\(\int x^3 \text {arcsinh}(a x)^{5/2} \, dx\) [87]

Optimal result

Integrand size = 12, antiderivative size = 247 \[ \int x^3 \text {arcsinh}(a x)^{5/2} \, dx=-\frac {225 \sqrt {\text {arcsinh}(a x)}}{2048 a^4}-\frac {45 x^2 \sqrt {\text {arcsinh}(a x)}}{256 a^2}+\frac {15}{256} x^4 \sqrt {\text {arcsinh}(a x)}+\frac {15 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{64 a^3}-\frac {5 x^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{32 a}-\frac {3 \text {arcsinh}(a x)^{5/2}}{32 a^4}+\frac {1}{4} x^4 \text {arcsinh}(a x)^{5/2}-\frac {15 \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{16384 a^4}+\frac {15 \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{512 a^4}-\frac {15 \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{16384 a^4}+\frac {15 \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{512 a^4} \]

[Out]

-3/32*arcsinh(a*x)^(5/2)/a^4+1/4*x^4*arcsinh(a*x)^(5/2)+15/1024*erf(2^(1/2)*arcsinh(a*x)^(1/2))*2^(1/2)*Pi^(1/
2)/a^4+15/1024*erfi(2^(1/2)*arcsinh(a*x)^(1/2))*2^(1/2)*Pi^(1/2)/a^4-15/16384*erf(2*arcsinh(a*x)^(1/2))*Pi^(1/
2)/a^4-15/16384*erfi(2*arcsinh(a*x)^(1/2))*Pi^(1/2)/a^4+15/64*x*arcsinh(a*x)^(3/2)*(a^2*x^2+1)^(1/2)/a^3-5/32*
x^3*arcsinh(a*x)^(3/2)*(a^2*x^2+1)^(1/2)/a-225/2048*arcsinh(a*x)^(1/2)/a^4-45/256*x^2*arcsinh(a*x)^(1/2)/a^2+1
5/256*x^4*arcsinh(a*x)^(1/2)

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.00, number of steps used = 27, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {5777, 5812, 5783, 5819, 3393, 3388, 2211, 2235, 2236} \[ \int x^3 \text {arcsinh}(a x)^{5/2} \, dx=-\frac {15 \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{16384 a^4}+\frac {15 \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{512 a^4}-\frac {15 \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{16384 a^4}+\frac {15 \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{512 a^4}-\frac {3 \text {arcsinh}(a x)^{5/2}}{32 a^4}-\frac {225 \sqrt {\text {arcsinh}(a x)}}{2048 a^4}-\frac {45 x^2 \sqrt {\text {arcsinh}(a x)}}{256 a^2}-\frac {5 x^3 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{32 a}+\frac {15 x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{64 a^3}+\frac {1}{4} x^4 \text {arcsinh}(a x)^{5/2}+\frac {15}{256} x^4 \sqrt {\text {arcsinh}(a x)} \]

[In]

Int[x^3*ArcSinh[a*x]^(5/2),x]

[Out]

(-225*Sqrt[ArcSinh[a*x]])/(2048*a^4) - (45*x^2*Sqrt[ArcSinh[a*x]])/(256*a^2) + (15*x^4*Sqrt[ArcSinh[a*x]])/256
+ (15*x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^(3/2))/(64*a^3) - (5*x^3*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^(3/2))/(32*a)
- (3*ArcSinh[a*x]^(5/2))/(32*a^4) + (x^4*ArcSinh[a*x]^(5/2))/4 - (15*Sqrt[Pi]*Erf[2*Sqrt[ArcSinh[a*x]]])/(1638
4*a^4) + (15*Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(512*a^4) - (15*Sqrt[Pi]*Erfi[2*Sqrt[ArcSinh[a*x]]])/
(16384*a^4) + (15*Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(512*a^4)

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 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 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 5783

