*e*x+e)^(3/2)*(a+b*a 
rcsin(c*x))*c^2*d*b/(-c^2*x^2+1)^(1/2)+3/4*x*(c*d*x+d)^(3/2)/(-c*e*x+e)^(1 
/2)*(a+b*arcsin(c*x))^2*c^2*e^2-6*x*(c*d*x+d)^(3/2)*(-c*e*x+e)^(1/2)*(a+b* 
arcsin(c*x))*c^2*e*b/(-c^2*x^2+1)^(1/2)+2*x*(c*d*x+d)^(3/2)*(-c*e*x+e)^(3/ 
2)*b^2*c^2/(-c^2*x^2+1)+2*x^2*(c*d*x+d)^(3/2)*(-c*e*x+e)^(3/2)*(a+b*arcsin 
(c*x))*b*c^3/(-c^2*x^2+1)^(3/2))
 

Fricas [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 302, normalized size of antiderivative = 0.89 \[ \int x (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arcsin (c x))^2 \, dx=-\frac {{\left (9 \, {\left (25 \, a^{2} - 2 \, b^{2}\right )} c^{6} d e x^{6} - {\left (675 \, a^{2} - 94 \, b^{2}\right )} c^{4} d e x^{4} + {\left (675 \, a^{2} - 374 \, b^{2}\right )} c^{2} d e x^{2} - {\left (225 \, a^{2} - 298 \, b^{2}\right )} d e + 225 \, {\left (b^{2} c^{6} d e x^{6} - 3 \, b^{2} c^{4} d e x^{4} + 3 \, b^{2} c^{2} d e x^{2} - b^{2} d e\right )} \arcsin \left (c x\right )^{2} + 450 \, {\left (a b c^{6} d e x^{6} - 3 \, a b c^{4} d e x^{4} + 3 \, a b c^{2} d e x^{2} - a b d e\right )} \arcsin \left (c x\right ) + 30 \, {\left (3 \, a b c^{5} d e x^{5} - 10 \, a b c^{3} d e x^{3} + 15 \, a b c d e x + {\left (3 \, b^{2} c^{5} d e x^{5} - 10 \, b^{2} c^{3} d e x^{3} + 15 \, b^{2} c d e x\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}\right )} \sqrt {c d x + d} \sqrt {-c e x + e}}{1125 \, {\left (c^{4} x^{2} - c^{2}\right )}} \] Input:

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

Output:

-1/1125*(9*(25*a^2 - 2*b^2)*c^6*d*e*x^6 - (675*a^2 - 94*b^2)*c^4*d*e*x^4 + 
 (675*a^2 - 374*b^2)*c^2*d*e*x^2 - (225*a^2 - 298*b^2)*d*e + 225*(b^2*c^6* 
d*e*x^6 - 3*b^2*c^4*d*e*x^4 + 3*b^2*c^2*d*e*x^2 - b^2*d*e)*arcsin(c*x)^2 + 
 450*(a*b*c^6*d*e*x^6 - 3*a*b*c^4*d*e*x^4 + 3*a*b*c^2*d*e*x^2 - a*b*d*e)*a 
rcsin(c*x) + 30*(3*a*b*c^5*d*e*x^5 - 10*a*b*c^3*d*e*x^3 + 15*a*b*c*d*e*x + 
 (3*b^2*c^5*d*e*x^5 - 10*b^2*c^3*d*e*x^3 + 15*b^2*c*d*e*x)*arcsin(c*x))*sq 
rt(-c^2*x^2 + 1))*sqrt(c*d*x + d)*sqrt(-c*e*x + e)/(c^4*x^2 - c^2)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

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

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(sqrt(e)*sqrt(d)*d*e*( - sqrt(c*x + 1)*sqrt( - c*x + 1)*a**2*c**4*x**4 + 2 
*sqrt(c*x + 1)*sqrt( - c*x + 1)*a**2*c**2*x**2 - sqrt(c*x + 1)*sqrt( - c*x 
 + 1)*a**2 - 10*int(sqrt(c*x + 1)*sqrt( - c*x + 1)*asin(c*x)*x**3,x)*a*b*c 
**4 + 10*int(sqrt(c*x + 1)*sqrt( - c*x + 1)*asin(c*x)*x,x)*a*b*c**2 - 5*in 
t(sqrt(c*x + 1)*sqrt( - c*x + 1)*asin(c*x)**2*x**3,x)*b**2*c**4 + 5*int(sq 
rt(c*x + 1)*sqrt( - c*x + 1)*asin(c*x)**2*x,x)*b**2*c**2))/(5*c**2)