\(\int x^4 \text {arcsinh}(a x) \, dx\) [1]

Optimal result

Integrand size = 8, antiderivative size = 72 \[ \int x^4 \text {arcsinh}(a x) \, dx=-\frac {\sqrt {1+a^2 x^2}}{5 a^5}+\frac {2 \left (1+a^2 x^2\right )^{3/2}}{15 a^5}-\frac {\left (1+a^2 x^2\right )^{5/2}}{25 a^5}+\frac {1}{5} x^5 \text {arcsinh}(a x) \]

[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*arcsinh(a*x)-1/5*(a^2*x^2+1)^(1/2)/a^5

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 72, 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 = {5776, 272, 45} \[ \int x^4 \text {arcsinh}(a x) \, dx=-\frac {\left (a^2 x^2+1\right )^{5/2}}{25 a^5}+\frac {2 \left (a^2 x^2+1\right )^{3/2}}{15 a^5}-\frac {\sqrt {a^2 x^2+1}}{5 a^5}+\frac {1}{5} x^5 \text {arcsinh}(a x) \]

[In]

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

[Out]

-1/5*Sqrt[1 + a^2*x^2]/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*ArcSinh[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 5776

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcS
inh[c*x])^n/(d*(m + 1))), x] - Dist[b*c*(n/(d*(m + 1))), Int[(d*x)^(m + 1)*((a + b*ArcSinh[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 \text {arcsinh}(a x)-\frac {1}{5} a \int \frac {x^5}{\sqrt {1+a^2 x^2}} \, dx \\ & = \frac {1}{5} x^5 \text {arcsinh}(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 \text {arcsinh}(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 \text {arcsinh}(a x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.69 \[ \int x^4 \text {arcsinh}(a x) \, dx=-\frac {\sqrt {1+a^2 x^2} \left (8-4 a^2 x^2+3 a^4 x^4\right )}{75 a^5}+\frac {1}{5} x^5 \text {arcsinh}(a x) \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.19 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.96

method	result	size
derivativedivides	\(\frac {\frac {a^{5} x^{5} \operatorname {arcsinh}\left (a x \right )}{5}-\frac {a^{4} x^{4} \sqrt {a^{2} x^{2}+1}}{25}+\frac {4 a^{2} x^{2} \sqrt {a^{2} x^{2}+1}}{75}-\frac {8 \sqrt {a^{2} x^{2}+1}}{75}}{a^{5}}\)	\(69\)
default	\(\frac {\frac {a^{5} x^{5} \operatorname {arcsinh}\left (a x \right )}{5}-\frac {a^{4} x^{4} \sqrt {a^{2} x^{2}+1}}{25}+\frac {4 a^{2} x^{2} \sqrt {a^{2} x^{2}+1}}{75}-\frac {8 \sqrt {a^{2} x^{2}+1}}{75}}{a^{5}}\)	\(69\)
parts	\(\frac {x^{5} \operatorname {arcsinh}\left (a x \right )}{5}-\frac {a \left (\frac {x^{4} \sqrt {a^{2} x^{2}+1}}{5 a^{2}}-\frac {4 \left (\frac {x^{2} \sqrt {a^{2} x^{2}+1}}{3 a^{2}}-\frac {2 \sqrt {a^{2} x^{2}+1}}{3 a^{4}}\right )}{5 a^{2}}\right )}{5}\)	\(75\)

[In]

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

[Out]

1/a^5*(1/5*a^5*x^5*arcsinh(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 = 0.26 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.85 \[ \int x^4 \text {arcsinh}(a x) \, dx=\frac {15 \, a^{5} x^{5} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) - {\left (3 \, a^{4} x^{4} - 4 \, a^{2} x^{2} + 8\right )} \sqrt {a^{2} x^{2} + 1}}{75 \, a^{5}} \]

[In]

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

[Out]

1/75*(15*a^5*x^5*log(a*x + sqrt(a^2*x^2 + 1)) - (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 = 0.37 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.97 \[ \int x^4 \text {arcsinh}(a x) \, dx=\begin {cases} \frac {x^{5} \operatorname {asinh}{\left (a x \right )}}{5} - \frac {x^{4} \sqrt {a^{2} x^{2} + 1}}{25 a} + \frac {4 x^{2} \sqrt {a^{2} x^{2} + 1}}{75 a^{3}} - \frac {8 \sqrt {a^{2} x^{2} + 1}}{75 a^{5}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((x**5*asinh(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*sqrt
(a**2*x**2 + 1)/(75*a**5), Ne(a, 0)), (0, True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.94 \[ \int x^4 \text {arcsinh}(a x) \, dx=\frac {1}{5} \, x^{5} \operatorname {arsinh}\left (a x\right ) - \frac {1}{75} \, {\left (\frac {3 \, \sqrt {a^{2} x^{2} + 1} x^{4}}{a^{2}} - \frac {4 \, \sqrt {a^{2} x^{2} + 1} x^{2}}{a^{4}} + \frac {8 \, \sqrt {a^{2} x^{2} + 1}}{a^{6}}\right )} a \]

[In]

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

[Out]

1/5*x^5*arcsinh(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 [F(-2)]

Exception generated. \[ \int x^4 \text {arcsinh}(a x) \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

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

[Out]

int(x^4*asinh(a*x), x)