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(1/(b*c*(n + 1)))*S
imp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSinh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ
[e, c^2*d] && NeQ[n, -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}{4} x^4 \text {arcsinh}(a x)^{5/2}-\frac {1}{8} (5 a) \int \frac {x^4 \text {arcsinh}(a x)^{3/2}}{\sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {5 x^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{32 a}+\frac {1}{4} x^4 \text {arcsinh}(a x)^{5/2}+\frac {15}{64} \int x^3 \sqrt {\text {arcsinh}(a x)} \, dx+\frac {15 \int \frac {x^2 \text {arcsinh}(a x)^{3/2}}{\sqrt {1+a^2 x^2}} \, dx}{32 a} \\ & = \frac {15}{256} x^4 \sqrt {\text {arcsinh}(a x)}+\frac {15 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{64 a^3}-\frac {5 x^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{32 a}+\frac {1}{4} x^4 \text {arcsinh}(a x)^{5/2}-\frac {15 \int \frac {\text {arcsinh}(a x)^{3/2}}{\sqrt {1+a^2 x^2}} \, dx}{64 a^3}-\frac {45 \int x \sqrt {\text {arcsinh}(a x)} \, dx}{128 a^2}-\frac {1}{512} (15 a) \int \frac {x^4}{\sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}} \, dx \\ & = -\frac {45 x^2 \sqrt {\text {arcsinh}(a x)}}{256 a^2}+\frac {15}{256} x^4 \sqrt {\text {arcsinh}(a x)}+\frac {15 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{64 a^3}-\frac {5 x^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{32 a}-\frac {3 \text {arcsinh}(a x)^{5/2}}{32 a^4}+\frac {1}{4} x^4 \text {arcsinh}(a x)^{5/2}-\frac {15 \text {Subst}\left (\int \frac {\sinh ^4(x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{512 a^4}+\frac {45 \int \frac {x^2}{\sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}} \, dx}{512 a} \\ & = -\frac {45 x^2 \sqrt {\text {arcsinh}(a x)}}{256 a^2}+\frac {15}{256} x^4 \sqrt {\text {arcsinh}(a x)}+\frac {15 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{64 a^3}-\frac {5 x^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{32 a}-\frac {3 \text {arcsinh}(a x)^{5/2}}{32 a^4}+\frac {1}{4} x^4 \text {arcsinh}(a x)^{5/2}-\frac {15 \text {Subst}\left (\int \left (\frac {3}{8 \sqrt {x}}-\frac {\cosh (2 x)}{2 \sqrt {x}}+\frac {\cosh (4 x)}{8 \sqrt {x}}\right ) \, dx,x,\text {arcsinh}(a x)\right )}{512 a^4}+\frac {45 \text {Subst}\left (\int \frac {\sinh ^2(x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{512 a^4} \\ & = -\frac {45 \sqrt {\text {arcsinh}(a x)}}{2048 a^4}-\frac {45 x^2 \sqrt {\text {arcsinh}(a x)}}{256 a^2}+\frac {15}{256} x^4 \sqrt {\text {arcsinh}(a x)}+\frac {15 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{64 a^3}-\frac {5 x^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{32 a}-\frac {3 \text {arcsinh}(a x)^{5/2}}{32 a^4}+\frac {1}{4} x^4 \text {arcsinh}(a x)^{5/2}-\frac {15 \text {Subst}\left (\int \frac {\cosh (4 x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{4096 a^4}+\frac {15 \text {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{1024 a^4}-\frac {45 \text {Subst}\left (\int \left (\frac {1}{2 \sqrt {x}}-\frac {\cosh (2 x)}{2 \sqrt {x}}\right ) \, dx,x,\text {arcsinh}(a x)\right )}{512 a^4} \\ & = -\frac {225 \sqrt {\text {arcsinh}(a x)}}{2048 a^4}-\frac {45 x^2 \sqrt {\text {arcsinh}(a x)}}{256 a^2}+\frac {15}{256} x^4 \sqrt {\text {arcsinh}(a x)}+\frac {15 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{64 a^3}-\frac {5 x^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{32 a}-\frac {3 \text {arcsinh}(a x)^{5/2}}{32 a^4}+\frac {1}{4} x^4 \text {arcsinh}(a x)^{5/2}-\frac {15 \text {Subst}\left (\int \frac {e^{-4 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{8192 a^4}-\frac {15 \text {Subst}\left (\int \frac {e^{4 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{8192 a^4}+\frac {15 \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{2048 a^4}+\frac {15 \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{2048 a^4}+\frac {45 \text {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{1024 a^4} \\ & = -\frac {225 \sqrt {\text {arcsinh}(a x)}}{2048 a^4}-\frac {45 x^2 \sqrt {\text {arcsinh}(a x)}}{256 a^2}+\frac {15}{256} x^4 \sqrt {\text {arcsinh}(a x)}+\frac {15 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{64 a^3}-\frac {5 x^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{32 a}-\frac {3 \text {arcsinh}(a x)^{5/2}}{32 a^4}+\frac {1}{4} x^4 \text {arcsinh}(a x)^{5/2}-\frac {15 \text {Subst}\left (\int e^{-4 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{4096 a^4}-\frac {15 \text {Subst}\left (\int e^{4 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{4096 a^4}+\frac {15 \text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{1024 a^4}+\frac {15 \text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{1024 a^4}+\frac {45 \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{2048 a^4}+\frac {45 \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{2048 a^4} \\ & = -\frac {225 \sqrt {\text {arcsinh}(a x)}}{2048 a^4}-\frac {45 x^2 \sqrt {\text {arcsinh}(a x)}}{256 a^2}+\frac {15}{256} x^4 \sqrt {\text {arcsinh}(a x)}+\frac {15 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{64 a^3}-\frac {5 x^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{32 a}-\frac {3 \text {arcsinh}(a x)^{5/2}}{32 a^4}+\frac {1}{4} x^4 \text {arcsinh}(a x)^{5/2}-\frac {15 \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{16384 a^4}+\frac {15 \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{2048 a^4}-\frac {15 \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{16384 a^4}+\frac {15 \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{2048 a^4}+\frac {45 \text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{1024 a^4}+\frac {45 \text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{1024 a^4} \\ & = -\frac {225 \sqrt {\text {arcsinh}(a x)}}{2048 a^4}-\frac {45 x^2 \sqrt {\text {arcsinh}(a x)}}{256 a^2}+\frac {15}{256} x^4 \sqrt {\text {arcsinh}(a x)}+\frac {15 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{64 a^3}-\frac {5 x^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{32 a}-\frac {3 \text {arcsinh}(a x)^{5/2}}{32 a^4}+\frac {1}{4} x^4 \text {arcsinh}(a x)^{5/2}-\frac {15 \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{16384 a^4}+\frac {15 \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{512 a^4}-\frac {15 \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{16384 a^4}+\frac {15 \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{512 a^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.40 \[ \int x^3 \text {arcsinh}(a x)^{5/2} \, dx=\frac {\frac {\sqrt {\text {arcsinh}(a x)} \Gamma \left (\frac {7}{2},-4 \text {arcsinh}(a x)\right )}{\sqrt {-\text {arcsinh}(a x)}}+\frac {16 \sqrt {2} \sqrt {-\text {arcsinh}(a x)} \Gamma \left (\frac {7}{2},-2 \text {arcsinh}(a x)\right )}{\sqrt {\text {arcsinh}(a x)}}-16 \sqrt {2} \Gamma \left (\frac {7}{2},2 \text {arcsinh}(a x)\right )+\Gamma \left (\frac {7}{2},4 \text {arcsinh}(a x)\right )}{2048 a^4} \]

