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

Optimal. Leaf size=365 \[ -\frac{2 \sqrt{\pi } c e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{3 b^{5/2} d^2}-\frac{2 \sqrt{2 \pi } e^{\frac{2 a}{b}} \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{3 b^{5/2} d^2}-\frac{2 \sqrt{\pi } c e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{3 b^{5/2} d^2}+\frac{2 \sqrt{2 \pi } e^{-\frac{2 a}{b}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{3 b^{5/2} d^2}-\frac{8 (c+d x)^2}{3 b^2 d^2 \sqrt{a+b \sinh ^{-1}(c+d x)}}+\frac{4 c (c+d x)}{3 b^2 d^2 \sqrt{a+b \sinh ^{-1}(c+d x)}}-\frac{4}{3 b^2 d^2 \sqrt{a+b \sinh ^{-1}(c+d x)}}-\frac{2 \sqrt{(c+d x)^2+1} (c+d x)}{3 b d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}+\frac{2 c \sqrt{(c+d x)^2+1}}{3 b d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}} \]

[Out]

(2*c*Sqrt[1 + (c + d*x)^2])/(3*b*d^2*(a + b*ArcSinh[c + d*x])^(3/2)) - (2*(c + d*x)*Sqrt[1 + (c + d*x)^2])/(3*
b*d^2*(a + b*ArcSinh[c + d*x])^(3/2)) - 4/(3*b^2*d^2*Sqrt[a + b*ArcSinh[c + d*x]]) + (4*c*(c + d*x))/(3*b^2*d^
2*Sqrt[a + b*ArcSinh[c + d*x]]) - (8*(c + d*x)^2)/(3*b^2*d^2*Sqrt[a + b*ArcSinh[c + d*x]]) - (2*c*E^(a/b)*Sqrt
[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(3*b^(5/2)*d^2) - (2*E^((2*a)/b)*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[
a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(3*b^(5/2)*d^2) - (2*c*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])
/(3*b^(5/2)*d^2*E^(a/b)) + (2*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(3*b^(5/2)*d^2*
E^((2*a)/b))

________________________________________________________________________________________

Rubi [A]  time = 0.882786, antiderivative size = 365, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 15, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.938, Rules used = {5865, 5803, 5655, 5774, 5657, 3307, 2180, 2205, 2204, 5667, 5669, 5448, 12, 3308, 5675} \[ -\frac{2 \sqrt{\pi } c e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{3 b^{5/2} d^2}-\frac{2 \sqrt{2 \pi } e^{\frac{2 a}{b}} \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{3 b^{5/2} d^2}-\frac{2 \sqrt{\pi } c e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{3 b^{5/2} d^2}+\frac{2 \sqrt{2 \pi } e^{-\frac{2 a}{b}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{3 b^{5/2} d^2}-\frac{8 (c+d x)^2}{3 b^2 d^2 \sqrt{a+b \sinh ^{-1}(c+d x)}}+\frac{4 c (c+d x)}{3 b^2 d^2 \sqrt{a+b \sinh ^{-1}(c+d x)}}-\frac{4}{3 b^2 d^2 \sqrt{a+b \sinh ^{-1}(c+d x)}}-\frac{2 \sqrt{(c+d x)^2+1} (c+d x)}{3 b d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}+\frac{2 c \sqrt{(c+d x)^2+1}}{3 b d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*c*Sqrt[1 + (c + d*x)^2])/(3*b*d^2*(a + b*ArcSinh[c + d*x])^(3/2)) - (2*(c + d*x)*Sqrt[1 + (c + d*x)^2])/(3*
b*d^2*(a + b*ArcSinh[c + d*x])^(3/2)) - 4/(3*b^2*d^2*Sqrt[a + b*ArcSinh[c + d*x]]) + (4*c*(c + d*x))/(3*b^2*d^
2*Sqrt[a + b*ArcSinh[c + d*x]]) - (8*(c + d*x)^2)/(3*b^2*d^2*Sqrt[a + b*ArcSinh[c + d*x]]) - (2*c*E^(a/b)*Sqrt
[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(3*b^(5/2)*d^2) - (2*E^((2*a)/b)*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[
a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(3*b^(5/2)*d^2) - (2*c*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])
/(3*b^(5/2)*d^2*E^(a/b)) + (2*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(3*b^(5/2)*d^2*
E^((2*a)/b))

