\(\int x^2 (a+b \arcsin (c x))^3 \, dx\) [22]

Optimal result

Integrand size = 14, antiderivative size = 170 \[ \int x^2 (a+b \arcsin (c x))^3 \, dx=-\frac {14 b^3 \sqrt {1-c^2 x^2}}{9 c^3}+\frac {2 b^3 \left (1-c^2 x^2\right )^{3/2}}{27 c^3}-\frac {4 b^2 x (a+b \arcsin (c x))}{3 c^2}-\frac {2}{9} b^2 x^3 (a+b \arcsin (c x))+\frac {2 b \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{3 c^3}+\frac {b x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{3 c}+\frac {1}{3} x^3 (a+b \arcsin (c x))^3 \] Output:

-14/9*b^3*(-c^2*x^2+1)^(1/2)/c^3+2/27*b^3*(-c^2*x^2+1)^(3/2)/c^3-4/3*b^2*x 
*(a+b*arcsin(c*x))/c^2-2/9*b^2*x^3*(a+b*arcsin(c*x))+2/3*b*(-c^2*x^2+1)^(1 
/2)*(a+b*arcsin(c*x))^2/c^3+1/3*b*x^2*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x)) 
^2/c+1/3*x^3*(a+b*arcsin(c*x))^3
 

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.96 \[ \int x^2 (a+b \arcsin (c x))^3 \, dx=\frac {1}{27} \left (9 x^3 (a+b \arcsin (c x))^3+\frac {b \left (9 c^2 x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2-2 b \left (b \sqrt {1-c^2 x^2} \left (2+c^2 x^2\right )+3 c^3 x^3 (a+b \arcsin (c x))\right )+18 \left (\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2-2 b \left (a c x+b \sqrt {1-c^2 x^2}+b c x \arcsin (c x)\right )\right )\right )}{c^3}\right ) \] Input:

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

Output:

(9*x^3*(a + b*ArcSin[c*x])^3 + (b*(9*c^2*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcS 
in[c*x])^2 - 2*b*(b*Sqrt[1 - c^2*x^2]*(2 + c^2*x^2) + 3*c^3*x^3*(a + b*Arc 
Sin[c*x])) + 18*(Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2 - 2*b*(a*c*x + b* 
Sqrt[1 - c^2*x^2] + b*c*x*ArcSin[c*x]))))/c^3)/27
 

Rubi [A] (verified)

Time = 0.85 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.18, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {5138, 5210, 5138, 243, 53, 2009, 5182, 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.

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

\(\Big \downarrow \) 5138

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

\(\Big \downarrow \) 5210

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

\(\Big \downarrow \) 5138

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 53

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{3} x^3 (a+b \arcsin (c x))^3-b c \left (\frac {2 \int \frac {x (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{3 c^2}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{3 c^2}+\frac {2 b \left (\frac {1}{3} x^3 (a+b \arcsin (c x))-\frac {1}{6} b c \left (\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {1-c^2 x^2}}{c^4}\right )\right )}{3 c}\right )\)

\(\Big \downarrow \) 5182

\(\displaystyle \frac {1}{3} x^3 (a+b \arcsin (c x))^3-b c \left (\frac {2 \left (\frac {2 b \int (a+b \arcsin (c x))dx}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{c^2}\right )}{3 c^2}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{3 c^2}+\frac {2 b \left (\frac {1}{3} x^3 (a+b \arcsin (c x))-\frac {1}{6} b c \left (\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {1-c^2 x^2}}{c^4}\right )\right )}{3 c}\right )\)

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 53

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int 
[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, 
x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) 
|| LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
 

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 2009

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

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 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 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 = 0.18 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.38

