3.56 \(\int (d+e x)^{3/2} (a+b \text{csch}^{-1}(c x)) \, dx\)

Optimal. Leaf size=486 \[ -\frac{4 b c \sqrt{c^2 x^2+1} \left (2 c^2 d^2-e^2\right ) \sqrt{\frac{d+e x}{\frac{e}{\sqrt{-c^2}}+d}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right ),-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{15 \left (-c^2\right )^{5/2} x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}}+\frac{2 (d+e x)^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e}-\frac{4 b d^3 \sqrt{c^2 x^2+1} \sqrt{\frac{\sqrt{-c^2} (d+e x)}{\sqrt{-c^2} d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|\frac{2 e}{\sqrt{-c^2} d+e}\right )}{5 c e x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}}+\frac{4 b e \left (c^2 x^2+1\right ) \sqrt{d+e x}}{15 c^3 x \sqrt{\frac{1}{c^2 x^2}+1}}+\frac{28 b c d \sqrt{c^2 x^2+1} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{15 \left (-c^2\right )^{3/2} x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{\frac{d+e x}{\frac{e}{\sqrt{-c^2}}+d}}} \]

[Out]

(4*b*e*Sqrt[d + e*x]*(1 + c^2*x^2))/(15*c^3*Sqrt[1 + 1/(c^2*x^2)]*x) + (2*(d + e*x)^(5/2)*(a + b*ArcCsch[c*x])
)/(5*e) + (28*b*c*d*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)])/(15*(-c^2)^(3/2)*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[(d + e*x)/(d + e/Sqrt[-c^2])]
) - (4*b*c*(2*c^2*d^2 - e^2)*Sqrt[(d + e*x)/(d + e/Sqrt[-c^2])]*Sqrt[1 + c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - Sq
rt[-c^2]*x]/Sqrt[2]], (-2*Sqrt[-c^2]*e)/(c^2*d - Sqrt[-c^2]*e)])/(15*(-c^2)^(5/2)*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt
[d + e*x]) - (4*b*d^3*Sqrt[(Sqrt[-c^2]*(d + e*x))/(Sqrt[-c^2]*d + e)]*Sqrt[1 + c^2*x^2]*EllipticPi[2, ArcSin[S
qrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], (2*e)/(Sqrt[-c^2]*d + e)])/(5*c*e*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x])

________________________________________________________________________________________

Rubi [A]  time = 1.02263, antiderivative size = 486, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 13, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.722, Rules used = {6290, 1574, 958, 719, 419, 933, 168, 538, 537, 844, 424, 931, 1584} \[ \frac{2 (d+e x)^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e}-\frac{4 b c \sqrt{c^2 x^2+1} \left (2 c^2 d^2-e^2\right ) \sqrt{\frac{d+e x}{\frac{e}{\sqrt{-c^2}}+d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{15 \left (-c^2\right )^{5/2} x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}}-\frac{4 b d^3 \sqrt{c^2 x^2+1} \sqrt{\frac{\sqrt{-c^2} (d+e x)}{\sqrt{-c^2} d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|\frac{2 e}{\sqrt{-c^2} d+e}\right )}{5 c e x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}}+\frac{4 b e \left (c^2 x^2+1\right ) \sqrt{d+e x}}{15 c^3 x \sqrt{\frac{1}{c^2 x^2}+1}}+\frac{28 b c d \sqrt{c^2 x^2+1} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{15 \left (-c^2\right )^{3/2} x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{\frac{d+e x}{\frac{e}{\sqrt{-c^2}}+d}}} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(3/2)*(a + b*ArcCsch[c*x]),x]

[Out]

