\(\int x^2 \sqrt {a+b \text {arcsinh}(c+d x)} \, dx\) [98]

Optimal result

Integrand size = 18, antiderivative size = 496 \[ \int x^2 \sqrt {a+b \text {arcsinh}(c+d x)} \, dx=\frac {c^2 (c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}}{d^3}+\frac {(c+d x)^3 \sqrt {a+b \text {arcsinh}(c+d x)}}{3 d^3}-\frac {c \sqrt {a+b \text {arcsinh}(c+d x)} \cosh (2 \text {arcsinh}(c+d x))}{2 d^3}-\frac {\sqrt {b} e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{16 d^3}+\frac {\sqrt {b} c^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{4 d^3}+\frac {\sqrt {b} c e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{8 d^3}+\frac {\sqrt {b} e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{48 d^3}+\frac {\sqrt {b} e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{16 d^3}-\frac {\sqrt {b} c^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{4 d^3}+\frac {\sqrt {b} c e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{8 d^3}-\frac {\sqrt {b} e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{48 d^3} \]

[Out]

1/144*exp(3*a/b)*erf(3^(1/2)*(a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))*b^(1/2)*3^(1/2)*Pi^(1/2)/d^3-1/144*erfi(3^(1/
2)*(a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))*b^(1/2)*3^(1/2)*Pi^(1/2)/d^3/exp(3*a/b)+1/16*c*exp(2*a/b)*erf(2^(1/2)*(
a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))*b^(1/2)*2^(1/2)*Pi^(1/2)/d^3+1/16*c*erfi(2^(1/2)*(a+b*arcsinh(d*x+c))^(1/2)
/b^(1/2))*b^(1/2)*2^(1/2)*Pi^(1/2)/d^3/exp(2*a/b)-1/16*exp(a/b)*erf((a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))*b^(1/2
)*Pi^(1/2)/d^3+1/4*c^2*exp(a/b)*erf((a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))*b^(1/2)*Pi^(1/2)/d^3+1/16*erfi((a+b*ar
csinh(d*x+c))^(1/2)/b^(1/2))*b^(1/2)*Pi^(1/2)/d^3/exp(a/b)-1/4*c^2*erfi((a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))*b^
(1/2)*Pi^(1/2)/d^3/exp(a/b)+c^2*(d*x+c)*(a+b*arcsinh(d*x+c))^(1/2)/d^3+1/3*(d*x+c)^3*(a+b*arcsinh(d*x+c))^(1/2
)/d^3-1/2*c*cosh(2*arcsinh(d*x+c))*(a+b*arcsinh(d*x+c))^(1/2)/d^3

Rubi [A] (verified)

Time = 1.36 (sec) , antiderivative size = 496, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {5859, 5830, 6873, 6874, 5433, 5406, 2236, 2235, 5432, 5407, 5480, 5408} \[ \int x^2 \sqrt {a+b \text {arcsinh}(c+d x)} \, dx=\frac {\sqrt {\pi } \sqrt {b} c^2 e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{4 d^3}-\frac {\sqrt {\pi } \sqrt {b} c^2 e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{4 d^3}+\frac {c^2 (c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}}{d^3}-\frac {\sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{16 d^3}+\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} c e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{8 d^3}+\frac {\sqrt {\frac {\pi }{3}} \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{48 d^3}+\frac {\sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{16 d^3}+\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} c e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{8 d^3}-\frac {\sqrt {\frac {\pi }{3}} \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{48 d^3}+\frac {(c+d x)^3 \sqrt {a+b \text {arcsinh}(c+d x)}}{3 d^3}-\frac {c \cosh (2 \text {arcsinh}(c+d x)) \sqrt {a+b \text {arcsinh}(c+d x)}}{2 d^3} \]

[In]

Int[x^2*Sqrt[a + b*ArcSinh[c + d*x]],x]

[Out]

