3.137 \(\int \frac{(d+e x^2)^{3/2} (a+b \text{csch}^{-1}(c x))}{x^8} \, dx\)

Optimal. Leaf size=643 \[ -\frac{b e x \left (-249 c^4 d^2 e+120 c^6 d^3+71 c^2 d e^2+210 e^3\right ) \sqrt{d+e x^2} \text{EllipticF}\left (\tan ^{-1}(c x),1-\frac{e}{c^2 d}\right )}{3675 d^3 \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1} \sqrt{\frac{d+e x^2}{d \left (c^2 x^2+1\right )}}}+\frac{2 e \left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{35 d^2 x^5}-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{7 d x^7}-\frac{b c^3 x^2 \left (-528 c^4 d^2 e+240 c^6 d^3+193 c^2 d e^2+247 e^3\right ) \sqrt{d+e x^2}}{3675 d^2 \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1}}-\frac{b c \sqrt{-c^2 x^2-1} \left (-528 c^4 d^2 e+240 c^6 d^3+193 c^2 d e^2+247 e^3\right ) \sqrt{d+e x^2}}{3675 d^2 \sqrt{-c^2 x^2}}+\frac{b c \sqrt{-c^2 x^2-1} \left (120 c^4 d^2-159 c^2 d e-37 e^2\right ) \sqrt{d+e x^2}}{3675 d x^2 \sqrt{-c^2 x^2}}+\frac{b c^2 x \left (-528 c^4 d^2 e+240 c^6 d^3+193 c^2 d e^2+247 e^3\right ) \sqrt{d+e x^2} E\left (\tan ^{-1}(c x)|1-\frac{e}{c^2 d}\right )}{3675 d^2 \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1} \sqrt{\frac{d+e x^2}{d \left (c^2 x^2+1\right )}}}+\frac{b c \sqrt{-c^2 x^2-1} \left (d+e x^2\right )^{5/2}}{49 d x^6 \sqrt{-c^2 x^2}}-\frac{b c \sqrt{-c^2 x^2-1} \left (30 c^2 d-11 e\right ) \left (d+e x^2\right )^{3/2}}{1225 d x^4 \sqrt{-c^2 x^2}} \]

[Out]

-(b*c^3*(240*c^6*d^3 - 528*c^4*d^2*e + 193*c^2*d*e^2 + 247*e^3)*x^2*Sqrt[d + e*x^2])/(3675*d^2*Sqrt[-(c^2*x^2)
]*Sqrt[-1 - c^2*x^2]) - (b*c*(240*c^6*d^3 - 528*c^4*d^2*e + 193*c^2*d*e^2 + 247*e^3)*Sqrt[-1 - c^2*x^2]*Sqrt[d
+ e*x^2])/(3675*d^2*Sqrt[-(c^2*x^2)]) + (b*c*(120*c^4*d^2 - 159*c^2*d*e - 37*e^2)*Sqrt[-1 - c^2*x^2]*Sqrt[d +
e*x^2])/(3675*d*x^2*Sqrt[-(c^2*x^2)]) - (b*c*(30*c^2*d - 11*e)*Sqrt[-1 - c^2*x^2]*(d + e*x^2)^(3/2))/(1225*d*
x^4*Sqrt[-(c^2*x^2)]) + (b*c*Sqrt[-1 - c^2*x^2]*(d + e*x^2)^(5/2))/(49*d*x^6*Sqrt[-(c^2*x^2)]) - ((d + e*x^2)^
(5/2)*(a + b*ArcCsch[c*x]))/(7*d*x^7) + (2*e*(d + e*x^2)^(5/2)*(a + b*ArcCsch[c*x]))/(35*d^2*x^5) + (b*c^2*(24
0*c^6*d^3 - 528*c^4*d^2*e + 193*c^2*d*e^2 + 247*e^3)*x*Sqrt[d + e*x^2]*EllipticE[ArcTan[c*x], 1 - e/(c^2*d)])/
(3675*d^2*Sqrt[-(c^2*x^2)]*Sqrt[-1 - c^2*x^2]*Sqrt[(d + e*x^2)/(d*(1 + c^2*x^2))]) - (b*e*(120*c^6*d^3 - 249*c
^4*d^2*e + 71*c^2*d*e^2 + 210*e^3)*x*Sqrt[d + e*x^2]*EllipticF[ArcTan[c*x], 1 - e/(c^2*d)])/(3675*d^3*Sqrt[-(c
^2*x^2)]*Sqrt[-1 - c^2*x^2]*Sqrt[(d + e*x^2)/(d*(1 + c^2*x^2))])

