Integrand size = 12, antiderivative size = 167 \[ \int \frac {x^3}{\text {arcsinh}(a x)^{5/2}} \, dx=-\frac {2 x^3 \sqrt {1+a^2 x^2}}{3 a \text {arcsinh}(a x)^{3/2}}-\frac {4 x^2}{a^2 \sqrt {\text {arcsinh}(a x)}}-\frac {16 x^4}{3 \sqrt {\text {arcsinh}(a x)}}-\frac {2 \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{3 a^4}+\frac {\sqrt {2 \pi } \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{3 a^4}+\frac {2 \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{3 a^4}-\frac {\sqrt {2 \pi } \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{3 a^4} \]
-2/3*erf(2*arcsinh(a*x)^(1/2))*Pi^(1/2)/a^4+2/3*erfi(2*arcsinh(a*x)^(1/2)) *Pi^(1/2)/a^4+1/3*erf(2^(1/2)*arcsinh(a*x)^(1/2))*2^(1/2)*Pi^(1/2)/a^4-1/3 *erfi(2^(1/2)*arcsinh(a*x)^(1/2))*2^(1/2)*Pi^(1/2)/a^4-2/3*x^3*(a^2*x^2+1) ^(1/2)/a/arcsinh(a*x)^(3/2)-4*x^2/a^2/arcsinh(a*x)^(1/2)-16/3*x^4/arcsinh( a*x)^(1/2)
Time = 0.30 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.10 \[ \int \frac {x^3}{\text {arcsinh}(a x)^{5/2}} \, dx=\frac {-4 e^{-4 \text {arcsinh}(a x)} \text {arcsinh}(a x)+4 e^{-2 \text {arcsinh}(a x)} \text {arcsinh}(a x)+4 e^{2 \text {arcsinh}(a x)} \text {arcsinh}(a x)-4 e^{4 \text {arcsinh}(a x)} \text {arcsinh}(a x)-8 (-\text {arcsinh}(a x))^{3/2} \Gamma \left (\frac {1}{2},-4 \text {arcsinh}(a x)\right )+4 \sqrt {2} (-\text {arcsinh}(a x))^{3/2} \Gamma \left (\frac {1}{2},-2 \text {arcsinh}(a x)\right )-4 \sqrt {2} \text {arcsinh}(a x)^{3/2} \Gamma \left (\frac {1}{2},2 \text {arcsinh}(a x)\right )+8 \text {arcsinh}(a x)^{3/2} \Gamma \left (\frac {1}{2},4 \text {arcsinh}(a x)\right )+2 \sinh (2 \text {arcsinh}(a x))-\sinh (4 \text {arcsinh}(a x))}{12 a^4 \text {arcsinh}(a x)^{3/2}} \]
((-4*ArcSinh[a*x])/E^(4*ArcSinh[a*x]) + (4*ArcSinh[a*x])/E^(2*ArcSinh[a*x] ) + 4*E^(2*ArcSinh[a*x])*ArcSinh[a*x] - 4*E^(4*ArcSinh[a*x])*ArcSinh[a*x] - 8*(-ArcSinh[a*x])^(3/2)*Gamma[1/2, -4*ArcSinh[a*x]] + 4*Sqrt[2]*(-ArcSin h[a*x])^(3/2)*Gamma[1/2, -2*ArcSinh[a*x]] - 4*Sqrt[2]*ArcSinh[a*x]^(3/2)*G amma[1/2, 2*ArcSinh[a*x]] + 8*ArcSinh[a*x]^(3/2)*Gamma[1/2, 4*ArcSinh[a*x] ] + 2*Sinh[2*ArcSinh[a*x]] - Sinh[4*ArcSinh[a*x]])/(12*a^4*ArcSinh[a*x]^(3 /2))
Result contains complex when optimal does not.
