\(\int x (a+b \arcsin (c x))^3 \, dx\) [23]

Optimal result

Integrand size = 12, antiderivative size = 125 \[ \int x (a+b \arcsin (c x))^3 \, dx=-\frac {3 b^3 x \sqrt {1-c^2 x^2}}{8 c}+\frac {3 b^3 \arcsin (c x)}{8 c^2}-\frac {3}{4} b^2 x^2 (a+b \arcsin (c x))+\frac {3 b x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{4 c}-\frac {(a+b \arcsin (c x))^3}{4 c^2}+\frac {1}{2} x^2 (a+b \arcsin (c x))^3 \] Output:

-3/8*b^3*x*(-c^2*x^2+1)^(1/2)/c+3/8*b^3*arcsin(c*x)/c^2-3/4*b^2*x^2*(a+b*a 
rcsin(c*x))+3/4*b*x*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))^2/c-1/4*(a+b*arcs 
in(c*x))^3/c^2+1/2*x^2*(a+b*arcsin(c*x))^3
 

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.91 \[ \int x (a+b \arcsin (c x))^3 \, dx=\frac {6 b c x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2-2 (a+b \arcsin (c x))^3+4 c^2 x^2 (a+b \arcsin (c x))^3-3 b^2 \left (c x \left (2 a c x+b \sqrt {1-c^2 x^2}\right )+b \left (-1+2 c^2 x^2\right ) \arcsin (c x)\right )}{8 c^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.10, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5138, 5210, 5138, 262, 223, 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.

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

\(\Big \downarrow \) 5138

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

\(\Big \downarrow \) 5210

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

\(\Big \downarrow \) 5138

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 223

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

\(\Big \downarrow \) 5152

\(\displaystyle \frac {1}{2} x^2 (a+b \arcsin (c x))^3-\frac {3}{2} b c \left (\frac {(a+b \arcsin (c x))^3}{6 b c^3}-\frac {x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{2 c^2}+\frac {b \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{c}\right )\)

Input:

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

Output:

(x^2*(a + b*ArcSin[c*x])^3)/2 - (3*b*c*(-1/2*(x*Sqrt[1 - c^2*x^2]*(a + b*A 
rcSin[c*x])^2)/c^2 + (a + b*ArcSin[c*x])^3/(6*b*c^3) + (b*((x^2*(a + b*Arc 
Sin[c*x]))/2 - (b*c*(-1/2*(x*Sqrt[1 - c^2*x^2])/c^2 + ArcSin[c*x]/(2*c^3)) 
)/2))/c))/2
 

Defintions of rubi rules used

rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt 
[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
 

rule 262

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) 
^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ 
(b*(m + 2*p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b 
, c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c 
, 2, m, p, 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]
 

Maple [A] (verified)

