\(\int x^2 \text {arcsinh}(a x)^{3/2} \, dx\) [114]

Optimal result

Integrand size = 12, antiderivative size = 179 \[ \int x^2 \text {arcsinh}(a x)^{3/2} \, dx=\frac {\sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}}{3 a^3}-\frac {x^2 \sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}}{6 a}+\frac {1}{3} x^3 \text {arcsinh}(a x)^{3/2}-\frac {3 \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )}{32 a^3}+\frac {\sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{96 a^3}-\frac {3 \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{32 a^3}+\frac {\sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{96 a^3} \] Output:

1/3*(a^2*x^2+1)^(1/2)*arcsinh(a*x)^(1/2)/a^3-1/6*x^2*(a^2*x^2+1)^(1/2)*arc 
sinh(a*x)^(1/2)/a+1/3*x^3*arcsinh(a*x)^(3/2)-3/32*Pi^(1/2)*erf(arcsinh(a*x 
)^(1/2))/a^3+1/288*3^(1/2)*Pi^(1/2)*erf(3^(1/2)*arcsinh(a*x)^(1/2))/a^3-3/ 
32*Pi^(1/2)*erfi(arcsinh(a*x)^(1/2))/a^3+1/288*3^(1/2)*Pi^(1/2)*erfi(3^(1/ 
2)*arcsinh(a*x)^(1/2))/a^3
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.55 \[ \int x^2 \text {arcsinh}(a x)^{3/2} \, dx=\frac {\frac {\sqrt {3} \sqrt {-\text {arcsinh}(a x)} \Gamma \left (\frac {5}{2},-3 \text {arcsinh}(a x)\right )}{\sqrt {\text {arcsinh}(a x)}}+\frac {27 \sqrt {\text {arcsinh}(a x)} \Gamma \left (\frac {5}{2},-\text {arcsinh}(a x)\right )}{\sqrt {-\text {arcsinh}(a x)}}+27 \Gamma \left (\frac {5}{2},\text {arcsinh}(a x)\right )-\sqrt {3} \Gamma \left (\frac {5}{2},3 \text {arcsinh}(a x)\right )}{216 a^3} \] Input:

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

Output:

((Sqrt[3]*Sqrt[-ArcSinh[a*x]]*Gamma[5/2, -3*ArcSinh[a*x]])/Sqrt[ArcSinh[a* 
x]] + (27*Sqrt[ArcSinh[a*x]]*Gamma[5/2, -ArcSinh[a*x]])/Sqrt[-ArcSinh[a*x] 
] + 27*Gamma[5/2, ArcSinh[a*x]] - Sqrt[3]*Gamma[5/2, 3*ArcSinh[a*x]])/(216 
*a^3)
 

Rubi [A] (verified)

Time = 1.78 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.28, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.083, Rules used = {6192, 6227, 6195, 5971, 2009, 6213, 6189, 3042, 3788, 26, 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.

\(\displaystyle \int x^2 \text {arcsinh}(a x)^{3/2} \, dx\)

\(\Big \downarrow \) 6192

\(\displaystyle \frac {1}{3} x^3 \text {arcsinh}(a x)^{3/2}-\frac {1}{2} a \int \frac {x^3 \sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx\)

\(\Big \downarrow \) 6227

\(\displaystyle \frac {1}{3} x^3 \text {arcsinh}(a x)^{3/2}-\frac {1}{2} a \left (-\frac {2 \int \frac {x \sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx}{3 a^2}-\frac {\int \frac {x^2}{\sqrt {\text {arcsinh}(a x)}}dx}{6 a}+\frac {x^2 \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{3 a^2}\right )\)

\(\Big \downarrow \) 6195

\(\displaystyle \frac {1}{3} x^3 \text {arcsinh}(a x)^{3/2}-\frac {1}{2} a \left (-\frac {2 \int \frac {x \sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx}{3 a^2}-\frac {\int \frac {a^2 x^2 \sqrt {a^2 x^2+1}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{6 a^4}+\frac {x^2 \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{3 a^2}\right )\)

\(\Big \downarrow \) 5971

\(\displaystyle \frac {1}{3} x^3 \text {arcsinh}(a x)^{3/2}-\frac {1}{2} a \left (-\frac {2 \int \frac {x \sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx}{3 a^2}-\frac {\int \left (\frac {\cosh (3 \text {arcsinh}(a x))}{4 \sqrt {\text {arcsinh}(a x)}}-\frac {\sqrt {a^2 x^2+1}}{4 \sqrt {\text {arcsinh}(a x)}}\right )d\text {arcsinh}(a x)}{6 a^4}+\frac {x^2 \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{3 a^2}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{3} x^3 \text {arcsinh}(a x)^{3/2}-\frac {1}{2} a \left (-\frac {2 \int \frac {x \sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx}{3 a^2}-\frac {-\frac {1}{8} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{8} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{6 a^4}+\frac {x^2 \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{3 a^2}\right )\)

\(\Big \downarrow \) 6213

\(\displaystyle \frac {1}{3} x^3 \text {arcsinh}(a x)^{3/2}-\frac {1}{2} a \left (-\frac {2 \left (\frac {\sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{a^2}-\frac {\int \frac {1}{\sqrt {\text {arcsinh}(a x)}}dx}{2 a}\right )}{3 a^2}-\frac {-\frac {1}{8} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{8} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{6 a^4}+\frac {x^2 \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{3 a^2}\right )\)

