\(\int x^4 \arcsin (a x) \, dx\) [1]

Optimal result

Integrand size = 8, antiderivative size = 75 \[ \int x^4 \arcsin (a x) \, dx=\frac {\sqrt {1-a^2 x^2}}{5 a^5}-\frac {2 \left (1-a^2 x^2\right )^{3/2}}{15 a^5}+\frac {\left (1-a^2 x^2\right )^{5/2}}{25 a^5}+\frac {1}{5} x^5 \arcsin (a x) \]

[Out]

-2/15*(-a^2*x^2+1)^(3/2)/a^5+1/25*(-a^2*x^2+1)^(5/2)/a^5+1/5*x^5*arcsin(a*x)+1/5*(-a^2*x^2+1)^(1/2)/a^5

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4723, 272, 45} \[ \int x^4 \arcsin (a x) \, dx=\frac {\left (1-a^2 x^2\right )^{5/2}}{25 a^5}-\frac {2 \left (1-a^2 x^2\right )^{3/2}}{15 a^5}+\frac {\sqrt {1-a^2 x^2}}{5 a^5}+\frac {1}{5} x^5 \arcsin (a x) \]

[In]

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

[Out]

Sqrt[1 - a^2*x^2]/(5*a^5) - (2*(1 - a^2*x^2)^(3/2))/(15*a^5) + (1 - a^2*x^2)^(5/2)/(25*a^5) + (x^5*ArcSin[a*x]
)/5

Rule 45

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] && NeQ[b*c - a*d, 0] && 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 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

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]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} x^5 \arcsin (a x)-\frac {1}{5} a \int \frac {x^5}{\sqrt {1-a^2 x^2}} \, dx \\ & = \frac {1}{5} x^5 \arcsin (a x)-\frac {1}{10} a \text {Subst}\left (\int \frac {x^2}{\sqrt {1-a^2 x}} \, dx,x,x^2\right ) \\ & = \frac {1}{5} x^5 \arcsin (a x)-\frac {1}{10} a \text {Subst}\left (\int \left (\frac {1}{a^4 \sqrt {1-a^2 x}}-\frac {2 \sqrt {1-a^2 x}}{a^4}+\frac {\left (1-a^2 x\right )^{3/2}}{a^4}\right ) \, dx,x,x^2\right ) \\ & = \frac {\sqrt {1-a^2 x^2}}{5 a^5}-\frac {2 \left (1-a^2 x^2\right )^{3/2}}{15 a^5}+\frac {\left (1-a^2 x^2\right )^{5/2}}{25 a^5}+\frac {1}{5} x^5 \arcsin (a x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.68 \[ \int x^4 \arcsin (a x) \, dx=\frac {\sqrt {1-a^2 x^2} \left (8+4 a^2 x^2+3 a^4 x^4\right )}{75 a^5}+\frac {1}{5} x^5 \arcsin (a x) \]

[In]

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

[Out]

(Sqrt[1 - a^2*x^2]*(8 + 4*a^2*x^2 + 3*a^4*x^4))/(75*a^5) + (x^5*ArcSin[a*x])/5

Maple [A] (verified)

Time = 0.56 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.96

method	result	size
derivativedivides	\(\frac {\frac {a^{5} x^{5} \arcsin \left (a x \right )}{5}+\frac {a^{4} x^{4} \sqrt {-a^{2} x^{2}+1}}{25}+\frac {4 a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}}{75}+\frac {8 \sqrt {-a^{2} x^{2}+1}}{75}}{a^{5}}\)	\(72\)
default	\(\frac {\frac {a^{5} x^{5} \arcsin \left (a x \right )}{5}+\frac {a^{4} x^{4} \sqrt {-a^{2} x^{2}+1}}{25}+\frac {4 a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}}{75}+\frac {8 \sqrt {-a^{2} x^{2}+1}}{75}}{a^{5}}\)	\(72\)
parts	\(\frac {x^{5} \arcsin \left (a x \right )}{5}-\frac {a \left (-\frac {x^{4} \sqrt {-a^{2} x^{2}+1}}{5 a^{2}}+\frac {-\frac {4 x^{2} \sqrt {-a^{2} x^{2}+1}}{15 a^{2}}-\frac {8 \sqrt {-a^{2} x^{2}+1}}{15 a^{4}}}{a^{2}}\right )}{5}\)	\(78\)