(4*b*e*Sqrt[d + e*x]*(1 + c^2*x^2))/(15*c^3*Sqrt[1 + 1/(c^2*x^2)]*x) + (2*(d + e*x)^(5/2)*(a + b*ArcCsch[c*x])
)/(5*e) + (28*b*c*d*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)])/(15*(-c^2)^(3/2)*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[(d + e*x)/(d + e/Sqrt[-c^2])]
) - (4*b*c*(2*c^2*d^2 - e^2)*Sqrt[(d + e*x)/(d + e/Sqrt[-c^2])]*Sqrt[1 + c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - Sq
rt[-c^2]*x]/Sqrt[2]], (-2*Sqrt[-c^2]*e)/(c^2*d - Sqrt[-c^2]*e)])/(15*(-c^2)^(5/2)*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt
[d + e*x]) - (4*b*d^3*Sqrt[(Sqrt[-c^2]*(d + e*x))/(Sqrt[-c^2]*d + e)]*Sqrt[1 + c^2*x^2]*EllipticPi[2, ArcSin[S
qrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], (2*e)/(Sqrt[-c^2]*d + e)])/(5*c*e*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[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]

Rule 931

Int[((d_.) + (e_.)*(x_))^(m_)/(Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> Simp[(2*e^2*(
d + e*x)^(m - 2)*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(c*g*(2*m - 1)), x] - Dist[1/(c*g*(2*m - 1)), Int[((d + e*x)^(
m - 3)*Simp[a*e^2*(d*g + 2*e*f*(m - 2)) - c*d^3*g*(2*m - 1) + e*(e*(a*e*g*(2*m - 3)) + c*d*(2*e*f - 3*d*g*(2*m
- 1)))*x + 2*e^2*(c*e*f - 3*c*d*g)*(m - 1)*x^2, 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] && IntegerQ[2*m] && GeQ[m, 2]