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 5803

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

Rule 5655

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

Rule 5774

Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp
[((f*x)^m*(a + b*ArcSinh[c*x])^(n + 1))/(b*c*Sqrt[d]*(n + 1)), x] - Dist[(f*m)/(b*c*Sqrt[d]*(n + 1)), Int[(f*x
)^(m - 1)*(a + b*ArcSinh[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && LtQ[n, -
1] && GtQ[d, 0]

Rule 5657

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 3307

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 2180

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] &&  !$UseGamma === True

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 5667

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[(x^m*Sqrt[1 + c^2*x^2]*(a + b*ArcSi
nh[c*x])^(n + 1))/(b*c*(n + 1)), x] + (-Dist[(c*(m + 1))/(b*(n + 1)), Int[(x^(m + 1)*(a + b*ArcSinh[c*x])^(n +
1))/Sqrt[1 + c^2*x^2], x], x] - Dist[m/(b*c*(n + 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] && LtQ[n, -2]

Rule 5669

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

Rule 5448

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 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 3308

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 5675

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

Rubi steps

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

Mathematica [A]  time = 2.6056, size = 375, normalized size = 1.03 \[ \frac{\frac{\sqrt{b} c e^{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \left (2 b e^{\sinh ^{-1}(c+d x)} \left (-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )+2 e^{\frac{2 a}{b}+\sinh ^{-1}(c+d x)} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \left (a+b \sinh ^{-1}(c+d x)\right ) \text{Gamma}\left (\frac{1}{2},\frac{a}{b}+\sinh ^{-1}(c+d x)\right )+e^{a/b} \left (2 a e^{2 \sinh ^{-1}(c+d x)}-2 a+b e^{2 \sinh ^{-1}(c+d x)}+2 b \left (e^{2 \sinh ^{-1}(c+d x)}-1\right ) \sinh ^{-1}(c+d x)+b\right )\right )}{\left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}-2 \sqrt{2 \pi } \left (\sinh \left (\frac{2 a}{b}\right )+\cosh \left (\frac{2 a}{b}\right )\right ) \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )+2 \sqrt{2 \pi } \left (\cosh \left (\frac{2 a}{b}\right )-\sinh \left (\frac{2 a}{b}\right )\right ) \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )-\frac{\sqrt{b} \left (4 \cosh \left (2 \sinh ^{-1}(c+d x)\right ) \left (a+b \sinh ^{-1}(c+d x)\right )+b \sinh \left (2 \sinh ^{-1}(c+d x)\right )\right )}{\left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}}{3 b^{5/2} d^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((Sqrt[b]*c*(E^(a/b)*(-2*a + b + 2*a*E^(2*ArcSinh[c + d*x]) + b*E^(2*ArcSinh[c + d*x]) + 2*b*(-1 + E^(2*ArcSin
h[c + d*x]))*ArcSinh[c + d*x]) + 2*E^((2*a)/b + ArcSinh[c + d*x])*Sqrt[a/b + ArcSinh[c + d*x]]*(a + b*ArcSinh[
c + d*x])*Gamma[1/2, a/b + ArcSinh[c + d*x]] + 2*b*E^ArcSinh[c + d*x]*(-((a + b*ArcSinh[c + d*x])/b))^(3/2)*Ga
mma[1/2, -((a + b*ArcSinh[c + d*x])/b)]))/(E^((a + b*ArcSinh[c + d*x])/b)*(a + b*ArcSinh[c + d*x])^(3/2)) + 2*
Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]]*(Cosh[(2*a)/b] - Sinh[(2*a)/b]) - 2*Sqrt[2*Pi]
*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]]*(Cosh[(2*a)/b] + Sinh[(2*a)/b]) - (Sqrt[b]*(4*(a + b*ArcS
inh[c + d*x])*Cosh[2*ArcSinh[c + d*x]] + b*Sinh[2*ArcSinh[c + d*x]]))/(a + b*ArcSinh[c + d*x])^(3/2))/(3*b^(5/
2)*d^2)

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Maple [F]  time = 0.165, 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 \frac{x}{{\left (b \operatorname{arsinh}\left (d x + c\right ) + a\right )}^{\frac{5}{2}}}\,{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(x/(b*arcsinh(d*x + c) + a)^(5/2), 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]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\left (a + b \operatorname{asinh}{\left (c + d x \right )}\right )^{\frac{5}{2}}}\, dx \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]

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

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