3.103 \(\int x (a+b \sinh ^{-1}(c+d x))^{5/2} \, dx\)

Optimal. Leaf size=389 \[ -\frac{15 \sqrt{\pi } b^{5/2} c e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{16 d^2}-\frac{15 \sqrt{\frac{\pi }{2}} b^{5/2} e^{\frac{2 a}{b}} \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{256 d^2}+\frac{15 \sqrt{\pi } b^{5/2} c e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{16 d^2}-\frac{15 \sqrt{\frac{\pi }{2}} b^{5/2} e^{-\frac{2 a}{b}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{256 d^2}-\frac{15 b^2 c (c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)}}{4 d^2}+\frac{15 b^2 \cosh \left (2 \sinh ^{-1}(c+d x)\right ) \sqrt{a+b \sinh ^{-1}(c+d x)}}{64 d^2}+\frac{5 b c \sqrt{(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{2 d^2}-\frac{5 b \sinh \left (2 \sinh ^{-1}(c+d x)\right ) \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{16 d^2}-\frac{c (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{d^2}+\frac{\cosh \left (2 \sinh ^{-1}(c+d x)\right ) \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{4 d^2} \]

[Out]

(-15*b^2*c*(c + d*x)*Sqrt[a + b*ArcSinh[c + d*x]])/(4*d^2) + (5*b*c*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d
*x])^(3/2))/(2*d^2) - (c*(c + d*x)*(a + b*ArcSinh[c + d*x])^(5/2))/d^2 + (15*b^2*Sqrt[a + b*ArcSinh[c + d*x]]*
Cosh[2*ArcSinh[c + d*x]])/(64*d^2) + ((a + b*ArcSinh[c + d*x])^(5/2)*Cosh[2*ArcSinh[c + d*x]])/(4*d^2) - (15*b
^(5/2)*c*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(16*d^2) - (15*b^(5/2)*E^((2*a)/b)*Sqrt[P
i/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(256*d^2) + (15*b^(5/2)*c*Sqrt[Pi]*Erfi[Sqrt[a + b*A
rcSinh[c + d*x]]/Sqrt[b]])/(16*d^2*E^(a/b)) - (15*b^(5/2)*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]
])/Sqrt[b]])/(256*d^2*E^((2*a)/b)) - (5*b*(a + b*ArcSinh[c + d*x])^(3/2)*Sinh[2*ArcSinh[c + d*x]])/(16*d^2)

________________________________________________________________________________________

Rubi [A]  time = 1.12634, antiderivative size = 389, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 10, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {5865, 5805, 6741, 6742, 5325, 5324, 5298, 2205, 2204, 5299} \[ -\frac{15 \sqrt{\pi } b^{5/2} c e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{16 d^2}-\frac{15 \sqrt{\frac{\pi }{2}} b^{5/2} e^{\frac{2 a}{b}} \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{256 d^2}+\frac{15 \sqrt{\pi } b^{5/2} c e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{16 d^2}-\frac{15 \sqrt{\frac{\pi }{2}} b^{5/2} e^{-\frac{2 a}{b}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{256 d^2}-\frac{15 b^2 c (c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)}}{4 d^2}+\frac{15 b^2 \cosh \left (2 \sinh ^{-1}(c+d x)\right ) \sqrt{a+b \sinh ^{-1}(c+d x)}}{64 d^2}+\frac{5 b c \sqrt{(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{2 d^2}-\frac{5 b \sinh \left (2 \sinh ^{-1}(c+d x)\right ) \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{16 d^2}-\frac{c (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{d^2}+\frac{\cosh \left (2 \sinh ^{-1}(c+d x)\right ) \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{4 d^2} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*ArcSinh[c + d*x])^(5/2),x]

[Out]

(-15*b^2*c*(c + d*x)*Sqrt[a + b*ArcSinh[c + d*x]])/(4*d^2) + (5*b*c*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d
*x])^(3/2))/(2*d^2) - (c*(c + d*x)*(a + b*ArcSinh[c + d*x])^(5/2))/d^2 + (15*b^2*Sqrt[a + b*ArcSinh[c + d*x]]*
Cosh[2*ArcSinh[c + d*x]])/(64*d^2) + ((a + b*ArcSinh[c + d*x])^(5/2)*Cosh[2*ArcSinh[c + d*x]])/(4*d^2) - (15*b
^(5/2)*c*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(16*d^2) - (15*b^(5/2)*E^((2*a)/b)*Sqrt[P
i/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(256*d^2) + (15*b^(5/2)*c*Sqrt[Pi]*Erfi[Sqrt[a + b*A
rcSinh[c + d*x]]/Sqrt[b]])/(16*d^2*E^(a/b)) - (15*b^(5/2)*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]
])/Sqrt[b]])/(256*d^2*E^((2*a)/b)) - (5*b*(a + b*ArcSinh[c + d*x])^(3/2)*Sinh[2*ArcSinh[c + d*x]])/(16*d^2)