Rule 1584

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rubi steps

\begin{align*} \int (d+e x)^{3/2} \left (a+b \text{csch}^{-1}(c x)\right ) \, dx &=\frac{2 (d+e x)^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e}+\frac{(2 b) \int \frac{(d+e x)^{5/2}}{\sqrt{1+\frac{1}{c^2 x^2}} x^2} \, dx}{5 c e}\\ &=\frac{2 (d+e x)^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e}+\frac{\left (2 b \sqrt{\frac{1}{c^2}+x^2}\right ) \int \frac{(d+e x)^{5/2}}{x \sqrt{\frac{1}{c^2}+x^2}} \, dx}{5 c e \sqrt{1+\frac{1}{c^2 x^2}} x}\\ &=\frac{2 (d+e x)^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e}+\frac{\left (2 b \sqrt{\frac{1}{c^2}+x^2}\right ) \int \left (\frac{3 d^2 e}{\sqrt{d+e x} \sqrt{\frac{1}{c^2}+x^2}}+\frac{d^3}{x \sqrt{d+e x} \sqrt{\frac{1}{c^2}+x^2}}+\frac{3 d e^2 x}{\sqrt{d+e x} \sqrt{\frac{1}{c^2}+x^2}}+\frac{e^3 x^2}{\sqrt{d+e x} \sqrt{\frac{1}{c^2}+x^2}}\right ) \, dx}{5 c e \sqrt{1+\frac{1}{c^2 x^2}} x}\\ &=\frac{2 (d+e x)^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e}+\frac{\left (6 b d^2 \sqrt{\frac{1}{c^2}+x^2}\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{\frac{1}{c^2}+x^2}} \, dx}{5 c \sqrt{1+\frac{1}{c^2 x^2}} x}+\frac{\left (2 b d^3 \sqrt{\frac{1}{c^2}+x^2}\right ) \int \frac{1}{x \sqrt{d+e x} \sqrt{\frac{1}{c^2}+x^2}} \, dx}{5 c e \sqrt{1+\frac{1}{c^2 x^2}} x}+\frac{\left (6 b d e \sqrt{\frac{1}{c^2}+x^2}\right ) \int \frac{x}{\sqrt{d+e x} \sqrt{\frac{1}{c^2}+x^2}} \, dx}{5 c \sqrt{1+\frac{1}{c^2 x^2}} x}+\frac{\left (2 b e^2 \sqrt{\frac{1}{c^2}+x^2}\right ) \int \frac{x^2}{\sqrt{d+e x} \sqrt{\frac{1}{c^2}+x^2}} \, dx}{5 c \sqrt{1+\frac{1}{c^2 x^2}} x}\\ &=\frac{4 b e \sqrt{d+e x} \left (1+c^2 x^2\right )}{15 c^3 \sqrt{1+\frac{1}{c^2 x^2}} x}+\frac{2 (d+e x)^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e}+\frac{\left (6 b d \sqrt{\frac{1}{c^2}+x^2}\right ) \int \frac{\sqrt{d+e x}}{\sqrt{\frac{1}{c^2}+x^2}} \, dx}{5 c \sqrt{1+\frac{1}{c^2 x^2}} x}-\frac{\left (6 b d^2 \sqrt{\frac{1}{c^2}+x^2}\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{\frac{1}{c^2}+x^2}} \, dx}{5 c \sqrt{1+\frac{1}{c^2 x^2}} x}-\frac{\left (2 b e \sqrt{\frac{1}{c^2}+x^2}\right ) \int \frac{\frac{e x}{c^2}+2 d x^2}{x \sqrt{d+e x} \sqrt{\frac{1}{c^2}+x^2}} \, dx}{15 c \sqrt{1+\frac{1}{c^2 x^2}} x}+\frac{\left (2 b d^3 \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}{5 c e \sqrt{1+\frac{1}{c^2 x^2}} x}+\frac{\left (12 b \sqrt{-c^2} d^2 \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 )}{5 c^3 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}\\ &=\frac{4 b e \sqrt{d+e x} \left (1+c^2 x^2\right )}{15 c^3 \sqrt{1+\frac{1}{c^2 x^2}} x}+\frac{2 (d+e x)^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e}+\frac{12 b \sqrt{-c^2} d^2 \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 )}{5 c^3 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}-\frac{\left (2 b e \sqrt{\frac{1}{c^2}+x^2}\right ) \int \frac{\frac{e}{c^2}+2 d x}{\sqrt{d+e x} \sqrt{\frac{1}{c^2}+x^2}} \, dx}{15 c \sqrt{1+\frac{1}{c^2 x^2}} x}-\frac{\left (4 b d^3 \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 )}{5 c e \sqrt{1+\frac{1}{c^2 x^2}} x}+\frac{\left (12 b \sqrt{-c^2} d \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 )}{5 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 (12 b \sqrt{-c^2} d^2 \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 )}{5 c^3 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}\\ &=\frac{4 b e \sqrt{d+e x} \left (1+c^2 x^2\right )}{15 c^3 \sqrt{1+\frac{1}{c^2 x^2}} x}+\frac{2 (d+e x)^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e}+\frac{12 b \sqrt{-c^2} d \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 )}{5 c^3 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{\frac{d+e x}{d+\frac{e}{\sqrt{-c^2}}}}}-\frac{\left (4 b d \sqrt{\frac{1}{c^2}+x^2}\right ) \int \frac{\sqrt{d+e x}}{\sqrt{\frac{1}{c^2}+x^2}} \, dx}{15 c \sqrt{1+\frac{1}{c^2 x^2}} x}-\frac{\left (2 b \left (-2 d^2+\frac{e^2}{c^2}\right ) \sqrt{\frac{1}{c^2}+x^2}\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{\frac{1}{c^2}+x^2}} \, dx}{15 c \sqrt{1+\frac{1}{c^2 x^2}} x}-\frac{\left (4 b d^3 \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 )}{5 c e \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}\\ &=\frac{4 b e \sqrt{d+e x} \left (1+c^2 x^2\right )}{15 c^3 \sqrt{1+\frac{1}{c^2 x^2}} x}+\frac{2 (d+e x)^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e}+\frac{12 b \sqrt{-c^2} d \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 )}{5 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 d^3 \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 )}{5 c e \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}-\frac{\left (8 b \sqrt{-c^2} d \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 )}{15 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} \left (-2 d^2+\frac{e^2}{c^2}\right ) \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 )}{15 c^3 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}\\ &=\frac{4 b e \sqrt{d+e x} \left (1+c^2 x^2\right )}{15 c^3 \sqrt{1+\frac{1}{c^2 x^2}} x}+\frac{2 (d+e x)^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e}+\frac{28 b \sqrt{-c^2} d \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 )}{15 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} \left (2 d^2-\frac{e^2}{c^2}\right ) \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 )}{15 c^3 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}-\frac{4 b d^3 \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 )}{5 c e \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}\\ \end{align*}

