\(\int x^2 \text {arcsinh}(a x)^{3/2} \, dx\) [82]

Optimal result

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

[Out]

1/3*x^3*arcsinh(a*x)^(3/2)+1/288*erf(3^(1/2)*arcsinh(a*x)^(1/2))*3^(1/2)*Pi^(1/2)/a^3+1/288*erfi(3^(1/2)*arcsi
nh(a*x)^(1/2))*3^(1/2)*Pi^(1/2)/a^3-3/32*erf(arcsinh(a*x)^(1/2))*Pi^(1/2)/a^3-3/32*erfi(arcsinh(a*x)^(1/2))*Pi
^(1/2)/a^3+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)*arcsinh(a*x)^(1/2)/a

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {5777, 5812, 5798, 5774, 3388, 2211, 2235, 2236, 5780, 5556} \[ \int x^2 \text {arcsinh}(a x)^{3/2} \, dx=-\frac {3 \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )}{32 a^3}+\frac {\sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{96 a^3}-\frac {3 \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{32 a^3}+\frac {\sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{96 a^3}-\frac {x^2 \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{6 a}+\frac {\sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{3 a^3}+\frac {1}{3} x^3 \text {arcsinh}(a x)^{3/2} \]

[In]

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

[Out]

(Sqrt[1 + a^2*x^2]*Sqrt[ArcSinh[a*x]])/(3*a^3) - (x^2*Sqrt[1 + a^2*x^2]*Sqrt[ArcSinh[a*x]])/(6*a) + (x^3*ArcSi
nh[a*x]^(3/2))/3 - (3*Sqrt[Pi]*Erf[Sqrt[ArcSinh[a*x]]])/(32*a^3) + (Sqrt[Pi/3]*Erf[Sqrt[3]*Sqrt[ArcSinh[a*x]]]
)/(96*a^3) - (3*Sqrt[Pi]*Erfi[Sqrt[ArcSinh[a*x]]])/(32*a^3) + (Sqrt[Pi/3]*Erfi[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/(9
6*a^3)

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 5774

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

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 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 5798

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/(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 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
]

Rubi steps \begin{align*} \text {integral}& = \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 {1+a^2 x^2}} \, dx \\ & = -\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 {1}{12} \int \frac {x^2}{\sqrt {\text {arcsinh}(a x)}} \, dx+\frac {\int \frac {x \sqrt {\text {arcsinh}(a x)}}{\sqrt {1+a^2 x^2}} \, dx}{3 a} \\ & = \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 {\text {Subst}\left (\int \frac {\cosh (x) \sinh ^2(x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{12 a^3}-\frac {\int \frac {1}{\sqrt {\text {arcsinh}(a x)}} \, dx}{6 a^2} \\ & = \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 {\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 )}{12 a^3}-\frac {\text {Subst}\left (\int \frac {\cosh (x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{6 a^3} \\ & = \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 {\text {Subst}\left (\int \frac {\cosh (x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{48 a^3}+\frac {\text {Subst}\left (\int \frac {\cosh (3 x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{48 a^3}-\frac {\text {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{12 a^3}-\frac {\text {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{12 a^3} \\ & = \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 {\text {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{96 a^3}-\frac {\text {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{96 a^3}-\frac {\text {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{96 a^3}+\frac {\text {Subst}\left (\int \frac {e^{3 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{96 a^3}-\frac {\text {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{6 a^3}-\frac {\text {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{6 a^3} \\ & = \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 {\sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )}{12 a^3}-\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{12 a^3}+\frac {\text {Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{48 a^3}-\frac {\text {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{48 a^3}-\frac {\text {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{48 a^3}+\frac {\text {Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{48 a^3} \\ & = \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} \\ \end{align*}

Mathematica [A] (verified)

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

[In]

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

[Out]

((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]])/(2
16*a^3)

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [F]

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

[In]

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

[Out]

Integral(x**2*asinh(a*x)**(3/2), x)

Maxima [F]

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

[In]

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

[Out]

integrate(x^2*arcsinh(a*x)^(3/2), x)

Giac [F]

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

[In]

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

[Out]

integrate(x^2*arcsinh(a*x)^(3/2), x)

Mupad [F(-1)]

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

[In]

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

[Out]

int(x^2*asinh(a*x)^(3/2), x)