3.59 \(\int \frac{x (a+b \text{csch}^{-1}(c x))}{\sqrt{d+e x}} \, dx\)

Optimal. Leaf size=474 \[ -\frac{8 b c d \sqrt{c^2 x^2+1} \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right ),-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{3 \left (-c^2\right )^{3/2} e x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}}-\frac{2 d \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right )}{e^2}+\frac{2 (d+e x)^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^2}+\frac{8 b d^2 \sqrt{c^2 x^2+1} \sqrt{\frac{\sqrt{-c^2} (d+e x)}{\sqrt{-c^2} d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|\frac{2 e}{\sqrt{-c^2} d+e}\right )}{3 c e^2 x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}}+\frac{4 b c \sqrt{c^2 x^2+1} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{3 \left (-c^2\right )^{3/2} e x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}}} \]

[Out]

(-2*d*Sqrt[d + e*x]*(a + b*ArcCsch[c*x]))/e^2 + (2*(d + e*x)^(3/2)*(a + b*ArcCsch[c*x]))/(3*e^2) + (4*b*c*Sqrt
[d + e*x]*Sqrt[1 + c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], (-2*Sqrt[-c^2]*e)/(c^2*d - Sqrt[
-c^2]*e)])/(3*(-c^2)^(3/2)*e*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[(c^2*(d + e*x))/(c^2*d - Sqrt[-c^2]*e)]) - (8*b*c*d*
Sqrt[(c^2*(d + e*x))/(c^2*d - Sqrt[-c^2]*e)]*Sqrt[1 + c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]
], (-2*Sqrt[-c^2]*e)/(c^2*d - Sqrt[-c^2]*e)])/(3*(-c^2)^(3/2)*e*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x]) + (8*b*
d^2*Sqrt[(Sqrt[-c^2]*(d + e*x))/(Sqrt[-c^2]*d + e)]*Sqrt[1 + c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - Sqrt[-c^2]
*x]/Sqrt[2]], (2*e)/(Sqrt[-c^2]*d + e)])/(3*c*e^2*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x])

________________________________________________________________________________________

Rubi [A]  time = 1.75063, antiderivative size = 474, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 13, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.684, Rules used = {43, 6310, 12, 6721, 6742, 719, 424, 944, 419, 932, 168, 538, 537} \[ -\frac{2 d \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right )}{e^2}+\frac{2 (d+e x)^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^2}+\frac{8 b d^2 \sqrt{c^2 x^2+1} \sqrt{\frac{\sqrt{-c^2} (d+e x)}{\sqrt{-c^2} d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|\frac{2 e}{\sqrt{-c^2} d+e}\right )}{3 c e^2 x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}}-\frac{8 b c d \sqrt{c^2 x^2+1} \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{3 \left (-c^2\right )^{3/2} e x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}}+\frac{4 b c \sqrt{c^2 x^2+1} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{3 \left (-c^2\right )^{3/2} e x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}}} \]

Antiderivative was successfully verified.

[In]

Int[(x*(a + b*ArcCsch[c*x]))/Sqrt[d + e*x],x]

[Out]

(-2*d*Sqrt[d + e*x]*(a + b*ArcCsch[c*x]))/e^2 + (2*(d + e*x)^(3/2)*(a + b*ArcCsch[c*x]))/(3*e^2) + (4*b*c*Sqrt
[d + e*x]*Sqrt[1 + c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], (-2*Sqrt[-c^2]*e)/(c^2*d - Sqrt[
-c^2]*e)])/(3*(-c^2)^(3/2)*e*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[(c^2*(d + e*x))/(c^2*d - Sqrt[-c^2]*e)]) - (8*b*c*d*
Sqrt[(c^2*(d + e*x))/(c^2*d - Sqrt[-c^2]*e)]*Sqrt[1 + c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]
], (-2*Sqrt[-c^2]*e)/(c^2*d - Sqrt[-c^2]*e)])/(3*(-c^2)^(3/2)*e*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x]) + (8*b*
d^2*Sqrt[(Sqrt[-c^2]*(d + e*x))/(Sqrt[-c^2]*d + e)]*Sqrt[1 + c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - Sqrt[-c^2]
*x]/Sqrt[2]], (2*e)/(Sqrt[-c^2]*d + e)])/(3*c*e^2*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x])

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 6310

Int[((a_.) + ArcCsch[(c_.)*(x_)]*(b_.))*(u_), x_Symbol] :> With[{v = IntHide[u, x]}, Dist[a + b*ArcCsch[c*x],
v, x] + Dist[b/c, Int[SimplifyIntegrand[v/(x^2*Sqrt[1 + 1/(c^2*x^2)]), x], x], x] /; InverseFunctionFreeQ[v, x
]] /; FreeQ[{a, b, c}, x]