Time = 1.07 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.47, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6194, 6233, 6195, 5971, 27, 2009, 3042, 26, 3789, 2611, 2633, 2634}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x^3}{\text {arcsinh}(a x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 6194 |
| \(\displaystyle \frac {2 \int \frac {x^2}{\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}dx}{a}+\frac {8}{3} a \int \frac {x^4}{\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}dx-\frac {2 x^3 \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^{3/2}}\) |
| \(\Big \downarrow \) 6233 |
| \(\displaystyle \frac {2 \left (\frac {4 \int \frac {x}{\sqrt {\text {arcsinh}(a x)}}dx}{a}-\frac {2 x^2}{a \sqrt {\text {arcsinh}(a x)}}\right )}{a}+\frac {8}{3} a \left (\frac {8 \int \frac {x^3}{\sqrt {\text {arcsinh}(a x)}}dx}{a}-\frac {2 x^4}{a \sqrt {\text {arcsinh}(a x)}}\right )-\frac {2 x^3 \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^{3/2}}\) |
| \(\Big \downarrow \) 6195 |
| \(\displaystyle \frac {2 \left (\frac {4 \int \frac {a x \sqrt {a^2 x^2+1}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{a^3}-\frac {2 x^2}{a \sqrt {\text {arcsinh}(a x)}}\right )}{a}+\frac {8}{3} a \left (\frac {8 \int \frac {a^3 x^3 \sqrt {a^2 x^2+1}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{a^5}-\frac {2 x^4}{a \sqrt {\text {arcsinh}(a x)}}\right )-\frac {2 x^3 \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^{3/2}}\) |
| \(\Big \downarrow \) 5971 |
| \(\displaystyle \frac {8}{3} a \left (\frac {8 \int \left (\frac {\sinh (4 \text {arcsinh}(a x))}{8 \sqrt {\text {arcsinh}(a x)}}-\frac {\sinh (2 \text {arcsinh}(a x))}{4 \sqrt {\text {arcsinh}(a x)}}\right )d\text {arcsinh}(a x)}{a^5}-\frac {2 x^4}{a \sqrt {\text {arcsinh}(a x)}}\right )+\frac {2 \left (\frac {4 \int \frac {\sinh (2 \text {arcsinh}(a x))}{2 \sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{a^3}-\frac {2 x^2}{a \sqrt {\text {arcsinh}(a x)}}\right )}{a}-\frac {2 x^3 \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {8}{3} a \left (\frac {8 \int \left (\frac {\sinh (4 \text {arcsinh}(a x))}{8 \sqrt {\text {arcsinh}(a x)}}-\frac {\sinh (2 \text {arcsinh}(a x))}{4 \sqrt {\text {arcsinh}(a x)}}\right )d\text {arcsinh}(a x)}{a^5}-\frac {2 x^4}{a \sqrt {\text {arcsinh}(a x)}}\right )+\frac {2 \left (\frac {2 \int \frac {\sinh (2 \text {arcsinh}(a x))}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{a^3}-\frac {2 x^2}{a \sqrt {\text {arcsinh}(a x)}}\right )}{a}-\frac {2 x^3 \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^{3/2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \left (\frac {2 \int \frac {\sinh (2 \text {arcsinh}(a x))}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{a^3}-\frac {2 x^2}{a \sqrt {\text {arcsinh}(a x)}}\right )}{a}+\frac {8}{3} a \left (\frac {8 \left (-\frac {1}{32} \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{32} \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{8} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )\right )}{a^5}-\frac {2 x^4}{a \sqrt {\text {arcsinh}(a x)}}\right )-\frac {2 x^3 \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \left (-\frac {2 x^2}{a \sqrt {\text {arcsinh}(a x)}}+\frac {2 \int -\frac {i \sin (2 i \text {arcsinh}(a x))}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{a^3}\right )}{a}+\frac {8}{3} a \left (\frac {8 \left (-\frac {1}{32} \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{32} \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{8} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )\right )}{a^5}-\frac {2 x^4}{a \sqrt {\text {arcsinh}(a x)}}\right )-\frac {2 x^3 \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^{3/2}}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {2 \left (-\frac {2 x^2}{a \sqrt {\text {arcsinh}(a x)}}-\frac {2 i \int \frac {\sin (2 i \text {arcsinh}(a x))}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{a^3}\right )}{a}+\frac {8}{3} a \left (\frac {8 \left (-\frac {1}{32} \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{32} \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{8} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )\right )}{a^5}-\frac {2 x^4}{a \sqrt {\text {arcsinh}(a x)}}\right )-\frac {2 x^3 \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^{3/2}}\) |
| \(\Big \downarrow \) 3789 |
