\(\int \frac {x^4 \arcsin (a x)^3}{\sqrt {1-a^2 x^2}} \, dx\) [282]

Optimal result

Integrand size = 24, antiderivative size = 191 \[ \int \frac {x^4 \arcsin (a x)^3}{\sqrt {1-a^2 x^2}} \, dx=-\frac {45 x^2}{128 a^3}-\frac {3 x^4}{128 a}+\frac {45 x \sqrt {1-a^2 x^2} \arcsin (a x)}{64 a^4}+\frac {3 x^3 \sqrt {1-a^2 x^2} \arcsin (a x)}{32 a^2}-\frac {45 \arcsin (a x)^2}{128 a^5}+\frac {9 x^2 \arcsin (a x)^2}{16 a^3}+\frac {3 x^4 \arcsin (a x)^2}{16 a}-\frac {3 x \sqrt {1-a^2 x^2} \arcsin (a x)^3}{8 a^4}-\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{4 a^2}+\frac {3 \arcsin (a x)^4}{32 a^5} \] Output:

-45/128*x^2/a^3-3/128*x^4/a+45/64*x*(-a^2*x^2+1)^(1/2)*arcsin(a*x)/a^4+3/3 
2*x^3*(-a^2*x^2+1)^(1/2)*arcsin(a*x)/a^2-45/128*arcsin(a*x)^2/a^5+9/16*x^2 
*arcsin(a*x)^2/a^3+3/16*x^4*arcsin(a*x)^2/a-3/8*x*(-a^2*x^2+1)^(1/2)*arcsi 
n(a*x)^3/a^4-1/4*x^3*(-a^2*x^2+1)^(1/2)*arcsin(a*x)^3/a^2+3/32*arcsin(a*x) 
^4/a^5
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.65 \[ \int \frac {x^4 \arcsin (a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\frac {-3 a^2 x^2 \left (15+a^2 x^2\right )+6 a x \sqrt {1-a^2 x^2} \left (15+2 a^2 x^2\right ) \arcsin (a x)+3 \left (-15+24 a^2 x^2+8 a^4 x^4\right ) \arcsin (a x)^2-16 a x \sqrt {1-a^2 x^2} \left (3+2 a^2 x^2\right ) \arcsin (a x)^3+12 \arcsin (a x)^4}{128 a^5} \] Input:

Integrate[(x^4*ArcSin[a*x]^3)/Sqrt[1 - a^2*x^2],x]
 

Output:

(-3*a^2*x^2*(15 + a^2*x^2) + 6*a*x*Sqrt[1 - a^2*x^2]*(15 + 2*a^2*x^2)*ArcS 
in[a*x] + 3*(-15 + 24*a^2*x^2 + 8*a^4*x^4)*ArcSin[a*x]^2 - 16*a*x*Sqrt[1 - 
 a^2*x^2]*(3 + 2*a^2*x^2)*ArcSin[a*x]^3 + 12*ArcSin[a*x]^4)/(128*a^5)
 

Rubi [A] (verified)

Time = 1.60 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.45, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5210, 5138, 5210, 15, 5138, 5152, 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.

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

\(\Big \downarrow \) 5210

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

\(\Big \downarrow \) 5138

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

\(\Big \downarrow \) 5210

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 5138

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

\(\Big \downarrow \) 5152

\(\displaystyle \frac {3 \left (\frac {1}{4} x^4 \arcsin (a x)^2-\frac {1}{2} a \left (\frac {3 \int \frac {x^2 \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)}{4 a^2}+\frac {x^4}{16 a}\right )\right )}{4 a}+\frac {3 \left (\frac {3 \left (\frac {1}{2} x^2 \arcsin (a x)^2-a \int \frac {x^2 \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx\right )}{2 a}+\frac {\arcsin (a x)^4}{8 a^3}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)^3}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{4 a^2}\)

\(\Big \downarrow \) 5210

\(\displaystyle \frac {3 \left (\frac {1}{4} x^4 \arcsin (a x)^2-\frac {1}{2} a \left (\frac {3 \left (\frac {\int \frac {\arcsin (a x)}{\sqrt {1-a^2 x^2}}dx}{2 a^2}+\frac {\int xdx}{2 a}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)}{4 a^2}+\frac {x^4}{16 a}\right )\right )}{4 a}+\frac {3 \left (\frac {3 \left (\frac {1}{2} x^2 \arcsin (a x)^2-a \left (\frac {\int \frac {\arcsin (a x)}{\sqrt {1-a^2 x^2}}dx}{2 a^2}+\frac {\int xdx}{2 a}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)}{2 a^2}\right )\right )}{2 a}+\frac {\arcsin (a x)^4}{8 a^3}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)^3}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{4 a^2}\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {3 \left (\frac {1}{4} x^4 \arcsin (a x)^2-\frac {1}{2} a \left (\frac {3 \left (\frac {\int \frac {\arcsin (a x)}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)}{2 a^2}+\frac {x^2}{4 a}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)}{4 a^2}+\frac {x^4}{16 a}\right )\right )}{4 a}+\frac {3 \left (\frac {3 \left (\frac {1}{2} x^2 \arcsin (a x)^2-a \left (\frac {\int \frac {\arcsin (a x)}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)}{2 a^2}+\frac {x^2}{4 a}\right )\right )}{2 a}+\frac {\arcsin (a x)^4}{8 a^3}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)^3}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{4 a^2}\)