Time = 0.19 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.27

method	result	size
derivativedivides	\(\frac {\frac {c^{2} x^{2} a^{3}}{2}+b^{3} \left (-\frac {\cos \left (2 \arcsin \left (c x \right )\right ) \arcsin \left (c x \right )^{3}}{4}+\frac {3 \sin \left (2 \arcsin \left (c x \right )\right ) \arcsin \left (c x \right )^{2}}{8}-\frac {3 \sin \left (2 \arcsin \left (c x \right )\right )}{16}+\frac {3 \cos \left (2 \arcsin \left (c x \right )\right ) \arcsin \left (c x \right )}{8}\right )+3 a \,b^{2} \left (-\frac {\cos \left (2 \arcsin \left (c x \right )\right ) \arcsin \left (c x \right )^{2}}{4}+\frac {\cos \left (2 \arcsin \left (c x \right )\right )}{8}+\frac {\sin \left (2 \arcsin \left (c x \right )\right ) \arcsin \left (c x \right )}{4}\right )+3 a^{2} b \left (\frac {c^{2} x^{2} \arcsin \left (c x \right )}{2}+\frac {c x \sqrt {-c^{2} x^{2}+1}}{4}-\frac {\arcsin \left (c x \right )}{4}\right )}{c^{2}}\)	\(159\)
default	\(\frac {\frac {c^{2} x^{2} a^{3}}{2}+b^{3} \left (-\frac {\cos \left (2 \arcsin \left (c x \right )\right ) \arcsin \left (c x \right )^{3}}{4}+\frac {3 \sin \left (2 \arcsin \left (c x \right )\right ) \arcsin \left (c x \right )^{2}}{8}-\frac {3 \sin \left (2 \arcsin \left (c x \right )\right )}{16}+\frac {3 \cos \left (2 \arcsin \left (c x \right )\right ) \arcsin \left (c x \right )}{8}\right )+3 a \,b^{2} \left (-\frac {\cos \left (2 \arcsin \left (c x \right )\right ) \arcsin \left (c x \right )^{2}}{4}+\frac {\cos \left (2 \arcsin \left (c x \right )\right )}{8}+\frac {\sin \left (2 \arcsin \left (c x \right )\right ) \arcsin \left (c x \right )}{4}\right )+3 a^{2} b \left (\frac {c^{2} x^{2} \arcsin \left (c x \right )}{2}+\frac {c x \sqrt {-c^{2} x^{2}+1}}{4}-\frac {\arcsin \left (c x \right )}{4}\right )}{c^{2}}\)	\(159\)
parts	\(\frac {a^{3} x^{2}}{2}+\frac {b^{3} \left (-\frac {\cos \left (2 \arcsin \left (c x \right )\right ) \arcsin \left (c x \right )^{3}}{4}+\frac {3 \sin \left (2 \arcsin \left (c x \right )\right ) \arcsin \left (c x \right )^{2}}{8}-\frac {3 \sin \left (2 \arcsin \left (c x \right )\right )}{16}+\frac {3 \cos \left (2 \arcsin \left (c x \right )\right ) \arcsin \left (c x \right )}{8}\right )}{c^{2}}+\frac {3 a^{2} b \left (\frac {c^{2} x^{2} \arcsin \left (c x \right )}{2}+\frac {c x \sqrt {-c^{2} x^{2}+1}}{4}-\frac {\arcsin \left (c x \right )}{4}\right )}{c^{2}}+\frac {3 a \,b^{2} \left (-\frac {\cos \left (2 \arcsin \left (c x \right )\right ) \arcsin \left (c x \right )^{2}}{4}+\frac {\cos \left (2 \arcsin \left (c x \right )\right )}{8}+\frac {\sin \left (2 \arcsin \left (c x \right )\right ) \arcsin \left (c x \right )}{4}\right )}{c^{2}}\)	\(161\)
orering	\(\frac {\left (15 c^{4} x^{4}-20 c^{2} x^{2}+8\right ) \left (a +b \arcsin \left (c x \right )\right )^{3}}{16 c^{4} x^{2}}-\frac {\left (7 c^{4} x^{4}-16 c^{2} x^{2}+8\right ) \left (\left (a +b \arcsin \left (c x \right )\right )^{3}+\frac {3 x \left (a +b \arcsin \left (c x \right )\right )^{2} b c}{\sqrt {-c^{2} x^{2}+1}}\right )}{16 c^{4} x^{2}}+\frac {\left (c x -1\right ) \left (c x +1\right ) \left (c^{2} x^{2}-2\right ) \left (\frac {6 \left (a +b \arcsin \left (c x \right )\right )^{2} b c}{\sqrt {-c^{2} x^{2}+1}}+\frac {6 x \left (a +b \arcsin \left (c x \right )\right ) b^{2} c^{2}}{-c^{2} x^{2}+1}+\frac {3 x^{2} \left (a +b \arcsin \left (c x \right )\right )^{2} b \,c^{3}}{\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{8 c^{4} x}-\frac {\left (c x -1\right )^{2} \left (c x +1\right )^{2} \left (\frac {18 \left (a +b \arcsin \left (c x \right )\right ) b^{2} c^{2}}{-c^{2} x^{2}+1}+\frac {12 \left (a +b \arcsin \left (c x \right )\right )^{2} b \,c^{3} x}{\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {6 x \,b^{3} c^{3}}{\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {18 x^{2} \left (a +b \arcsin \left (c x \right )\right ) b^{2} c^{4}}{\left (-c^{2} x^{2}+1\right )^{2}}+\frac {9 x^{3} \left (a +b \arcsin \left (c x \right )\right )^{2} b \,c^{5}}{\left (-c^{2} x^{2}+1\right )^{\frac {5}{2}}}\right )}{16 c^{4}}\)	\(376\)