________________________________________________________________________________________

Rubi [A]  time = 0.866062, antiderivative size = 643, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 10, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.435, Rules used = {271, 264, 6302, 12, 580, 583, 531, 418, 492, 411} \[ \frac{2 e \left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{35 d^2 x^5}-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{7 d x^7}-\frac{b c^3 x^2 \left (-528 c^4 d^2 e+240 c^6 d^3+193 c^2 d e^2+247 e^3\right ) \sqrt{d+e x^2}}{3675 d^2 \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1}}-\frac{b c \sqrt{-c^2 x^2-1} \left (-528 c^4 d^2 e+240 c^6 d^3+193 c^2 d e^2+247 e^3\right ) \sqrt{d+e x^2}}{3675 d^2 \sqrt{-c^2 x^2}}+\frac{b c \sqrt{-c^2 x^2-1} \left (120 c^4 d^2-159 c^2 d e-37 e^2\right ) \sqrt{d+e x^2}}{3675 d x^2 \sqrt{-c^2 x^2}}-\frac{b e x \left (-249 c^4 d^2 e+120 c^6 d^3+71 c^2 d e^2+210 e^3\right ) \sqrt{d+e x^2} F\left (\tan ^{-1}(c x)|1-\frac{e}{c^2 d}\right )}{3675 d^3 \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1} \sqrt{\frac{d+e x^2}{d \left (c^2 x^2+1\right )}}}+\frac{b c^2 x \left (-528 c^4 d^2 e+240 c^6 d^3+193 c^2 d e^2+247 e^3\right ) \sqrt{d+e x^2} E\left (\tan ^{-1}(c x)|1-\frac{e}{c^2 d}\right )}{3675 d^2 \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1} \sqrt{\frac{d+e x^2}{d \left (c^2 x^2+1\right )}}}+\frac{b c \sqrt{-c^2 x^2-1} \left (d+e x^2\right )^{5/2}}{49 d x^6 \sqrt{-c^2 x^2}}-\frac{b c \sqrt{-c^2 x^2-1} \left (30 c^2 d-11 e\right ) \left (d+e x^2\right )^{3/2}}{1225 d x^4 \sqrt{-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x^2)^(3/2)*(a + b*ArcCsch[c*x]))/x^8,x]

[Out]

-(b*c^3*(240*c^6*d^3 - 528*c^4*d^2*e + 193*c^2*d*e^2 + 247*e^3)*x^2*Sqrt[d + e*x^2])/(3675*d^2*Sqrt[-(c^2*x^2)
]*Sqrt[-1 - c^2*x^2]) - (b*c*(240*c^6*d^3 - 528*c^4*d^2*e + 193*c^2*d*e^2 + 247*e^3)*Sqrt[-1 - c^2*x^2]*Sqrt[d
+ e*x^2])/(3675*d^2*Sqrt[-(c^2*x^2)]) + (b*c*(120*c^4*d^2 - 159*c^2*d*e - 37*e^2)*Sqrt[-1 - c^2*x^2]*Sqrt[d +
e*x^2])/(3675*d*x^2*Sqrt[-(c^2*x^2)]) - (b*c*(30*c^2*d - 11*e)*Sqrt[-1 - c^2*x^2]*(d + e*x^2)^(3/2))/(1225*d*
x^4*Sqrt[-(c^2*x^2)]) + (b*c*Sqrt[-1 - c^2*x^2]*(d + e*x^2)^(5/2))/(49*d*x^6*Sqrt[-(c^2*x^2)]) - ((d + e*x^2)^
(5/2)*(a + b*ArcCsch[c*x]))/(7*d*x^7) + (2*e*(d + e*x^2)^(5/2)*(a + b*ArcCsch[c*x]))/(35*d^2*x^5) + (b*c^2*(24
0*c^6*d^3 - 528*c^4*d^2*e + 193*c^2*d*e^2 + 247*e^3)*x*Sqrt[d + e*x^2]*EllipticE[ArcTan[c*x], 1 - e/(c^2*d)])/
(3675*d^2*Sqrt[-(c^2*x^2)]*Sqrt[-1 - c^2*x^2]*Sqrt[(d + e*x^2)/(d*(1 + c^2*x^2))]) - (b*e*(120*c^6*d^3 - 249*c
^4*d^2*e + 71*c^2*d*e^2 + 210*e^3)*x*Sqrt[d + e*x^2]*EllipticF[ArcTan[c*x], 1 - e/(c^2*d)])/(3675*d^3*Sqrt[-(c
^2*x^2)]*Sqrt[-1 - c^2*x^2]*Sqrt[(d + e*x^2)/(d*(1 + c^2*x^2))])

