\(\int x^5 \arcsin (a x)^4 \, dx\) [32]

Optimal result

Integrand size = 10, antiderivative size = 282 \[ \int x^5 \arcsin (a x)^4 \, dx=\frac {245 x^2}{1152 a^4}+\frac {65 x^4}{3456 a^2}+\frac {x^6}{324}-\frac {245 x \sqrt {1-a^2 x^2} \arcsin (a x)}{576 a^5}-\frac {65 x^3 \sqrt {1-a^2 x^2} \arcsin (a x)}{864 a^3}-\frac {x^5 \sqrt {1-a^2 x^2} \arcsin (a x)}{54 a}+\frac {245 \arcsin (a x)^2}{1152 a^6}-\frac {5 x^2 \arcsin (a x)^2}{16 a^4}-\frac {5 x^4 \arcsin (a x)^2}{48 a^2}-\frac {1}{18} x^6 \arcsin (a x)^2+\frac {5 x \sqrt {1-a^2 x^2} \arcsin (a x)^3}{24 a^5}+\frac {5 x^3 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{36 a^3}+\frac {x^5 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{9 a}-\frac {5 \arcsin (a x)^4}{96 a^6}+\frac {1}{6} x^6 \arcsin (a x)^4 \]

[Out]

245/1152*x^2/a^4+65/3456*x^4/a^2+1/324*x^6+245/1152*arcsin(a*x)^2/a^6-5/16*x^2*arcsin(a*x)^2/a^4-5/48*x^4*arcs
in(a*x)^2/a^2-1/18*x^6*arcsin(a*x)^2-5/96*arcsin(a*x)^4/a^6+1/6*x^6*arcsin(a*x)^4-245/576*x*arcsin(a*x)*(-a^2*
x^2+1)^(1/2)/a^5-65/864*x^3*arcsin(a*x)*(-a^2*x^2+1)^(1/2)/a^3-1/54*x^5*arcsin(a*x)*(-a^2*x^2+1)^(1/2)/a+5/24*
x*arcsin(a*x)^3*(-a^2*x^2+1)^(1/2)/a^5+5/36*x^3*arcsin(a*x)^3*(-a^2*x^2+1)^(1/2)/a^3+1/9*x^5*arcsin(a*x)^3*(-a
^2*x^2+1)^(1/2)/a

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4723, 4795, 4737, 30} \[ \int x^5 \arcsin (a x)^4 \, dx=-\frac {5 \arcsin (a x)^4}{96 a^6}+\frac {245 \arcsin (a x)^2}{1152 a^6}-\frac {5 x^2 \arcsin (a x)^2}{16 a^4}+\frac {245 x^2}{1152 a^4}-\frac {5 x^4 \arcsin (a x)^2}{48 a^2}+\frac {x^5 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{9 a}-\frac {x^5 \sqrt {1-a^2 x^2} \arcsin (a x)}{54 a}+\frac {65 x^4}{3456 a^2}+\frac {5 x \sqrt {1-a^2 x^2} \arcsin (a x)^3}{24 a^5}-\frac {245 x \sqrt {1-a^2 x^2} \arcsin (a x)}{576 a^5}+\frac {5 x^3 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{36 a^3}-\frac {65 x^3 \sqrt {1-a^2 x^2} \arcsin (a x)}{864 a^3}+\frac {1}{6} x^6 \arcsin (a x)^4-\frac {1}{18} x^6 \arcsin (a x)^2+\frac {x^6}{324} \]

[In]

Int[x^5*ArcSin[a*x]^4,x]

[Out]

