\(\int \frac {x^4}{\text {arcsinh}(a x)^{5/2}} \, dx\) [105]

Optimal result

Integrand size = 12, antiderivative size = 223 \[ \int \frac {x^4}{\text {arcsinh}(a x)^{5/2}} \, dx=-\frac {2 x^4 \sqrt {1+a^2 x^2}}{3 a \text {arcsinh}(a x)^{3/2}}-\frac {16 x^3}{3 a^2 \sqrt {\text {arcsinh}(a x)}}-\frac {20 x^5}{3 \sqrt {\text {arcsinh}(a x)}}+\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )}{12 a^5}-\frac {3 \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{8 a^5}+\frac {5 \sqrt {5 \pi } \text {erf}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )}{24 a^5}+\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{12 a^5}-\frac {3 \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{8 a^5}+\frac {5 \sqrt {5 \pi } \text {erfi}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )}{24 a^5} \]

[Out]

1/12*erf(arcsinh(a*x)^(1/2))*Pi^(1/2)/a^5+1/12*erfi(arcsinh(a*x)^(1/2))*Pi^(1/2)/a^5-3/8*erf(3^(1/2)*arcsinh(a
*x)^(1/2))*3^(1/2)*Pi^(1/2)/a^5-3/8*erfi(3^(1/2)*arcsinh(a*x)^(1/2))*3^(1/2)*Pi^(1/2)/a^5+5/24*erf(5^(1/2)*arc
sinh(a*x)^(1/2))*5^(1/2)*Pi^(1/2)/a^5+5/24*erfi(5^(1/2)*arcsinh(a*x)^(1/2))*5^(1/2)*Pi^(1/2)/a^5-2/3*x^4*(a^2*
x^2+1)^(1/2)/a/arcsinh(a*x)^(3/2)-16/3*x^3/a^2/arcsinh(a*x)^(1/2)-20/3*x^5/arcsinh(a*x)^(1/2)

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 34, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {5779, 5818, 5780, 5556, 3388, 2211, 2235, 2236} \[ \int \frac {x^4}{\text {arcsinh}(a x)^{5/2}} \, dx=\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )}{12 a^5}-\frac {3 \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{8 a^5}+\frac {5 \sqrt {5 \pi } \text {erf}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )}{24 a^5}+\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{12 a^5}-\frac {3 \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{8 a^5}+\frac {5 \sqrt {5 \pi } \text {erfi}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )}{24 a^5}-\frac {16 x^3}{3 a^2 \sqrt {\text {arcsinh}(a x)}}-\frac {2 x^4 \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^{3/2}}-\frac {20 x^5}{3 \sqrt {\text {arcsinh}(a x)}} \]

[In]

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

[Out]

(-2*x^4*Sqrt[1 + a^2*x^2])/(3*a*ArcSinh[a*x]^(3/2)) - (16*x^3)/(3*a^2*Sqrt[ArcSinh[a*x]]) - (20*x^5)/(3*Sqrt[A
rcSinh[a*x]]) + (Sqrt[Pi]*Erf[Sqrt[ArcSinh[a*x]]])/(12*a^5) - (3*Sqrt[3*Pi]*Erf[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/(
8*a^5) + (5*Sqrt[5*Pi]*Erf[Sqrt[5]*Sqrt[ArcSinh[a*x]]])/(24*a^5) + (Sqrt[Pi]*Erfi[Sqrt[ArcSinh[a*x]]])/(12*a^5
) - (3*Sqrt[3*Pi]*Erfi[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/(8*a^5) + (5*Sqrt[5*Pi]*Erfi[Sqrt[5]*Sqrt[ArcSinh[a*x]]])/
(24*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 3388

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

Rule 5556

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int
[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n,
0] && IGtQ[p, 0]

Rule 5779

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[x^m*Sqrt[1 + c^2*x^2]*((a + b*ArcSi
nh[c*x])^(n + 1)/(b*c*(n + 1))), x] + (-Dist[c*((m + 1)/(b*(n + 1))), Int[x^(m + 1)*((a + b*ArcSinh[c*x])^(n +
1)/Sqrt[1 + c^2*x^2]), x], x] - Dist[m/(b*c*(n + 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] && LtQ[n, -2]

