\(\int (c e+d e x)^3 (a+b \arcsin (c+d x))^2 \, dx\) [68]

Optimal result

Integrand size = 23, antiderivative size = 176 \[ \int (c e+d e x)^3 (a+b \arcsin (c+d x))^2 \, dx=-\frac {3 b^2 e^3 (c+d x)^2}{32 d}-\frac {b^2 e^3 (c+d x)^4}{32 d}+\frac {3 b e^3 (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))}{16 d}+\frac {b e^3 (c+d x)^3 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))}{8 d}-\frac {3 e^3 (a+b \arcsin (c+d x))^2}{32 d}+\frac {e^3 (c+d x)^4 (a+b \arcsin (c+d x))^2}{4 d} \] Output:

-3/32*b^2*e^3*(d*x+c)^2/d-1/32*b^2*e^3*(d*x+c)^4/d+3/16*b*e^3*(d*x+c)*(1-( 
d*x+c)^2)^(1/2)*(a+b*arcsin(d*x+c))/d+1/8*b*e^3*(d*x+c)^3*(1-(d*x+c)^2)^(1 
/2)*(a+b*arcsin(d*x+c))/d-3/32*e^3*(a+b*arcsin(d*x+c))^2/d+1/4*e^3*(d*x+c) 
^4*(a+b*arcsin(d*x+c))^2/d
 

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.81 \[ \int (c e+d e x)^3 (a+b \arcsin (c+d x))^2 \, dx=\frac {e^3 \left ((c+d x)^4 (a+b \arcsin (c+d x))^2+\frac {1}{8} \left (-b^2 (c+d x)^4+4 b (c+d x)^3 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))-3 \left (b^2 (c+d x)^2-2 b (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))+(a+b \arcsin (c+d x))^2\right )\right )\right )}{4 d} \] Input:

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

Output:

(e^3*((c + d*x)^4*(a + b*ArcSin[c + d*x])^2 + (-(b^2*(c + d*x)^4) + 4*b*(c 
 + d*x)^3*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x]) - 3*(b^2*(c + d*x) 
^2 - 2*b*(c + d*x)*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x]) + (a + b* 
ArcSin[c + d*x])^2))/8))/(4*d)
 

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.88, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {5304, 27, 5138, 5210, 15, 5210, 15, 5152}

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.

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

\(\Big \downarrow \) 5304

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5138

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

\(\Big \downarrow \) 5210

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 5210

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 5152

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

Input:

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

Output:

(e^3*(((c + d*x)^4*(a + b*ArcSin[c + d*x])^2)/4 - (b*((b*(c + d*x)^4)/16 - 
 ((c + d*x)^3*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x]))/4 + (3*((b*(c 
 + d*x)^2)/4 - ((c + d*x)*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x]))/2 
 + (a + b*ArcSin[c + d*x])^2/(4*b)))/4))/2))/d
 

Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

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 5138

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] 
:> Simp[(d*x)^(m + 1)*((a + b*ArcSin[c*x])^n/(d*(m + 1))), x] - Simp[b*c*(n 
/(d*(m + 1)))   Int[(d*x)^(m + 1)*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2 
*x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
 

rule 5152

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

rule 5210

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. 
)*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + 
 b*ArcSin[c*x])^n/(e*(m + 2*p + 1))), x] + (Simp[f^2*((m - 1)/(c^2*(m + 2*p 
 + 1)))   Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] + S 
imp[b*f*(n/(c*(m + 2*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, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && IGtQ[m 
, 1] && NeQ[m + 2*p + 1, 0]
 

rule 5304

Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m 
_.), x_Symbol] :> Simp[1/d   Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b*A 
rcSin[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
 

Maple [A] (verified)