(245*x^2)/(1152*a^4) + (65*x^4)/(3456*a^2) + x^6/324 - (245*x*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(576*a^5) - (65*x
^3*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(864*a^3) - (x^5*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(54*a) + (245*ArcSin[a*x]^2)
/(1152*a^6) - (5*x^2*ArcSin[a*x]^2)/(16*a^4) - (5*x^4*ArcSin[a*x]^2)/(48*a^2) - (x^6*ArcSin[a*x]^2)/18 + (5*x*
Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/(24*a^5) + (5*x^3*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/(36*a^3) + (x^5*Sqrt[1 - a
^2*x^2]*ArcSin[a*x]^3)/(9*a) - (5*ArcSin[a*x]^4)/(96*a^6) + (x^6*ArcSin[a*x]^4)/6

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 4723

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcSi
n[c*x])^n/(d*(m + 1))), x] - Dist[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 4737

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(1/(b*c*(n + 1)))*Si
mp[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 4795

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] + (Dist[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] + Dist[b*f*(n/(c*(m + 2*p + 1)))*S
imp[(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]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} x^6 \arcsin (a x)^4-\frac {1}{3} (2 a) \int \frac {x^6 \arcsin (a x)^3}{\sqrt {1-a^2 x^2}} \, dx \\ & = \frac {x^5 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{9 a}+\frac {1}{6} x^6 \arcsin (a x)^4-\frac {1}{3} \int x^5 \arcsin (a x)^2 \, dx-\frac {5 \int \frac {x^4 \arcsin (a x)^3}{\sqrt {1-a^2 x^2}} \, dx}{9 a} \\ & = -\frac {1}{18} x^6 \arcsin (a x)^2+\frac {5 x^3 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{36 a^3}+\frac {x^5 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{9 a}+\frac {1}{6} x^6 \arcsin (a x)^4-\frac {5 \int \frac {x^2 \arcsin (a x)^3}{\sqrt {1-a^2 x^2}} \, dx}{12 a^3}-\frac {5 \int x^3 \arcsin (a x)^2 \, dx}{12 a^2}+\frac {1}{9} a \int \frac {x^6 \arcsin (a x)}{\sqrt {1-a^2 x^2}} \, dx \\ & = -\frac {x^5 \sqrt {1-a^2 x^2} \arcsin (a x)}{54 a}-\frac {5 x^4 \arcsin (a x)^2}{48 a^2}-\frac {1}{18} x^6 \arcsin (a x)^2+\frac {5 x \sqrt {1-a^2 x^2} \arcsin (a x)^3}{24 a^5}+\frac {5 x^3 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{36 a^3}+\frac {x^5 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{9 a}+\frac {1}{6} x^6 \arcsin (a x)^4+\frac {\int x^5 \, dx}{54}-\frac {5 \int \frac {\arcsin (a x)^3}{\sqrt {1-a^2 x^2}} \, dx}{24 a^5}-\frac {5 \int x \arcsin (a x)^2 \, dx}{8 a^4}+\frac {5 \int \frac {x^4 \arcsin (a x)}{\sqrt {1-a^2 x^2}} \, dx}{54 a}+\frac {5 \int \frac {x^4 \arcsin (a x)}{\sqrt {1-a^2 x^2}} \, dx}{24 a} \\ & = \frac {x^6}{324}-\frac {65 x^3 \sqrt {1-a^2 x^2} \arcsin (a x)}{864 a^3}-\frac {x^5 \sqrt {1-a^2 x^2} \arcsin (a x)}{54 a}-\frac {5 x^2 \arcsin (a x)^2}{16 a^4}-\frac {5 x^4 \arcsin (a x)^2}{48 a^2}-\frac {1}{18} x^6 \arcsin (a x)^2+\frac {5 x \sqrt {1-a^2 x^2} \arcsin (a x)^3}{24 a^5}+\frac {5 x^3 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{36 a^3}+\frac {x^5 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{9 a}-\frac {5 \arcsin (a x)^4}{96 a^6}+\frac {1}{6} x^6 \arcsin (a x)^4+\frac {5 \int \frac {x^2 \arcsin (a x)}{\sqrt {1-a^2 x^2}} \, dx}{72 a^3}+\frac {5 \int \frac {x^2 \arcsin (a x)}{\sqrt {1-a^2 x^2}} \, dx}{32 a^3}+\frac {5 \int \frac {x^2 \arcsin (a x)}{\sqrt {1-a^2 x^2}} \, dx}{8 a^3}+\frac {5 \int x^3 \, dx}{216 a^2}+\frac {5 \int x^3 \, dx}{96 a^2} \\ & = \frac {65 x^4}{3456 a^2}+\frac {x^6}{324}-\frac {245 x \sqrt {1-a^2 x^2} \arcsin (a x)}{576 a^5}-\frac {65 x^3 \sqrt {1-a^2 x^2} \arcsin (a x)}{864 a^3}-\frac {x^5 \sqrt {1-a^2 x^2} \arcsin (a x)}{54 a}-\frac {5 x^2 \arcsin (a x)^2}{16 a^4}-\frac {5 x^4 \arcsin (a x)^2}{48 a^2}-\frac {1}{18} x^6 \arcsin (a x)^2+\frac {5 x \sqrt {1-a^2 x^2} \arcsin (a x)^3}{24 a^5}+\frac {5 x^3 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{36 a^3}+\frac {x^5 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{9 a}-\frac {5 \arcsin (a x)^4}{96 a^6}+\frac {1}{6} x^6 \arcsin (a x)^4+\frac {5 \int \frac {\arcsin (a x)}{\sqrt {1-a^2 x^2}} \, dx}{144 a^5}+\frac {5 \int \frac {\arcsin (a x)}{\sqrt {1-a^2 x^2}} \, dx}{64 a^5}+\frac {5 \int \frac {\arcsin (a x)}{\sqrt {1-a^2 x^2}} \, dx}{16 a^5}+\frac {5 \int x \, dx}{144 a^4}+\frac {5 \int x \, dx}{64 a^4}+\frac {5 \int x \, dx}{16 a^4} \\ & = \frac {245 x^2}{1152 a^4}+\frac {65 x^4}{3456 a^2}+\frac {x^6}{324}-\frac {245 x \sqrt {1-a^2 x^2} \arcsin (a x)}{576 a^5}-\frac {65 x^3 \sqrt {1-a^2 x^2} \arcsin (a x)}{864 a^3}-\frac {x^5 \sqrt {1-a^2 x^2} \arcsin (a x)}{54 a}+\frac {245 \arcsin (a x)^2}{1152 a^6}-\frac {5 x^2 \arcsin (a x)^2}{16 a^4}-\frac {5 x^4 \arcsin (a x)^2}{48 a^2}-\frac {1}{18} x^6 \arcsin (a x)^2+\frac {5 x \sqrt {1-a^2 x^2} \arcsin (a x)^3}{24 a^5}+\frac {5 x^3 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{36 a^3}+\frac {x^5 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{9 a}-\frac {5 \arcsin (a x)^4}{96 a^6}+\frac {1}{6} x^6 \arcsin (a x)^4 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.59 \[ \int x^5 \arcsin (a x)^4 \, dx=\frac {a^2 x^2 \left (2205+195 a^2 x^2+32 a^4 x^4\right )-6 a x \sqrt {1-a^2 x^2} \left (735+130 a^2 x^2+32 a^4 x^4\right ) \arcsin (a x)-9 \left (-245+360 a^2 x^2+120 a^4 x^4+64 a^6 x^6\right ) \arcsin (a x)^2+144 a x \sqrt {1-a^2 x^2} \left (15+10 a^2 x^2+8 a^4 x^4\right ) \arcsin (a x)^3+108 \left (-5+16 a^6 x^6\right ) \arcsin (a x)^4}{10368 a^6} \]

[In]

Integrate[x^5*ArcSin[a*x]^4,x]

[Out]

