\(\int x^4 \text {arcsinh}(a x)^{5/2} \, dx\) [86]

Optimal result

Integrand size = 12, antiderivative size = 379 \[ \int x^4 \text {arcsinh}(a x)^{5/2} \, dx=\frac {2 x \sqrt {\text {arcsinh}(a x)}}{5 a^4}-\frac {x^3 \sqrt {\text {arcsinh}(a x)}}{15 a^2}+\frac {3}{100} x^5 \sqrt {\text {arcsinh}(a x)}-\frac {4 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{15 a^5}+\frac {2 x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{15 a^3}-\frac {x^4 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{10 a}+\frac {1}{5} x^5 \text {arcsinh}(a x)^{5/2}+\frac {15 \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )}{128 a^5}-\frac {\sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{240 a^5}-\frac {\sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{1280 a^5}+\frac {3 \sqrt {\frac {\pi }{5}} \text {erf}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )}{6400 a^5}-\frac {15 \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{128 a^5}+\frac {\sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{240 a^5}+\frac {\sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{1280 a^5}-\frac {3 \sqrt {\frac {\pi }{5}} \text {erfi}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )}{6400 a^5} \]

[Out]

1/5*x^5*arcsinh(a*x)^(5/2)+3/32000*erf(5^(1/2)*arcsinh(a*x)^(1/2))*5^(1/2)*Pi^(1/2)/a^5-3/32000*erfi(5^(1/2)*a
rcsinh(a*x)^(1/2))*5^(1/2)*Pi^(1/2)/a^5-5/2304*erf(3^(1/2)*arcsinh(a*x)^(1/2))*3^(1/2)*Pi^(1/2)/a^5+5/2304*erf
i(3^(1/2)*arcsinh(a*x)^(1/2))*3^(1/2)*Pi^(1/2)/a^5+15/128*erf(arcsinh(a*x)^(1/2))*Pi^(1/2)/a^5-15/128*erfi(arc
sinh(a*x)^(1/2))*Pi^(1/2)/a^5-4/15*arcsinh(a*x)^(3/2)*(a^2*x^2+1)^(1/2)/a^5+2/15*x^2*arcsinh(a*x)^(3/2)*(a^2*x
^2+1)^(1/2)/a^3-1/10*x^4*arcsinh(a*x)^(3/2)*(a^2*x^2+1)^(1/2)/a+2/5*x*arcsinh(a*x)^(1/2)/a^4-1/15*x^3*arcsinh(
a*x)^(1/2)/a^2+3/100*x^5*arcsinh(a*x)^(1/2)

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 379, normalized size of antiderivative = 1.00, number of steps used = 44, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {5777, 5812, 5798, 5772, 5819, 3389, 2211, 2235, 2236, 3393} \[ \int x^4 \text {arcsinh}(a x)^{5/2} \, dx=\frac {15 \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )}{128 a^5}-\frac {\sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{1280 a^5}-\frac {\sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{240 a^5}+\frac {3 \sqrt {\frac {\pi }{5}} \text {erf}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )}{6400 a^5}-\frac {15 \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{128 a^5}+\frac {\sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{1280 a^5}+\frac {\sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{240 a^5}-\frac {3 \sqrt {\frac {\pi }{5}} \text {erfi}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )}{6400 a^5}+\frac {2 x \sqrt {\text {arcsinh}(a x)}}{5 a^4}-\frac {x^3 \sqrt {\text {arcsinh}(a x)}}{15 a^2}-\frac {x^4 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{10 a}-\frac {4 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{15 a^5}+\frac {2 x^2 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{15 a^3}+\frac {1}{5} x^5 \text {arcsinh}(a x)^{5/2}+\frac {3}{100} x^5 \sqrt {\text {arcsinh}(a x)} \]

[In]

Int[x^4*ArcSinh[a*x]^(5/2),x]

[Out]

(2*x*Sqrt[ArcSinh[a*x]])/(5*a^4) - (x^3*Sqrt[ArcSinh[a*x]])/(15*a^2) + (3*x^5*Sqrt[ArcSinh[a*x]])/100 - (4*Sqr
t[1 + a^2*x^2]*ArcSinh[a*x]^(3/2))/(15*a^5) + (2*x^2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^(3/2))/(15*a^3) - (x^4*Sqr
t[1 + a^2*x^2]*ArcSinh[a*x]^(3/2))/(10*a) + (x^5*ArcSinh[a*x]^(5/2))/5 + (15*Sqrt[Pi]*Erf[Sqrt[ArcSinh[a*x]]])
/(128*a^5) - (Sqrt[Pi/3]*Erf[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/(240*a^5) - (Sqrt[3*Pi]*Erf[Sqrt[3]*Sqrt[ArcSinh[a*x
]]])/(1280*a^5) + (3*Sqrt[Pi/5]*Erf[Sqrt[5]*Sqrt[ArcSinh[a*x]]])/(6400*a^5) - (15*Sqrt[Pi]*Erfi[Sqrt[ArcSinh[a
*x]]])/(128*a^5) + (Sqrt[Pi/3]*Erfi[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/(240*a^5) + (Sqrt[3*Pi]*Erfi[Sqrt[3]*Sqrt[Arc
Sinh[a*x]]])/(1280*a^5) - (3*Sqrt[Pi/5]*Erfi[Sqrt[5]*Sqrt[ArcSinh[a*x]]])/(6400*a^5)