Mathematica [C]  time = 1.52612, size = 380, normalized size = 0.78 \[ \frac{2 \left (\frac{2 i b \sqrt{-\frac{e (c x-i)}{c d+i e}} \sqrt{-\frac{e (c x+i)}{c d-i e}} \left (\left (-9 c^2 d^2-7 i c d e+e^2\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{-\frac{c}{c d-i e}} \sqrt{d+e x}\right ),\frac{c d-i e}{c d+i e}\right )+3 c^2 d^2 \Pi \left (1-\frac{i e}{c d};i \sinh ^{-1}\left (\sqrt{-\frac{c}{c d-i e}} \sqrt{d+e x}\right )|\frac{c d-i e}{c d+i e}\right )+7 c d (c d+i e) E\left (i \sinh ^{-1}\left (\sqrt{-\frac{c}{c d-i e}} \sqrt{d+e x}\right )|\frac{c d-i e}{c d+i e}\right )\right )}{c^3 x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{-\frac{c}{c d-i e}}}+3 a (d+e x)^{5/2}+\frac{2 b e^2 x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}}{c}+3 b \text{csch}^{-1}(c x) (d+e x)^{5/2}\right )}{15 e} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(3/2)*(a + b*ArcCsch[c*x]),x]

[Out]

(2*((2*b*e^2*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x])/c + 3*a*(d + e*x)^(5/2) + 3*b*(d + e*x)^(5/2)*ArcCsch[c*x]
+ ((2*I)*b*Sqrt[-((e*(-I + c*x))/(c*d + I*e))]*Sqrt[-((e*(I + c*x))/(c*d - I*e))]*(7*c*d*(c*d + I*e)*Elliptic
E[I*ArcSinh[Sqrt[-(c/(c*d - I*e))]*Sqrt[d + e*x]], (c*d - I*e)/(c*d + I*e)] + (-9*c^2*d^2 - (7*I)*c*d*e + e^2)
*EllipticF[I*ArcSinh[Sqrt[-(c/(c*d - I*e))]*Sqrt[d + e*x]], (c*d - I*e)/(c*d + I*e)] + 3*c^2*d^2*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^3*Sqrt[-(c/(c*d
- I*e))]*Sqrt[1 + 1/(c^2*x^2)]*x)))/(15*e)

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Maple [C]  time = 0.293, size = 1939, normalized size = 4. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(3/2)*(a+b*arccsch(c*x)),x)

[Out]

2/e*(1/5*(e*x+d)^(5/2)*a+b*(1/5*arccsch(c*x)*(e*x+d)^(5/2)+2/15/c^3*(I*((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2)*(e*x+
d)^(1/2)*c^2*d^2*e-((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2)*(e*x+d)^(5/2)*c^3*d+I*((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2)*(
e*x+d)^(5/2)*c^2*e+2*((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2)*(e*x+d)^(3/2)*c^3*d^2-2*I*((I*e+c*d)*c/(c^2*d^2+e^2))^(
1/2)*(e*x+d)^(3/2)*c^2*d*e-9*(-(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)*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^3*d^3+7*(-(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)*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^3*d^3+2*I*(-(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)*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*e+3*(-
(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)*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^3*d^3-I*(-(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)*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^3-((I*e+c*d)*c/(c
^2*d^2+e^2))^(1/2)*(e*x+d)^(1/2)*c^3*d^3-3*I*(-(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)*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*e-6*(-(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)*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+7*(-(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)*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*d*e^2+I*((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2)*(e*x+d)^(1/
2)*e^3-((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2)*(e*x+d)^(1/2)*c*d*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((e*x+d)^(3/2)*(a+b*arccsch(c*x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)*(a+b*arccsch(c*x)),x, algorithm="fricas")

[Out]

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

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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((e*x+d)**(3/2)*(a+b*acsch(c*x)),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (e x + d\right )}^{\frac{3}{2}}{\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((e*x+d)^(3/2)*(a+b*arccsch(c*x)),x, algorithm="giac")

[Out]

integrate((e*x + d)^(3/2)*(b*arccsch(c*x) + a), x)