Integrand size = 10, antiderivative size = 201 \[ \int x^4 \arcsin (a x)^3 \, dx=-\frac {298 \sqrt {1-a^2 x^2}}{375 a^5}+\frac {76 \left (1-a^2 x^2\right )^{3/2}}{1125 a^5}-\frac {6 \left (1-a^2 x^2\right )^{5/2}}{625 a^5}-\frac {16 x \arcsin (a x)}{25 a^4}-\frac {8 x^3 \arcsin (a x)}{75 a^2}-\frac {6}{125} x^5 \arcsin (a x)+\frac {8 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{25 a^5}+\frac {4 x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{25 a^3}+\frac {3 x^4 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{25 a}+\frac {1}{5} x^5 \arcsin (a x)^3 \]
76/1125*(-a^2*x^2+1)^(3/2)/a^5-6/625*(-a^2*x^2+1)^(5/2)/a^5-16/25*x*arcsin (a*x)/a^4-8/75*x^3*arcsin(a*x)/a^2-6/125*x^5*arcsin(a*x)+1/5*x^5*arcsin(a* x)^3-298/375*(-a^2*x^2+1)^(1/2)/a^5+8/25*arcsin(a*x)^2*(-a^2*x^2+1)^(1/2)/ a^5+4/25*x^2*arcsin(a*x)^2*(-a^2*x^2+1)^(1/2)/a^3+3/25*x^4*arcsin(a*x)^2*( -a^2*x^2+1)^(1/2)/a
Time = 0.07 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.61 \[ \int x^4 \arcsin (a x)^3 \, dx=\frac {-2 \sqrt {1-a^2 x^2} \left (2072+136 a^2 x^2+27 a^4 x^4\right )-30 a x \left (120+20 a^2 x^2+9 a^4 x^4\right ) \arcsin (a x)+225 \sqrt {1-a^2 x^2} \left (8+4 a^2 x^2+3 a^4 x^4\right ) \arcsin (a x)^2+1125 a^5 x^5 \arcsin (a x)^3}{5625 a^5} \]
(-2*Sqrt[1 - a^2*x^2]*(2072 + 136*a^2*x^2 + 27*a^4*x^4) - 30*a*x*(120 + 20 *a^2*x^2 + 9*a^4*x^4)*ArcSin[a*x] + 225*Sqrt[1 - a^2*x^2]*(8 + 4*a^2*x^2 + 3*a^4*x^4)*ArcSin[a*x]^2 + 1125*a^5*x^5*ArcSin[a*x]^3)/(5625*a^5)
Time = 1.22 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.50, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.400, Rules used = {5138, 5210, 5138, 243, 53, 2009, 5210, 5138, 243, 53, 2009, 5182, 5130, 241}
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.
| \(\displaystyle \int x^4 \arcsin (a x)^3 \, dx\) |
| \(\Big \downarrow \) 5138 |
| \(\displaystyle \frac {1}{5} x^5 \arcsin (a x)^3-\frac {3}{5} a \int \frac {x^5 \arcsin (a x)^2}{\sqrt {1-a^2 x^2}}dx\) |
| \(\Big \downarrow \) 5210 |
| \(\displaystyle \frac {1}{5} x^5 \arcsin (a x)^3-\frac {3}{5} a \left (\frac {4 \int \frac {x^3 \arcsin (a x)^2}{\sqrt {1-a^2 x^2}}dx}{5 a^2}+\frac {2 \int x^4 \arcsin (a x)dx}{5 a}-\frac {x^4 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{5 a^2}\right )\) |
| \(\Big \downarrow \) 5138 |
| \(\displaystyle \frac {1}{5} x^5 \arcsin (a x)^3-\frac {3}{5} a \left (\frac {2 \left (\frac {1}{5} x^5 \arcsin (a x)-\frac {1}{5} a \int \frac {x^5}{\sqrt {1-a^2 x^2}}dx\right )}{5 a}+\frac {4 \int \frac {x^3 \arcsin (a x)^2}{\sqrt {1-a^2 x^2}}dx}{5 a^2}-\frac {x^4 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{5 a^2}\right )\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {1}{5} x^5 \arcsin (a x)^3-\frac {3}{5} a \left (\frac {4 \int \frac {x^3 \arcsin (a x)^2}{\sqrt {1-a^2 x^2}}dx}{5 a^2}+\frac {2 \left (\frac {1}{5} x^5 \arcsin (a x)-\frac {1}{10} a \int \frac {x^4}{\sqrt {1-a^2 x^2}}dx^2\right )}{5 a}-\frac {x^4 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{5 a^2}\right )\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \frac {1}{5} x^5 \arcsin (a x)^3-\frac {3}{5} a \left (\frac {4 \int \frac {x^3 \arcsin (a x)^2}{\sqrt {1-a^2 x^2}}dx}{5 a^2}+\frac {2 \left (\frac {1}{5} x^5 \arcsin (a x)-\frac {1}{10} a \int \left (\frac {\left (1-a^2 x^2\right )^{3/2}}{a^4}-\frac {2 \sqrt {1-a^2 x^2}}{a^4}+\frac {1}{a^4 \sqrt {1-a^2 x^2}}\right )dx^2\right )}{5 a}-\frac {x^4 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{5 a^2}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{5} x^5 \arcsin (a x)^3-\frac {3}{5} a \left (\frac {4 \int \frac {x^3 \arcsin (a x)^2}{\sqrt {1-a^2 x^2}}dx}{5 a^2}-\frac {x^4 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{5 a^2}+\frac {2 \left (\frac {1}{5} x^5 \arcsin (a x)-\frac {1}{10} a \left (-\frac {2 \left (1-a^2 x^2\right )^{5/2}}{5 a^6}+\frac {4 \left (1-a^2 x^2\right )^{3/2}}{3 a^6}-\frac {2 \sqrt {1-a^2 x^2}}{a^6}\right )\right )}{5 a}\right )\) |
| \(\Big \downarrow \) 5210 |
| \(\displaystyle \frac {1}{5} x^5 \arcsin (a x)^3-\frac {3}{5} a \left (\frac {4 \left (\frac {2 \int \frac {x \arcsin (a x)^2}{\sqrt {1-a^2 x^2}}dx}{3 a^2}+\frac {2 \int x^2 \arcsin (a x)dx}{3 a}-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{3 a^2}\right )}{5 a^2}-\frac {x^4 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{5 a^2}+\frac {2 \left (\frac {1}{5} x^5 \arcsin (a x)-\frac {1}{10} a \left (-\frac {2 \left (1-a^2 x^2\right )^{5/2}}{5 a^6}+\frac {4 \left (1-a^2 x^2\right )^{3/2}}{3 a^6}-\frac {2 \sqrt {1-a^2 x^2}}{a^6}\right )\right )}{5 a}\right )\) |
| \(\Big \downarrow \) 5138 |
| \(\displaystyle \frac {1}{5} x^5 \arcsin (a x)^3-\frac {3}{5} a \left (\frac {4 \left (\frac {2 \int \frac {x \arcsin (a x)^2}{\sqrt {1-a^2 x^2}}dx}{3 a^2}+\frac {2 \left (\frac {1}{3} x^3 \arcsin (a x)-\frac {1}{3} a \int \frac {x^3}{\sqrt {1-a^2 x^2}}dx\right )}{3 a}-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{3 a^2}\right )}{5 a^2}-\frac {x^4 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{5 a^2}+\frac {2 \left (\frac {1}{5} x^5 \arcsin (a x)-\frac {1}{10} a \left (-\frac {2 \left (1-a^2 x^2\right )^{5/2}}{5 a^6}+\frac {4 \left (1-a^2 x^2\right )^{3/2}}{3 a^6}-\frac {2 \sqrt {1-a^2 x^2}}{a^6}\right )\right )}{5 a}\right )\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {1}{5} x^5 \arcsin (a x)^3-\frac {3}{5} a \left (\frac {4 \left (\frac {2 \int \frac {x \arcsin (a x)^2}{\sqrt {1-a^2 x^2}}dx}{3 a^2}+\frac {2 \left (\frac {1}{3} x^3 \arcsin (a x)-\frac {1}{6} a \int \frac {x^2}{\sqrt {1-a^2 x^2}}dx^2\right )}{3 a}-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{3 a^2}\right )}{5 a^2}-\frac {x^4 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{5 a^2}+\frac {2 \left (\frac {1}{5} x^5 \arcsin (a x)-\frac {1}{10} a \left (-\frac {2 \left (1-a^2 x^2\right )^{5/2}}{5 a^6}+\frac {4 \left (1-a^2 x^2\right )^{3/2}}{3 a^6}-\frac {2 \sqrt {1-a^2 x^2}}{a^6}\right )\right )}{5 a}\right )\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \frac {1}{5} x^5 \arcsin (a x)^3-\frac {3}{5} a \left (\frac {4 \left (\frac {2 \int \frac {x \arcsin (a x)^2}{\sqrt {1-a^2 x^2}}dx}{3 a^2}+\frac {2 \left (\frac {1}{3} x^3 \arcsin (a x)-\frac {1}{6} a \int \left (\frac {1}{a^2 \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{a^2}\right )dx^2\right )}{3 a}-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{3 a^2}\right )}{5 a^2}-\frac {x^4 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{5 a^2}+\frac {2 \left (\frac {1}{5} x^5 \arcsin (a x)-\frac {1}{10} a \left (-\frac {2 \left (1-a^2 x^2\right )^{5/2}}{5 a^6}+\frac {4 \left (1-a^2 x^2\right )^{3/2}}{3 a^6}-\frac {2 \sqrt {1-a^2 x^2}}{a^6}\right )\right )}{5 a}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{5} x^5 \arcsin (a x)^3-\frac {3}{5} a \left (\frac {4 \left (\frac {2 \int \frac {x \arcsin (a x)^2}{\sqrt {1-a^2 x^2}}dx}{3 a^2}-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{3 a^2}+\frac {2 \left (\frac {1}{3} x^3 \arcsin (a x)-\frac {1}{6} a \left (\frac {2 \left (1-a^2 x^2\right )^{3/2}}{3 a^4}-\frac {2 \sqrt {1-a^2 x^2}}{a^4}\right )\right )}{3 a}\right )}{5 a^2}-\frac {x^4 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{5 a^2}+\frac {2 \left (\frac {1}{5} x^5 \arcsin (a x)-\frac {1}{10} a \left (-\frac {2 \left (1-a^2 x^2\right )^{5/2}}{5 a^6}+\frac {4 \left (1-a^2 x^2\right )^{3/2}}{3 a^6}-\frac {2 \sqrt {1-a^2 x^2}}{a^6}\right )\right )}{5 a}\right )\) |
| \(\Big \downarrow \) 5182 |
| \(\displaystyle \frac {1}{5} x^5 \arcsin (a x)^3-\frac {3}{5} a \left (\frac {4 \left (\frac {2 \left (\frac {2 \int \arcsin (a x)dx}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^2}{a^2}\right )}{3 a^2}-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{3 a^2}+\frac {2 \left (\frac {1}{3} x^3 \arcsin (a x)-\frac {1}{6} a \left (\frac {2 \left (1-a^2 x^2\right )^{3/2}}{3 a^4}-\frac {2 \sqrt {1-a^2 x^2}}{a^4}\right )\right )}{3 a}\right )}{5 a^2}-\frac {x^4 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{5 a^2}+\frac {2 \left (\frac {1}{5} x^5 \arcsin (a x)-\frac {1}{10} a \left (-\frac {2 \left (1-a^2 x^2\right )^{5/2}}{5 a^6}+\frac {4 \left (1-a^2 x^2\right )^{3/2}}{3 a^6}-\frac {2 \sqrt {1-a^2 x^2}}{a^6}\right )\right )}{5 a}\right )\) |
| \(\Big \downarrow \) 5130 |