Input:

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

Output:

1/c^2*(1/2*c^2*x^2*a^3+b^3*(-1/4*cos(2*arcsin(c*x))*arcsin(c*x)^3+3/8*sin( 
2*arcsin(c*x))*arcsin(c*x)^2-3/16*sin(2*arcsin(c*x))+3/8*cos(2*arcsin(c*x) 
)*arcsin(c*x))+3*a*b^2*(-1/4*cos(2*arcsin(c*x))*arcsin(c*x)^2+1/8*cos(2*ar 
csin(c*x))+1/4*sin(2*arcsin(c*x))*arcsin(c*x))+3*a^2*b*(1/2*c^2*x^2*arcsin 
(c*x)+1/4*c*x*(-c^2*x^2+1)^(1/2)-1/4*arcsin(c*x)))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.35 \[ \int x (a+b \arcsin (c x))^3 \, dx=\frac {2 \, {\left (2 \, a^{3} - 3 \, a b^{2}\right )} c^{2} x^{2} + 2 \, {\left (2 \, b^{3} c^{2} x^{2} - b^{3}\right )} \arcsin \left (c x\right )^{3} + 6 \, {\left (2 \, a b^{2} c^{2} x^{2} - a b^{2}\right )} \arcsin \left (c x\right )^{2} + 3 \, {\left (2 \, {\left (2 \, a^{2} b - b^{3}\right )} c^{2} x^{2} - 2 \, a^{2} b + b^{3}\right )} \arcsin \left (c x\right ) + 3 \, {\left (2 \, b^{3} c x \arcsin \left (c x\right )^{2} + 4 \, a b^{2} c x \arcsin \left (c x\right ) + {\left (2 \, a^{2} b - b^{3}\right )} c x\right )} \sqrt {-c^{2} x^{2} + 1}}{8 \, c^{2}} \] Input:

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

Output:

1/8*(2*(2*a^3 - 3*a*b^2)*c^2*x^2 + 2*(2*b^3*c^2*x^2 - b^3)*arcsin(c*x)^3 + 
 6*(2*a*b^2*c^2*x^2 - a*b^2)*arcsin(c*x)^2 + 3*(2*(2*a^2*b - b^3)*c^2*x^2 
- 2*a^2*b + b^3)*arcsin(c*x) + 3*(2*b^3*c*x*arcsin(c*x)^2 + 4*a*b^2*c*x*ar 
csin(c*x) + (2*a^2*b - b^3)*c*x)*sqrt(-c^2*x^2 + 1))/c^2
                                                                                    
                                                                                    
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 264 vs. \(2 (116) = 232\).