method	result	size
derivativedivides	\(\frac {\frac {c^{3} x^{3} a^{3}}{3}+b^{3} \left (\frac {c^{3} x^{3} \arcsin \left (c x \right )^{3}}{3}+\frac {\arcsin \left (c x \right )^{2} \left (c^{2} x^{2}+2\right ) \sqrt {-c^{2} x^{2}+1}}{3}-\frac {4 \sqrt {-c^{2} x^{2}+1}}{3}-\frac {4 c x \arcsin \left (c x \right )}{3}-\frac {2 c^{3} x^{3} \arcsin \left (c x \right )}{9}-\frac {2 \left (c^{2} x^{2}+2\right ) \sqrt {-c^{2} x^{2}+1}}{27}\right )+3 a \,b^{2} \left (\frac {c^{3} x^{3} \arcsin \left (c x \right )^{2}}{3}+\frac {2 \arcsin \left (c x \right ) \left (c^{2} x^{2}+2\right ) \sqrt {-c^{2} x^{2}+1}}{9}-\frac {2 c^{3} x^{3}}{27}-\frac {4 c x}{9}\right )+3 a^{2} b \left (\frac {c^{3} x^{3} \arcsin \left (c x \right )}{3}+\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{9}+\frac {2 \sqrt {-c^{2} x^{2}+1}}{9}\right )}{c^{3}}\)	\(235\)
default	\(\frac {\frac {c^{3} x^{3} a^{3}}{3}+b^{3} \left (\frac {c^{3} x^{3} \arcsin \left (c x \right )^{3}}{3}+\frac {\arcsin \left (c x \right )^{2} \left (c^{2} x^{2}+2\right ) \sqrt {-c^{2} x^{2}+1}}{3}-\frac {4 \sqrt {-c^{2} x^{2}+1}}{3}-\frac {4 c x \arcsin \left (c x \right )}{3}-\frac {2 c^{3} x^{3} \arcsin \left (c x \right )}{9}-\frac {2 \left (c^{2} x^{2}+2\right ) \sqrt {-c^{2} x^{2}+1}}{27}\right )+3 a \,b^{2} \left (\frac {c^{3} x^{3} \arcsin \left (c x \right )^{2}}{3}+\frac {2 \arcsin \left (c x \right ) \left (c^{2} x^{2}+2\right ) \sqrt {-c^{2} x^{2}+1}}{9}-\frac {2 c^{3} x^{3}}{27}-\frac {4 c x}{9}\right )+3 a^{2} b \left (\frac {c^{3} x^{3} \arcsin \left (c x \right )}{3}+\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{9}+\frac {2 \sqrt {-c^{2} x^{2}+1}}{9}\right )}{c^{3}}\)	\(235\)
parts	\(\frac {a^{3} x^{3}}{3}+\frac {b^{3} \left (\frac {c^{3} x^{3} \arcsin \left (c x \right )^{3}}{3}+\frac {\arcsin \left (c x \right )^{2} \left (c^{2} x^{2}+2\right ) \sqrt {-c^{2} x^{2}+1}}{3}-\frac {4 \sqrt {-c^{2} x^{2}+1}}{3}-\frac {4 c x \arcsin \left (c x \right )}{3}-\frac {2 c^{3} x^{3} \arcsin \left (c x \right )}{9}-\frac {2 \left (c^{2} x^{2}+2\right ) \sqrt {-c^{2} x^{2}+1}}{27}\right )}{c^{3}}+\frac {3 a^{2} b \left (\frac {c^{3} x^{3} \arcsin \left (c x \right )}{3}+\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{9}+\frac {2 \sqrt {-c^{2} x^{2}+1}}{9}\right )}{c^{3}}+\frac {3 a \,b^{2} \left (\frac {c^{3} x^{3} \arcsin \left (c x \right )^{2}}{3}+\frac {2 \arcsin \left (c x \right ) \left (c^{2} x^{2}+2\right ) \sqrt {-c^{2} x^{2}+1}}{9}-\frac {2 c^{3} x^{3}}{27}-\frac {4 c x}{9}\right )}{c^{3}}\)	\(237\)
orering	\(\frac {5 \left (13 c^{6} x^{6}+40 c^{4} x^{4}-152 c^{2} x^{2}+96\right ) \left (a +b \arcsin \left (c x \right )\right )^{3}}{81 c^{6} x^{3}}-\frac {\left (25 c^{6} x^{6}+166 c^{4} x^{4}-572 c^{2} x^{2}+360\right ) \left (2 x \left (a +b \arcsin \left (c x \right )\right )^{3}+\frac {3 x^{2} \left (a +b \arcsin \left (c x \right )\right )^{2} b c}{\sqrt {-c^{2} x^{2}+1}}\right )}{81 c^{6} x^{4}}+\frac {2 \left (c x -1\right ) \left (c x +1\right ) \left (c^{4} x^{4}+12 c^{2} x^{2}-20\right ) \left (2 \left (a +b \arcsin \left (c x \right )\right )^{3}+\frac {12 x \left (a +b \arcsin \left (c x \right )\right )^{2} b c}{\sqrt {-c^{2} x^{2}+1}}+\frac {6 x^{2} \left (a +b \arcsin \left (c x \right )\right ) b^{2} c^{2}}{-c^{2} x^{2}+1}+\frac {3 x^{3} \left (a +b \arcsin \left (c x \right )\right )^{2} b \,c^{3}}{\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{27 c^{6} x^{3}}-\frac {\left (c^{2} x^{2}+20\right ) \left (c x -1\right )^{2} \left (c x +1\right )^{2} \left (\frac {18 \left (a +b \arcsin \left (c x \right )\right )^{2} b c}{\sqrt {-c^{2} x^{2}+1}}+\frac {36 x \left (a +b \arcsin \left (c x \right )\right ) b^{2} c^{2}}{-c^{2} x^{2}+1}+\frac {21 x^{2} \left (a +b \arcsin \left (c x \right )\right )^{2} b \,c^{3}}{\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {6 x^{2} b^{3} c^{3}}{\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {18 x^{3} \left (a +b \arcsin \left (c x \right )\right ) b^{2} c^{4}}{\left (-c^{2} x^{2}+1\right )^{2}}+\frac {9 x^{4} \left (a +b \arcsin \left (c x \right )\right )^{2} b \,c^{5}}{\left (-c^{2} x^{2}+1\right )^{\frac {5}{2}}}\right )}{81 c^{6} x^{2}}\)	\(463\)