| \(\displaystyle \frac {1}{5} x^5 \arcsin (a x)^3-\frac {3}{5} a \left (\frac {4 \left (\frac {2 \left (\frac {2 \left (x \arcsin (a x)-a \int \frac {x}{\sqrt {1-a^2 x^2}}dx\right )}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^2}{a^2}\right )}{3 a^2}-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{3 a^2}+\frac {2 \left (\frac {1}{3} x^3 \arcsin (a x)-\frac {1}{6} a \left (\frac {2 \left (1-a^2 x^2\right )^{3/2}}{3 a^4}-\frac {2 \sqrt {1-a^2 x^2}}{a^4}\right )\right )}{3 a}\right )}{5 a^2}-\frac {x^4 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{5 a^2}+\frac {2 \left (\frac {1}{5} x^5 \arcsin (a x)-\frac {1}{10} a \left (-\frac {2 \left (1-a^2 x^2\right )^{5/2}}{5 a^6}+\frac {4 \left (1-a^2 x^2\right )^{3/2}}{3 a^6}-\frac {2 \sqrt {1-a^2 x^2}}{a^6}\right )\right )}{5 a}\right )\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle \frac {1}{5} x^5 \arcsin (a x)^3-\frac {3}{5} a \left (-\frac {x^4 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{5 a^2}+\frac {2 \left (\frac {1}{5} x^5 \arcsin (a x)-\frac {1}{10} a \left (-\frac {2 \left (1-a^2 x^2\right )^{5/2}}{5 a^6}+\frac {4 \left (1-a^2 x^2\right )^{3/2}}{3 a^6}-\frac {2 \sqrt {1-a^2 x^2}}{a^6}\right )\right )}{5 a}+\frac {4 \left (-\frac {x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{3 a^2}+\frac {2 \left (\frac {2 \left (\frac {\sqrt {1-a^2 x^2}}{a}+x \arcsin (a x)\right )}{a}-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)^2}{a^2}\right )}{3 a^2}+\frac {2 \left (\frac {1}{3} x^3 \arcsin (a x)-\frac {1}{6} a \left (\frac {2 \left (1-a^2 x^2\right )^{3/2}}{3 a^4}-\frac {2 \sqrt {1-a^2 x^2}}{a^4}\right )\right )}{3 a}\right )}{5 a^2}\right )\) |
(x^5*ArcSin[a*x]^3)/5 - (3*a*(-1/5*(x^4*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2)/a ^2 + (2*(-1/10*(a*((-2*Sqrt[1 - a^2*x^2])/a^6 + (4*(1 - a^2*x^2)^(3/2))/(3 *a^6) - (2*(1 - a^2*x^2)^(5/2))/(5*a^6))) + (x^5*ArcSin[a*x])/5))/(5*a) + (4*(-1/3*(x^2*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2)/a^2 + (2*(-1/6*(a*((-2*Sqrt [1 - a^2*x^2])/a^4 + (2*(1 - a^2*x^2)^(3/2))/(3*a^4))) + (x^3*ArcSin[a*x]) /3))/(3*a) + (2*(-((Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2)/a^2) + (2*(Sqrt[1 - a ^2*x^2]/a + x*ArcSin[a*x]))/a))/(3*a^2)))/(5*a^2)))/5
3.1.22.3.1 Defintions of rubi rules used
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])
Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ (2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -1]
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]
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]
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]
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]
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]
Time = 0.05 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.79
| method | result | size |
| derivativedivides | \(\frac {\frac {a^{5} x^{5} \arcsin \left (a x \right )^{3}}{5}+\frac {\arcsin \left (a x \right )^{2} \left (3 a^{4} x^{4}+4 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{25}-\frac {6 a^{5} x^{5} \arcsin \left (a x \right )}{125}-\frac {2 \left (3 a^{4} x^{4}+4 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{625}-\frac {8 a^{3} x^{3} \arcsin \left (a x \right )}{75}-\frac {8 \left (a^{2} x^{2}+2\right ) \sqrt {-a^{2} x^{2}+1}}{225}-\frac {16 \sqrt {-a^{2} x^{2}+1}}{25}-\frac {16 a x \arcsin \left (a x \right )}{25}}{a^{5}}\) | \(159\) |
