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

Optimal. Leaf size=429 \[ \frac{4 b c d \sqrt{c^2 x^2+1} \sqrt{\frac{d+e x}{\frac{e}{\sqrt{-c^2}}+d}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right ),-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{3 \left (-c^2\right )^{3/2} x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}}+\frac{2 (d+e x)^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e}-\frac{4 b d^2 \sqrt{c^2 x^2+1} \sqrt{\frac{\sqrt{-c^2} (d+e x)}{\sqrt{-c^2} d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|\frac{2 e}{\sqrt{-c^2} d+e}\right )}{3 c e x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}}+\frac{4 b c \sqrt{c^2 x^2+1} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{3 \left (-c^2\right )^{3/2} x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{\frac{d+e x}{\frac{e}{\sqrt{-c^2}}+d}}} \]

[Out]

(2*(d + e*x)^(3/2)*(a + b*ArcCsch[c*x]))/(3*e) + (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)*Sqrt[1 + 1/(c^2*x^2)]*x
*Sqrt[(d + e*x)/(d + e/Sqrt[-c^2])]) + (4*b*c*d*Sqrt[(d + e*x)/(d + e/Sqrt[-c^2])]*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)*Sqrt[1 + 1
/(c^2*x^2)]*x*Sqrt[d + e*x]) - (4*b*d^2*Sqrt[(Sqrt[-c^2]*(d + e*x))/(Sqrt[-c^2]*d + e)]*Sqrt[1 + c^2*x^2]*Elli
pticPi[2, ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], (2*e)/(Sqrt[-c^2]*d + e)])/(3*c*e*Sqrt[1 + 1/(c^2*x^2)]*x*Sq
rt[d + e*x])

________________________________________________________________________________________

Rubi [A]  time = 0.708368, antiderivative size = 429, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 11, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.611, Rules used = {6290, 1574, 958, 719, 419, 933, 168, 538, 537, 844, 424} \[ \frac{2 (d+e x)^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e}-\frac{4 b d^2 \sqrt{c^2 x^2+1} \sqrt{\frac{\sqrt{-c^2} (d+e x)}{\sqrt{-c^2} d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|\frac{2 e}{\sqrt{-c^2} d+e}\right )}{3 c e x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}}+\frac{4 b c d \sqrt{c^2 x^2+1} \sqrt{\frac{d+e x}{\frac{e}{\sqrt{-c^2}}+d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{3 \left (-c^2\right )^{3/2} x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}}+\frac{4 b c \sqrt{c^2 x^2+1} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{3 \left (-c^2\right )^{3/2} x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{\frac{d+e x}{\frac{e}{\sqrt{-c^2}}+d}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(3/2)*(a + b*ArcCsch[c*x]))/(3*e) + (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)*Sqrt[1 + 1/(c^2*x^2)]*x
*Sqrt[(d + e*x)/(d + e/Sqrt[-c^2])]) + (4*b*c*d*Sqrt[(d + e*x)/(d + e/Sqrt[-c^2])]*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)*Sqrt[1 + 1
/(c^2*x^2)]*x*Sqrt[d + e*x]) - (4*b*d^2*Sqrt[(Sqrt[-c^2]*(d + e*x))/(Sqrt[-c^2]*d + e)]*Sqrt[1 + c^2*x^2]*Elli
pticPi[2, ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], (2*e)/(Sqrt[-c^2]*d + e)])/(3*c*e*Sqrt[1 + 1/(c^2*x^2)]*x*Sq
rt[d + e*x])

Rule 6290

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

Rule 1574

Int[(x_)^(m_.)*((a_.) + (c_.)*(x_)^(mn2_.))^(p_)*((d_) + (e_.)*(x_)^(n_.))^(q_.), x_Symbol] :> Dist[(x^(2*n*Fr
acPart[p])*(a + c/x^(2*n))^FracPart[p])/(c + a*x^(2*n))^FracPart[p], Int[x^(m - 2*n*p)*(d + e*x^n)^q*(c + a*x^
(2*n))^p, x], x] /; FreeQ[{a, c, d, e, m, n, p, q}, x] && EqQ[mn2, -2*n] &&  !IntegerQ[p] &&  !IntegerQ[q] &&
PosQ[n]