\(\Big \downarrow \) 5152

\(\displaystyle -\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{4 a^2}+\frac {3 \left (\frac {\arcsin (a x)^4}{8 a^3}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)^3}{2 a^2}+\frac {3 \left (\frac {1}{2} x^2 \arcsin (a x)^2-a \left (\frac {\arcsin (a x)^2}{4 a^3}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)}{2 a^2}+\frac {x^2}{4 a}\right )\right )}{2 a}\right )}{4 a^2}+\frac {3 \left (\frac {1}{4} x^4 \arcsin (a x)^2-\frac {1}{2} a \left (-\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)}{4 a^2}+\frac {3 \left (\frac {\arcsin (a x)^2}{4 a^3}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)}{2 a^2}+\frac {x^2}{4 a}\right )}{4 a^2}+\frac {x^4}{16 a}\right )\right )}{4 a}\)

Input:

Int[(x^4*ArcSin[a*x]^3)/Sqrt[1 - a^2*x^2],x]
 

Output:

-1/4*(x^3*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/a^2 + (3*((x^4*ArcSin[a*x]^2)/4 
 - (a*(x^4/(16*a) - (x^3*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(4*a^2) + (3*(x^2/ 
(4*a) - (x*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(2*a^2) + ArcSin[a*x]^2/(4*a^3)) 
)/(4*a^2)))/2))/(4*a) + (3*(-1/2*(x*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/a^2 + 
 ArcSin[a*x]^4/(8*a^3) + (3*((x^2*ArcSin[a*x]^2)/2 - a*(x^2/(4*a) - (x*Sqr 
t[1 - a^2*x^2]*ArcSin[a*x])/(2*a^2) + ArcSin[a*x]^2/(4*a^3))))/(2*a)))/(4* 
a^2)
 

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 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 = 0.44 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.84

method	result	size
default	\(\frac {-128 \arcsin \left (a x \right )^{3} \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}+96 \arcsin \left (a x \right )^{2} a^{4} x^{4}+48 \arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}-12 a^{4} x^{4}-192 \arcsin \left (a x \right )^{3} \sqrt {-a^{2} x^{2}+1}\, a x +288 x^{2} \arcsin \left (a x \right )^{2} a^{2}+48 \arcsin \left (a x \right )^{4}+360 \arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}\, a x -180 a^{2} x^{2}-180 \arcsin \left (a x \right )^{2}-27}{512 a^{5}}\)	\(160\)

Input:

int(x^4*arcsin(a*x)^3/(-a^2*x^2+1)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

1/512*(-128*arcsin(a*x)^3*(-a^2*x^2+1)^(1/2)*a^3*x^3+96*arcsin(a*x)^2*a^4* 
x^4+48*arcsin(a*x)*(-a^2*x^2+1)^(1/2)*a^3*x^3-12*a^4*x^4-192*arcsin(a*x)^3 
*(-a^2*x^2+1)^(1/2)*a*x+288*x^2*arcsin(a*x)^2*a^2+48*arcsin(a*x)^4+360*arc 
sin(a*x)*(-a^2*x^2+1)^(1/2)*a*x-180*a^2*x^2-180*arcsin(a*x)^2-27)/a^5
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.58 \[ \int \frac {x^4 \arcsin (a x)^3}{\sqrt {1-a^2 x^2}} \, dx=-\frac {3 \, a^{4} x^{4} + 45 \, a^{2} x^{2} - 12 \, \arcsin \left (a x\right )^{4} - 3 \, {\left (8 \, a^{4} x^{4} + 24 \, a^{2} x^{2} - 15\right )} \arcsin \left (a x\right )^{2} + 2 \, \sqrt {-a^{2} x^{2} + 1} {\left (8 \, {\left (2 \, a^{3} x^{3} + 3 \, a x\right )} \arcsin \left (a x\right )^{3} - 3 \, {\left (2 \, a^{3} x^{3} + 15 \, a x\right )} \arcsin \left (a x\right )\right )}}{128 \, a^{5}} \] Input:

integrate(x^4*arcsin(a*x)^3/(-a^2*x^2+1)^(1/2),x, algorithm="fricas")
 

Output:

-1/128*(3*a^4*x^4 + 45*a^2*x^2 - 12*arcsin(a*x)^4 - 3*(8*a^4*x^4 + 24*a^2* 
x^2 - 15)*arcsin(a*x)^2 + 2*sqrt(-a^2*x^2 + 1)*(8*(2*a^3*x^3 + 3*a*x)*arcs 
in(a*x)^3 - 3*(2*a^3*x^3 + 15*a*x)*arcsin(a*x)))/a^5
 

Sympy [A] (verification not implemented)