(c^2*(c + d*x)*Sqrt[a + b*ArcSinh[c + d*x]])/d^3 + ((c + d*x)^3*Sqrt[a + b*ArcSinh[c + d*x]])/(3*d^3) - (c*Sqr
t[a + b*ArcSinh[c + d*x]]*Cosh[2*ArcSinh[c + d*x]])/(2*d^3) - (Sqrt[b]*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh
[c + d*x]]/Sqrt[b]])/(16*d^3) + (Sqrt[b]*c^2*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(4*d^
3) + (Sqrt[b]*c*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(8*d^3) + (Sqrt[b]
*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(48*d^3) + (Sqrt[b]*Sqrt[Pi]*Erfi
[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(16*d^3*E^(a/b)) - (Sqrt[b]*c^2*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d
*x]]/Sqrt[b]])/(4*d^3*E^(a/b)) + (Sqrt[b]*c*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(
8*d^3*E^((2*a)/b)) - (Sqrt[b]*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(48*d^3*E^((3*a
)/b))

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 5406

Int[Sinh[(c_.) + (d_.)*(x_)^(n_)], x_Symbol] :> Dist[1/2, Int[E^(c + d*x^n), x], x] - Dist[1/2, Int[E^(-c - d*
x^n), x], x] /; FreeQ[{c, d}, x] && IGtQ[n, 1]

Rule 5407

Int[Cosh[(c_.) + (d_.)*(x_)^(n_)], x_Symbol] :> Dist[1/2, Int[E^(c + d*x^n), x], x] + Dist[1/2, Int[E^(-c - d*
x^n), x], x] /; FreeQ[{c, d}, x] && IGtQ[n, 1]

Rule 5408

Int[((a_.) + (b_.)*Sinh[(c_.) + (d_.)*(x_)^(n_)])^(p_), x_Symbol] :> Int[ExpandTrigReduce[(a + b*Sinh[c + d*x^
n])^p, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 1] && IGtQ[p, 1]

Rule 5432

Int[((e_.)*(x_))^(m_.)*Sinh[(c_.) + (d_.)*(x_)^(n_)], x_Symbol] :> Simp[e^(n - 1)*(e*x)^(m - n + 1)*(Cosh[c +
d*x^n]/(d*n)), x] - Dist[e^n*((m - n + 1)/(d*n)), Int[(e*x)^(m - n)*Cosh[c + d*x^n], x], x] /; FreeQ[{c, d, e}
, x] && IGtQ[n, 0] && LtQ[0, n, m + 1]

Rule 5433

Int[Cosh[(c_.) + (d_.)*(x_)^(n_)]*((e_.)*(x_))^(m_.), x_Symbol] :> Simp[e^(n - 1)*(e*x)^(m - n + 1)*(Sinh[c +
d*x^n]/(d*n)), x] - Dist[e^n*((m - n + 1)/(d*n)), Int[(e*x)^(m - n)*Sinh[c + d*x^n], x], x] /; FreeQ[{c, d, e}
, x] && IGtQ[n, 0] && LtQ[0, n, m + 1]

Rule 5480

Int[Cosh[(a_.) + (b_.)*(x_)^(n_.)]*(x_)^(m_.)*Sinh[(a_.) + (b_.)*(x_)^(n_.)]^(p_.), x_Symbol] :> Simp[x^(m - n
+ 1)*(Sinh[a + b*x^n]^(p + 1)/(b*n*(p + 1))), x] - Dist[(m - n + 1)/(b*n*(p + 1)), Int[x^(m - n)*Sinh[a + b*x
^n]^(p + 1), x], x] /; FreeQ[{a, b, p}, x] && LtQ[0, n, m + 1] && NeQ[p, -1]

Rule 5830

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Dist[1/c^(m + 1), Subst
[Int[(a + b*x)^n*Cosh[x]*(c*d + e*Sinh[x])^m, x], x, ArcSinh[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ
[m, 0]

