\(\int x^2 \arcsin (a x)^4 \, dx\) [32]

Optimal result

Integrand size = 10, antiderivative size = 166 \[ \int x^2 \arcsin (a x)^4 \, dx=\frac {160 x}{27 a^2}+\frac {8 x^3}{81}-\frac {160 \sqrt {1-a^2 x^2} \arcsin (a x)}{27 a^3}-\frac {8 x^2 \sqrt {1-a^2 x^2} \arcsin (a x)}{27 a}-\frac {8 x \arcsin (a x)^2}{3 a^2}-\frac {4}{9} x^3 \arcsin (a x)^2+\frac {8 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{9 a^3}+\frac {4 x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{9 a}+\frac {1}{3} x^3 \arcsin (a x)^4 \] Output:

160/27*x/a^2+8/81*x^3-160/27*(-a^2*x^2+1)^(1/2)*arcsin(a*x)/a^3-8/27*x^2*( 
-a^2*x^2+1)^(1/2)*arcsin(a*x)/a-8/3*x*arcsin(a*x)^2/a^2-4/9*x^3*arcsin(a*x 
)^2+8/9*(-a^2*x^2+1)^(1/2)*arcsin(a*x)^3/a^3+4/9*x^2*(-a^2*x^2+1)^(1/2)*ar 
csin(a*x)^3/a+1/3*x^3*arcsin(a*x)^4
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.69 \[ \int x^2 \arcsin (a x)^4 \, dx=\frac {8 a x \left (60+a^2 x^2\right )-24 \sqrt {1-a^2 x^2} \left (20+a^2 x^2\right ) \arcsin (a x)-36 a x \left (6+a^2 x^2\right ) \arcsin (a x)^2+36 \sqrt {1-a^2 x^2} \left (2+a^2 x^2\right ) \arcsin (a x)^3+27 a^3 x^3 \arcsin (a x)^4}{81 a^3} \] Input:

Integrate[x^2*ArcSin[a*x]^4,x]
 

Output:

(8*a*x*(60 + a^2*x^2) - 24*Sqrt[1 - a^2*x^2]*(20 + a^2*x^2)*ArcSin[a*x] - 
36*a*x*(6 + a^2*x^2)*ArcSin[a*x]^2 + 36*Sqrt[1 - a^2*x^2]*(2 + a^2*x^2)*Ar 
cSin[a*x]^3 + 27*a^3*x^3*ArcSin[a*x]^4)/(81*a^3)
 

Rubi [A] (verified)

Time = 1.36 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.37, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.100, Rules used = {5138, 5210, 5138, 5182, 5130, 5182, 24, 5210, 15, 5182, 24}

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.

\(\displaystyle \int x^2 \arcsin (a x)^4 \, dx\)

\(\Big \downarrow \) 5138

\(\displaystyle \frac {1}{3} x^3 \arcsin (a x)^4-\frac {4}{3} a \int \frac {x^3 \arcsin (a x)^3}{\sqrt {1-a^2 x^2}}dx\)

\(\Big \downarrow \) 5210

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

\(\Big \downarrow \) 5138

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

\(\Big \downarrow \) 5182

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

\(\Big \downarrow \) 5130

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

\(\Big \downarrow \) 5182

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

\(\Big \downarrow \) 24

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

\(\Big \downarrow \) 5210

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

\(\Big \downarrow \) 15

\(\displaystyle \frac {1}{3} x^3 \arcsin (a x)^4-\frac {4}{3} a \left (\frac {\frac {1}{3} x^3 \arcsin (a x)^2-\frac {2}{3} a \left (\frac {2 \int \frac {x \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx}{3 a^2}-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)}{3 a^2}+\frac {x^3}{9 a}\right )}{a}-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{3 a^2}+\frac {2 \left (\frac {3 \left (x \arcsin (a x)^2-2 a \left (\frac {x}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)}{a^2}\right )\right )}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^3}{a^2}\right )}{3 a^2}\right )\)

\(\Big \downarrow \) 5182

\(\displaystyle \frac {1}{3} x^3 \arcsin (a x)^4-\frac {4}{3} a \left (\frac {\frac {1}{3} x^3 \arcsin (a x)^2-\frac {2}{3} a \left (\frac {2 \left (\frac {\int 1dx}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)}{a^2}\right )}{3 a^2}-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)}{3 a^2}+\frac {x^3}{9 a}\right )}{a}-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{3 a^2}+\frac {2 \left (\frac {3 \left (x \arcsin (a x)^2-2 a \left (\frac {x}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)}{a^2}\right )\right )}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^3}{a^2}\right )}{3 a^2}\right )\)