Rule 12

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

Rule 6721

Int[(u_.)*((a_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(b^IntPart[p]*(a + b*x^n)^FracPart[p])/(x^(n*FracP
art[p])*(1 + a/(x^n*b))^FracPart[p]), Int[u*x^(n*p)*(1 + a/(x^n*b))^p, x], x] /; FreeQ[{a, b, p}, x] &&  !Inte
gerQ[p] && ILtQ[n, 0] &&  !RationalFunctionQ[u, x] && IntegerQ[p + 1/2]

Rule 6742

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

Rule 719

Int[((d_) + (e_.)*(x_))^(m_)/Sqrt[(a_) + (c_.)*(x_)^2], x_Symbol] :> Dist[(2*a*Rt[-(c/a), 2]*(d + e*x)^m*Sqrt[
1 + (c*x^2)/a])/(c*Sqrt[a + c*x^2]*((c*(d + e*x))/(c*d - a*e*Rt[-(c/a), 2]))^m), Subst[Int[(1 + (2*a*e*Rt[-(c/
a), 2]*x^2)/(c*d - a*e*Rt[-(c/a), 2]))^m/Sqrt[1 - x^2], x], x, Sqrt[(1 - Rt[-(c/a), 2]*x)/2]], x] /; FreeQ[{a,
c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m^2, 1/4]

Rule 424

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[ArcSin[Rt[-(d/c)
, 2]*x], (b*c)/(a*d)])/(Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[
a, 0]

Rule 944

Int[Sqrt[(f_.) + (g_.)*(x_)]/(((d_.) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[g/e, Int[1/(S
qrt[f + g*x]*Sqrt[a + c*x^2]), x], x] + Dist[(e*f - d*g)/e, Int[1/((d + e*x)*Sqrt[f + g*x]*Sqrt[a + c*x^2]), x
], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0]

Rule 419

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1*EllipticF[ArcSin[Rt[-(d/c),
2]*x], (b*c)/(a*d)])/(Sqrt[a]*Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &
& GtQ[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-(b/a), -(d/c)])

Rule 932

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[-(c
/a), 2]}, Dist[1/Sqrt[a], Int[1/((d + e*x)*Sqrt[f + g*x]*Sqrt[1 - q*x]*Sqrt[1 + q*x]), x], x]] /; FreeQ[{a, c,
d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] && GtQ[a, 0]

Rule 168

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Sym
bol] :> Dist[-2, Subst[Int[1/(Simp[b*c - a*d - b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + (f*x^2)/d, x]]*Sqrt[Simp[(d
*g - c*h)/d + (h*x^2)/d, x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && GtQ[(d*e - c
*f)/d, 0]

Rule 538

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[1 +
(d*x^2)/c]/Sqrt[c + d*x^2], Int[1/((a + b*x^2)*Sqrt[1 + (d*x^2)/c]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c,
d, e, f}, x] &&  !GtQ[c, 0]

Rule 537

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1*Ellipt
icPi[(b*c)/(a*d), ArcSin[Rt[-(d/c), 2]*x], (c*f)/(d*e)])/(a*Sqrt[c]*Sqrt[e]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b,
c, d, e, f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && SimplerSqrtQ[-(f/e), -(d/c)
])