Input:

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

Output:

1/c^3*(1/3*c^3*x^3*a^3+b^3*(1/3*c^3*x^3*arcsin(c*x)^3+1/3*arcsin(c*x)^2*(c 
^2*x^2+2)*(-c^2*x^2+1)^(1/2)-4/3*(-c^2*x^2+1)^(1/2)-4/3*c*x*arcsin(c*x)-2/ 
9*c^3*x^3*arcsin(c*x)-2/27*(c^2*x^2+2)*(-c^2*x^2+1)^(1/2))+3*a*b^2*(1/3*c^ 
3*x^3*arcsin(c*x)^2+2/9*arcsin(c*x)*(c^2*x^2+2)*(-c^2*x^2+1)^(1/2)-2/27*c^ 
3*x^3-4/9*c*x)+3*a^2*b*(1/3*c^3*x^3*arcsin(c*x)+1/9*c^2*x^2*(-c^2*x^2+1)^( 
1/2)+2/9*(-c^2*x^2+1)^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.14 \[ \int x^2 (a+b \arcsin (c x))^3 \, dx=\frac {9 \, b^{3} c^{3} x^{3} \arcsin \left (c x\right )^{3} + 27 \, a b^{2} c^{3} x^{3} \arcsin \left (c x\right )^{2} + 3 \, {\left (3 \, a^{3} - 2 \, a b^{2}\right )} c^{3} x^{3} - 36 \, a b^{2} c x + 3 \, {\left ({\left (9 \, a^{2} b - 2 \, b^{3}\right )} c^{3} x^{3} - 12 \, b^{3} c x\right )} \arcsin \left (c x\right ) + {\left ({\left (9 \, a^{2} b - 2 \, b^{3}\right )} c^{2} x^{2} + 18 \, a^{2} b - 40 \, b^{3} + 9 \, {\left (b^{3} c^{2} x^{2} + 2 \, b^{3}\right )} \arcsin \left (c x\right )^{2} + 18 \, {\left (a b^{2} c^{2} x^{2} + 2 \, a b^{2}\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{27 \, c^{3}} \] Input:

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

Output:

1/27*(9*b^3*c^3*x^3*arcsin(c*x)^3 + 27*a*b^2*c^3*x^3*arcsin(c*x)^2 + 3*(3* 
a^3 - 2*a*b^2)*c^3*x^3 - 36*a*b^2*c*x + 3*((9*a^2*b - 2*b^3)*c^3*x^3 - 12* 
b^3*c*x)*arcsin(c*x) + ((9*a^2*b - 2*b^3)*c^2*x^2 + 18*a^2*b - 40*b^3 + 9* 
(b^3*c^2*x^2 + 2*b^3)*arcsin(c*x)^2 + 18*(a*b^2*c^2*x^2 + 2*a*b^2)*arcsin( 
c*x))*sqrt(-c^2*x^2 + 1))/c^3
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 328 vs. \(2 (158) = 316\).