Time = 0.35 (sec) , antiderivative size = 264, normalized size of antiderivative = 2.11 \[ \int x (a+b \arcsin (c x))^3 \, dx=\begin {cases} \frac {a^{3} x^{2}}{2} + \frac {3 a^{2} b x^{2} \operatorname {asin}{\left (c x \right )}}{2} + \frac {3 a^{2} b x \sqrt {- c^{2} x^{2} + 1}}{4 c} - \frac {3 a^{2} b \operatorname {asin}{\left (c x \right )}}{4 c^{2}} + \frac {3 a b^{2} x^{2} \operatorname {asin}^{2}{\left (c x \right )}}{2} - \frac {3 a b^{2} x^{2}}{4} + \frac {3 a b^{2} x \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{2 c} - \frac {3 a b^{2} \operatorname {asin}^{2}{\left (c x \right )}}{4 c^{2}} + \frac {b^{3} x^{2} \operatorname {asin}^{3}{\left (c x \right )}}{2} - \frac {3 b^{3} x^{2} \operatorname {asin}{\left (c x \right )}}{4} + \frac {3 b^{3} x \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (c x \right )}}{4 c} - \frac {3 b^{3} x \sqrt {- c^{2} x^{2} + 1}}{8 c} - \frac {b^{3} \operatorname {asin}^{3}{\left (c x \right )}}{4 c^{2}} + \frac {3 b^{3} \operatorname {asin}{\left (c x \right )}}{8 c^{2}} & \text {for}\: c \neq 0 \\\frac {a^{3} x^{2}}{2} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((a**3*x**2/2 + 3*a**2*b*x**2*asin(c*x)/2 + 3*a**2*b*x*sqrt(-c**2 
*x**2 + 1)/(4*c) - 3*a**2*b*asin(c*x)/(4*c**2) + 3*a*b**2*x**2*asin(c*x)** 
2/2 - 3*a*b**2*x**2/4 + 3*a*b**2*x*sqrt(-c**2*x**2 + 1)*asin(c*x)/(2*c) - 
3*a*b**2*asin(c*x)**2/(4*c**2) + b**3*x**2*asin(c*x)**3/2 - 3*b**3*x**2*as 
in(c*x)/4 + 3*b**3*x*sqrt(-c**2*x**2 + 1)*asin(c*x)**2/(4*c) - 3*b**3*x*sq 
rt(-c**2*x**2 + 1)/(8*c) - b**3*asin(c*x)**3/(4*c**2) + 3*b**3*asin(c*x)/( 
8*c**2), Ne(c, 0)), (a**3*x**2/2, True))
 

Maxima [F]

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

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

Output:

1/2*b^3*x^2*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^3 + 1/2*a^3*x^2 + 3 
/4*(2*x^2*arcsin(c*x) + c*(sqrt(-c^2*x^2 + 1)*x/c^2 - arcsin(c*x)/c^3))*a^ 
2*b + integrate(3/2*(sqrt(c*x + 1)*sqrt(-c*x + 1)*b^3*c*x^2*arctan2(c*x, s 
qrt(c*x + 1)*sqrt(-c*x + 1))^2 + 2*(a*b^2*c^2*x^3 - a*b^2*x)*arctan2(c*x, 
sqrt(c*x + 1)*sqrt(-c*x + 1))^2)/(c^2*x^2 - 1), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 285 vs. \(2 (109) = 218\).

Time = 0.14 (sec) , antiderivative size = 285, normalized size of antiderivative = 2.28 \[ \int x (a+b \arcsin (c x))^3 \, dx=\frac {3 \, \sqrt {-c^{2} x^{2} + 1} b^{3} x \arcsin \left (c x\right )^{2}}{4 \, c} + \frac {{\left (c^{2} x^{2} - 1\right )} b^{3} \arcsin \left (c x\right )^{3}}{2 \, c^{2}} + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} a b^{2} x \arcsin \left (c x\right )}{2 \, c} + \frac {3 \, {\left (c^{2} x^{2} - 1\right )} a b^{2} \arcsin \left (c x\right )^{2}}{2 \, c^{2}} + \frac {b^{3} \arcsin \left (c x\right )^{3}}{4 \, c^{2}} + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} a^{2} b x}{4 \, c} - \frac {3 \, \sqrt {-c^{2} x^{2} + 1} b^{3} x}{8 \, c} + \frac {3 \, {\left (c^{2} x^{2} - 1\right )} a^{2} b \arcsin \left (c x\right )}{2 \, c^{2}} - \frac {3 \, {\left (c^{2} x^{2} - 1\right )} b^{3} \arcsin \left (c x\right )}{4 \, c^{2}} + \frac {3 \, a b^{2} \arcsin \left (c x\right )^{2}}{4 \, c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )} a^{3}}{2 \, c^{2}} - \frac {3 \, {\left (c^{2} x^{2} - 1\right )} a b^{2}}{4 \, c^{2}} + \frac {3 \, a^{2} b \arcsin \left (c x\right )}{4 \, c^{2}} - \frac {3 \, b^{3} \arcsin \left (c x\right )}{8 \, c^{2}} - \frac {3 \, a b^{2}}{8 \, c^{2}} \] Input:

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

Output:

3/4*sqrt(-c^2*x^2 + 1)*b^3*x*arcsin(c*x)^2/c + 1/2*(c^2*x^2 - 1)*b^3*arcsi 
n(c*x)^3/c^2 + 3/2*sqrt(-c^2*x^2 + 1)*a*b^2*x*arcsin(c*x)/c + 3/2*(c^2*x^2 
 - 1)*a*b^2*arcsin(c*x)^2/c^2 + 1/4*b^3*arcsin(c*x)^3/c^2 + 3/4*sqrt(-c^2* 
x^2 + 1)*a^2*b*x/c - 3/8*sqrt(-c^2*x^2 + 1)*b^3*x/c + 3/2*(c^2*x^2 - 1)*a^ 
2*b*arcsin(c*x)/c^2 - 3/4*(c^2*x^2 - 1)*b^3*arcsin(c*x)/c^2 + 3/4*a*b^2*ar 
csin(c*x)^2/c^2 + 1/2*(c^2*x^2 - 1)*a^3/c^2 - 3/4*(c^2*x^2 - 1)*a*b^2/c^2 
+ 3/4*a^2*b*arcsin(c*x)/c^2 - 3/8*b^3*arcsin(c*x)/c^2 - 3/8*a*b^2/c^2
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.77 \[ \int x (a+b \arcsin (c x))^3 \, dx=\frac {4 \mathit {asin} \left (c x \right )^{3} b^{3} c^{2} x^{2}-2 \mathit {asin} \left (c x \right )^{3} b^{3}+6 \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right )^{2} b^{3} c x +12 \mathit {asin} \left (c x \right )^{2} a \,b^{2} c^{2} x^{2}-6 \mathit {asin} \left (c x \right )^{2} a \,b^{2}+12 \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) a \,b^{2} c x +12 \mathit {asin} \left (c x \right ) a^{2} b \,c^{2} x^{2}-6 \mathit {asin} \left (c x \right ) a^{2} b -6 \mathit {asin} \left (c x \right ) b^{3} c^{2} x^{2}+3 \mathit {asin} \left (c x \right ) b^{3}+6 \sqrt {-c^{2} x^{2}+1}\, a^{2} b c x -3 \sqrt {-c^{2} x^{2}+1}\, b^{3} c x +4 a^{3} c^{2} x^{2}-6 a \,b^{2} c^{2} x^{2}}{8 c^{2}} \] Input:

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

Output:

(4*asin(c*x)**3*b**3*c**2*x**2 - 2*asin(c*x)**3*b**3 + 6*sqrt( - c**2*x**2 
 + 1)*asin(c*x)**2*b**3*c*x + 12*asin(c*x)**2*a*b**2*c**2*x**2 - 6*asin(c* 
x)**2*a*b**2 + 12*sqrt( - c**2*x**2 + 1)*asin(c*x)*a*b**2*c*x + 12*asin(c* 
x)*a**2*b*c**2*x**2 - 6*asin(c*x)*a**2*b - 6*asin(c*x)*b**3*c**2*x**2 + 3* 
asin(c*x)*b**3 + 6*sqrt( - c**2*x**2 + 1)*a**2*b*c*x - 3*sqrt( - c**2*x**2 
 + 1)*b**3*c*x + 4*a**3*c**2*x**2 - 6*a*b**2*c**2*x**2)/(8*c**2)