Rubi steps

\begin{align*} \int \frac{x \left (a+b \text{csch}^{-1}(c x)\right )}{\sqrt{d+e x}} \, dx &=-\frac{2 d \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right )}{e^2}+\frac{2 (d+e x)^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^2}+\frac{b \int \frac{2 (-2 d+e x) \sqrt{d+e x}}{3 e^2 \sqrt{1+\frac{1}{c^2 x^2}} x^2} \, dx}{c}\\ &=-\frac{2 d \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right )}{e^2}+\frac{2 (d+e x)^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^2}+\frac{(2 b) \int \frac{(-2 d+e x) \sqrt{d+e x}}{\sqrt{1+\frac{1}{c^2 x^2}} x^2} \, dx}{3 c e^2}\\ &=-\frac{2 d \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right )}{e^2}+\frac{2 (d+e x)^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^2}+\frac{\left (2 b \sqrt{1+c^2 x^2}\right ) \int \frac{(-2 d+e x) \sqrt{d+e x}}{x \sqrt{1+c^2 x^2}} \, dx}{3 c e^2 \sqrt{1+\frac{1}{c^2 x^2}} x}\\ &=-\frac{2 d \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right )}{e^2}+\frac{2 (d+e x)^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^2}+\frac{\left (2 b \sqrt{1+c^2 x^2}\right ) \int \left (\frac{e \sqrt{d+e x}}{\sqrt{1+c^2 x^2}}-\frac{2 d \sqrt{d+e x}}{x \sqrt{1+c^2 x^2}}\right ) \, dx}{3 c e^2 \sqrt{1+\frac{1}{c^2 x^2}} x}\\ &=-\frac{2 d \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right )}{e^2}+\frac{2 (d+e x)^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^2}-\frac{\left (4 b d \sqrt{1+c^2 x^2}\right ) \int \frac{\sqrt{d+e x}}{x \sqrt{1+c^2 x^2}} \, dx}{3 c e^2 \sqrt{1+\frac{1}{c^2 x^2}} x}+\frac{\left (2 b \sqrt{1+c^2 x^2}\right ) \int \frac{\sqrt{d+e x}}{\sqrt{1+c^2 x^2}} \, dx}{3 c e \sqrt{1+\frac{1}{c^2 x^2}} x}\\ &=-\frac{2 d \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right )}{e^2}+\frac{2 (d+e x)^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^2}-\frac{\left (4 b d^2 \sqrt{1+c^2 x^2}\right ) \int \frac{1}{x \sqrt{d+e x} \sqrt{1+c^2 x^2}} \, dx}{3 c e^2 \sqrt{1+\frac{1}{c^2 x^2}} x}-\frac{\left (4 b d \sqrt{1+c^2 x^2}\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{1+c^2 x^2}} \, dx}{3 c e \sqrt{1+\frac{1}{c^2 x^2}} x}+\frac{\left (4 b \sqrt{-c^2} \sqrt{d+e x} \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{2 \sqrt{-c^2} e x^2}{c^2 d-\sqrt{-c^2} e}}}{\sqrt{1-x^2}} \, dx,x,\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )}{3 c^3 e \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}}}\\ &=-\frac{2 d \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right )}{e^2}+\frac{2 (d+e x)^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^2}+\frac{4 b \sqrt{-c^2} \sqrt{d+e x} \sqrt{1+c^2 x^2} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{3 c^3 e \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}}}-\frac{\left (4 b d^2 \sqrt{1+c^2 x^2}\right ) \int \frac{1}{x \sqrt{1-\sqrt{-c^2} x} \sqrt{1+\sqrt{-c^2} x} \sqrt{d+e x}} \, dx}{3 c e^2 \sqrt{1+\frac{1}{c^2 x^2}} x}-\frac{\left (8 b \sqrt{-c^2} d \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}} \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1+\frac{2 \sqrt{-c^2} e x^2}{c^2 d-\sqrt{-c^2} e}}} \, dx,x,\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )}{3 c^3 e \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}\\ &=-\frac{2 d \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right )}{e^2}+\frac{2 (d+e x)^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^2}+\frac{4 b \sqrt{-c^2} \sqrt{d+e x} \sqrt{1+c^2 x^2} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{3 c^3 e \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}}}-\frac{8 b \sqrt{-c^2} d \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}} \sqrt{1+c^2 x^2} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{3 c^3 e \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}+\frac{\left (8 b d^2 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right ) \sqrt{2-x^2} \sqrt{d+\frac{e}{\sqrt{-c^2}}-\frac{e x^2}{\sqrt{-c^2}}}} \, dx,x,\sqrt{1-\sqrt{-c^2} x}\right )}{3 c e^2 \sqrt{1+\frac{1}{c^2 x^2}} x}\\ &=-\frac{2 d \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right )}{e^2}+\frac{2 (d+e x)^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^2}+\frac{4 b \sqrt{-c^2} \sqrt{d+e x} \sqrt{1+c^2 x^2} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{3 c^3 e \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}}}-\frac{8 b \sqrt{-c^2} d \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}} \sqrt{1+c^2 x^2} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{3 c^3 e \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}+\frac{\left (8 b d^2 \sqrt{1+c^2 x^2} \sqrt{1+\frac{e \left (-1+\sqrt{-c^2} x\right )}{\sqrt{-c^2} d+e}}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right ) \sqrt{2-x^2} \sqrt{1-\frac{e x^2}{\sqrt{-c^2} \left (d+\frac{e}{\sqrt{-c^2}}\right )}}} \, dx,x,\sqrt{1-\sqrt{-c^2} x}\right )}{3 c e^2 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}\\ &=-\frac{2 d \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right )}{e^2}+\frac{2 (d+e x)^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^2}+\frac{4 b \sqrt{-c^2} \sqrt{d+e x} \sqrt{1+c^2 x^2} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{3 c^3 e \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}}}-\frac{8 b \sqrt{-c^2} d \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}} \sqrt{1+c^2 x^2} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{3 c^3 e \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}+\frac{8 b d^2 \sqrt{1+c^2 x^2} \sqrt{1-\frac{e \left (1-\sqrt{-c^2} x\right )}{\sqrt{-c^2} d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|\frac{2 e}{\sqrt{-c^2} d+e}\right )}{3 c e^2 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}\\ \end{align*}