\(\Big \downarrow \) 6189

\(\displaystyle \frac {1}{3} x^3 \text {arcsinh}(a x)^{3/2}-\frac {1}{2} a \left (-\frac {2 \left (\frac {\sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{a^2}-\frac {\int \frac {\sqrt {a^2 x^2+1}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{2 a^2}\right )}{3 a^2}-\frac {-\frac {1}{8} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{8} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{6 a^4}+\frac {x^2 \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{3 a^2}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{3} x^3 \text {arcsinh}(a x)^{3/2}-\frac {1}{2} a \left (-\frac {2 \left (\frac {\sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{a^2}-\frac {\int \frac {\sin \left (i \text {arcsinh}(a x)+\frac {\pi }{2}\right )}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{2 a^2}\right )}{3 a^2}-\frac {-\frac {1}{8} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{8} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{6 a^4}+\frac {x^2 \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{3 a^2}\right )\)

\(\Big \downarrow \) 3788

\(\displaystyle \frac {1}{3} x^3 \text {arcsinh}(a x)^{3/2}-\frac {1}{2} a \left (-\frac {2 \left (\frac {\sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{a^2}-\frac {\frac {1}{2} i \int -\frac {i e^{\text {arcsinh}(a x)}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)-\frac {1}{2} i \int \frac {i e^{-\text {arcsinh}(a x)}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{2 a^2}\right )}{3 a^2}-\frac {-\frac {1}{8} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{8} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{6 a^4}+\frac {x^2 \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{3 a^2}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {1}{3} x^3 \text {arcsinh}(a x)^{3/2}-\frac {1}{2} a \left (-\frac {2 \left (\frac {\sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{a^2}-\frac {\frac {1}{2} \int \frac {e^{-\text {arcsinh}(a x)}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)+\frac {1}{2} \int \frac {e^{\text {arcsinh}(a x)}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{2 a^2}\right )}{3 a^2}-\frac {-\frac {1}{8} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{8} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{6 a^4}+\frac {x^2 \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{3 a^2}\right )\)

\(\Big \downarrow \) 2611

\(\displaystyle \frac {1}{3} x^3 \text {arcsinh}(a x)^{3/2}-\frac {1}{2} a \left (-\frac {2 \left (\frac {\sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{a^2}-\frac {\int e^{-\text {arcsinh}(a x)}d\sqrt {\text {arcsinh}(a x)}+\int e^{\text {arcsinh}(a x)}d\sqrt {\text {arcsinh}(a x)}}{2 a^2}\right )}{3 a^2}-\frac {-\frac {1}{8} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{8} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{6 a^4}+\frac {x^2 \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{3 a^2}\right )\)

\(\Big \downarrow \) 2633

\(\displaystyle \frac {1}{3} x^3 \text {arcsinh}(a x)^{3/2}-\frac {1}{2} a \left (-\frac {2 \left (\frac {\sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{a^2}-\frac {\int e^{-\text {arcsinh}(a x)}d\sqrt {\text {arcsinh}(a x)}+\frac {1}{2} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{2 a^2}\right )}{3 a^2}-\frac {-\frac {1}{8} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{8} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{6 a^4}+\frac {x^2 \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{3 a^2}\right )\)

\(\Big \downarrow \) 2634

\(\displaystyle \frac {1}{3} x^3 \text {arcsinh}(a x)^{3/2}-\frac {1}{2} a \left (-\frac {-\frac {1}{8} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{8} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{6 a^4}-\frac {2 \left (\frac {\sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{a^2}-\frac {\frac {1}{2} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{2} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{2 a^2}\right )}{3 a^2}+\frac {x^2 \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{3 a^2}\right )\)

Input:

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

Output:

(x^3*ArcSinh[a*x]^(3/2))/3 - (a*((x^2*Sqrt[1 + a^2*x^2]*Sqrt[ArcSinh[a*x]] 
)/(3*a^2) - (2*((Sqrt[1 + a^2*x^2]*Sqrt[ArcSinh[a*x]])/a^2 - ((Sqrt[Pi]*Er 
f[Sqrt[ArcSinh[a*x]]])/2 + (Sqrt[Pi]*Erfi[Sqrt[ArcSinh[a*x]]])/2)/(2*a^2)) 
)/(3*a^2) - (-1/8*(Sqrt[Pi]*Erf[Sqrt[ArcSinh[a*x]]]) + (Sqrt[Pi/3]*Erf[Sqr 
t[3]*Sqrt[ArcSinh[a*x]]])/8 - (Sqrt[Pi]*Erfi[Sqrt[ArcSinh[a*x]]])/8 + (Sqr 
t[Pi/3]*Erfi[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/8)/(6*a^4)))/2
 

Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2611

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]
 

rule 2633

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 2634

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]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3788

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol 
] :> Simp[I/2   Int[(c + d*x)^m/(E^(I*k*Pi)*E^(I*(e + f*x))), x], x] - Simp 
[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 5971

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 6189

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[1/(b*c)   S 
ubst[Int[x^n*Cosh[-a/b + x/b], x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, 
b, c, n}, x]
 

rule 6192

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

rule 6195

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]
 

rule 6213

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] - Simp[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 6227

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] + (-Simp[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] 
 - Simp[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]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-2)]

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

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

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (constant residues)
 

Sympy [F]

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

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

Output:

Integral(x**2*asinh(a*x)**(3/2), x)
 

Maxima [F]

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

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

Output:

integrate(x^2*arcsinh(a*x)^(3/2), x)
 

Giac [F]

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

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

Output:

integrate(x^2*arcsinh(a*x)^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int x^2 \text {arcsinh}(a x)^{3/2} \, dx=\int x^2\,{\mathrm {asinh}\left (a\,x\right )}^{3/2} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int x^2 \text {arcsinh}(a x)^{3/2} \, dx=\int \sqrt {\mathit {asinh} \left (a x \right )}\, \mathit {asinh} \left (a x \right ) x^{2}d x \] Input:

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

Output:

int(sqrt(asinh(a*x))*asinh(a*x)*x**2,x)