Rule 5859

Int[((a_.) + ArcSinh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[
Int[((d*e - c*f)/d + f*(x/d))^m*(a + b*ArcSinh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x
]

Rule 6873

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \left (-\frac {c}{d}+\frac {x}{d}\right )^2 \sqrt {a+b \text {arcsinh}(x)} \, dx,x,c+d x\right )}{d} \\ & = \frac {\text {Subst}\left (\int \sqrt {a+b x} \cosh (x) \left (-\frac {c}{d}+\frac {\sinh (x)}{d}\right )^2 \, dx,x,\text {arcsinh}(c+d x)\right )}{d} \\ & = \frac {2 \text {Subst}\left (\int x^2 \cosh \left (\frac {a-x^2}{b}\right ) \left (c+\sinh \left (\frac {a-x^2}{b}\right )\right )^2 \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{b d^3} \\ & = \frac {2 \text {Subst}\left (\int x^2 \cosh \left (\frac {a}{b}-\frac {x^2}{b}\right ) \left (c+\sinh \left (\frac {a-x^2}{b}\right )\right )^2 \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{b d^3} \\ & = \frac {2 \text {Subst}\left (\int \left (c^2 x^2 \cosh \left (\frac {a}{b}-\frac {x^2}{b}\right )+c x^2 \sinh \left (\frac {2 a}{b}-\frac {2 x^2}{b}\right )+x^2 \cosh \left (\frac {a}{b}-\frac {x^2}{b}\right ) \sinh ^2\left (\frac {a}{b}-\frac {x^2}{b}\right )\right ) \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{b d^3} \\ & = \frac {2 \text {Subst}\left (\int x^2 \cosh \left (\frac {a}{b}-\frac {x^2}{b}\right ) \sinh ^2\left (\frac {a}{b}-\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{b d^3}+\frac {(2 c) \text {Subst}\left (\int x^2 \sinh \left (\frac {2 a}{b}-\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{b d^3}+\frac {\left (2 c^2\right ) \text {Subst}\left (\int x^2 \cosh \left (\frac {a}{b}-\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{b d^3} \\ & = \frac {c^2 (c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}}{d^3}+\frac {(c+d x)^3 \sqrt {a+b \text {arcsinh}(c+d x)}}{3 d^3}-\frac {c \sqrt {a+b \text {arcsinh}(c+d x)} \cosh (2 \text {arcsinh}(c+d x))}{2 d^3}+\frac {\text {Subst}\left (\int \sinh ^3\left (\frac {a}{b}-\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{3 d^3}+\frac {c \text {Subst}\left (\int \cosh \left (\frac {2 a}{b}-\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{2 d^3}+\frac {c^2 \text {Subst}\left (\int \sinh \left (\frac {a}{b}-\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{d^3} \\ & = \frac {c^2 (c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}}{d^3}+\frac {(c+d x)^3 \sqrt {a+b \text {arcsinh}(c+d x)}}{3 d^3}-\frac {c \sqrt {a+b \text {arcsinh}(c+d x)} \cosh (2 \text {arcsinh}(c+d x))}{2 d^3}+\frac {\text {Subst}\left (\int \left (\frac {1}{4} \sinh \left (\frac {3 a}{b}-\frac {3 x^2}{b}\right )-\frac {3}{4} \sinh \left (\frac {a}{b}-\frac {x^2}{b}\right )\right ) \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{3 d^3}+\frac {c \text {Subst}\left (\int e^{\frac {2 a}{b}-\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{4 d^3}+\frac {c \text {Subst}\left (\int e^{-\frac {2 a}{b}+\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{4 d^3}+\frac {c^2 \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{2 d^3}-\frac {c^2 \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{2 d^3} \\ & = \frac {c^2 (c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}}{d^3}+\frac {(c+d x)^3 \sqrt {a+b \text {arcsinh}(c+d x)}}{3 d^3}-\frac {c \sqrt {a+b \text {arcsinh}(c+d x)} \cosh (2 \text {arcsinh}(c+d x))}{2 d^3}+\frac {\sqrt {b} c^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{4 d^3}+\frac {\sqrt {b} c e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{8 d^3}-\frac {\sqrt {b} c^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{4 d^3}+\frac {\sqrt {b} c e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{8 d^3}+\frac {\text {Subst}\left (\int \sinh \left (\frac {3 a}{b}-\frac {3 x^2}{b}\right ) \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{12 d^3}-\frac {\text {Subst}\left (\int \sinh \left (\frac {a}{b}-\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{4 d^3} \\ & = \frac {c^2 (c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}}{d^3}+\frac {(c+d x)^3 \sqrt {a+b \text {arcsinh}(c+d x)}}{3 d^3}-\frac {c \sqrt {a+b \text {arcsinh}(c+d x)} \cosh (2 \text {arcsinh}(c+d x))}{2 d^3}+\frac {\sqrt {b} c^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{4 d^3}+\frac {\sqrt {b} c e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{8 d^3}-\frac {\sqrt {b} c^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{4 d^3}+\frac {\sqrt {b} c e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{8 d^3}+\frac {\text {Subst}\left (\int e^{\frac {3 a}{b}-\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{24 d^3}-\frac {\text {Subst}\left (\int e^{-\frac {3 a}{b}+\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{24 d^3}-\frac {\text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{8 d^3}+\frac {\text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{8 d^3} \\ & = \frac {c^2 (c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}}{d^3}+\frac {(c+d x)^3 \sqrt {a+b \text {arcsinh}(c+d x)}}{3 d^3}-\frac {c \sqrt {a+b \text {arcsinh}(c+d x)} \cosh (2 \text {arcsinh}(c+d x))}{2 d^3}-\frac {\sqrt {b} e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{16 d^3}+\frac {\sqrt {b} c^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{4 d^3}+\frac {\sqrt {b} c e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{8 d^3}+\frac {\sqrt {b} e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{48 d^3}+\frac {\sqrt {b} e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{16 d^3}-\frac {\sqrt {b} c^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{4 d^3}+\frac {\sqrt {b} c e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{8 d^3}-\frac {\sqrt {b} e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{48 d^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.53 (sec) , antiderivative size = 656, normalized size of antiderivative = 1.32 \[ \int x^2 \sqrt {a+b \text {arcsinh}(c+d x)} \, dx=\frac {-36 (c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}+144 c^2 (c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}-72 c \sqrt {a+b \text {arcsinh}(c+d x)} \cosh (2 \text {arcsinh}(c+d x))+\sqrt {b} \sqrt {3 \pi } \cosh \left (\frac {3 a}{b}\right ) \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+9 \sqrt {b} \sqrt {\pi } \cosh \left (\frac {a}{b}\right ) \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-36 \sqrt {b} c^2 \sqrt {\pi } \cosh \left (\frac {a}{b}\right ) \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+9 \sqrt {b} c \sqrt {2 \pi } \cosh \left (\frac {2 a}{b}\right ) \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\sqrt {b} \sqrt {3 \pi } \cosh \left (\frac {3 a}{b}\right ) \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-9 \sqrt {b} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \sinh \left (\frac {a}{b}\right )+36 \sqrt {b} c^2 \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \sinh \left (\frac {a}{b}\right )+9 \sqrt {b} \left (-1+4 c^2\right ) \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )+\sinh \left (\frac {a}{b}\right )\right )-9 \sqrt {b} c \sqrt {2 \pi } \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \sinh \left (\frac {2 a}{b}\right )+9 \sqrt {b} c \sqrt {2 \pi } \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {2 a}{b}\right )+\sinh \left (\frac {2 a}{b}\right )\right )+\sqrt {b} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \sinh \left (\frac {3 a}{b}\right )+\sqrt {b} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \sinh \left (\frac {3 a}{b}\right )+12 \sqrt {a+b \text {arcsinh}(c+d x)} \sinh (3 \text {arcsinh}(c+d x))}{144 d^3} \]

