3.86 \(\int x^4 \sinh ^{-1}(a x)^{5/2} \, dx\)

Optimal. Leaf size=379 \[ \frac {15 \sqrt {\pi } \text {erf}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{128 a^5}-\frac {\sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\sinh ^{-1}(a x)}\right )}{1280 a^5}-\frac {\sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\sinh ^{-1}(a x)}\right )}{240 a^5}+\frac {3 \sqrt {\frac {\pi }{5}} \text {erf}\left (\sqrt {5} \sqrt {\sinh ^{-1}(a x)}\right )}{6400 a^5}-\frac {15 \sqrt {\pi } \text {erfi}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{128 a^5}+\frac {\sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\sinh ^{-1}(a x)}\right )}{1280 a^5}+\frac {\sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\sinh ^{-1}(a x)}\right )}{240 a^5}-\frac {3 \sqrt {\frac {\pi }{5}} \text {erfi}\left (\sqrt {5} \sqrt {\sinh ^{-1}(a x)}\right )}{6400 a^5}+\frac {2 x \sqrt {\sinh ^{-1}(a x)}}{5 a^4}-\frac {x^3 \sqrt {\sinh ^{-1}(a x)}}{15 a^2}-\frac {x^4 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^{3/2}}{10 a}-\frac {4 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^{3/2}}{15 a^5}+\frac {2 x^2 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^{3/2}}{15 a^3}+\frac {1}{5} x^5 \sinh ^{-1}(a x)^{5/2}+\frac {3}{100} x^5 \sqrt {\sinh ^{-1}(a x)} \]

[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]  time = 1.00, antiderivative size = 379, normalized size of antiderivative = 1.00, number of steps used = 44, number of rules used = 10, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {5663, 5758, 5717, 5653, 5779, 3308, 2180, 2204, 2205, 3312} \[ \frac {15 \sqrt {\pi } \text {Erf}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{128 a^5}-\frac {\sqrt {3 \pi } \text {Erf}\left (\sqrt {3} \sqrt {\sinh ^{-1}(a x)}\right )}{1280 a^5}-\frac {\sqrt {\frac {\pi }{3}} \text {Erf}\left (\sqrt {3} \sqrt {\sinh ^{-1}(a x)}\right )}{240 a^5}+\frac {3 \sqrt {\frac {\pi }{5}} \text {Erf}\left (\sqrt {5} \sqrt {\sinh ^{-1}(a x)}\right )}{6400 a^5}-\frac {15 \sqrt {\pi } \text {Erfi}\left (\sqrt {\sinh ^{-1}(a x)}\right )}{128 a^5}+\frac {\sqrt {3 \pi } \text {Erfi}\left (\sqrt {3} \sqrt {\sinh ^{-1}(a x)}\right )}{1280 a^5}+\frac {\sqrt {\frac {\pi }{3}} \text {Erfi}\left (\sqrt {3} \sqrt {\sinh ^{-1}(a x)}\right )}{240 a^5}-\frac {3 \sqrt {\frac {\pi }{5}} \text {Erfi}\left (\sqrt {5} \sqrt {\sinh ^{-1}(a x)}\right )}{6400 a^5}-\frac {x^4 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^{3/2}}{10 a}-\frac {x^3 \sqrt {\sinh ^{-1}(a x)}}{15 a^2}+\frac {2 x^2 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^{3/2}}{15 a^3}-\frac {4 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^{3/2}}{15 a^5}+\frac {2 x \sqrt {\sinh ^{-1}(a x)}}{5 a^4}+\frac {1}{5} x^5 \sinh ^{-1}(a x)^{5/2}+\frac {3}{100} x^5 \sqrt {\sinh ^{-1}(a x)} \]

Antiderivative was successfully verified.

[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 2180

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] &&  !$UseGamma === True

Rule 2204

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 2205

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 3308

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 3312

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 5653

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 5663

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 5717

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*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*c*(p +
1)*(1 + c^2*x^2)^FracPart[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 5758

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

Rule 5779

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

Rubi steps

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

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Mathematica [A]  time = 0.13, size = 152, normalized size = 0.40 \[ \frac {\frac {27 \sqrt {5} \sqrt {\sinh ^{-1}(a x)} \Gamma \left (\frac {7}{2},-5 \sinh ^{-1}(a x)\right )}{\sqrt {-\sinh ^{-1}(a x)}}+\frac {625 \sqrt {3} \sqrt {-\sinh ^{-1}(a x)} \Gamma \left (\frac {7}{2},-3 \sinh ^{-1}(a x)\right )}{\sqrt {\sinh ^{-1}(a x)}}+\frac {33750 \sqrt {\sinh ^{-1}(a x)} \Gamma \left (\frac {7}{2},-\sinh ^{-1}(a x)\right )}{\sqrt {-\sinh ^{-1}(a x)}}-33750 \Gamma \left (\frac {7}{2},\sinh ^{-1}(a x)\right )+625 \sqrt {3} \Gamma \left (\frac {7}{2},3 \sinh ^{-1}(a x)\right )-27 \sqrt {5} \Gamma \left (\frac {7}{2},5 \sinh ^{-1}(a x)\right )}{540000 a^5} \]

Warning: Unable to verify antiderivative.

[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)

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[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:Simplification as
suming a near 0Simplification assuming a near 0Simplification assuming a near 0Simplification assuming a near
0Simplification assuming a near 0Simplification assuming t_nostep near 0Simplification assuming a near 0Simpli
fication assuming a near 0Simplification assuming a near 0Simplification assuming a near 0Simplification assum
ing a near 0Simplification assuming t_nostep near 0Simplification assuming a near 0Simplification assuming a n
ear 0Simplification assuming a near 0Simplification assuming a near 0Simplification assuming a near 0Simplific
ation assuming t_nostep near 0Simplification assuming a near 0Simplification assuming a near 0Simplification a
ssuming a near 0Simplification assuming a near 0Simplification assuming a near 0Simplification assuming t_nost
ep near 0sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [F(-2)]  time = 180.00, size = 0, normalized size = 0.00 \[ \int x^{4} \arcsinh \left (a x \right )^{\frac {5}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{4} \operatorname {arsinh}\left (a x\right )^{\frac {5}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^4*arcsinh(a*x)^(5/2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int x^4\,{\mathrm {asinh}\left (a\,x\right )}^{5/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^4*asinh(a*x)^(5/2), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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