Time = 0.40 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.93 \[ \int x^2 (a+b \arcsin (c x))^3 \, dx=\begin {cases} \frac {a^{3} x^{3}}{3} + a^{2} b x^{3} \operatorname {asin}{\left (c x \right )} + \frac {a^{2} b x^{2} \sqrt {- c^{2} x^{2} + 1}}{3 c} + \frac {2 a^{2} b \sqrt {- c^{2} x^{2} + 1}}{3 c^{3}} + a b^{2} x^{3} \operatorname {asin}^{2}{\left (c x \right )} - \frac {2 a b^{2} x^{3}}{9} + \frac {2 a b^{2} x^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{3 c} - \frac {4 a b^{2} x}{3 c^{2}} + \frac {4 a b^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{3 c^{3}} + \frac {b^{3} x^{3} \operatorname {asin}^{3}{\left (c x \right )}}{3} - \frac {2 b^{3} x^{3} \operatorname {asin}{\left (c x \right )}}{9} + \frac {b^{3} x^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (c x \right )}}{3 c} - \frac {2 b^{3} x^{2} \sqrt {- c^{2} x^{2} + 1}}{27 c} - \frac {4 b^{3} x \operatorname {asin}{\left (c x \right )}}{3 c^{2}} + \frac {2 b^{3} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (c x \right )}}{3 c^{3}} - \frac {40 b^{3} \sqrt {- c^{2} x^{2} + 1}}{27 c^{3}} & \text {for}\: c \neq 0 \\\frac {a^{3} x^{3}}{3} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((a**3*x**3/3 + a**2*b*x**3*asin(c*x) + a**2*b*x**2*sqrt(-c**2*x* 
*2 + 1)/(3*c) + 2*a**2*b*sqrt(-c**2*x**2 + 1)/(3*c**3) + a*b**2*x**3*asin( 
c*x)**2 - 2*a*b**2*x**3/9 + 2*a*b**2*x**2*sqrt(-c**2*x**2 + 1)*asin(c*x)/( 
3*c) - 4*a*b**2*x/(3*c**2) + 4*a*b**2*sqrt(-c**2*x**2 + 1)*asin(c*x)/(3*c* 
*3) + b**3*x**3*asin(c*x)**3/3 - 2*b**3*x**3*asin(c*x)/9 + b**3*x**2*sqrt( 
-c**2*x**2 + 1)*asin(c*x)**2/(3*c) - 2*b**3*x**2*sqrt(-c**2*x**2 + 1)/(27* 
c) - 4*b**3*x*asin(c*x)/(3*c**2) + 2*b**3*sqrt(-c**2*x**2 + 1)*asin(c*x)** 
2/(3*c**3) - 40*b**3*sqrt(-c**2*x**2 + 1)/(27*c**3), Ne(c, 0)), (a**3*x**3 
/3, True))
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.61 \[ \int x^2 (a+b \arcsin (c x))^3 \, dx=\frac {1}{3} \, b^{3} x^{3} \arcsin \left (c x\right )^{3} + a b^{2} x^{3} \arcsin \left (c x\right )^{2} + \frac {1}{3} \, a^{3} x^{3} + \frac {1}{3} \, {\left (3 \, x^{3} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} a^{2} b + \frac {2}{9} \, {\left (3 \, c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )} \arcsin \left (c x\right ) - \frac {c^{2} x^{3} + 6 \, x}{c^{2}}\right )} a b^{2} + \frac {1}{27} \, {\left (9 \, c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )} \arcsin \left (c x\right )^{2} - 2 \, c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2} + \frac {20 \, \sqrt {-c^{2} x^{2} + 1}}{c^{2}}}{c^{2}} + \frac {3 \, {\left (c^{2} x^{3} + 6 \, x\right )} \arcsin \left (c x\right )}{c^{3}}\right )}\right )} b^{3} \] Input:

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

Output:

1/3*b^3*x^3*arcsin(c*x)^3 + a*b^2*x^3*arcsin(c*x)^2 + 1/3*a^3*x^3 + 1/3*(3 
*x^3*arcsin(c*x) + c*(sqrt(-c^2*x^2 + 1)*x^2/c^2 + 2*sqrt(-c^2*x^2 + 1)/c^ 
4))*a^2*b + 2/9*(3*c*(sqrt(-c^2*x^2 + 1)*x^2/c^2 + 2*sqrt(-c^2*x^2 + 1)/c^ 
4)*arcsin(c*x) - (c^2*x^3 + 6*x)/c^2)*a*b^2 + 1/27*(9*c*(sqrt(-c^2*x^2 + 1 
)*x^2/c^2 + 2*sqrt(-c^2*x^2 + 1)/c^4)*arcsin(c*x)^2 - 2*c*((sqrt(-c^2*x^2 
+ 1)*x^2 + 20*sqrt(-c^2*x^2 + 1)/c^2)/c^2 + 3*(c^2*x^3 + 6*x)*arcsin(c*x)/ 
c^3))*b^3
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 368 vs. \(2 (148) = 296\).