\(\Big \downarrow \) 24

\(\displaystyle \frac {1}{3} x^3 \arcsin (a x)^4-\frac {4}{3} a \left (-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{3 a^2}+\frac {2 \left (\frac {3 \left (x \arcsin (a x)^2-2 a \left (\frac {x}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)}{a^2}\right )\right )}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^3}{a^2}\right )}{3 a^2}+\frac {\frac {1}{3} x^3 \arcsin (a x)^2-\frac {2}{3} a \left (-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)}{3 a^2}+\frac {2 \left (\frac {x}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)}{a^2}\right )}{3 a^2}+\frac {x^3}{9 a}\right )}{a}\right )\)

Input:

Int[x^2*ArcSin[a*x]^4,x]
 

Output:

(x^3*ArcSin[a*x]^4)/3 - (4*a*(-1/3*(x^2*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/a 
^2 + ((x^3*ArcSin[a*x]^2)/3 - (2*a*(x^3/(9*a) - (x^2*Sqrt[1 - a^2*x^2]*Arc 
Sin[a*x])/(3*a^2) + (2*(x/a - (Sqrt[1 - a^2*x^2]*ArcSin[a*x])/a^2))/(3*a^2 
)))/3)/a + (2*(-((Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/a^2) + (3*(x*ArcSin[a*x 
]^2 - 2*a*(x/a - (Sqrt[1 - a^2*x^2]*ArcSin[a*x])/a^2)))/a))/(3*a^2)))/3
 

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 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
 

rule 5130

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

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 = 0.24 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.78

method	result	size
derivativedivides	\(\frac {\frac {a^{3} x^{3} \arcsin \left (a x \right )^{4}}{3}+\frac {4 \arcsin \left (a x \right )^{3} \left (a^{2} x^{2}+2\right ) \sqrt {-a^{2} x^{2}+1}}{9}-\frac {8 \arcsin \left (a x \right )^{2} a x}{3}+\frac {160 a x}{27}-\frac {16 \arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}}{3}-\frac {4 \arcsin \left (a x \right )^{2} a^{3} x^{3}}{9}-\frac {8 \arcsin \left (a x \right ) \left (a^{2} x^{2}+2\right ) \sqrt {-a^{2} x^{2}+1}}{27}+\frac {8 a^{3} x^{3}}{81}}{a^{3}}\)	\(130\)
default	\(\frac {\frac {a^{3} x^{3} \arcsin \left (a x \right )^{4}}{3}+\frac {4 \arcsin \left (a x \right )^{3} \left (a^{2} x^{2}+2\right ) \sqrt {-a^{2} x^{2}+1}}{9}-\frac {8 \arcsin \left (a x \right )^{2} a x}{3}+\frac {160 a x}{27}-\frac {16 \arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}}{3}-\frac {4 \arcsin \left (a x \right )^{2} a^{3} x^{3}}{9}-\frac {8 \arcsin \left (a x \right ) \left (a^{2} x^{2}+2\right ) \sqrt {-a^{2} x^{2}+1}}{27}+\frac {8 a^{3} x^{3}}{81}}{a^{3}}\)	\(130\)
orering	\(\frac {\left (211 a^{6} x^{6}+1440 a^{4} x^{4}-9360 a^{2} x^{2}+8640\right ) \arcsin \left (a x \right )^{4}}{243 a^{6} x^{3}}-\frac {2 \left (45 a^{6} x^{6}+649 a^{4} x^{4}-3810 a^{2} x^{2}+3420\right ) \left (2 x \arcsin \left (a x \right )^{4}+\frac {4 x^{2} \arcsin \left (a x \right )^{3} a}{\sqrt {-a^{2} x^{2}+1}}\right )}{243 a^{6} x^{4}}+\frac {\left (25 a^{6} x^{6}+578 a^{4} x^{4}-2940 a^{2} x^{2}+2520\right ) \left (2 \arcsin \left (a x \right )^{4}+\frac {16 x \arcsin \left (a x \right )^{3} a}{\sqrt {-a^{2} x^{2}+1}}+\frac {12 x^{2} \arcsin \left (a x \right )^{2} a^{2}}{-a^{2} x^{2}+1}+\frac {4 x^{3} \arcsin \left (a x \right )^{3} a^{3}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{243 x^{3} a^{6}}-\frac {\left (a x -1\right ) \left (a x +1\right ) \left (5 a^{4} x^{4}+169 a^{2} x^{2}-540\right ) \left (\frac {24 \arcsin \left (a x \right )^{3} a}{\sqrt {-a^{2} x^{2}+1}}+\frac {72 x \arcsin \left (a x \right )^{2} a^{2}}{-a^{2} x^{2}+1}+\frac {28 x^{2} \arcsin \left (a x \right )^{3} a^{3}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {24 x^{2} \arcsin \left (a x \right ) a^{3}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {36 x^{3} \arcsin \left (a x \right )^{2} a^{4}}{\left (-a^{2} x^{2}+1\right )^{2}}+\frac {12 x^{4} \arcsin \left (a x \right )^{3} a^{5}}{\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}\right )}{243 x^{2} a^{6}}+\frac {\left (a^{2} x^{2}+60\right ) \left (a x +1\right )^{2} \left (a x -1\right )^{2} \left (\frac {144 \arcsin \left (a x \right )^{2} a^{2}}{-a^{2} x^{2}+1}+\frac {80 \arcsin \left (a x \right )^{3} a^{3} x}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {192 x \arcsin \left (a x \right ) a^{3}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {336 x^{2} \arcsin \left (a x \right )^{2} a^{4}}{\left (-a^{2} x^{2}+1\right )^{2}}+\frac {132 x^{3} \arcsin \left (a x \right )^{3} a^{5}}{\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}+\frac {24 x^{2} a^{4}}{\left (-a^{2} x^{2}+1\right )^{2}}+\frac {144 x^{3} \arcsin \left (a x \right ) a^{5}}{\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}+\frac {180 x^{4} \arcsin \left (a x \right )^{2} a^{6}}{\left (-a^{2} x^{2}+1\right )^{3}}+\frac {60 x^{5} \arcsin \left (a x \right )^{3} a^{7}}{\left (-a^{2} x^{2}+1\right )^{\frac {7}{2}}}\right )}{243 x \,a^{6}}\)	\(660\)