(a^2*x^2*(2205 + 195*a^2*x^2 + 32*a^4*x^4) - 6*a*x*Sqrt[1 - a^2*x^2]*(735 + 130*a^2*x^2 + 32*a^4*x^4)*ArcSin[a
*x] - 9*(-245 + 360*a^2*x^2 + 120*a^4*x^4 + 64*a^6*x^6)*ArcSin[a*x]^2 + 144*a*x*Sqrt[1 - a^2*x^2]*(15 + 10*a^2
*x^2 + 8*a^4*x^4)*ArcSin[a*x]^3 + 108*(-5 + 16*a^6*x^6)*ArcSin[a*x]^4)/(10368*a^6)

Maple [A] (verified)

Time = 0.11 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.22

method	result	size
derivativedivides	\(\frac {\frac {a^{6} x^{6} \arcsin \left (a x \right )^{4}}{6}-\frac {\arcsin \left (a x \right )^{3} \left (-8 \sqrt {-a^{2} x^{2}+1}\, a^{5} x^{5}-10 a^{3} x^{3} \sqrt {-a^{2} x^{2}+1}-15 a x \sqrt {-a^{2} x^{2}+1}+15 \arcsin \left (a x \right )\right )}{72}-\frac {\arcsin \left (a x \right )^{2} a^{6} x^{6}}{18}+\frac {\arcsin \left (a x \right ) \left (-8 \sqrt {-a^{2} x^{2}+1}\, a^{5} x^{5}-10 a^{3} x^{3} \sqrt {-a^{2} x^{2}+1}-15 a x \sqrt {-a^{2} x^{2}+1}+15 \arcsin \left (a x \right )\right )}{432}+\frac {115 \arcsin \left (a x \right )^{2}}{1152}+\frac {\left (a^{2} x^{2}-1\right )^{3}}{324}+\frac {13 \left (a^{2} x^{2}-1\right )^{2}}{864}+\frac {7 a^{2} x^{2}}{36}-\frac {11}{288}-\frac {5 a^{4} x^{4} \arcsin \left (a x \right )^{2}}{48}+\frac {5 \arcsin \left (a x \right ) \left (-2 a^{3} x^{3} \sqrt {-a^{2} x^{2}+1}-3 a x \sqrt {-a^{2} x^{2}+1}+3 \arcsin \left (a x \right )\right )}{192}+\frac {5 \left (2 a^{2} x^{2}+3\right )^{2}}{1536}-\frac {5 \arcsin \left (a x \right )^{2} \left (a^{2} x^{2}-1\right )}{16}-\frac {5 \arcsin \left (a x \right ) \left (a x \sqrt {-a^{2} x^{2}+1}+\arcsin \left (a x \right )\right )}{16}+\frac {5 \arcsin \left (a x \right )^{4}}{32}}{a^{6}}\)	\(345\)
default	\(\frac {\frac {a^{6} x^{6} \arcsin \left (a x \right )^{4}}{6}-\frac {\arcsin \left (a x \right )^{3} \left (-8 \sqrt {-a^{2} x^{2}+1}\, a^{5} x^{5}-10 a^{3} x^{3} \sqrt {-a^{2} x^{2}+1}-15 a x \sqrt {-a^{2} x^{2}+1}+15 \arcsin \left (a x \right )\right )}{72}-\frac {\arcsin \left (a x \right )^{2} a^{6} x^{6}}{18}+\frac {\arcsin \left (a x \right ) \left (-8 \sqrt {-a^{2} x^{2}+1}\, a^{5} x^{5}-10 a^{3} x^{3} \sqrt {-a^{2} x^{2}+1}-15 a x \sqrt {-a^{2} x^{2}+1}+15 \arcsin \left (a x \right )\right )}{432}+\frac {115 \arcsin \left (a x \right )^{2}}{1152}+\frac {\left (a^{2} x^{2}-1\right )^{3}}{324}+\frac {13 \left (a^{2} x^{2}-1\right )^{2}}{864}+\frac {7 a^{2} x^{2}}{36}-\frac {11}{288}-\frac {5 a^{4} x^{4} \arcsin \left (a x \right )^{2}}{48}+\frac {5 \arcsin \left (a x \right ) \left (-2 a^{3} x^{3} \sqrt {-a^{2} x^{2}+1}-3 a x \sqrt {-a^{2} x^{2}+1}+3 \arcsin \left (a x \right )\right )}{192}+\frac {5 \left (2 a^{2} x^{2}+3\right )^{2}}{1536}-\frac {5 \arcsin \left (a x \right )^{2} \left (a^{2} x^{2}-1\right )}{16}-\frac {5 \arcsin \left (a x \right ) \left (a x \sqrt {-a^{2} x^{2}+1}+\arcsin \left (a x \right )\right )}{16}+\frac {5 \arcsin \left (a x \right )^{4}}{32}}{a^{6}}\)	\(345\)