[In]

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

[Out]

1/a^5*(1/5*a^5*x^5*arcsin(a*x)+1/25*a^4*x^4*(-a^2*x^2+1)^(1/2)+4/75*a^2*x^2*(-a^2*x^2+1)^(1/2)+8/75*(-a^2*x^2+
1)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.65 \[ \int x^4 \arcsin (a x) \, dx=\frac {15 \, a^{5} x^{5} \arcsin \left (a x\right ) + {\left (3 \, a^{4} x^{4} + 4 \, a^{2} x^{2} + 8\right )} \sqrt {-a^{2} x^{2} + 1}}{75 \, a^{5}} \]

[In]

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

[Out]

1/75*(15*a^5*x^5*arcsin(a*x) + (3*a^4*x^4 + 4*a^2*x^2 + 8)*sqrt(-a^2*x^2 + 1))/a^5

Sympy [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.93 \[ \int x^4 \arcsin (a x) \, dx=\begin {cases} \frac {x^{5} \operatorname {asin}{\left (a x \right )}}{5} + \frac {x^{4} \sqrt {- a^{2} x^{2} + 1}}{25 a} + \frac {4 x^{2} \sqrt {- a^{2} x^{2} + 1}}{75 a^{3}} + \frac {8 \sqrt {- a^{2} x^{2} + 1}}{75 a^{5}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((x**5*asin(a*x)/5 + x**4*sqrt(-a**2*x**2 + 1)/(25*a) + 4*x**2*sqrt(-a**2*x**2 + 1)/(75*a**3) + 8*sqr
t(-a**2*x**2 + 1)/(75*a**5), Ne(a, 0)), (0, True))

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.95 \[ \int x^4 \arcsin (a x) \, dx=\frac {1}{5} \, x^{5} \arcsin \left (a x\right ) + \frac {1}{75} \, {\left (\frac {3 \, \sqrt {-a^{2} x^{2} + 1} x^{4}}{a^{2}} + \frac {4 \, \sqrt {-a^{2} x^{2} + 1} x^{2}}{a^{4}} + \frac {8 \, \sqrt {-a^{2} x^{2} + 1}}{a^{6}}\right )} a \]

[In]

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

[Out]

1/5*x^5*arcsin(a*x) + 1/75*(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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.51 \[ \int x^4 \arcsin (a x) \, dx=\frac {{\left (a^{2} x^{2} - 1\right )}^{2} x \arcsin \left (a x\right )}{5 \, a^{4}} + \frac {2 \, {\left (a^{2} x^{2} - 1\right )} x \arcsin \left (a x\right )}{5 \, a^{4}} + \frac {x \arcsin \left (a x\right )}{5 \, a^{4}} + \frac {{\left (a^{2} x^{2} - 1\right )}^{2} \sqrt {-a^{2} x^{2} + 1}}{25 \, a^{5}} - \frac {2 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{15 \, a^{5}} + \frac {\sqrt {-a^{2} x^{2} + 1}}{5 \, a^{5}} \]

[In]

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

[Out]

1/5*(a^2*x^2 - 1)^2*x*arcsin(a*x)/a^4 + 2/5*(a^2*x^2 - 1)*x*arcsin(a*x)/a^4 + 1/5*x*arcsin(a*x)/a^4 + 1/25*(a^
2*x^2 - 1)^2*sqrt(-a^2*x^2 + 1)/a^5 - 2/15*(-a^2*x^2 + 1)^(3/2)/a^5 + 1/5*sqrt(-a^2*x^2 + 1)/a^5

Mupad [F(-1)]

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

[In]

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

[Out]

int(x^4*asin(a*x), x)