Rule 2211

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[F^(g*(e - c*(
f/d)) + f*g*(x^2/d)), x], x, Sqrt[c + d*x]], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]

Rule 2235

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt[Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2
]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 2236

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt[Pi]*(Erf[(c + d*x)*Rt[(-b)*Log[F],
2]]/(2*d*Rt[(-b)*Log[F], 2])), x] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rule 3389

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Dist[I/2, Int[(c + d*x)^m/E^(I*(e + f*x))
, x], x] - Dist[I/2, Int[(c + d*x)^m*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e, f, m}, x]

Rule 3393

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

Rule 5772

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcSinh[c*x])^n, x] - Dist[b*c*n, In
t[x*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 5777

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcSinh[c*x])^n/(
m + 1)), x] - Dist[b*c*(n/(m + 1)), Int[x^(m + 1)*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + c^2*x^2]), x], x] /;
FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GtQ[n, 0]

Rule 5798

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x^2)^
(p + 1)*((a + b*ArcSinh[c*x])^n/(2*e*(p + 1))), x] - Dist[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)
^p], Int[(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e
, c^2*d] && GtQ[n, 0] && NeQ[p, -1]

Rule 5812

Int[((a_.) + ArcSinh[(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*ArcSinh[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*ArcSinh[c*x])^n, x], x] - Dist[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*ArcSinh[c*x])^(n - 1)
, x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && IGtQ[m, 1] && NeQ[m + 2*p + 1, 0
]