| default | \(\frac {\frac {a^{5} x^{5} \arcsin \left (a x \right )^{3}}{5}+\frac {\arcsin \left (a x \right )^{2} \left (3 a^{4} x^{4}+4 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{25}-\frac {6 a^{5} x^{5} \arcsin \left (a x \right )}{125}-\frac {2 \left (3 a^{4} x^{4}+4 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{625}-\frac {8 a^{3} x^{3} \arcsin \left (a x \right )}{75}-\frac {8 \left (a^{2} x^{2}+2\right ) \sqrt {-a^{2} x^{2}+1}}{225}-\frac {16 \sqrt {-a^{2} x^{2}+1}}{25}-\frac {16 a x \arcsin \left (a x \right )}{25}}{a^{5}}\) | \(159\) |
1/a^5*(1/5*a^5*x^5*arcsin(a*x)^3+1/25*arcsin(a*x)^2*(3*a^4*x^4+4*a^2*x^2+8 )*(-a^2*x^2+1)^(1/2)-6/125*a^5*x^5*arcsin(a*x)-2/625*(3*a^4*x^4+4*a^2*x^2+ 8)*(-a^2*x^2+1)^(1/2)-8/75*a^3*x^3*arcsin(a*x)-8/225*(a^2*x^2+2)*(-a^2*x^2 +1)^(1/2)-16/25*(-a^2*x^2+1)^(1/2)-16/25*a*x*arcsin(a*x))
Time = 0.26 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.52 \[ \int x^4 \arcsin (a x)^3 \, dx=\frac {1125 \, a^{5} x^{5} \arcsin \left (a x\right )^{3} - 30 \, {\left (9 \, a^{5} x^{5} + 20 \, a^{3} x^{3} + 120 \, a x\right )} \arcsin \left (a x\right ) - {\left (54 \, a^{4} x^{4} + 272 \, a^{2} x^{2} - 225 \, {\left (3 \, a^{4} x^{4} + 4 \, a^{2} x^{2} + 8\right )} \arcsin \left (a x\right )^{2} + 4144\right )} \sqrt {-a^{2} x^{2} + 1}}{5625 \, a^{5}} \]
1/5625*(1125*a^5*x^5*arcsin(a*x)^3 - 30*(9*a^5*x^5 + 20*a^3*x^3 + 120*a*x) *arcsin(a*x) - (54*a^4*x^4 + 272*a^2*x^2 - 225*(3*a^4*x^4 + 4*a^2*x^2 + 8) *arcsin(a*x)^2 + 4144)*sqrt(-a^2*x^2 + 1))/a^5
Time = 0.54 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.98 \[ \int x^4 \arcsin (a x)^3 \, dx=\begin {cases} \frac {x^{5} \operatorname {asin}^{3}{\left (a x \right )}}{5} - \frac {6 x^{5} \operatorname {asin}{\left (a x \right )}}{125} + \frac {3 x^{4} \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a x \right )}}{25 a} - \frac {6 x^{4} \sqrt {- a^{2} x^{2} + 1}}{625 a} - \frac {8 x^{3} \operatorname {asin}{\left (a x \right )}}{75 a^{2}} + \frac {4 x^{2} \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a x \right )}}{25 a^{3}} - \frac {272 x^{2} \sqrt {- a^{2} x^{2} + 1}}{5625 a^{3}} - \frac {16 x \operatorname {asin}{\left (a x \right )}}{25 a^{4}} + \frac {8 \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a x \right )}}{25 a^{5}} - \frac {4144 \sqrt {- a^{2} x^{2} + 1}}{5625 a^{5}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Piecewise((x**5*asin(a*x)**3/5 - 6*x**5*asin(a*x)/125 + 3*x**4*sqrt(-a**2* x**2 + 1)*asin(a*x)**2/(25*a) - 6*x**4*sqrt(-a**2*x**2 + 1)/(625*a) - 8*x* *3*asin(a*x)/(75*a**2) + 4*x**2*sqrt(-a**2*x**2 + 1)*asin(a*x)**2/(25*a**3 ) - 272*x**2*sqrt(-a**2*x**2 + 1)/(5625*a**3) - 16*x*asin(a*x)/(25*a**4) + 8*sqrt(-a**2*x**2 + 1)*asin(a*x)**2/(25*a**5) - 4144*sqrt(-a**2*x**2 + 1) /(5625*a**5), Ne(a, 0)), (0, True))
