\(\int x^2 \text {arcsinh}(a x)^{5/2} \, dx\) [88]

Optimal result

Integrand size = 12, antiderivative size = 210 \[ \int x^2 \text {arcsinh}(a x)^{5/2} \, dx=-\frac {5 x \sqrt {\text {arcsinh}(a x)}}{6 a^2}+\frac {5}{36} x^3 \sqrt {\text {arcsinh}(a x)}+\frac {5 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{9 a^3}-\frac {5 x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{18 a}+\frac {1}{3} x^3 \text {arcsinh}(a x)^{5/2}-\frac {15 \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )}{64 a^3}+\frac {5 \sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{576 a^3}+\frac {15 \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{64 a^3}-\frac {5 \sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{576 a^3} \]

[Out]

1/3*x^3*arcsinh(a*x)^(5/2)+5/1728*erf(3^(1/2)*arcsinh(a*x)^(1/2))*3^(1/2)*Pi^(1/2)/a^3-5/1728*erfi(3^(1/2)*arc
sinh(a*x)^(1/2))*3^(1/2)*Pi^(1/2)/a^3-15/64*erf(arcsinh(a*x)^(1/2))*Pi^(1/2)/a^3+15/64*erfi(arcsinh(a*x)^(1/2)
)*Pi^(1/2)/a^3+5/9*arcsinh(a*x)^(3/2)*(a^2*x^2+1)^(1/2)/a^3-5/18*x^2*arcsinh(a*x)^(3/2)*(a^2*x^2+1)^(1/2)/a-5/
6*x*arcsinh(a*x)^(1/2)/a^2+5/36*x^3*arcsinh(a*x)^(1/2)

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {5777, 5812, 5798, 5772, 5819, 3389, 2211, 2235, 2236, 3393} \[ \int x^2 \text {arcsinh}(a x)^{5/2} \, dx=-\frac {15 \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )}{64 a^3}+\frac {5 \sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{576 a^3}+\frac {15 \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{64 a^3}-\frac {5 \sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{576 a^3}-\frac {5 x^2 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{18 a}-\frac {5 x \sqrt {\text {arcsinh}(a x)}}{6 a^2}+\frac {5 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{9 a^3}+\frac {1}{3} x^3 \text {arcsinh}(a x)^{5/2}+\frac {5}{36} x^3 \sqrt {\text {arcsinh}(a x)} \]

[In]

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

[Out]

(-5*x*Sqrt[ArcSinh[a*x]])/(6*a^2) + (5*x^3*Sqrt[ArcSinh[a*x]])/36 + (5*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^(3/2))/(
9*a^3) - (5*x^2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^(3/2))/(18*a) + (x^3*ArcSinh[a*x]^(5/2))/3 - (15*Sqrt[Pi]*Erf[S
qrt[ArcSinh[a*x]]])/(64*a^3) + (5*Sqrt[Pi/3]*Erf[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/(576*a^3) + (15*Sqrt[Pi]*Erfi[Sq
rt[ArcSinh[a*x]]])/(64*a^3) - (5*Sqrt[Pi/3]*Erfi[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/(576*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 3389

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

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 5772

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

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 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
]