Rule 5819

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[(1/(b*
c^(m + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p], Subst[Int[x^n*Sinh[-a/b + x/b]^m*Cosh[-a/b + x/b]^(2*p + 1),
x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c^2*d] && IGtQ[2*p + 2, 0] && IGtQ[m,
0]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} x^5 \text {arcsinh}(a x)^{5/2}-\frac {1}{2} a \int \frac {x^5 \text {arcsinh}(a x)^{3/2}}{\sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {x^4 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{10 a}+\frac {1}{5} x^5 \text {arcsinh}(a x)^{5/2}+\frac {3}{20} \int x^4 \sqrt {\text {arcsinh}(a x)} \, dx+\frac {2 \int \frac {x^3 \text {arcsinh}(a x)^{3/2}}{\sqrt {1+a^2 x^2}} \, dx}{5 a} \\ & = \frac {3}{100} x^5 \sqrt {\text {arcsinh}(a x)}+\frac {2 x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{15 a^3}-\frac {x^4 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{10 a}+\frac {1}{5} x^5 \text {arcsinh}(a x)^{5/2}-\frac {4 \int \frac {x \text {arcsinh}(a x)^{3/2}}{\sqrt {1+a^2 x^2}} \, dx}{15 a^3}-\frac {\int x^2 \sqrt {\text {arcsinh}(a x)} \, dx}{5 a^2}-\frac {1}{200} (3 a) \int \frac {x^5}{\sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}} \, dx \\ & = -\frac {x^3 \sqrt {\text {arcsinh}(a x)}}{15 a^2}+\frac {3}{100} x^5 \sqrt {\text {arcsinh}(a x)}-\frac {4 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{15 a^5}+\frac {2 x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{15 a^3}-\frac {x^4 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{10 a}+\frac {1}{5} x^5 \text {arcsinh}(a x)^{5/2}-\frac {3 \text {Subst}\left (\int \frac {\sinh ^5(x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{200 a^5}+\frac {2 \int \sqrt {\text {arcsinh}(a x)} \, dx}{5 a^4}+\frac {\int \frac {x^3}{\sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}} \, dx}{30 a} \\ & = \frac {2 x \sqrt {\text {arcsinh}(a x)}}{5 a^4}-\frac {x^3 \sqrt {\text {arcsinh}(a x)}}{15 a^2}+\frac {3}{100} x^5 \sqrt {\text {arcsinh}(a x)}-\frac {4 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{15 a^5}+\frac {2 x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{15 a^3}-\frac {x^4 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{10 a}+\frac {1}{5} x^5 \text {arcsinh}(a x)^{5/2}+\frac {(3 i) \text {Subst}\left (\int \left (\frac {5 i \sinh (x)}{8 \sqrt {x}}-\frac {5 i \sinh (3 x)}{16 \sqrt {x}}+\frac {i \sinh (5 x)}{16 \sqrt {x}}\right ) \, dx,x,\text {arcsinh}(a x)\right )}{200 a^5}+\frac {\text {Subst}\left (\int \frac {\sinh ^3(x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{30 a^5}-\frac {\int \frac {x}{\sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}} \, dx}{5 a^3} \\ & = \frac {2 x \sqrt {\text {arcsinh}(a x)}}{5 a^4}-\frac {x^3 \sqrt {\text {arcsinh}(a x)}}{15 a^2}+\frac {3}{100} x^5 \sqrt {\text {arcsinh}(a x)}-\frac {4 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{15 a^5}+\frac {2 x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{15 a^3}-\frac {x^4 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{10 a}+\frac {1}{5} x^5 \text {arcsinh}(a x)^{5/2}+\frac {i \text {Subst}\left (\int \left (\frac {3 i \sinh (x)}{4 \sqrt {x}}-\frac {i \sinh (3 x)}{4 \sqrt {x}}\right ) \, dx,x,\text {arcsinh}(a x)\right )}{30 a^5}-\frac {3 \text {Subst}\left (\int \frac {\sinh (5 x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{3200 a^5}+\frac {3 \text {Subst}\left (\int \frac {\sinh (3 x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{640 a^5}-\frac {3 \text {Subst}\left (\int \frac {\sinh (x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{320 a^5}-\frac {\text {Subst}\left (\int \frac {\sinh (x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{5 a^5} \\ & = \frac {2 x \sqrt {\text {arcsinh}(a x)}}{5 a^4}-\frac {x^3 \sqrt {\text {arcsinh}(a x)}}{15 a^2}+\frac {3}{100} x^5 \sqrt {\text {arcsinh}(a x)}-\frac {4 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{15 a^5}+\frac {2 x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{15 a^3}-\frac {x^4 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{10 a}+\frac {1}{5} x^5 \text {arcsinh}(a x)^{5/2}+\frac {3 \text {Subst}\left (\int \frac {e^{-5 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{6400 a^5}-\frac {3 \text {Subst}\left (\int \frac {e^{5 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{6400 a^5}-\frac {3 \text {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{1280 a^5}+\frac {3 \text {Subst}\left (\int \frac {e^{3 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{1280 a^5}+\frac {3 \text {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{640 a^5}-\frac {3 \text {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{640 a^5}+\frac {\text {Subst}\left (\int \frac {\sinh (3 x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{120 a^5}-\frac {\text {Subst}\left (\int \frac {\sinh (x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{40 a^5}+\frac {\text {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{10 a^5}-\frac {\text {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{10 a^5} \\ & = \frac {2 x \sqrt {\text {arcsinh}(a x)}}{5 a^4}-\frac {x^3 \sqrt {\text {arcsinh}(a x)}}{15 a^2}+\frac {3}{100} x^5 \sqrt {\text {arcsinh}(a x)}-\frac {4 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{15 a^5}+\frac {2 x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{15 a^3}-\frac {x^4 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{10 a}+\frac {1}{5} x^5 \text {arcsinh}(a x)^{5/2}+\frac {3 \text {Subst}\left (\int e^{-5 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{3200 a^5}-\frac {3 \text {Subst}\left (\int e^{5 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{3200 a^5}-\frac {\text {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{240 a^5}+\frac {\text {Subst}\left (\int \frac {e^{3 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{240 a^5}-\frac {3 \text {Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{640 a^5}+\frac {3 \text {Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{640 a^5}+\frac {3 \text {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{320 a^5}-\frac {3 \text {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{320 a^5}+\frac {\text {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{80 a^5}-\frac {\text {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{80 a^5}+\frac {\text {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{5 a^5}-\frac {\text {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{5 a^5} \\ & = \frac {2 x \sqrt {\text {arcsinh}(a x)}}{5 a^4}-\frac {x^3 \sqrt {\text {arcsinh}(a x)}}{15 a^2}+\frac {3}{100} x^5 \sqrt {\text {arcsinh}(a x)}-\frac {4 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{15 a^5}+\frac {2 x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{15 a^3}-\frac {x^4 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{10 a}+\frac {1}{5} x^5 \text {arcsinh}(a x)^{5/2}+\frac {67 \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )}{640 a^5}-\frac {\sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{1280 a^5}+\frac {3 \sqrt {\frac {\pi }{5}} \text {erf}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )}{6400 a^5}-\frac {67 \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{640 a^5}+\frac {\sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{1280 a^5}-\frac {3 \sqrt {\frac {\pi }{5}} \text {erfi}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )}{6400 a^5}-\frac {\text {Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{120 a^5}+\frac {\text {Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{120 a^5}+\frac {\text {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{40 a^5}-\frac {\text {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{40 a^5} \\ & = \frac {2 x \sqrt {\text {arcsinh}(a x)}}{5 a^4}-\frac {x^3 \sqrt {\text {arcsinh}(a x)}}{15 a^2}+\frac {3}{100} x^5 \sqrt {\text {arcsinh}(a x)}-\frac {4 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{15 a^5}+\frac {2 x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{15 a^3}-\frac {x^4 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{10 a}+\frac {1}{5} x^5 \text {arcsinh}(a x)^{5/2}+\frac {15 \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )}{128 a^5}-\frac {\sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{240 a^5}-\frac {\sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{1280 a^5}+\frac {3 \sqrt {\frac {\pi }{5}} \text {erf}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )}{6400 a^5}-\frac {15 \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{128 a^5}+\frac {\sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{240 a^5}+\frac {\sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{1280 a^5}-\frac {3 \sqrt {\frac {\pi }{5}} \text {erfi}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )}{6400 a^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.40 \[ \int x^4 \text {arcsinh}(a x)^{5/2} \, dx=\frac {\frac {27 \sqrt {5} \sqrt {\text {arcsinh}(a x)} \Gamma \left (\frac {7}{2},-5 \text {arcsinh}(a x)\right )}{\sqrt {-\text {arcsinh}(a x)}}+\frac {625 \sqrt {3} \sqrt {-\text {arcsinh}(a x)} \Gamma \left (\frac {7}{2},-3 \text {arcsinh}(a x)\right )}{\sqrt {\text {arcsinh}(a x)}}+\frac {33750 \sqrt {\text {arcsinh}(a x)} \Gamma \left (\frac {7}{2},-\text {arcsinh}(a x)\right )}{\sqrt {-\text {arcsinh}(a x)}}-33750 \Gamma \left (\frac {7}{2},\text {arcsinh}(a x)\right )+625 \sqrt {3} \Gamma \left (\frac {7}{2},3 \text {arcsinh}(a x)\right )-27 \sqrt {5} \Gamma \left (\frac {7}{2},5 \text {arcsinh}(a x)\right )}{540000 a^5} \]