Time = 0.16 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.15

method	result	size
derivativedivides	\(\frac {\frac {e^{3} a^{2} \left (d x +c \right )^{4}}{4}+e^{3} b^{2} \left (\frac {\left (d x +c \right )^{4} \arcsin \left (d x +c \right )^{2}}{4}-\frac {\arcsin \left (d x +c \right ) \left (-2 \left (d x +c \right )^{3} \sqrt {1-\left (d x +c \right )^{2}}-3 \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}+3 \arcsin \left (d x +c \right )\right )}{16}+\frac {3 \arcsin \left (d x +c \right )^{2}}{32}-\frac {\left (2 \left (d x +c \right )^{2}+3\right )^{2}}{128}\right )+2 e^{3} a b \left (\frac {\left (d x +c \right )^{4} \arcsin \left (d x +c \right )}{4}+\frac {\left (d x +c \right )^{3} \sqrt {1-\left (d x +c \right )^{2}}}{16}+\frac {3 \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}}{32}-\frac {3 \arcsin \left (d x +c \right )}{32}\right )}{d}\)	\(203\)
default	\(\frac {\frac {e^{3} a^{2} \left (d x +c \right )^{4}}{4}+e^{3} b^{2} \left (\frac {\left (d x +c \right )^{4} \arcsin \left (d x +c \right )^{2}}{4}-\frac {\arcsin \left (d x +c \right ) \left (-2 \left (d x +c \right )^{3} \sqrt {1-\left (d x +c \right )^{2}}-3 \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}+3 \arcsin \left (d x +c \right )\right )}{16}+\frac {3 \arcsin \left (d x +c \right )^{2}}{32}-\frac {\left (2 \left (d x +c \right )^{2}+3\right )^{2}}{128}\right )+2 e^{3} a b \left (\frac {\left (d x +c \right )^{4} \arcsin \left (d x +c \right )}{4}+\frac {\left (d x +c \right )^{3} \sqrt {1-\left (d x +c \right )^{2}}}{16}+\frac {3 \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}}{32}-\frac {3 \arcsin \left (d x +c \right )}{32}\right )}{d}\)	\(203\)
parts	\(\frac {e^{3} a^{2} \left (d x +c \right )^{4}}{4 d}+\frac {e^{3} b^{2} \left (\frac {\left (d x +c \right )^{4} \arcsin \left (d x +c \right )^{2}}{4}-\frac {\arcsin \left (d x +c \right ) \left (-2 \left (d x +c \right )^{3} \sqrt {1-\left (d x +c \right )^{2}}-3 \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}+3 \arcsin \left (d x +c \right )\right )}{16}+\frac {3 \arcsin \left (d x +c \right )^{2}}{32}-\frac {\left (2 \left (d x +c \right )^{2}+3\right )^{2}}{128}\right )}{d}+\frac {2 e^{3} a b \left (\frac {\left (d x +c \right )^{4} \arcsin \left (d x +c \right )}{4}+\frac {\left (d x +c \right )^{3} \sqrt {1-\left (d x +c \right )^{2}}}{16}+\frac {3 \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}}{32}-\frac {3 \arcsin \left (d x +c \right )}{32}\right )}{d}\)	\(208\)
orering	\(\frac {\left (37 d^{6} x^{6}+222 c \,d^{5} x^{5}+555 c^{2} d^{4} x^{4}+740 c^{3} d^{3} x^{3}+546 c^{4} d^{2} x^{2}+21 d^{4} x^{4}+204 c^{5} d x +84 c \,d^{3} x^{3}+28 c^{6}+99 c^{2} d^{2} x^{2}+30 c^{3} d x +6 c^{4}-60 d^{2} x^{2}-120 c d x -24 c^{2}\right ) \left (d e x +c e \right )^{3} \left (a +b \arcsin \left (d x +c \right )\right )^{2}}{64 d \left (d x +c \right )^{5}}-\frac {\left (9 d^{6} x^{6}+54 c \,d^{5} x^{5}+135 c^{2} d^{4} x^{4}+180 c^{3} d^{3} x^{3}+130 c^{4} d^{2} x^{2}+11 d^{4} x^{4}+44 c^{5} d x +44 c \,d^{3} x^{3}+4 c^{6}+51 c^{2} d^{2} x^{2}+14 c^{3} d x +2 c^{4}-24 d^{2} x^{2}-48 c d x -6 c^{2}\right ) \left (3 \left (d e x +c e \right )^{2} \left (a +b \arcsin \left (d x +c \right )\right )^{2} d e +\frac {2 \left (d e x +c e \right )^{3} \left (a +b \arcsin \left (d x +c \right )\right ) b d}{\sqrt {1-\left (d x +c \right )^{2}}}\right )}{64 d^{2} \left (d x +c \right )^{4}}+\frac {x \left (d^{3} x^{3}+4 c \,d^{2} x^{2}+6 c^{2} d x +4 c^{3}+3 d x +6 c \right ) \left (d x +c +1\right ) \left (d x +c -1\right ) \left (6 \left (d e x +c e \right ) \left (a +b \arcsin \left (d x +c \right )\right )^{2} d^{2} e^{2}+\frac {12 \left (d e x +c e \right )^{2} \left (a +b \arcsin \left (d x +c \right )\right ) d^{2} e b}{\sqrt {1-\left (d x +c \right )^{2}}}+\frac {2 \left (d e x +c e \right )^{3} b^{2} d^{2}}{1-\left (d x +c \right )^{2}}+\frac {2 \left (d e x +c e \right )^{3} \left (a +b \arcsin \left (d x +c \right )\right ) b \,d^{2} \left (d x +c \right )}{\left (1-\left (d x +c \right )^{2}\right )^{\frac {3}{2}}}\right )}{64 d^{2} \left (d x +c \right )^{3}}\)	\(559\)