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}{3} x^3 \text {arcsinh}(a x)^{5/2}-\frac {1}{6} (5 a) \int \frac {x^3 \text {arcsinh}(a x)^{3/2}}{\sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {5 x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{18 a}+\frac {1}{3} x^3 \text {arcsinh}(a x)^{5/2}+\frac {5}{12} \int x^2 \sqrt {\text {arcsinh}(a x)} \, dx+\frac {5 \int \frac {x \text {arcsinh}(a x)^{3/2}}{\sqrt {1+a^2 x^2}} \, dx}{9 a} \\ & = \frac {5}{36} x^3 \sqrt {\text {arcsinh}(a x)}+\frac {5 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{9 a^3}-\frac {5 x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{18 a}+\frac {1}{3} x^3 \text {arcsinh}(a x)^{5/2}-\frac {5 \int \sqrt {\text {arcsinh}(a x)} \, dx}{6 a^2}-\frac {1}{72} (5 a) \int \frac {x^3}{\sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}} \, dx \\ & = -\frac {5 x \sqrt {\text {arcsinh}(a x)}}{6 a^2}+\frac {5}{36} x^3 \sqrt {\text {arcsinh}(a x)}+\frac {5 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{9 a^3}-\frac {5 x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{18 a}+\frac {1}{3} x^3 \text {arcsinh}(a x)^{5/2}-\frac {5 \text {Subst}\left (\int \frac {\sinh ^3(x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{72 a^3}+\frac {5 \int \frac {x}{\sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}} \, dx}{12 a} \\ & = -\frac {5 x \sqrt {\text {arcsinh}(a x)}}{6 a^2}+\frac {5}{36} x^3 \sqrt {\text {arcsinh}(a x)}+\frac {5 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{9 a^3}-\frac {5 x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{18 a}+\frac {1}{3} x^3 \text {arcsinh}(a x)^{5/2}-\frac {(5 i) \text {Subst}\left (\int \left (\frac {3 i \sinh (x)}{4 \sqrt {x}}-\frac {i \sinh (3 x)}{4 \sqrt {x}}\right ) \, dx,x,\text {arcsinh}(a x)\right )}{72 a^3}+\frac {5 \text {Subst}\left (\int \frac {\sinh (x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{12 a^3} \\ & = -\frac {5 x \sqrt {\text {arcsinh}(a x)}}{6 a^2}+\frac {5}{36} x^3 \sqrt {\text {arcsinh}(a x)}+\frac {5 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{9 a^3}-\frac {5 x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{18 a}+\frac {1}{3} x^3 \text {arcsinh}(a x)^{5/2}-\frac {5 \text {Subst}\left (\int \frac {\sinh (3 x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{288 a^3}+\frac {5 \text {Subst}\left (\int \frac {\sinh (x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{96 a^3}-\frac {5 \text {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{24 a^3}+\frac {5 \text {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{24 a^3} \\ & = -\frac {5 x \sqrt {\text {arcsinh}(a x)}}{6 a^2}+\frac {5}{36} x^3 \sqrt {\text {arcsinh}(a x)}+\frac {5 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{9 a^3}-\frac {5 x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{18 a}+\frac {1}{3} x^3 \text {arcsinh}(a x)^{5/2}+\frac {5 \text {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{576 a^3}-\frac {5 \text {Subst}\left (\int \frac {e^{3 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{576 a^3}-\frac {5 \text {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{192 a^3}+\frac {5 \text {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{192 a^3}-\frac {5 \text {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{12 a^3}+\frac {5 \text {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{12 a^3} \\ & = -\frac {5 x \sqrt {\text {arcsinh}(a x)}}{6 a^2}+\frac {5}{36} x^3 \sqrt {\text {arcsinh}(a x)}+\frac {5 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{9 a^3}-\frac {5 x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{18 a}+\frac {1}{3} x^3 \text {arcsinh}(a x)^{5/2}-\frac {5 \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )}{24 a^3}+\frac {5 \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{24 a^3}+\frac {5 \text {Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{288 a^3}-\frac {5 \text {Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{288 a^3}-\frac {5 \text {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{96 a^3}+\frac {5 \text {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{96 a^3} \\ & = -\frac {5 x \sqrt {\text {arcsinh}(a x)}}{6 a^2}+\frac {5}{36} x^3 \sqrt {\text {arcsinh}(a x)}+\frac {5 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{9 a^3}-\frac {5 x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{18 a}+\frac {1}{3} x^3 \text {arcsinh}(a x)^{5/2}-\frac {15 \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )}{64 a^3}+\frac {5 \sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{576 a^3}+\frac {15 \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{64 a^3}-\frac {5 \sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{576 a^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.47 \[ \int x^2 \text {arcsinh}(a x)^{5/2} \, dx=\frac {\frac {\sqrt {3} \sqrt {\text {arcsinh}(a x)} \Gamma \left (\frac {7}{2},-3 \text {arcsinh}(a x)\right )}{\sqrt {-\text {arcsinh}(a x)}}+\frac {81 \sqrt {-\text {arcsinh}(a x)} \Gamma \left (\frac {7}{2},-\text {arcsinh}(a x)\right )}{\sqrt {\text {arcsinh}(a x)}}+81 \Gamma \left (\frac {7}{2},\text {arcsinh}(a x)\right )-\sqrt {3} \Gamma \left (\frac {7}{2},3 \text {arcsinh}(a x)\right )}{648 a^3} \]

[In]

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

[Out]

((Sqrt[3]*Sqrt[ArcSinh[a*x]]*Gamma[7/2, -3*ArcSinh[a*x]])/Sqrt[-ArcSinh[a*x]] + (81*Sqrt[-ArcSinh[a*x]]*Gamma[
7/2, -ArcSinh[a*x]])/Sqrt[ArcSinh[a*x]] + 81*Gamma[7/2, ArcSinh[a*x]] - Sqrt[3]*Gamma[7/2, 3*ArcSinh[a*x]])/(6
48*a^3)

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

integrate(x^2*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]

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

[In]

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

[Out]

Integral(x**2*asinh(a*x)**(5/2), x)

Maxima [F]

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

[In]

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

[Out]

integrate(x^2*arcsinh(a*x)^(5/2), x)

Giac [F(-2)]

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

[In]

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

[Out]

Exception raised: RuntimeError >> an error occurred running a Giac command:INPUT:sage2OUTPUT:sym2poly/r2sym(co
nst gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

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

[Out]

int(x^2*asinh(a*x)^(5/2), x)