Rule 5780

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

Rule 5818

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

Rubi steps \begin{align*} \text {integral}& = -\frac {2 x^4 \sqrt {1+a^2 x^2}}{3 a \text {arcsinh}(a x)^{3/2}}+\frac {8 \int \frac {x^3}{\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}} \, dx}{3 a}+\frac {1}{3} (10 a) \int \frac {x^5}{\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}} \, dx \\ & = -\frac {2 x^4 \sqrt {1+a^2 x^2}}{3 a \text {arcsinh}(a x)^{3/2}}-\frac {16 x^3}{3 a^2 \sqrt {\text {arcsinh}(a x)}}-\frac {20 x^5}{3 \sqrt {\text {arcsinh}(a x)}}+\frac {100}{3} \int \frac {x^4}{\sqrt {\text {arcsinh}(a x)}} \, dx+\frac {16 \int \frac {x^2}{\sqrt {\text {arcsinh}(a x)}} \, dx}{a^2} \\ & = -\frac {2 x^4 \sqrt {1+a^2 x^2}}{3 a \text {arcsinh}(a x)^{3/2}}-\frac {16 x^3}{3 a^2 \sqrt {\text {arcsinh}(a x)}}-\frac {20 x^5}{3 \sqrt {\text {arcsinh}(a x)}}+\frac {16 \text {Subst}\left (\int \frac {\cosh (x) \sinh ^2(x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{a^5}+\frac {100 \text {Subst}\left (\int \frac {\cosh (x) \sinh ^4(x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{3 a^5} \\ & = -\frac {2 x^4 \sqrt {1+a^2 x^2}}{3 a \text {arcsinh}(a x)^{3/2}}-\frac {16 x^3}{3 a^2 \sqrt {\text {arcsinh}(a x)}}-\frac {20 x^5}{3 \sqrt {\text {arcsinh}(a x)}}+\frac {16 \text {Subst}\left (\int \left (-\frac {\cosh (x)}{4 \sqrt {x}}+\frac {\cosh (3 x)}{4 \sqrt {x}}\right ) \, dx,x,\text {arcsinh}(a x)\right )}{a^5}+\frac {100 \text {Subst}\left (\int \left (\frac {\cosh (x)}{8 \sqrt {x}}-\frac {3 \cosh (3 x)}{16 \sqrt {x}}+\frac {\cosh (5 x)}{16 \sqrt {x}}\right ) \, dx,x,\text {arcsinh}(a x)\right )}{3 a^5} \\ & = -\frac {2 x^4 \sqrt {1+a^2 x^2}}{3 a \text {arcsinh}(a x)^{3/2}}-\frac {16 x^3}{3 a^2 \sqrt {\text {arcsinh}(a x)}}-\frac {20 x^5}{3 \sqrt {\text {arcsinh}(a x)}}+\frac {25 \text {Subst}\left (\int \frac {\cosh (5 x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{12 a^5}-\frac {4 \text {Subst}\left (\int \frac {\cosh (x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{a^5}+\frac {4 \text {Subst}\left (\int \frac {\cosh (3 x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{a^5}+\frac {25 \text {Subst}\left (\int \frac {\cosh (x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{6 a^5}-\frac {25 \text {Subst}\left (\int \frac {\cosh (3 x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{4 a^5} \\ & = -\frac {2 x^4 \sqrt {1+a^2 x^2}}{3 a \text {arcsinh}(a x)^{3/2}}-\frac {16 x^3}{3 a^2 \sqrt {\text {arcsinh}(a x)}}-\frac {20 x^5}{3 \sqrt {\text {arcsinh}(a x)}}+\frac {25 \text {Subst}\left (\int \frac {e^{-5 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{24 a^5}+\frac {25 \text {Subst}\left (\int \frac {e^{5 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{24 a^5}+\frac {2 \text {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{a^5}-\frac {2 \text {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{a^5}-\frac {2 \text {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{a^5}+\frac {2 \text {Subst}\left (\int \frac {e^{3 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{a^5}+\frac {25 \text {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{12 a^5}+\frac {25 \text {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{12 a^5}-\frac {25 \text {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{8 a^5}-\frac {25 \text {Subst}\left (\int \frac {e^{3 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{8 a^5} \\ & = -\frac {2 x^4 \sqrt {1+a^2 x^2}}{3 a \text {arcsinh}(a x)^{3/2}}-\frac {16 x^3}{3 a^2 \sqrt {\text {arcsinh}(a x)}}-\frac {20 x^5}{3 \sqrt {\text {arcsinh}(a x)}}+\frac {25 \text {Subst}\left (\int e^{-5 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{12 a^5}+\frac {25 \text {Subst}\left (\int e^{5 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{12 a^5}+\frac {4 \text {Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{a^5}-\frac {4 \text {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{a^5}-\frac {4 \text {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{a^5}+\frac {4 \text {Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{a^5}+\frac {25 \text {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{6 a^5}+\frac {25 \text {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{6 a^5}-\frac {25 \text {Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{4 a^5}-\frac {25 \text {Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{4 a^5} \\ & = -\frac {2 x^4 \sqrt {1+a^2 x^2}}{3 a \text {arcsinh}(a x)^{3/2}}-\frac {16 x^3}{3 a^2 \sqrt {\text {arcsinh}(a x)}}-\frac {20 x^5}{3 \sqrt {\text {arcsinh}(a x)}}+\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )}{12 a^5}-\frac {3 \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{8 a^5}+\frac {5 \sqrt {5 \pi } \text {erf}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )}{24 a^5}+\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{12 a^5}-\frac {3 \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{8 a^5}+\frac {5 \sqrt {5 \pi } \text {erfi}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )}{24 a^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.52 \[ \int \frac {x^4}{\text {arcsinh}(a x)^{5/2}} \, dx=\frac {-\frac {e^{5 \text {arcsinh}(a x)} (1+10 \text {arcsinh}(a x))+10 \sqrt {5} (-\text {arcsinh}(a x))^{3/2} \Gamma \left (\frac {1}{2},-5 \text {arcsinh}(a x)\right )}{48 \text {arcsinh}(a x)^{3/2}}+\frac {e^{3 \text {arcsinh}(a x)} (1+6 \text {arcsinh}(a x))+6 \sqrt {3} (-\text {arcsinh}(a x))^{3/2} \Gamma \left (\frac {1}{2},-3 \text {arcsinh}(a x)\right )}{16 \text {arcsinh}(a x)^{3/2}}-\frac {e^{\text {arcsinh}(a x)} (1+2 \text {arcsinh}(a x))+2 (-\text {arcsinh}(a x))^{3/2} \Gamma \left (\frac {1}{2},-\text {arcsinh}(a x)\right )}{24 \text {arcsinh}(a x)^{3/2}}-\frac {e^{-\text {arcsinh}(a x)} \left (1-2 \text {arcsinh}(a x)+2 e^{\text {arcsinh}(a x)} \text {arcsinh}(a x)^{3/2} \Gamma \left (\frac {1}{2},\text {arcsinh}(a x)\right )\right )}{24 \text {arcsinh}(a x)^{3/2}}+\frac {1}{16} \left (\frac {e^{-3 \text {arcsinh}(a x)}}{\text {arcsinh}(a x)^{3/2}}-\frac {6 e^{-3 \text {arcsinh}(a x)}}{\sqrt {\text {arcsinh}(a x)}}+6 \sqrt {3} \Gamma \left (\frac {1}{2},3 \text {arcsinh}(a x)\right )\right )-\frac {e^{-5 \text {arcsinh}(a x)} \left (1-10 \text {arcsinh}(a x)+10 \sqrt {5} e^{5 \text {arcsinh}(a x)} \text {arcsinh}(a x)^{3/2} \Gamma \left (\frac {1}{2},5 \text {arcsinh}(a x)\right )\right )}{48 \text {arcsinh}(a x)^{3/2}}}{a^5} \]