[In]

int(x^5*arcsin(a*x)^4,x,method=_RETURNVERBOSE)

[Out]

1/a^6*(1/6*a^6*x^6*arcsin(a*x)^4-1/72*arcsin(a*x)^3*(-8*(-a^2*x^2+1)^(1/2)*a^5*x^5-10*a^3*x^3*(-a^2*x^2+1)^(1/
2)-15*a*x*(-a^2*x^2+1)^(1/2)+15*arcsin(a*x))-1/18*arcsin(a*x)^2*a^6*x^6+1/432*arcsin(a*x)*(-8*(-a^2*x^2+1)^(1/
2)*a^5*x^5-10*a^3*x^3*(-a^2*x^2+1)^(1/2)-15*a*x*(-a^2*x^2+1)^(1/2)+15*arcsin(a*x))+115/1152*arcsin(a*x)^2+1/32
4*(a^2*x^2-1)^3+13/864*(a^2*x^2-1)^2+7/36*a^2*x^2-11/288-5/48*a^4*x^4*arcsin(a*x)^2+5/192*arcsin(a*x)*(-2*a^3*
x^3*(-a^2*x^2+1)^(1/2)-3*a*x*(-a^2*x^2+1)^(1/2)+3*arcsin(a*x))+5/1536*(2*a^2*x^2+3)^2-5/16*arcsin(a*x)^2*(a^2*
x^2-1)-5/16*arcsin(a*x)*(a*x*(-a^2*x^2+1)^(1/2)+arcsin(a*x))+5/32*arcsin(a*x)^4)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.54 \[ \int x^5 \arcsin (a x)^4 \, dx=\frac {32 \, a^{6} x^{6} + 195 \, a^{4} x^{4} + 108 \, {\left (16 \, a^{6} x^{6} - 5\right )} \arcsin \left (a x\right )^{4} + 2205 \, a^{2} x^{2} - 9 \, {\left (64 \, a^{6} x^{6} + 120 \, a^{4} x^{4} + 360 \, a^{2} x^{2} - 245\right )} \arcsin \left (a x\right )^{2} + 6 \, \sqrt {-a^{2} x^{2} + 1} {\left (24 \, {\left (8 \, a^{5} x^{5} + 10 \, a^{3} x^{3} + 15 \, a x\right )} \arcsin \left (a x\right )^{3} - {\left (32 \, a^{5} x^{5} + 130 \, a^{3} x^{3} + 735 \, a x\right )} \arcsin \left (a x\right )\right )}}{10368 \, a^{6}} \]

[In]

integrate(x^5*arcsin(a*x)^4,x, algorithm="fricas")

[Out]

1/10368*(32*a^6*x^6 + 195*a^4*x^4 + 108*(16*a^6*x^6 - 5)*arcsin(a*x)^4 + 2205*a^2*x^2 - 9*(64*a^6*x^6 + 120*a^
4*x^4 + 360*a^2*x^2 - 245)*arcsin(a*x)^2 + 6*sqrt(-a^2*x^2 + 1)*(24*(8*a^5*x^5 + 10*a^3*x^3 + 15*a*x)*arcsin(a
*x)^3 - (32*a^5*x^5 + 130*a^3*x^3 + 735*a*x)*arcsin(a*x)))/a^6