| \(\displaystyle \frac {2 \left (-\frac {2 x^2}{a \sqrt {\text {arcsinh}(a x)}}-\frac {2 i \left (\frac {1}{2} i \int \frac {e^{2 \text {arcsinh}(a x)}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)-\frac {1}{2} i \int \frac {e^{-2 \text {arcsinh}(a x)}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)\right )}{a^3}\right )}{a}+\frac {8}{3} a \left (\frac {8 \left (-\frac {1}{32} \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{32} \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{8} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )\right )}{a^5}-\frac {2 x^4}{a \sqrt {\text {arcsinh}(a x)}}\right )-\frac {2 x^3 \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^{3/2}}\) |
| \(\Big \downarrow \) 2611 |
| \(\displaystyle \frac {2 \left (-\frac {2 x^2}{a \sqrt {\text {arcsinh}(a x)}}-\frac {2 i \left (i \int e^{2 \text {arcsinh}(a x)}d\sqrt {\text {arcsinh}(a x)}-i \int e^{-2 \text {arcsinh}(a x)}d\sqrt {\text {arcsinh}(a x)}\right )}{a^3}\right )}{a}+\frac {8}{3} a \left (\frac {8 \left (-\frac {1}{32} \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{32} \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{8} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )\right )}{a^5}-\frac {2 x^4}{a \sqrt {\text {arcsinh}(a x)}}\right )-\frac {2 x^3 \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^{3/2}}\) |
| \(\Big \downarrow \) 2633 |
| \(\displaystyle \frac {2 \left (-\frac {2 x^2}{a \sqrt {\text {arcsinh}(a x)}}-\frac {2 i \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )-i \int e^{-2 \text {arcsinh}(a x)}d\sqrt {\text {arcsinh}(a x)}\right )}{a^3}\right )}{a}+\frac {8}{3} a \left (\frac {8 \left (-\frac {1}{32} \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{32} \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{8} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )\right )}{a^5}-\frac {2 x^4}{a \sqrt {\text {arcsinh}(a x)}}\right )-\frac {2 x^3 \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^{3/2}}\) |
| \(\Big \downarrow \) 2634 |
| \(\displaystyle \frac {8}{3} a \left (\frac {8 \left (-\frac {1}{32} \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{32} \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{8} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )\right )}{a^5}-\frac {2 x^4}{a \sqrt {\text {arcsinh}(a x)}}\right )+\frac {2 \left (-\frac {2 x^2}{a \sqrt {\text {arcsinh}(a x)}}-\frac {2 i \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )\right )}{a^3}\right )}{a}-\frac {2 x^3 \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^{3/2}}\) |
(-2*x^3*Sqrt[1 + a^2*x^2])/(3*a*ArcSinh[a*x]^(3/2)) + (8*a*((-2*x^4)/(a*Sq rt[ArcSinh[a*x]]) + (8*(-1/32*(Sqrt[Pi]*Erf[2*Sqrt[ArcSinh[a*x]]]) + (Sqrt [Pi/2]*Erf[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/8 + (Sqrt[Pi]*Erfi[2*Sqrt[ArcSinh[ a*x]]])/32 - (Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/8))/a^5))/3 + ( 2*((-2*x^2)/(a*Sqrt[ArcSinh[a*x]]) - ((2*I)*((-1/2*I)*Sqrt[Pi/2]*Erf[Sqrt[ 2]*Sqrt[ArcSinh[a*x]]] + (I/2)*Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ArcSinh[a*x]]] ))/a^3))/a
3.2.6.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] : > Simp[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]
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]
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] /; Fr eeQ[{F, a, b, c, d}, x] && NegQ[b]
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I /2 Int[(c + d*x)^m/E^(I*(e + f*x)), x], x] - Simp[I/2 Int[(c + d*x)^m*E ^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e, f, m}, x]
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]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ x^m*Sqrt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])^(n + 1)/(b*c*(n + 1))), x] + (- Simp[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] - Simp[m/(b*c*(n + 1)) Int[x^(m - 1)*((a + b*A rcSinh[c*x])^(n + 1)/Sqrt[1 + c^2*x^2]), x], x]) /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && LtQ[n, -2]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ 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]
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] - Simp[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]
\[\int \frac {x^{3}}{\operatorname {arcsinh}\left (a x \right )^{\frac {5}{2}}}d x\]
Exception generated. \[ \int \frac {x^3}{\text {arcsinh}(a x)^{5/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {x^3}{\text {arcsinh}(a x)^{5/2}} \, dx=\int \frac {x^{3}}{\operatorname {asinh}^{\frac {5}{2}}{\left (a x \right )}}\, dx \]
\[ \int \frac {x^3}{\text {arcsinh}(a x)^{5/2}} \, dx=\int { \frac {x^{3}}{\operatorname {arsinh}\left (a x\right )^{\frac {5}{2}}} \,d x } \]
Exception generated. \[ \int \frac {x^3}{\text {arcsinh}(a x)^{5/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {x^3}{\text {arcsinh}(a x)^{5/2}} \, dx=\int \frac {x^3}{{\mathrm {asinh}\left (a\,x\right )}^{5/2}} \,d x \]