Time = 0.28 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.85 \[ \int x^4 \arcsin (a x)^3 \, dx=\frac {1}{5} \, x^{5} \arcsin \left (a x\right )^{3} + \frac {1}{25} \, {\left (\frac {3 \, \sqrt {-a^{2} x^{2} + 1} x^{4}}{a^{2}} + \frac {4 \, \sqrt {-a^{2} x^{2} + 1} x^{2}}{a^{4}} + \frac {8 \, \sqrt {-a^{2} x^{2} + 1}}{a^{6}}\right )} a \arcsin \left (a x\right )^{2} - \frac {2}{5625} \, a {\left (\frac {27 \, \sqrt {-a^{2} x^{2} + 1} a^{2} x^{4} + 136 \, \sqrt {-a^{2} x^{2} + 1} x^{2} + \frac {2072 \, \sqrt {-a^{2} x^{2} + 1}}{a^{2}}}{a^{4}} + \frac {15 \, {\left (9 \, a^{4} x^{5} + 20 \, a^{2} x^{3} + 120 \, x\right )} \arcsin \left (a x\right )}{a^{5}}\right )} \]
1/5*x^5*arcsin(a*x)^3 + 1/25*(3*sqrt(-a^2*x^2 + 1)*x^4/a^2 + 4*sqrt(-a^2*x ^2 + 1)*x^2/a^4 + 8*sqrt(-a^2*x^2 + 1)/a^6)*a*arcsin(a*x)^2 - 2/5625*a*((2 7*sqrt(-a^2*x^2 + 1)*a^2*x^4 + 136*sqrt(-a^2*x^2 + 1)*x^2 + 2072*sqrt(-a^2 *x^2 + 1)/a^2)/a^4 + 15*(9*a^4*x^5 + 20*a^2*x^3 + 120*x)*arcsin(a*x)/a^5)
Time = 0.27 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.24 \[ \int x^4 \arcsin (a x)^3 \, dx=\frac {{\left (a^{2} x^{2} - 1\right )}^{2} x \arcsin \left (a x\right )^{3}}{5 \, a^{4}} + \frac {2 \, {\left (a^{2} x^{2} - 1\right )} x \arcsin \left (a x\right )^{3}}{5 \, a^{4}} - \frac {6 \, {\left (a^{2} x^{2} - 1\right )}^{2} x \arcsin \left (a x\right )}{125 \, a^{4}} + \frac {x \arcsin \left (a x\right )^{3}}{5 \, a^{4}} + \frac {3 \, {\left (a^{2} x^{2} - 1\right )}^{2} \sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )^{2}}{25 \, a^{5}} - \frac {76 \, {\left (a^{2} x^{2} - 1\right )} x \arcsin \left (a x\right )}{375 \, a^{4}} - \frac {2 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} \arcsin \left (a x\right )^{2}}{5 \, a^{5}} - \frac {298 \, x \arcsin \left (a x\right )}{375 \, a^{4}} - \frac {6 \, {\left (a^{2} x^{2} - 1\right )}^{2} \sqrt {-a^{2} x^{2} + 1}}{625 \, a^{5}} + \frac {3 \, \sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )^{2}}{5 \, a^{5}} + \frac {76 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{1125 \, a^{5}} - \frac {298 \, \sqrt {-a^{2} x^{2} + 1}}{375 \, a^{5}} \]
1/5*(a^2*x^2 - 1)^2*x*arcsin(a*x)^3/a^4 + 2/5*(a^2*x^2 - 1)*x*arcsin(a*x)^ 3/a^4 - 6/125*(a^2*x^2 - 1)^2*x*arcsin(a*x)/a^4 + 1/5*x*arcsin(a*x)^3/a^4 + 3/25*(a^2*x^2 - 1)^2*sqrt(-a^2*x^2 + 1)*arcsin(a*x)^2/a^5 - 76/375*(a^2* x^2 - 1)*x*arcsin(a*x)/a^4 - 2/5*(-a^2*x^2 + 1)^(3/2)*arcsin(a*x)^2/a^5 - 298/375*x*arcsin(a*x)/a^4 - 6/625*(a^2*x^2 - 1)^2*sqrt(-a^2*x^2 + 1)/a^5 + 3/5*sqrt(-a^2*x^2 + 1)*arcsin(a*x)^2/a^5 + 76/1125*(-a^2*x^2 + 1)^(3/2)/a ^5 - 298/375*sqrt(-a^2*x^2 + 1)/a^5
Timed out. \[ \int x^4 \arcsin (a x)^3 \, dx=\int x^4\,{\mathrm {asin}\left (a\,x\right )}^3 \,d x \]