\(\int x \sqrt {a+b \text {arcsinh}(c+d x)} \, dx\) [99]

Optimal result

Integrand size = 16, antiderivative size = 259 \[ \int x \sqrt {a+b \text {arcsinh}(c+d x)} \, dx=-\frac {c (c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}}{d^2}+\frac {\sqrt {a+b \text {arcsinh}(c+d x)} \cosh (2 \text {arcsinh}(c+d x))}{4 d^2}-\frac {\sqrt {b} c e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{4 d^2}-\frac {\sqrt {b} e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{16 d^2}+\frac {\sqrt {b} c e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{4 d^2}-\frac {\sqrt {b} e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{16 d^2} \]

[Out]

-1/32*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^2-1/32*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^2/exp(2*a/b)-1/4*c*exp(a/b)*erf((a+b*arcsinh(
d*x+c))^(1/2)/b^(1/2))*b^(1/2)*Pi^(1/2)/d^2+1/4*c*erfi((a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))*b^(1/2)*Pi^(1/2)/d^
2/exp(a/b)-c*(d*x+c)*(a+b*arcsinh(d*x+c))^(1/2)/d^2+1/4*cosh(2*arcsinh(d*x+c))*(a+b*arcsinh(d*x+c))^(1/2)/d^2

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {5859, 5830, 6873, 6874, 5433, 5406, 2236, 2235, 5432, 5407} \[ \int x \sqrt {a+b \text {arcsinh}(c+d x)} \, dx=-\frac {\sqrt {\pi } \sqrt {b} c e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{4 d^2}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{16 d^2}+\frac {\sqrt {\pi } \sqrt {b} c e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{4 d^2}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{16 d^2}-\frac {c (c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}}{d^2}+\frac {\cosh (2 \text {arcsinh}(c+d x)) \sqrt {a+b \text {arcsinh}(c+d x)}}{4 d^2} \]

[In]

Int[x*Sqrt[a + b*ArcSinh[c + d*x]],x]

[Out]

-((c*(c + d*x)*Sqrt[a + b*ArcSinh[c + d*x]])/d^2) + (Sqrt[a + b*ArcSinh[c + d*x]]*Cosh[2*ArcSinh[c + d*x]])/(4
*d^2) - (Sqrt[b]*c*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(4*d^2) - (Sqrt[b]*E^((2*a)/b)*
Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(16*d^2) + (Sqrt[b]*c*Sqrt[Pi]*Erfi[Sqrt[a + b
*ArcSinh[c + d*x]]/Sqrt[b]])/(4*d^2*E^(a/b)) - (Sqrt[b]*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])
/Sqrt[b]])/(16*d^2*E^((2*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 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 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 ) \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 ) \, 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 ) \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{b d^2} \\ & = -\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 ) \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{b d^2} \\ & = -\frac {2 \text {Subst}\left (\int \left (c x^2 \cosh \left (\frac {a}{b}-\frac {x^2}{b}\right )+\frac {1}{2} x^2 \sinh \left (\frac {2 a}{b}-\frac {2 x^2}{b}\right )\right ) \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{b d^2} \\ & = -\frac {\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^2}-\frac {(2 c) \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^2} \\ & = -\frac {c (c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}}{d^2}+\frac {\sqrt {a+b \text {arcsinh}(c+d x)} \cosh (2 \text {arcsinh}(c+d x))}{4 d^2}-\frac {\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 )}{4 d^2}-\frac {c \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^2} \\ & = -\frac {c (c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}}{d^2}+\frac {\sqrt {a+b \text {arcsinh}(c+d x)} \cosh (2 \text {arcsinh}(c+d x))}{4 d^2}-\frac {\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 )}{8 d^2}-\frac {\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 )}{8 d^2}-\frac {c \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^2}+\frac {c \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^2} \\ & = -\frac {c (c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}}{d^2}+\frac {\sqrt {a+b \text {arcsinh}(c+d x)} \cosh (2 \text {arcsinh}(c+d x))}{4 d^2}-\frac {\sqrt {b} c e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{4 d^2}-\frac {\sqrt {b} 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 )}{16 d^2}+\frac {\sqrt {b} c e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{4 d^2}-\frac {\sqrt {b} 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 )}{16 d^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.27 (sec) , antiderivative size = 250, normalized size of antiderivative = 0.97 \[ \int x \sqrt {a+b \text {arcsinh}(c+d x)} \, dx=-\frac {-8 \sqrt {a+b \text {arcsinh}(c+d x)} \cosh (2 \text {arcsinh}(c+d x))+16 c e^{-\frac {a}{b}} \sqrt {a+b \text {arcsinh}(c+d x)} \left (-\frac {e^{\frac {2 a}{b}} \Gamma \left (\frac {3}{2},\frac {a}{b}+\text {arcsinh}(c+d x)\right )}{\sqrt {\frac {a}{b}+\text {arcsinh}(c+d x)}}+\frac {\Gamma \left (\frac {3}{2},-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{\sqrt {-\frac {a+b \text {arcsinh}(c+d x)}{b}}}\right )+\sqrt {b} \sqrt {2 \pi } \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {2 a}{b}\right )-\sinh \left (\frac {2 a}{b}\right )\right )+\sqrt {b} \sqrt {2 \pi } \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {2 a}{b}\right )+\sinh \left (\frac {2 a}{b}\right )\right )}{32 d^2} \]

[In]

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

[Out]

-1/32*(-8*Sqrt[a + b*ArcSinh[c + d*x]]*Cosh[2*ArcSinh[c + d*x]] + (16*c*Sqrt[a + b*ArcSinh[c + d*x]]*(-((E^((2
*a)/b)*Gamma[3/2, a/b + ArcSinh[c + d*x]])/Sqrt[a/b + ArcSinh[c + d*x]]) + Gamma[3/2, -((a + b*ArcSinh[c + d*x
])/b)]/Sqrt[-((a + b*ArcSinh[c + d*x])/b)]))/E^(a/b) + Sqrt[b]*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c +
d*x]])/Sqrt[b]]*(Cosh[(2*a)/b] - Sinh[(2*a)/b]) + Sqrt[b]*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]
])/Sqrt[b]]*(Cosh[(2*a)/b] + Sinh[(2*a)/b]))/d^2

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

integrate(x*(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]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(x*(a + b*asinh(c + d*x))^(1/2),x)

[Out]

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