Rule 5865

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 5805

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 6741

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

Rule 6742

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

Rule 5325

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 5324

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 5298

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 2205

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 2204

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 5299

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]

Rubi steps

\begin{align*} \int x \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2} \, dx &=\frac{\operatorname{Subst}\left (\int \left (-\frac{c}{d}+\frac{x}{d}\right ) \left (a+b \sinh ^{-1}(x)\right )^{5/2} \, dx,x,c+d x\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int (a+b x)^{5/2} \cosh (x) \left (-\frac{c}{d}+\frac{\sinh (x)}{d}\right ) \, dx,x,\sinh ^{-1}(c+d x)\right )}{d}\\ &=-\frac{2 \operatorname{Subst}\left (\int x^6 \cosh \left (\frac{a-x^2}{b}\right ) \left (c+\sinh \left (\frac{a-x^2}{b}\right )\right ) \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{b d^2}\\ &=-\frac{2 \operatorname{Subst}\left (\int x^6 \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 \sinh ^{-1}(c+d x)}\right )}{b d^2}\\ &=-\frac{2 \operatorname{Subst}\left (\int \left (c x^6 \cosh \left (\frac{a}{b}-\frac{x^2}{b}\right )+\frac{1}{2} x^6 \sinh \left (\frac{2 a}{b}-\frac{2 x^2}{b}\right )\right ) \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{b d^2}\\ &=-\frac{\operatorname{Subst}\left (\int x^6 \sinh \left (\frac{2 a}{b}-\frac{2 x^2}{b}\right ) \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{b d^2}-\frac{(2 c) \operatorname{Subst}\left (\int x^6 \cosh \left (\frac{a}{b}-\frac{x^2}{b}\right ) \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{b d^2}\\ &=-\frac{c (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{d^2}+\frac{\left (a+b \sinh ^{-1}(c+d x)\right )^{5/2} \cosh \left (2 \sinh ^{-1}(c+d x)\right )}{4 d^2}-\frac{5 \operatorname{Subst}\left (\int x^4 \cosh \left (\frac{2 a}{b}-\frac{2 x^2}{b}\right ) \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{4 d^2}-\frac{(5 c) \operatorname{Subst}\left (\int x^4 \sinh \left (\frac{a}{b}-\frac{x^2}{b}\right ) \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{d^2}\\ &=\frac{5 b c \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{2 d^2}-\frac{c (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{d^2}+\frac{\left (a+b \sinh ^{-1}(c+d x)\right )^{5/2} \cosh \left (2 \sinh ^{-1}(c+d x)\right )}{4 d^2}-\frac{5 b \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2} \sinh \left (2 \sinh ^{-1}(c+d x)\right )}{16 d^2}-\frac{(15 b) \operatorname{Subst}\left (\int x^2 \sinh \left (\frac{2 a}{b}-\frac{2 x^2}{b}\right ) \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{16 d^2}-\frac{(15 b c) \operatorname{Subst}\left (\int x^2 \cosh \left (\frac{a}{b}-\frac{x^2}{b}\right ) \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{2 d^2}\\ &=-\frac{15 b^2 c (c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)}}{4 d^2}+\frac{5 b c \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{2 d^2}-\frac{c (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{d^2}+\frac{15 b^2 \sqrt{a+b \sinh ^{-1}(c+d x)} \cosh \left (2 \sinh ^{-1}(c+d x)\right )}{64 d^2}+\frac{\left (a+b \sinh ^{-1}(c+d x)\right )^{5/2} \cosh \left (2 \sinh ^{-1}(c+d x)\right )}{4 d^2}-\frac{5 b \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2} \sinh \left (2 \sinh ^{-1}(c+d x)\right )}{16 d^2}-\frac{\left (15 b^2\right ) \operatorname{Subst}\left (\int \cosh \left (\frac{2 a}{b}-\frac{2 x^2}{b}\right ) \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{64 d^2}-\frac{\left (15 b^2 c\right ) \operatorname{Subst}\left (\int \sinh \left (\frac{a}{b}-\frac{x^2}{b}\right ) \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{4 d^2}\\ &=-\frac{15 b^2 c (c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)}}{4 d^2}+\frac{5 b c \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{2 d^2}-\frac{c (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{d^2}+\frac{15 b^2 \sqrt{a+b \sinh ^{-1}(c+d x)} \cosh \left (2 \sinh ^{-1}(c+d x)\right )}{64 d^2}+\frac{\left (a+b \sinh ^{-1}(c+d x)\right )^{5/2} \cosh \left (2 \sinh ^{-1}(c+d x)\right )}{4 d^2}-\frac{5 b \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2} \sinh \left (2 \sinh ^{-1}(c+d x)\right )}{16 d^2}-\frac{\left (15 b^2\right ) \operatorname{Subst}\left (\int e^{\frac{2 a}{b}-\frac{2 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{128 d^2}-\frac{\left (15 b^2\right ) \operatorname{Subst}\left (\int e^{-\frac{2 a}{b}+\frac{2 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{128 d^2}-\frac{\left (15 b^2 c\right ) \operatorname{Subst}\left (\int e^{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{8 d^2}+\frac{\left (15 b^2 c\right ) \operatorname{Subst}\left (\int e^{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{8 d^2}\\ &=-\frac{15 b^2 c (c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)}}{4 d^2}+\frac{5 b c \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{2 d^2}-\frac{c (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{d^2}+\frac{15 b^2 \sqrt{a+b \sinh ^{-1}(c+d x)} \cosh \left (2 \sinh ^{-1}(c+d x)\right )}{64 d^2}+\frac{\left (a+b \sinh ^{-1}(c+d x)\right )^{5/2} \cosh \left (2 \sinh ^{-1}(c+d x)\right )}{4 d^2}-\frac{15 b^{5/2} c e^{a/b} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{16 d^2}-\frac{15 b^{5/2} e^{\frac{2 a}{b}} \sqrt{\frac{\pi }{2}} \text{erf}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{256 d^2}+\frac{15 b^{5/2} c e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{16 d^2}-\frac{15 b^{5/2} e^{-\frac{2 a}{b}} \sqrt{\frac{\pi }{2}} \text{erfi}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{256 d^2}-\frac{5 b \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2} \sinh \left (2 \sinh ^{-1}(c+d x)\right )}{16 d^2}\\ \end{align*}