Sympy [A] (verification not implemented)

Time = 1.04 (sec) , antiderivative size = 269, normalized size of antiderivative = 0.95 \[ \int x^5 \arcsin (a x)^4 \, dx=\begin {cases} \frac {x^{6} \operatorname {asin}^{4}{\left (a x \right )}}{6} - \frac {x^{6} \operatorname {asin}^{2}{\left (a x \right )}}{18} + \frac {x^{6}}{324} + \frac {x^{5} \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{3}{\left (a x \right )}}{9 a} - \frac {x^{5} \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}{\left (a x \right )}}{54 a} - \frac {5 x^{4} \operatorname {asin}^{2}{\left (a x \right )}}{48 a^{2}} + \frac {65 x^{4}}{3456 a^{2}} + \frac {5 x^{3} \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{3}{\left (a x \right )}}{36 a^{3}} - \frac {65 x^{3} \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}{\left (a x \right )}}{864 a^{3}} - \frac {5 x^{2} \operatorname {asin}^{2}{\left (a x \right )}}{16 a^{4}} + \frac {245 x^{2}}{1152 a^{4}} + \frac {5 x \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{3}{\left (a x \right )}}{24 a^{5}} - \frac {245 x \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}{\left (a x \right )}}{576 a^{5}} - \frac {5 \operatorname {asin}^{4}{\left (a x \right )}}{96 a^{6}} + \frac {245 \operatorname {asin}^{2}{\left (a x \right )}}{1152 a^{6}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]

[In]

integrate(x**5*asin(a*x)**4,x)

[Out]

Piecewise((x**6*asin(a*x)**4/6 - x**6*asin(a*x)**2/18 + x**6/324 + x**5*sqrt(-a**2*x**2 + 1)*asin(a*x)**3/(9*a
) - x**5*sqrt(-a**2*x**2 + 1)*asin(a*x)/(54*a) - 5*x**4*asin(a*x)**2/(48*a**2) + 65*x**4/(3456*a**2) + 5*x**3*
sqrt(-a**2*x**2 + 1)*asin(a*x)**3/(36*a**3) - 65*x**3*sqrt(-a**2*x**2 + 1)*asin(a*x)/(864*a**3) - 5*x**2*asin(
a*x)**2/(16*a**4) + 245*x**2/(1152*a**4) + 5*x*sqrt(-a**2*x**2 + 1)*asin(a*x)**3/(24*a**5) - 245*x*sqrt(-a**2*
x**2 + 1)*asin(a*x)/(576*a**5) - 5*asin(a*x)**4/(96*a**6) + 245*asin(a*x)**2/(1152*a**6), Ne(a, 0)), (0, True)
)

Maxima [F]

\[ \int x^5 \arcsin (a x)^4 \, dx=\int { x^{5} \arcsin \left (a x\right )^{4} \,d x } \]

[In]

integrate(x^5*arcsin(a*x)^4,x, algorithm="maxima")

[Out]