Time = 0.70 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.97 \[ \int \frac {x^4 \arcsin (a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\begin {cases} \frac {3 x^{4} \operatorname {asin}^{2}{\left (a x \right )}}{16 a} - \frac {3 x^{4}}{128 a} - \frac {x^{3} \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{3}{\left (a x \right )}}{4 a^{2}} + \frac {3 x^{3} \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}{\left (a x \right )}}{32 a^{2}} + \frac {9 x^{2} \operatorname {asin}^{2}{\left (a x \right )}}{16 a^{3}} - \frac {45 x^{2}}{128 a^{3}} - \frac {3 x \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{3}{\left (a x \right )}}{8 a^{4}} + \frac {45 x \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}{\left (a x \right )}}{64 a^{4}} + \frac {3 \operatorname {asin}^{4}{\left (a x \right )}}{32 a^{5}} - \frac {45 \operatorname {asin}^{2}{\left (a x \right )}}{128 a^{5}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \] Input:

integrate(x**4*asin(a*x)**3/(-a**2*x**2+1)**(1/2),x)
 

Output:

Piecewise((3*x**4*asin(a*x)**2/(16*a) - 3*x**4/(128*a) - x**3*sqrt(-a**2*x 
**2 + 1)*asin(a*x)**3/(4*a**2) + 3*x**3*sqrt(-a**2*x**2 + 1)*asin(a*x)/(32 
*a**2) + 9*x**2*asin(a*x)**2/(16*a**3) - 45*x**2/(128*a**3) - 3*x*sqrt(-a* 
*2*x**2 + 1)*asin(a*x)**3/(8*a**4) + 45*x*sqrt(-a**2*x**2 + 1)*asin(a*x)/( 
64*a**4) + 3*asin(a*x)**4/(32*a**5) - 45*asin(a*x)**2/(128*a**5), Ne(a, 0) 
), (0, True))
 

Maxima [F]

\[ \int \frac {x^4 \arcsin (a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\int { \frac {x^{4} \arcsin \left (a x\right )^{3}}{\sqrt {-a^{2} x^{2} + 1}} \,d x } \] Input:

integrate(x^4*arcsin(a*x)^3/(-a^2*x^2+1)^(1/2),x, algorithm="maxima")
 

Output:

integrate(x^4*arcsin(a*x)^3/sqrt(-a^2*x^2 + 1), x)
 

Giac [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.01 \[ \int \frac {x^4 \arcsin (a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x \arcsin \left (a x\right )^{3}}{4 \, a^{4}} - \frac {5 \, \sqrt {-a^{2} x^{2} + 1} x \arcsin \left (a x\right )^{3}}{8 \, a^{4}} - \frac {3 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x \arcsin \left (a x\right )}{32 \, a^{4}} + \frac {3 \, {\left (a^{2} x^{2} - 1\right )}^{2} \arcsin \left (a x\right )^{2}}{16 \, a^{5}} + \frac {3 \, \arcsin \left (a x\right )^{4}}{32 \, a^{5}} + \frac {51 \, \sqrt {-a^{2} x^{2} + 1} x \arcsin \left (a x\right )}{64 \, a^{4}} + \frac {15 \, {\left (a^{2} x^{2} - 1\right )} \arcsin \left (a x\right )^{2}}{16 \, a^{5}} - \frac {3 \, {\left (a^{2} x^{2} - 1\right )}^{2}}{128 \, a^{5}} + \frac {51 \, \arcsin \left (a x\right )^{2}}{128 \, a^{5}} - \frac {51 \, {\left (a^{2} x^{2} - 1\right )}}{128 \, a^{5}} - \frac {195}{1024 \, a^{5}} \] Input:

integrate(x^4*arcsin(a*x)^3/(-a^2*x^2+1)^(1/2),x, algorithm="giac")
 

Output:

1/4*(-a^2*x^2 + 1)^(3/2)*x*arcsin(a*x)^3/a^4 - 5/8*sqrt(-a^2*x^2 + 1)*x*ar 
csin(a*x)^3/a^4 - 3/32*(-a^2*x^2 + 1)^(3/2)*x*arcsin(a*x)/a^4 + 3/16*(a^2* 
x^2 - 1)^2*arcsin(a*x)^2/a^5 + 3/32*arcsin(a*x)^4/a^5 + 51/64*sqrt(-a^2*x^ 
2 + 1)*x*arcsin(a*x)/a^4 + 15/16*(a^2*x^2 - 1)*arcsin(a*x)^2/a^5 - 3/128*( 
a^2*x^2 - 1)^2/a^5 + 51/128*arcsin(a*x)^2/a^5 - 51/128*(a^2*x^2 - 1)/a^5 - 
 195/1024/a^5
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 \arcsin (a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {x^4\,{\mathrm {asin}\left (a\,x\right )}^3}{\sqrt {1-a^2\,x^2}} \,d x \] Input:

int((x^4*asin(a*x)^3)/(1 - a^2*x^2)^(1/2),x)
 

Output:

int((x^4*asin(a*x)^3)/(1 - a^2*x^2)^(1/2), x)
 

Reduce [F]

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

int(x^4*asin(a*x)^3/(-a^2*x^2+1)^(1/2),x)
 

Output:

int((asin(a*x)**3*x**4)/sqrt( - a**2*x**2 + 1),x)