Input:

int((d*e*x+c*e)^3*(a+b*arcsin(d*x+c))^2,x,method=_RETURNVERBOSE)
 

Output:

1/d*(1/4*e^3*a^2*(d*x+c)^4+e^3*b^2*(1/4*(d*x+c)^4*arcsin(d*x+c)^2-1/16*arc 
sin(d*x+c)*(-2*(d*x+c)^3*(1-(d*x+c)^2)^(1/2)-3*(d*x+c)*(1-(d*x+c)^2)^(1/2) 
+3*arcsin(d*x+c))+3/32*arcsin(d*x+c)^2-1/128*(2*(d*x+c)^2+3)^2)+2*e^3*a*b* 
(1/4*(d*x+c)^4*arcsin(d*x+c)+1/16*(d*x+c)^3*(1-(d*x+c)^2)^(1/2)+3/32*(d*x+ 
c)*(1-(d*x+c)^2)^(1/2)-3/32*arcsin(d*x+c)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 442 vs. \(2 (160) = 320\).

Time = 0.10 (sec) , antiderivative size = 442, normalized size of antiderivative = 2.51 \[ \int (c e+d e x)^3 (a+b \arcsin (c+d x))^2 \, dx=\frac {{\left (8 \, a^{2} - b^{2}\right )} d^{4} e^{3} x^{4} + 4 \, {\left (8 \, a^{2} - b^{2}\right )} c d^{3} e^{3} x^{3} + 3 \, {\left (2 \, {\left (8 \, a^{2} - b^{2}\right )} c^{2} - b^{2}\right )} d^{2} e^{3} x^{2} + 2 \, {\left (2 \, {\left (8 \, a^{2} - b^{2}\right )} c^{3} - 3 \, b^{2} c\right )} d e^{3} x + {\left (8 \, b^{2} d^{4} e^{3} x^{4} + 32 \, b^{2} c d^{3} e^{3} x^{3} + 48 \, b^{2} c^{2} d^{2} e^{3} x^{2} + 32 \, b^{2} c^{3} d e^{3} x + {\left (8 \, b^{2} c^{4} - 3 \, b^{2}\right )} e^{3}\right )} \arcsin \left (d x + c\right )^{2} + 2 \, {\left (8 \, a b d^{4} e^{3} x^{4} + 32 \, a b c d^{3} e^{3} x^{3} + 48 \, a b c^{2} d^{2} e^{3} x^{2} + 32 \, a b c^{3} d e^{3} x + {\left (8 \, a b c^{4} - 3 \, a b\right )} e^{3}\right )} \arcsin \left (d x + c\right ) + 2 \, {\left (2 \, a b d^{3} e^{3} x^{3} + 6 \, a b c d^{2} e^{3} x^{2} + 3 \, {\left (2 \, a b c^{2} + a b\right )} d e^{3} x + {\left (2 \, a b c^{3} + 3 \, a b c\right )} e^{3} + {\left (2 \, b^{2} d^{3} e^{3} x^{3} + 6 \, b^{2} c d^{2} e^{3} x^{2} + 3 \, {\left (2 \, b^{2} c^{2} + b^{2}\right )} d e^{3} x + {\left (2 \, b^{2} c^{3} + 3 \, b^{2} c\right )} e^{3}\right )} \arcsin \left (d x + c\right )\right )} \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}{32 \, d} \] Input:

integrate((d*e*x+c*e)^3*(a+b*arcsin(d*x+c))^2,x, algorithm="fricas")
 

Output:

1/32*((8*a^2 - b^2)*d^4*e^3*x^4 + 4*(8*a^2 - b^2)*c*d^3*e^3*x^3 + 3*(2*(8* 
a^2 - b^2)*c^2 - b^2)*d^2*e^3*x^2 + 2*(2*(8*a^2 - b^2)*c^3 - 3*b^2*c)*d*e^ 
3*x + (8*b^2*d^4*e^3*x^4 + 32*b^2*c*d^3*e^3*x^3 + 48*b^2*c^2*d^2*e^3*x^2 + 
 32*b^2*c^3*d*e^3*x + (8*b^2*c^4 - 3*b^2)*e^3)*arcsin(d*x + c)^2 + 2*(8*a* 
b*d^4*e^3*x^4 + 32*a*b*c*d^3*e^3*x^3 + 48*a*b*c^2*d^2*e^3*x^2 + 32*a*b*c^3 
*d*e^3*x + (8*a*b*c^4 - 3*a*b)*e^3)*arcsin(d*x + c) + 2*(2*a*b*d^3*e^3*x^3 
 + 6*a*b*c*d^2*e^3*x^2 + 3*(2*a*b*c^2 + a*b)*d*e^3*x + (2*a*b*c^3 + 3*a*b* 
c)*e^3 + (2*b^2*d^3*e^3*x^3 + 6*b^2*c*d^2*e^3*x^2 + 3*(2*b^2*c^2 + b^2)*d* 
e^3*x + (2*b^2*c^3 + 3*b^2*c)*e^3)*arcsin(d*x + c))*sqrt(-d^2*x^2 - 2*c*d* 
x - c^2 + 1))/d
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 916 vs. \(2 (155) = 310\).

Time = 0.47 (sec) , antiderivative size = 916, normalized size of antiderivative = 5.20 \[ \int (c e+d e x)^3 (a+b \arcsin (c+d x))^2 \, dx =\text {Too large to display} \] Input:

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

Output:

Piecewise((a**2*c**3*e**3*x + 3*a**2*c**2*d*e**3*x**2/2 + a**2*c*d**2*e**3 
*x**3 + a**2*d**3*e**3*x**4/4 + a*b*c**4*e**3*asin(c + d*x)/(2*d) + 2*a*b* 
c**3*e**3*x*asin(c + d*x) + a*b*c**3*e**3*sqrt(-c**2 - 2*c*d*x - d**2*x**2 
 + 1)/(8*d) + 3*a*b*c**2*d*e**3*x**2*asin(c + d*x) + 3*a*b*c**2*e**3*x*sqr 
t(-c**2 - 2*c*d*x - d**2*x**2 + 1)/8 + 2*a*b*c*d**2*e**3*x**3*asin(c + d*x 
) + 3*a*b*c*d*e**3*x**2*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/8 + 3*a*b*c* 
e**3*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/(16*d) + a*b*d**3*e**3*x**4*asi 
n(c + d*x)/2 + a*b*d**2*e**3*x**3*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/8 
+ 3*a*b*e**3*x*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/16 - 3*a*b*e**3*asin( 
c + d*x)/(16*d) + b**2*c**4*e**3*asin(c + d*x)**2/(4*d) + b**2*c**3*e**3*x 
*asin(c + d*x)**2 - b**2*c**3*e**3*x/8 + b**2*c**3*e**3*sqrt(-c**2 - 2*c*d 
*x - d**2*x**2 + 1)*asin(c + d*x)/(8*d) + 3*b**2*c**2*d*e**3*x**2*asin(c + 
 d*x)**2/2 - 3*b**2*c**2*d*e**3*x**2/16 + 3*b**2*c**2*e**3*x*sqrt(-c**2 - 
2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)/8 + b**2*c*d**2*e**3*x**3*asin(c + 
d*x)**2 - b**2*c*d**2*e**3*x**3/8 + 3*b**2*c*d*e**3*x**2*sqrt(-c**2 - 2*c* 
d*x - d**2*x**2 + 1)*asin(c + d*x)/8 - 3*b**2*c*e**3*x/16 + 3*b**2*c*e**3* 
sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)/(16*d) + b**2*d**3*e** 
3*x**4*asin(c + d*x)**2/4 - b**2*d**3*e**3*x**4/32 + b**2*d**2*e**3*x**3*s 
qrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)/8 - 3*b**2*d*e**3*x**2/ 
32 + 3*b**2*e**3*x*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)/...
 