1/6*x^6*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))^4 + 2*a*integrate(1/3*sqrt(a*x + 1)*sqrt(-a*x + 1)*x^6*arct
an2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))^3/(a^2*x^2 - 1), x)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.28 \[ \int x^5 \arcsin (a x)^4 \, dx=\frac {{\left (a^{2} x^{2} - 1\right )}^{2} \sqrt {-a^{2} x^{2} + 1} x \arcsin \left (a x\right )^{3}}{9 \, a^{5}} + \frac {{\left (a^{2} x^{2} - 1\right )}^{3} \arcsin \left (a x\right )^{4}}{6 \, a^{6}} - \frac {13 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x \arcsin \left (a x\right )^{3}}{36 \, a^{5}} + \frac {{\left (a^{2} x^{2} - 1\right )}^{2} \arcsin \left (a x\right )^{4}}{2 \, a^{6}} - \frac {{\left (a^{2} x^{2} - 1\right )}^{2} \sqrt {-a^{2} x^{2} + 1} x \arcsin \left (a x\right )}{54 \, a^{5}} + \frac {11 \, \sqrt {-a^{2} x^{2} + 1} x \arcsin \left (a x\right )^{3}}{24 \, a^{5}} - \frac {{\left (a^{2} x^{2} - 1\right )}^{3} \arcsin \left (a x\right )^{2}}{18 \, a^{6}} + \frac {{\left (a^{2} x^{2} - 1\right )} \arcsin \left (a x\right )^{4}}{2 \, a^{6}} + \frac {97 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x \arcsin \left (a x\right )}{864 \, a^{5}} - \frac {13 \, {\left (a^{2} x^{2} - 1\right )}^{2} \arcsin \left (a x\right )^{2}}{48 \, a^{6}} + \frac {11 \, \arcsin \left (a x\right )^{4}}{96 \, a^{6}} - \frac {299 \, \sqrt {-a^{2} x^{2} + 1} x \arcsin \left (a x\right )}{576 \, a^{5}} + \frac {{\left (a^{2} x^{2} - 1\right )}^{3}}{324 \, a^{6}} - \frac {11 \, {\left (a^{2} x^{2} - 1\right )} \arcsin \left (a x\right )^{2}}{16 \, a^{6}} + \frac {97 \, {\left (a^{2} x^{2} - 1\right )}^{2}}{3456 \, a^{6}} - \frac {299 \, \arcsin \left (a x\right )^{2}}{1152 \, a^{6}} + \frac {299 \, {\left (a^{2} x^{2} - 1\right )}}{1152 \, a^{6}} + \frac {9971}{82944 \, a^{6}} \]

[In]

integrate(x^5*arcsin(a*x)^4,x, algorithm="giac")

[Out]

1/9*(a^2*x^2 - 1)^2*sqrt(-a^2*x^2 + 1)*x*arcsin(a*x)^3/a^5 + 1/6*(a^2*x^2 - 1)^3*arcsin(a*x)^4/a^6 - 13/36*(-a
^2*x^2 + 1)^(3/2)*x*arcsin(a*x)^3/a^5 + 1/2*(a^2*x^2 - 1)^2*arcsin(a*x)^4/a^6 - 1/54*(a^2*x^2 - 1)^2*sqrt(-a^2
*x^2 + 1)*x*arcsin(a*x)/a^5 + 11/24*sqrt(-a^2*x^2 + 1)*x*arcsin(a*x)^3/a^5 - 1/18*(a^2*x^2 - 1)^3*arcsin(a*x)^
2/a^6 + 1/2*(a^2*x^2 - 1)*arcsin(a*x)^4/a^6 + 97/864*(-a^2*x^2 + 1)^(3/2)*x*arcsin(a*x)/a^5 - 13/48*(a^2*x^2 -
1)^2*arcsin(a*x)^2/a^6 + 11/96*arcsin(a*x)^4/a^6 - 299/576*sqrt(-a^2*x^2 + 1)*x*arcsin(a*x)/a^5 + 1/324*(a^2*
x^2 - 1)^3/a^6 - 11/16*(a^2*x^2 - 1)*arcsin(a*x)^2/a^6 + 97/3456*(a^2*x^2 - 1)^2/a^6 - 299/1152*arcsin(a*x)^2/
a^6 + 299/1152*(a^2*x^2 - 1)/a^6 + 9971/82944/a^6

Mupad [F(-1)]

Timed out. \[ \int x^5 \arcsin (a x)^4 \, dx=\int x^5\,{\mathrm {asin}\left (a\,x\right )}^4 \,d x \]

[In]

int(x^5*asin(a*x)^4,x)

[Out]

int(x^5*asin(a*x)^4, x)