[In]

Integrate[x^3*ArcSinh[a*x]^(5/2),x]

[Out]

((Sqrt[ArcSinh[a*x]]*Gamma[7/2, -4*ArcSinh[a*x]])/Sqrt[-ArcSinh[a*x]] + (16*Sqrt[2]*Sqrt[-ArcSinh[a*x]]*Gamma[
7/2, -2*ArcSinh[a*x]])/Sqrt[ArcSinh[a*x]] - 16*Sqrt[2]*Gamma[7/2, 2*ArcSinh[a*x]] + Gamma[7/2, 4*ArcSinh[a*x]]
)/(2048*a^4)

Maple [F]

\[\int x^{3} \operatorname {arcsinh}\left (a x \right )^{\frac {5}{2}}d x\]

[In]

int(x^3*arcsinh(a*x)^(5/2),x)

[Out]

int(x^3*arcsinh(a*x)^(5/2),x)

Fricas [F(-2)]

Exception generated. \[ \int x^3 \text {arcsinh}(a x)^{5/2} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x^3*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]

\[ \int x^3 \text {arcsinh}(a x)^{5/2} \, dx=\int x^{3} \operatorname {asinh}^{\frac {5}{2}}{\left (a x \right )}\, dx \]

[In]

integrate(x**3*asinh(a*x)**(5/2),x)

[Out]

Integral(x**3*asinh(a*x)**(5/2), x)

Maxima [F]

\[ \int x^3 \text {arcsinh}(a x)^{5/2} \, dx=\int { x^{3} \operatorname {arsinh}\left (a x\right )^{\frac {5}{2}} \,d x } \]

[In]

integrate(x^3*arcsinh(a*x)^(5/2),x, algorithm="maxima")

[Out]

integrate(x^3*arcsinh(a*x)^(5/2), x)

Giac [F(-2)]

Exception generated. \[ \int x^3 \text {arcsinh}(a x)^{5/2} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x^3*arcsinh(a*x)^(5/2),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. \[ \int x^3 \text {arcsinh}(a x)^{5/2} \, dx=\int x^3\,{\mathrm {asinh}\left (a\,x\right )}^{5/2} \,d x \]

[In]

int(x^3*asinh(a*x)^(5/2),x)

[Out]

int(x^3*asinh(a*x)^(5/2), x)