Maxima [F]

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

integrate((d*e*x+c*e)^3*(a+b*arcsin(d*x+c))^2,x, algorithm="maxima")
                                                                                    
                                                                                    
 

Output:

1/4*a^2*d^3*e^3*x^4 + a^2*c*d^2*e^3*x^3 + 3/2*a^2*c^2*d*e^3*x^2 + 3/2*(2*x 
^2*arcsin(d*x + c) + d*(3*c^2*arcsin(-(d^2*x + c*d)/sqrt(c^2*d^2 - (c^2 - 
1)*d^2))/d^3 + sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*x/d^2 - (c^2 - 1)*arcsin 
(-(d^2*x + c*d)/sqrt(c^2*d^2 - (c^2 - 1)*d^2))/d^3 - 3*sqrt(-d^2*x^2 - 2*c 
*d*x - c^2 + 1)*c/d^3))*a*b*c^2*d*e^3 + 1/3*(6*x^3*arcsin(d*x + c) + d*(2* 
sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*x^2/d^2 - 15*c^3*arcsin(-(d^2*x + c*d)/ 
sqrt(c^2*d^2 - (c^2 - 1)*d^2))/d^4 - 5*sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)* 
c*x/d^3 + 9*(c^2 - 1)*c*arcsin(-(d^2*x + c*d)/sqrt(c^2*d^2 - (c^2 - 1)*d^2 
))/d^4 + 15*sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*c^2/d^4 - 4*sqrt(-d^2*x^2 - 
 2*c*d*x - c^2 + 1)*(c^2 - 1)/d^4))*a*b*c*d^2*e^3 + 1/48*(24*x^4*arcsin(d* 
x + c) + (6*sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*x^3/d^2 - 14*sqrt(-d^2*x^2 
- 2*c*d*x - c^2 + 1)*c*x^2/d^3 + 105*c^4*arcsin(-(d^2*x + c*d)/sqrt(c^2*d^ 
2 - (c^2 - 1)*d^2))/d^5 + 35*sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*c^2*x/d^4 
- 90*(c^2 - 1)*c^2*arcsin(-(d^2*x + c*d)/sqrt(c^2*d^2 - (c^2 - 1)*d^2))/d^ 
5 - 105*sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*c^3/d^5 - 9*sqrt(-d^2*x^2 - 2*c 
*d*x - c^2 + 1)*(c^2 - 1)*x/d^4 + 9*(c^2 - 1)^2*arcsin(-(d^2*x + c*d)/sqrt 
(c^2*d^2 - (c^2 - 1)*d^2))/d^5 + 55*sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*(c^ 
2 - 1)*c/d^5)*d)*a*b*d^3*e^3 + a^2*c^3*e^3*x + 2*((d*x + c)*arcsin(d*x + c 
) + sqrt(-(d*x + c)^2 + 1))*a*b*c^3*e^3/d + 1/4*(b^2*d^3*e^3*x^4 + 4*b^2*c 
*d^2*e^3*x^3 + 6*b^2*c^2*d*e^3*x^2 + 4*b^2*c^3*e^3*x)*arctan2(d*x + c, ...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 342 vs. \(2 (160) = 320\).