Rule 958

Int[((f_.) + (g_.)*(x_))^(n_)/(((d_.) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> Int[ExpandIntegra
nd[1/(Sqrt[f + g*x]*Sqrt[a + c*x^2]), (f + g*x)^(n + 1/2)/(d + e*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] && IntegerQ[n + 1/2]

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 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 933

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[-(c
/a), 2]}, Dist[Sqrt[1 + (c*x^2)/a]/Sqrt[a + c*x^2], 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)
])

Rule 844

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

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]

Rubi steps

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

Mathematica [C]  time = 13.9916, size = 926, normalized size = 2.16 \[ \frac{2 a (d+e x)^{3/2}}{3 e}+\frac{b \left (-\frac{(c d+c e x) \left (-\frac{2}{3} c x \text{csch}^{-1}(c x)-\frac{2 c d \text{csch}^{-1}(c x)}{3 e}-\frac{4}{3} \sqrt{1+\frac{1}{c^2 x^2}}\right )}{\sqrt{d+e x}}-\frac{2 (c d+c e x) \left (-\frac{\sqrt{2} c d e \sqrt{i c x+1} (c x+i) \sqrt{\frac{c d+c e x}{c d-i e}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{-\frac{e (c x+i)}{c d-i e}}\right ),\frac{i c d+e}{2 e}\right )}{\sqrt{1+\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} (c x)^{3/2} \sqrt{\frac{e (1-i c x)}{i c d+e}}}+\frac{i \sqrt{2} (c d-i e) \left (c^2 d^2+e^2\right ) \sqrt{i c x+1} \sqrt{\frac{e (c x+i) (c d+c e x)}{(i c d+e)^2}} \Pi \left (\frac{i c d}{e}+1;\sin ^{-1}\left (\sqrt{-\frac{e (c x+i)}{c d-i e}}\right )|\frac{i c d+e}{2 e}\right )}{e \sqrt{1+\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} (c x)^{3/2}}-\frac{2 e \cosh \left (2 \text{csch}^{-1}(c x)\right ) \left (\frac{c x \left (c d \sqrt{2 i c x+2} (c x+i) \sqrt{\frac{c d+c e x}{c d-i e}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{-\frac{e (c x+i)}{c d-i e}}\right ),\frac{i c d+e}{2 e}\right )+2 \sqrt{-\frac{e (c x-i)}{c d+i e}} (c x+i) \sqrt{\frac{c d+c e x}{c d-i e}} \left ((c d+i e) E\left (\sin ^{-1}\left (\sqrt{\frac{c d+c e x}{c d-i e}}\right )|\frac{c d-i e}{c d+i e}\right )-i e \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{c d+c e x}{c d-i e}}\right ),\frac{c d-i e}{c d+i e}\right )\right )+(i c d+e) \sqrt{2 i c x+2} \sqrt{-\frac{e (c x+i)}{c d-i e}} \sqrt{\frac{e (c x+i) (c d+c e x)}{(i c d+e)^2}} \Pi \left (\frac{i c d}{e}+1;\sin ^{-1}\left (\sqrt{-\frac{e (c x+i)}{c d-i e}}\right )|\frac{i c d+e}{2 e}\right )\right )}{2 \sqrt{-\frac{e (c x+i)}{c d-i e}}}-(c d+c e x) \left (c^2 x^2+1\right )\right )}{\sqrt{1+\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} \sqrt{c x} \left (c^2 x^2+2\right )}\right )}{3 e \sqrt{\frac{d}{x}+e} \sqrt{c x} \sqrt{d+e x}}\right )}{c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.301, size = 840, normalized size = 2. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/e*(1/3*a*(e*x+d)^(3/2)+b*(1/3*(e*x+d)^(3/2)*arccsch(c*x)+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)*(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-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))*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-I*Ellipti
cPi((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+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^2*d^2-Ellip
ticF((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+Ellip
ticE((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((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((a+b*arccsch(c*x))*(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

Timed out

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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