[In]

Integrate[x^4*ArcSinh[a*x]^(5/2),x]

[Out]

((27*Sqrt[5]*Sqrt[ArcSinh[a*x]]*Gamma[7/2, -5*ArcSinh[a*x]])/Sqrt[-ArcSinh[a*x]] + (625*Sqrt[3]*Sqrt[-ArcSinh[
a*x]]*Gamma[7/2, -3*ArcSinh[a*x]])/Sqrt[ArcSinh[a*x]] + (33750*Sqrt[ArcSinh[a*x]]*Gamma[7/2, -ArcSinh[a*x]])/S
qrt[-ArcSinh[a*x]] - 33750*Gamma[7/2, ArcSinh[a*x]] + 625*Sqrt[3]*Gamma[7/2, 3*ArcSinh[a*x]] - 27*Sqrt[5]*Gamm
a[7/2, 5*ArcSinh[a*x]])/(540000*a^5)

Maple [F]

\[\int x^{4} \operatorname {arcsinh}\left (a x \right )^{\frac {5}{2}}d x\]

[In]

int(x^4*arcsinh(a*x)^(5/2),x)

[Out]

int(x^4*arcsinh(a*x)^(5/2),x)

Fricas [F(-2)]

Exception generated. \[ \int x^4 \text {arcsinh}(a x)^{5/2} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x^4*arcsinh(a*x)^(5/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [F(-1)]

Timed out. \[ \int x^4 \text {arcsinh}(a x)^{5/2} \, dx=\text {Timed out} \]

[In]

integrate(x**4*asinh(a*x)**(5/2),x)

[Out]

Timed out

Maxima [F]

\[ \int x^4 \text {arcsinh}(a x)^{5/2} \, dx=\int { x^{4} \operatorname {arsinh}\left (a x\right )^{\frac {5}{2}} \,d x } \]

[In]

integrate(x^4*arcsinh(a*x)^(5/2),x, algorithm="maxima")

[Out]

integrate(x^4*arcsinh(a*x)^(5/2), x)

Giac [F(-2)]

Exception generated. \[ \int x^4 \text {arcsinh}(a x)^{5/2} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate(x^4*arcsinh(a*x)^(5/2),x, algorithm="giac")

[Out]

Exception raised: RuntimeError >> an error occurred running a Giac command:INPUT:sage2OUTPUT:sym2poly/r2sym(co
nst gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int x^4 \text {arcsinh}(a x)^{5/2} \, dx=\int x^4\,{\mathrm {asinh}\left (a\,x\right )}^{5/2} \,d x \]

[In]

int(x^4*asinh(a*x)^(5/2),x)

[Out]

int(x^4*asinh(a*x)^(5/2), x)