Time = 0.17 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.94 \[ \int (c e+d e x)^3 (a+b \arcsin (c+d x))^2 \, dx=\frac {{\left (d x + c\right )}^{4} a^{2} e^{3}}{4 \, d} + \frac {{\left ({\left (d x + c\right )}^{2} - 1\right )}^{2} b^{2} e^{3} \arcsin \left (d x + c\right )^{2}}{4 \, d} - \frac {{\left (-{\left (d x + c\right )}^{2} + 1\right )}^{\frac {3}{2}} {\left (d x + c\right )} b^{2} e^{3} \arcsin \left (d x + c\right )}{8 \, d} + \frac {{\left ({\left (d x + c\right )}^{2} - 1\right )}^{2} a b e^{3} \arcsin \left (d x + c\right )}{2 \, d} + \frac {{\left ({\left (d x + c\right )}^{2} - 1\right )} b^{2} e^{3} \arcsin \left (d x + c\right )^{2}}{2 \, d} - \frac {{\left (-{\left (d x + c\right )}^{2} + 1\right )}^{\frac {3}{2}} {\left (d x + c\right )} a b e^{3}}{8 \, d} + \frac {5 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} {\left (d x + c\right )} b^{2} e^{3} \arcsin \left (d x + c\right )}{16 \, d} - \frac {{\left ({\left (d x + c\right )}^{2} - 1\right )}^{2} b^{2} e^{3}}{32 \, d} + \frac {{\left ({\left (d x + c\right )}^{2} - 1\right )} a b e^{3} \arcsin \left (d x + c\right )}{d} + \frac {5 \, b^{2} e^{3} \arcsin \left (d x + c\right )^{2}}{32 \, d} + \frac {5 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} {\left (d x + c\right )} a b e^{3}}{16 \, d} - \frac {5 \, {\left ({\left (d x + c\right )}^{2} - 1\right )} b^{2} e^{3}}{32 \, d} + \frac {5 \, a b e^{3} \arcsin \left (d x + c\right )}{16 \, d} - \frac {17 \, b^{2} e^{3}}{256 \, d} \] Input:

integrate((d*e*x+c*e)^3*(a+b*arcsin(d*x+c))^2,x, algorithm="giac")
 

Output:

1/4*(d*x + c)^4*a^2*e^3/d + 1/4*((d*x + c)^2 - 1)^2*b^2*e^3*arcsin(d*x + c 
)^2/d - 1/8*(-(d*x + c)^2 + 1)^(3/2)*(d*x + c)*b^2*e^3*arcsin(d*x + c)/d + 
 1/2*((d*x + c)^2 - 1)^2*a*b*e^3*arcsin(d*x + c)/d + 1/2*((d*x + c)^2 - 1) 
*b^2*e^3*arcsin(d*x + c)^2/d - 1/8*(-(d*x + c)^2 + 1)^(3/2)*(d*x + c)*a*b* 
e^3/d + 5/16*sqrt(-(d*x + c)^2 + 1)*(d*x + c)*b^2*e^3*arcsin(d*x + c)/d - 
1/32*((d*x + c)^2 - 1)^2*b^2*e^3/d + ((d*x + c)^2 - 1)*a*b*e^3*arcsin(d*x 
+ c)/d + 5/32*b^2*e^3*arcsin(d*x + c)^2/d + 5/16*sqrt(-(d*x + c)^2 + 1)*(d 
*x + c)*a*b*e^3/d - 5/32*((d*x + c)^2 - 1)*b^2*e^3/d + 5/16*a*b*e^3*arcsin 
(d*x + c)/d - 17/256*b^2*e^3/d
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (c e+d e x)^3 (a+b \arcsin (c+d x))^2 \, dx=\frac {e^{3} \left (48 \left (\int \mathit {asin} \left (d x +c \right )^{2} x^{3}d x \right ) b^{2} d^{4}+48 \mathit {asin} \left (d x +c \right )^{2} b^{2} c^{3} d x +72 \mathit {asin} \left (d x +c \right )^{2} b^{2} c^{2} d^{2} x^{2}+24 \mathit {asin} \left (d x +c \right ) a b \,d^{4} x^{4}+6 \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}\, a b \,d^{3} x^{3}+9 \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}\, a b d x +156 b^{2} c^{4}+36 b^{2} c^{2}+72 \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}\, \mathit {asin} \left (d x +c \right ) b^{2} c^{2} d x +96 \mathit {asin} \left (d x +c \right ) a b \,c^{3} d x +144 \mathit {asin} \left (d x +c \right ) a b \,c^{2} d^{2} x^{2}+96 \mathit {asin} \left (d x +c \right ) a b c \,d^{3} x^{3}+18 \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}\, a b \,c^{2} d x +18 \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}\, a b c \,d^{2} x^{2}-120 \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}\, \mathit {asin} \left (d x +c \right ) b^{2} c^{3}+24 \mathit {asin} \left (d x +c \right ) a b \,c^{4}+6 \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}\, a b \,c^{3}+9 \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}\, a b c +48 a^{2} c^{3} d x +72 a^{2} c^{2} d^{2} x^{2}+48 a^{2} c \,d^{3} x^{3}+120 b^{2} c^{3} d x -36 b^{2} c^{2} d^{2} x^{2}+144 \left (\int \mathit {asin} \left (d x +c \right )^{2} x^{2}d x \right ) b^{2} c \,d^{3}-9 \mathit {asin} \left (d x +c \right ) a b -24 \mathit {asin} \left (d x +c \right )^{2} b^{2} c^{4}-36 \mathit {asin} \left (d x +c \right )^{2} b^{2} c^{2}+12 a^{2} d^{4} x^{4}+384 a b \,c^{3}+64 a b c \right )}{48 d} \] Input:

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

Output:

(e**3*( - 24*asin(c + d*x)**2*b**2*c**4 + 48*asin(c + d*x)**2*b**2*c**3*d* 
x + 72*asin(c + d*x)**2*b**2*c**2*d**2*x**2 - 36*asin(c + d*x)**2*b**2*c** 
2 - 120*sqrt( - c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)*b**2*c**3 + 
72*sqrt( - c**2 - 2*c*d*x - d**2*x**2 + 1)*asin(c + d*x)*b**2*c**2*d*x + 2 
4*asin(c + d*x)*a*b*c**4 + 96*asin(c + d*x)*a*b*c**3*d*x + 144*asin(c + d* 
x)*a*b*c**2*d**2*x**2 + 96*asin(c + d*x)*a*b*c*d**3*x**3 + 24*asin(c + d*x 
)*a*b*d**4*x**4 - 9*asin(c + d*x)*a*b + 6*sqrt( - c**2 - 2*c*d*x - d**2*x* 
*2 + 1)*a*b*c**3 + 18*sqrt( - c**2 - 2*c*d*x - d**2*x**2 + 1)*a*b*c**2*d*x 
 + 18*sqrt( - c**2 - 2*c*d*x - d**2*x**2 + 1)*a*b*c*d**2*x**2 + 9*sqrt( - 
c**2 - 2*c*d*x - d**2*x**2 + 1)*a*b*c + 6*sqrt( - c**2 - 2*c*d*x - d**2*x* 
*2 + 1)*a*b*d**3*x**3 + 9*sqrt( - c**2 - 2*c*d*x - d**2*x**2 + 1)*a*b*d*x 
+ 48*int(asin(c + d*x)**2*x**3,x)*b**2*d**4 + 144*int(asin(c + d*x)**2*x** 
2,x)*b**2*c*d**3 + 48*a**2*c**3*d*x + 72*a**2*c**2*d**2*x**2 + 48*a**2*c*d 
**3*x**3 + 12*a**2*d**4*x**4 + 384*a*b*c**3 + 64*a*b*c + 156*b**2*c**4 + 1 
20*b**2*c**3*d*x - 36*b**2*c**2*d**2*x**2 + 36*b**2*c**2))/(48*d)