Mathematica [C]  time = 1.27859, size = 343, normalized size = 0.72 \[ \frac{2 \left (\frac{2 b \sqrt{-\frac{e (c x-i)}{c d+i e}} \sqrt{-\frac{e (c x+i)}{c d-i e}} \left ((e+i c d) \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{-\frac{c}{c d-i e}} \sqrt{d+e x}\right ),\frac{c d-i e}{c d+i e}\right )+(-e+i c d) E\left (i \sinh ^{-1}\left (\sqrt{-\frac{c}{c d-i e}} \sqrt{d+e x}\right )|\frac{c d-i e}{c d+i e}\right )-2 i c d \Pi \left (1-\frac{i e}{c d};i \sinh ^{-1}\left (\sqrt{-\frac{c}{c d-i e}} \sqrt{d+e x}\right )|\frac{c d-i e}{c d+i e}\right )\right )}{c^2 x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{-\frac{c}{c d-i e}}}+a \sqrt{d+e x} (e x-2 d)+b \text{csch}^{-1}(c x) \sqrt{d+e x} (e x-2 d)\right )}{3 e^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(x*(a + b*ArcCsch[c*x]))/Sqrt[d + e*x],x]

[Out]

(2*(a*(-2*d + e*x)*Sqrt[d + e*x] + b*(-2*d + e*x)*Sqrt[d + e*x]*ArcCsch[c*x] + (2*b*Sqrt[-((e*(-I + c*x))/(c*d
+ I*e))]*Sqrt[-((e*(I + c*x))/(c*d - I*e))]*((I*c*d - e)*EllipticE[I*ArcSinh[Sqrt[-(c/(c*d - I*e))]*Sqrt[d +
e*x]], (c*d - I*e)/(c*d + I*e)] + (I*c*d + e)*EllipticF[I*ArcSinh[Sqrt[-(c/(c*d - I*e))]*Sqrt[d + e*x]], (c*d
- I*e)/(c*d + I*e)] - (2*I)*c*d*EllipticPi[1 - (I*e)/(c*d), I*ArcSinh[Sqrt[-(c/(c*d - I*e))]*Sqrt[d + e*x]], (
c*d - I*e)/(c*d + I*e)]))/(c^2*Sqrt[-(c/(c*d - I*e))]*Sqrt[1 + 1/(c^2*x^2)]*x)))/(3*e^2)

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Maple [C]  time = 0.299, size = 868, normalized size = 1.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/e^2*(a*(1/3*(e*x+d)^(3/2)-d*(e*x+d)^(1/2))+b*(1/3*(e*x+d)^(3/2)*arccsch(c*x)-arccsch(c*x)*d*(e*x+d)^(1/2)-2/
3/c^2*(-(I*(e*x+d)*c*e+(e*x+d)*c^2*d-c^2*d^2-e^2)/(c^2*d^2+e^2))^(1/2)*((I*(e*x+d)*c*e-(e*x+d)*c^2*d+c^2*d^2+e
^2)/(c^2*d^2+e^2))^(1/2)*(2*I*EllipticF((e*x+d)^(1/2)*((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2),(-(2*I*c*d*e-c^2*d^2+e
^2)/(c^2*d^2+e^2))^(1/2))*c*d*e-EllipticF((e*x+d)^(1/2)*((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2),(-(2*I*c*d*e-c^2*d^2
+e^2)/(c^2*d^2+e^2))^(1/2))*c^2*d^2-EllipticE((e*x+d)^(1/2)*((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2),(-(2*I*c*d*e-c^2
*d^2+e^2)/(c^2*d^2+e^2))^(1/2))*c^2*d^2-2*I*EllipticPi((e*x+d)^(1/2)*((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2),1/(I*e+
c*d)/c*(c^2*d^2+e^2)/d,(-(I*e-c*d)*c/(c^2*d^2+e^2))^(1/2)/((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2))*c*d*e+2*EllipticP
i((e*x+d)^(1/2)*((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2),1/(I*e+c*d)/c*(c^2*d^2+e^2)/d,(-(I*e-c*d)*c/(c^2*d^2+e^2))^(
1/2)/((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2))*c^2*d^2+EllipticF((e*x+d)^(1/2)*((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2),(-(2
*I*c*d*e-c^2*d^2+e^2)/(c^2*d^2+e^2))^(1/2))*e^2-EllipticE((e*x+d)^(1/2)*((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2),(-(2
*I*c*d*e-c^2*d^2+e^2)/(c^2*d^2+e^2))^(1/2))*e^2)/(((e*x+d)^2*c^2-2*(e*x+d)*c^2*d+c^2*d^2+e^2)/c^2/x^2/e^2)^(1/
2)/x/((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2)/(I*e-c*d)))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [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*arccsch(c*x))/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \left (a + b \operatorname{acsch}{\left (c x \right )}\right )}{\sqrt{d + e x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x*(a + b*acsch(c*x))/sqrt(d + e*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*arccsch(c*x) + a)*x/sqrt(e*x + d), x)