Mathematica [B]  time = 9.2564, size = 939, normalized size = 2.41 \[ \frac{480 c \sqrt{\pi } \cosh \left (\frac{a}{b}\right ) \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right ) b^{5/2}-15 \sqrt{2 \pi } \cosh \left (\frac{2 a}{b}\right ) \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right ) b^{5/2}-480 c \sqrt{\pi } \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right ) \sinh \left (\frac{a}{b}\right ) b^{5/2}+15 \sqrt{2 \pi } \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right ) \sinh \left (\frac{2 a}{b}\right ) b^{5/2}-15 \sqrt{2 \pi } \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right ) \left (\cosh \left (\frac{2 a}{b}\right )+\sinh \left (\frac{2 a}{b}\right )\right ) b^{5/2}+128 \sinh ^{-1}(c+d x)^2 \sqrt{a+b \sinh ^{-1}(c+d x)} \cosh \left (2 \sinh ^{-1}(c+d x)\right ) b^2+120 \sqrt{a+b \sinh ^{-1}(c+d x)} \cosh \left (2 \sinh ^{-1}(c+d x)\right ) b^2-160 \sinh ^{-1}(c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)} \sinh \left (2 \sinh ^{-1}(c+d x)\right ) b^2-1920 c^2 \sqrt{a+b \sinh ^{-1}(c+d x)} b^2-512 c^2 \sinh ^{-1}(c+d x)^2 \sqrt{a+b \sinh ^{-1}(c+d x)} b^2-512 c d x \sinh ^{-1}(c+d x)^2 \sqrt{a+b \sinh ^{-1}(c+d x)} b^2-1920 c d x \sqrt{a+b \sinh ^{-1}(c+d x)} b^2+1280 c \sqrt{c^2+2 d x c+d^2 x^2+1} \sinh ^{-1}(c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)} b^2+256 a \sinh ^{-1}(c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)} \cosh \left (2 \sinh ^{-1}(c+d x)\right ) b+\frac{256 a^2 c e^{a/b} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \text{Gamma}\left (\frac{3}{2},\frac{a}{b}+\sinh ^{-1}(c+d x)\right ) b}{\sqrt{a+b \sinh ^{-1}(c+d x)}}+\frac{256 a^2 c e^{-\frac{a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \text{Gamma}\left (\frac{3}{2},-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right ) b}{\sqrt{a+b \sinh ^{-1}(c+d x)}}-160 a \sqrt{a+b \sinh ^{-1}(c+d x)} \sinh \left (2 \sinh ^{-1}(c+d x)\right ) b-1024 a c^2 \sinh ^{-1}(c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)} b-1024 a c d x \sinh ^{-1}(c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)} b+1280 a c \sqrt{c^2+2 d x c+d^2 x^2+1} \sqrt{a+b \sinh ^{-1}(c+d x)} b-128 a^2 c \sqrt{\pi } \cosh \left (\frac{a}{b}\right ) \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right ) \sqrt{b}+128 a^2 c \sqrt{\pi } \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right ) \sinh \left (\frac{a}{b}\right ) \sqrt{b}+32 \left (4 a^2-15 b^2\right ) c \sqrt{\pi } \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right ) \left (\cosh \left (\frac{a}{b}\right )+\sinh \left (\frac{a}{b}\right )\right ) \sqrt{b}+128 a^2 \sqrt{a+b \sinh ^{-1}(c+d x)} \cosh \left (2 \sinh ^{-1}(c+d x)\right )}{512 d^2} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x*(a + b*ArcSinh[c + d*x])^(5/2),x]