Time = 0.14 (sec) , antiderivative size = 368, normalized size of antiderivative = 2.16 \[ \int x^2 (a+b \arcsin (c x))^3 \, dx=\frac {1}{3} \, a^{3} x^{3} + \frac {{\left (c^{2} x^{2} - 1\right )} b^{3} x \arcsin \left (c x\right )^{3}}{3 \, c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )} a b^{2} x \arcsin \left (c x\right )^{2}}{c^{2}} + \frac {b^{3} x \arcsin \left (c x\right )^{3}}{3 \, c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )} a^{2} b x \arcsin \left (c x\right )}{c^{2}} - \frac {2 \, {\left (c^{2} x^{2} - 1\right )} b^{3} x \arcsin \left (c x\right )}{9 \, c^{2}} + \frac {a b^{2} x \arcsin \left (c x\right )^{2}}{c^{2}} - \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b^{3} \arcsin \left (c x\right )^{2}}{3 \, c^{3}} - \frac {2 \, {\left (c^{2} x^{2} - 1\right )} a b^{2} x}{9 \, c^{2}} + \frac {a^{2} b x \arcsin \left (c x\right )}{c^{2}} - \frac {14 \, b^{3} x \arcsin \left (c x\right )}{9 \, c^{2}} - \frac {2 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} a b^{2} \arcsin \left (c x\right )}{3 \, c^{3}} + \frac {\sqrt {-c^{2} x^{2} + 1} b^{3} \arcsin \left (c x\right )^{2}}{c^{3}} - \frac {14 \, a b^{2} x}{9 \, c^{2}} - \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{2} b}{3 \, c^{3}} + \frac {2 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b^{3}}{27 \, c^{3}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} a b^{2} \arcsin \left (c x\right )}{c^{3}} + \frac {\sqrt {-c^{2} x^{2} + 1} a^{2} b}{c^{3}} - \frac {14 \, \sqrt {-c^{2} x^{2} + 1} b^{3}}{9 \, c^{3}} \] Input:

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

Output:

1/3*a^3*x^3 + 1/3*(c^2*x^2 - 1)*b^3*x*arcsin(c*x)^3/c^2 + (c^2*x^2 - 1)*a* 
b^2*x*arcsin(c*x)^2/c^2 + 1/3*b^3*x*arcsin(c*x)^3/c^2 + (c^2*x^2 - 1)*a^2* 
b*x*arcsin(c*x)/c^2 - 2/9*(c^2*x^2 - 1)*b^3*x*arcsin(c*x)/c^2 + a*b^2*x*ar 
csin(c*x)^2/c^2 - 1/3*(-c^2*x^2 + 1)^(3/2)*b^3*arcsin(c*x)^2/c^3 - 2/9*(c^ 
2*x^2 - 1)*a*b^2*x/c^2 + a^2*b*x*arcsin(c*x)/c^2 - 14/9*b^3*x*arcsin(c*x)/ 
c^2 - 2/3*(-c^2*x^2 + 1)^(3/2)*a*b^2*arcsin(c*x)/c^3 + sqrt(-c^2*x^2 + 1)* 
b^3*arcsin(c*x)^2/c^3 - 14/9*a*b^2*x/c^2 - 1/3*(-c^2*x^2 + 1)^(3/2)*a^2*b/ 
c^3 + 2/27*(-c^2*x^2 + 1)^(3/2)*b^3/c^3 + 2*sqrt(-c^2*x^2 + 1)*a*b^2*arcsi 
n(c*x)/c^3 + sqrt(-c^2*x^2 + 1)*a^2*b/c^3 - 14/9*sqrt(-c^2*x^2 + 1)*b^3/c^ 
3
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int x^2 (a+b \arcsin (c x))^3 \, dx=\frac {3 \mathit {asin} \left (c x \right ) a^{2} b \,c^{3} x^{3}+\sqrt {-c^{2} x^{2}+1}\, a^{2} b \,c^{2} x^{2}+2 \sqrt {-c^{2} x^{2}+1}\, a^{2} b +3 \left (\int \mathit {asin} \left (c x \right )^{3} x^{2}d x \right ) b^{3} c^{3}+9 \left (\int \mathit {asin} \left (c x \right )^{2} x^{2}d x \right ) a \,b^{2} c^{3}+a^{3} c^{3} x^{3}}{3 c^{3}} \] Input:

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

Output:

(3*asin(c*x)*a**2*b*c**3*x**3 + sqrt( - c**2*x**2 + 1)*a**2*b*c**2*x**2 + 
2*sqrt( - c**2*x**2 + 1)*a**2*b + 3*int(asin(c*x)**3*x**2,x)*b**3*c**3 + 9 
*int(asin(c*x)**2*x**2,x)*a*b**2*c**3 + a**3*c**3*x**3)/(3*c**3)