[In]

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

[Out]

(-1/48*(E^(5*ArcSinh[a*x])*(1 + 10*ArcSinh[a*x]) + 10*Sqrt[5]*(-ArcSinh[a*x])^(3/2)*Gamma[1/2, -5*ArcSinh[a*x]
])/ArcSinh[a*x]^(3/2) + (E^(3*ArcSinh[a*x])*(1 + 6*ArcSinh[a*x]) + 6*Sqrt[3]*(-ArcSinh[a*x])^(3/2)*Gamma[1/2,
-3*ArcSinh[a*x]])/(16*ArcSinh[a*x]^(3/2)) - (E^ArcSinh[a*x]*(1 + 2*ArcSinh[a*x]) + 2*(-ArcSinh[a*x])^(3/2)*Gam
ma[1/2, -ArcSinh[a*x]])/(24*ArcSinh[a*x]^(3/2)) - (1 - 2*ArcSinh[a*x] + 2*E^ArcSinh[a*x]*ArcSinh[a*x]^(3/2)*Ga
mma[1/2, ArcSinh[a*x]])/(24*E^ArcSinh[a*x]*ArcSinh[a*x]^(3/2)) + (1/(E^(3*ArcSinh[a*x])*ArcSinh[a*x]^(3/2)) -
6/(E^(3*ArcSinh[a*x])*Sqrt[ArcSinh[a*x]]) + 6*Sqrt[3]*Gamma[1/2, 3*ArcSinh[a*x]])/16 - (1 - 10*ArcSinh[a*x] +
10*Sqrt[5]*E^(5*ArcSinh[a*x])*ArcSinh[a*x]^(3/2)*Gamma[1/2, 5*ArcSinh[a*x]])/(48*E^(5*ArcSinh[a*x])*ArcSinh[a*
x]^(3/2)))/a^5

Maple [F]

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

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

[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]

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

[In]

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

[Out]

Integral(x**4/asinh(a*x)**(5/2), x)

Maxima [F]

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

[In]

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

[Out]

integrate(x^4/arcsinh(a*x)^(5/2), x)

Giac [F]

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

[In]

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

[Out]

integrate(x^4/arcsinh(a*x)^(5/2), x)

Mupad [F(-1)]

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

[In]

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

[Out]

int(x^4/asinh(a*x)^(5/2), x)