[Out]

(-1920*b^2*c^2*Sqrt[a + b*ArcSinh[c + d*x]] - 1920*b^2*c*d*x*Sqrt[a + b*ArcSinh[c + d*x]] + 1280*a*b*c*Sqrt[1
+ c^2 + 2*c*d*x + d^2*x^2]*Sqrt[a + b*ArcSinh[c + d*x]] - 1024*a*b*c^2*ArcSinh[c + d*x]*Sqrt[a + b*ArcSinh[c +
d*x]] - 1024*a*b*c*d*x*ArcSinh[c + d*x]*Sqrt[a + b*ArcSinh[c + d*x]] + 1280*b^2*c*Sqrt[1 + c^2 + 2*c*d*x + d^
2*x^2]*ArcSinh[c + d*x]*Sqrt[a + b*ArcSinh[c + d*x]] - 512*b^2*c^2*ArcSinh[c + d*x]^2*Sqrt[a + b*ArcSinh[c + d
*x]] - 512*b^2*c*d*x*ArcSinh[c + d*x]^2*Sqrt[a + b*ArcSinh[c + d*x]] + 128*a^2*Sqrt[a + b*ArcSinh[c + d*x]]*Co
sh[2*ArcSinh[c + d*x]] + 120*b^2*Sqrt[a + b*ArcSinh[c + d*x]]*Cosh[2*ArcSinh[c + d*x]] + 256*a*b*ArcSinh[c + d
*x]*Sqrt[a + b*ArcSinh[c + d*x]]*Cosh[2*ArcSinh[c + d*x]] + 128*b^2*ArcSinh[c + d*x]^2*Sqrt[a + b*ArcSinh[c +
d*x]]*Cosh[2*ArcSinh[c + d*x]] - 128*a^2*Sqrt[b]*c*Sqrt[Pi]*Cosh[a/b]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b
]] + 480*b^(5/2)*c*Sqrt[Pi]*Cosh[a/b]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]] - 15*b^(5/2)*Sqrt[2*Pi]*Cosh[
(2*a)/b]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]] + (256*a^2*b*c*E^(a/b)*Sqrt[a/b + ArcSinh[c + d*
x]]*Gamma[3/2, a/b + ArcSinh[c + d*x]])/Sqrt[a + b*ArcSinh[c + d*x]] + (256*a^2*b*c*Sqrt[-((a + b*ArcSinh[c +
d*x])/b)]*Gamma[3/2, -((a + b*ArcSinh[c + d*x])/b)])/(E^(a/b)*Sqrt[a + b*ArcSinh[c + d*x]]) + 128*a^2*Sqrt[b]*
c*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]]*Sinh[a/b] - 480*b^(5/2)*c*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSi
nh[c + d*x]]/Sqrt[b]]*Sinh[a/b] + 32*Sqrt[b]*(4*a^2 - 15*b^2)*c*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt
[b]]*(Cosh[a/b] + Sinh[a/b]) + 15*b^(5/2)*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]]*Sinh
[(2*a)/b] - 15*b^(5/2)*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]]*(Cosh[(2*a)/b] + Sinh[(2
*a)/b]) - 160*a*b*Sqrt[a + b*ArcSinh[c + d*x]]*Sinh[2*ArcSinh[c + d*x]] - 160*b^2*ArcSinh[c + d*x]*Sqrt[a + b*
ArcSinh[c + d*x]]*Sinh[2*ArcSinh[c + d*x]])/(512*d^2)

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Maple [F]  time = 0.166, size = 0, normalized size = 0. \begin{align*} \int x \left ( a+b{\it Arcsinh} \left ( dx+c \right ) \right ) ^{{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a+b*arcsinh(d*x+c))^(5/2),x)

[Out]

int(x*(a+b*arcsinh(d*x+c))^(5/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \operatorname{arsinh}\left (d x + c\right ) + a\right )}^{\frac{5}{2}} x\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*arcsinh(d*x + c) + a)^(5/2)*x, x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*arcsinh(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*asinh(d*x+c))**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \operatorname{arsinh}\left (d x + c\right ) + a\right )}^{\frac{5}{2}} x\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*arcsinh(d*x + c) + a)^(5/2)*x, x)