[In]

Integrate[x^2*Sqrt[a + b*ArcSinh[c + d*x]],x]

[Out]

(-36*(c + d*x)*Sqrt[a + b*ArcSinh[c + d*x]] + 144*c^2*(c + d*x)*Sqrt[a + b*ArcSinh[c + d*x]] - 72*c*Sqrt[a + b
*ArcSinh[c + d*x]]*Cosh[2*ArcSinh[c + d*x]] + Sqrt[b]*Sqrt[3*Pi]*Cosh[(3*a)/b]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh
[c + d*x]])/Sqrt[b]] + 9*Sqrt[b]*Sqrt[Pi]*Cosh[a/b]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]] - 36*Sqrt[b]*c^
2*Sqrt[Pi]*Cosh[a/b]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]] + 9*Sqrt[b]*c*Sqrt[2*Pi]*Cosh[(2*a)/b]*Erfi[(S
qrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]] - Sqrt[b]*Sqrt[3*Pi]*Cosh[(3*a)/b]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcS
inh[c + d*x]])/Sqrt[b]] - 9*Sqrt[b]*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]]*Sinh[a/b] + 36*Sqrt[b]
*c^2*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]]*Sinh[a/b] + 9*Sqrt[b]*(-1 + 4*c^2)*Sqrt[Pi]*Erf[Sqrt[
a + b*ArcSinh[c + d*x]]/Sqrt[b]]*(Cosh[a/b] + Sinh[a/b]) - 9*Sqrt[b]*c*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[a + b*Arc
Sinh[c + d*x]])/Sqrt[b]]*Sinh[(2*a)/b] + 9*Sqrt[b]*c*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqr
t[b]]*(Cosh[(2*a)/b] + Sinh[(2*a)/b]) + Sqrt[b]*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]]
*Sinh[(3*a)/b] + Sqrt[b]*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]]*Sinh[(3*a)/b] + 12*Sq
rt[a + b*ArcSinh[c + d*x]]*Sinh[3*ArcSinh[c + d*x]])/(144*d^3)

Maple [F]

\[\int x^{2} \sqrt {a +b \,\operatorname {arcsinh}\left (d x +c \right )}d x\]

[In]

int(x^2*(a+b*arcsinh(d*x+c))^(1/2),x)

[Out]

int(x^2*(a+b*arcsinh(d*x+c))^(1/2),x)

Fricas [F(-2)]

Exception generated. \[ \int x^2 \sqrt {a+b \text {arcsinh}(c+d x)} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x^2*(a+b*arcsinh(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int x^2 \sqrt {a+b \text {arcsinh}(c+d x)} \, dx=\int x^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}}\, dx \]

[In]

integrate(x**2*(a+b*asinh(d*x+c))**(1/2),x)

[Out]

Integral(x**2*sqrt(a + b*asinh(c + d*x)), x)

Maxima [F]

\[ \int x^2 \sqrt {a+b \text {arcsinh}(c+d x)} \, dx=\int { \sqrt {b \operatorname {arsinh}\left (d x + c\right ) + a} x^{2} \,d x } \]

[In]

integrate(x^2*(a+b*arcsinh(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(b*arcsinh(d*x + c) + a)*x^2, x)

Giac [F]

\[ \int x^2 \sqrt {a+b \text {arcsinh}(c+d x)} \, dx=\int { \sqrt {b \operatorname {arsinh}\left (d x + c\right ) + a} x^{2} \,d x } \]

[In]

integrate(x^2*(a+b*arcsinh(d*x+c))^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(b*arcsinh(d*x + c) + a)*x^2, x)

Mupad [F(-1)]

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

[In]

int(x^2*(a + b*asinh(c + d*x))^(1/2),x)

[Out]

int(x^2*(a + b*asinh(c + d*x))^(1/2), x)