Input:

int(x^2*arcsin(a*x)^4,x,method=_RETURNVERBOSE)
 

Output:

1/a^3*(1/3*a^3*x^3*arcsin(a*x)^4+4/9*arcsin(a*x)^3*(a^2*x^2+2)*(-a^2*x^2+1 
)^(1/2)-8/3*arcsin(a*x)^2*a*x+160/27*a*x-16/3*arcsin(a*x)*(-a^2*x^2+1)^(1/ 
2)-4/9*arcsin(a*x)^2*a^3*x^3-8/27*arcsin(a*x)*(a^2*x^2+2)*(-a^2*x^2+1)^(1/ 
2)+8/81*a^3*x^3)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.60 \[ \int x^2 \arcsin (a x)^4 \, dx=\frac {27 \, a^{3} x^{3} \arcsin \left (a x\right )^{4} + 8 \, a^{3} x^{3} - 36 \, {\left (a^{3} x^{3} + 6 \, a x\right )} \arcsin \left (a x\right )^{2} + 480 \, a x + 12 \, \sqrt {-a^{2} x^{2} + 1} {\left (3 \, {\left (a^{2} x^{2} + 2\right )} \arcsin \left (a x\right )^{3} - 2 \, {\left (a^{2} x^{2} + 20\right )} \arcsin \left (a x\right )\right )}}{81 \, a^{3}} \] Input:

integrate(x^2*arcsin(a*x)^4,x, algorithm="fricas")
 

Output:

1/81*(27*a^3*x^3*arcsin(a*x)^4 + 8*a^3*x^3 - 36*(a^3*x^3 + 6*a*x)*arcsin(a 
*x)^2 + 480*a*x + 12*sqrt(-a^2*x^2 + 1)*(3*(a^2*x^2 + 2)*arcsin(a*x)^3 - 2 
*(a^2*x^2 + 20)*arcsin(a*x)))/a^3
 

Sympy [A] (verification not implemented)

Time = 0.51 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.95 \[ \int x^2 \arcsin (a x)^4 \, dx=\begin {cases} \frac {x^{3} \operatorname {asin}^{4}{\left (a x \right )}}{3} - \frac {4 x^{3} \operatorname {asin}^{2}{\left (a x \right )}}{9} + \frac {8 x^{3}}{81} + \frac {4 x^{2} \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{3}{\left (a x \right )}}{9 a} - \frac {8 x^{2} \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}{\left (a x \right )}}{27 a} - \frac {8 x \operatorname {asin}^{2}{\left (a x \right )}}{3 a^{2}} + \frac {160 x}{27 a^{2}} + \frac {8 \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{3}{\left (a x \right )}}{9 a^{3}} - \frac {160 \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}{\left (a x \right )}}{27 a^{3}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \] Input:

integrate(x**2*asin(a*x)**4,x)
 

Output:

Piecewise((x**3*asin(a*x)**4/3 - 4*x**3*asin(a*x)**2/9 + 8*x**3/81 + 4*x** 
2*sqrt(-a**2*x**2 + 1)*asin(a*x)**3/(9*a) - 8*x**2*sqrt(-a**2*x**2 + 1)*as 
in(a*x)/(27*a) - 8*x*asin(a*x)**2/(3*a**2) + 160*x/(27*a**2) + 8*sqrt(-a** 
2*x**2 + 1)*asin(a*x)**3/(9*a**3) - 160*sqrt(-a**2*x**2 + 1)*asin(a*x)/(27 
*a**3), Ne(a, 0)), (0, True))
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.89 \[ \int x^2 \arcsin (a x)^4 \, dx=\frac {1}{3} \, x^{3} \arcsin \left (a x\right )^{4} + \frac {4}{9} \, a {\left (\frac {\sqrt {-a^{2} x^{2} + 1} x^{2}}{a^{2}} + \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{a^{4}}\right )} \arcsin \left (a x\right )^{3} - \frac {4}{81} \, {\left (2 \, a {\left (\frac {3 \, {\left (\sqrt {-a^{2} x^{2} + 1} x^{2} + \frac {20 \, \sqrt {-a^{2} x^{2} + 1}}{a^{2}}\right )} \arcsin \left (a x\right )}{a^{3}} - \frac {a^{2} x^{3} + 60 \, x}{a^{4}}\right )} + \frac {9 \, {\left (a^{2} x^{3} + 6 \, x\right )} \arcsin \left (a x\right )^{2}}{a^{3}}\right )} a \] Input:

integrate(x^2*arcsin(a*x)^4,x, algorithm="maxima")
 

Output:

1/3*x^3*arcsin(a*x)^4 + 4/9*a*(sqrt(-a^2*x^2 + 1)*x^2/a^2 + 2*sqrt(-a^2*x^ 
2 + 1)/a^4)*arcsin(a*x)^3 - 4/81*(2*a*(3*(sqrt(-a^2*x^2 + 1)*x^2 + 20*sqrt 
(-a^2*x^2 + 1)/a^2)*arcsin(a*x)/a^3 - (a^2*x^3 + 60*x)/a^4) + 9*(a^2*x^3 + 
 6*x)*arcsin(a*x)^2/a^3)*a
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.06 \[ \int x^2 \arcsin (a x)^4 \, dx=\frac {{\left (a^{2} x^{2} - 1\right )} x \arcsin \left (a x\right )^{4}}{3 \, a^{2}} + \frac {x \arcsin \left (a x\right )^{4}}{3 \, a^{2}} - \frac {4 \, {\left (a^{2} x^{2} - 1\right )} x \arcsin \left (a x\right )^{2}}{9 \, a^{2}} - \frac {4 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} \arcsin \left (a x\right )^{3}}{9 \, a^{3}} - \frac {28 \, x \arcsin \left (a x\right )^{2}}{9 \, a^{2}} + \frac {4 \, \sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )^{3}}{3 \, a^{3}} + \frac {8 \, {\left (a^{2} x^{2} - 1\right )} x}{81 \, a^{2}} + \frac {8 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} \arcsin \left (a x\right )}{27 \, a^{3}} + \frac {488 \, x}{81 \, a^{2}} - \frac {56 \, \sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )}{9 \, a^{3}} \] Input:

integrate(x^2*arcsin(a*x)^4,x, algorithm="giac")
 

Output:

1/3*(a^2*x^2 - 1)*x*arcsin(a*x)^4/a^2 + 1/3*x*arcsin(a*x)^4/a^2 - 4/9*(a^2 
*x^2 - 1)*x*arcsin(a*x)^2/a^2 - 4/9*(-a^2*x^2 + 1)^(3/2)*arcsin(a*x)^3/a^3 
 - 28/9*x*arcsin(a*x)^2/a^2 + 4/3*sqrt(-a^2*x^2 + 1)*arcsin(a*x)^3/a^3 + 8 
/81*(a^2*x^2 - 1)*x/a^2 + 8/27*(-a^2*x^2 + 1)^(3/2)*arcsin(a*x)/a^3 + 488/ 
81*x/a^2 - 56/9*sqrt(-a^2*x^2 + 1)*arcsin(a*x)/a^3
 

Mupad [F(-1)]

Timed out. \[ \int x^2 \arcsin (a x)^4 \, dx=\int x^2\,{\mathrm {asin}\left (a\,x\right )}^4 \,d x \] Input:

int(x^2*asin(a*x)^4,x)
 

Output:

int(x^2*asin(a*x)^4, x)
 

Reduce [F]

\[ \int x^2 \arcsin (a x)^4 \, dx=\int \mathit {asin} \left (a x \right )^{4} x^{2}d x \] Input:

int(x^2*asin(a*x)^4,x)
 

Output:

int(asin(a*x)**4*x**2,x)