Rule 271

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x^(m + 1)*(a + b*x^n)^(p + 1))/(a*(m + 1)), x]
- Dist[(b*(m + n*(p + 1) + 1))/(a*(m + 1)), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 6302

Int[((a_.) + ArcCsch[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u
= IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcCsch[c*x], u, x] - Dist[(b*c*x)/Sqrt[-(c^2*x^2)], Int[Simp
lifyIntegrand[u/(x*Sqrt[-1 - c^2*x^2]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && ((IGtQ[p, 0] &&
!(ILtQ[(m - 1)/2, 0] && GtQ[m + 2*p + 3, 0])) || (IGtQ[(m + 1)/2, 0] &&  !(ILtQ[p, 0] && GtQ[m + 2*p + 3, 0]))
|| (ILtQ[(m + 2*p + 1)/2, 0] &&  !ILtQ[(m - 1)/2, 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 580

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[(e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q)/(a*g*(m + 1)), x] - Dist[1/(a*g^n*(m + 1
)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*(b*e - a*f)*(m + 1) + e*n*(b*c*(p + 1) + a*d*q)
+ d*((b*e - a*f)*(m + 1) + b*e*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && IGtQ[n
, 0] && GtQ[q, 0] && LtQ[m, -1] &&  !(EqQ[q, 1] && SimplerQ[e + f*x^n, c + d*x^n])

Rule 583

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

Rule 531

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Dist[
e, Int[(a + b*x^n)^p*(c + d*x^n)^q, x], x] + Dist[f, Int[x^n*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, n, p, q}, x]

Rule 418

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

Rule 492

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(x*Sqrt[a + b*x^2])/(b*Sqr
t[c + d*x^2]), x] - Dist[c/b, Int[Sqrt[a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b
*c - a*d, 0] && PosQ[b/a] && PosQ[d/c] &&  !SimplerSqrtQ[b/a, d/c]

Rule 411

Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(Sqrt[a + b*x^2]*EllipticE[ArcTan
[Rt[d/c, 2]*x], 1 - (b*c)/(a*d)])/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]), x] /;
FreeQ[{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]

Rubi steps

\begin{align*} \int \frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{x^8} \, dx &=-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{7 d x^7}+\frac{2 e \left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{35 d^2 x^5}-\frac{(b c x) \int \frac{\left (d+e x^2\right )^{5/2} \left (-5 d+2 e x^2\right )}{35 d^2 x^8 \sqrt{-1-c^2 x^2}} \, dx}{\sqrt{-c^2 x^2}}\\ &=-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{7 d x^7}+\frac{2 e \left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{35 d^2 x^5}-\frac{(b c x) \int \frac{\left (d+e x^2\right )^{5/2} \left (-5 d+2 e x^2\right )}{x^8 \sqrt{-1-c^2 x^2}} \, dx}{35 d^2 \sqrt{-c^2 x^2}}\\ &=\frac{b c \sqrt{-1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{49 d x^6 \sqrt{-c^2 x^2}}-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{7 d x^7}+\frac{2 e \left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{35 d^2 x^5}+\frac{(b c x) \int \frac{\left (d+e x^2\right )^{3/2} \left (-d \left (30 c^2 d-11 e\right )-e \left (5 c^2 d+14 e\right ) x^2\right )}{x^6 \sqrt{-1-c^2 x^2}} \, dx}{245 d^2 \sqrt{-c^2 x^2}}\\ &=-\frac{b c \left (30 c^2 d-11 e\right ) \sqrt{-1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{1225 d x^4 \sqrt{-c^2 x^2}}+\frac{b c \sqrt{-1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{49 d x^6 \sqrt{-c^2 x^2}}-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{7 d x^7}+\frac{2 e \left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{35 d^2 x^5}-\frac{(b c x) \int \frac{\sqrt{d+e x^2} \left (-d \left (120 c^4 d^2-159 c^2 d e-37 e^2\right )-2 e \left (15 c^4 d^2-18 c^2 d e-35 e^2\right ) x^2\right )}{x^4 \sqrt{-1-c^2 x^2}} \, dx}{1225 d^2 \sqrt{-c^2 x^2}}\\ &=\frac{b c \left (120 c^4 d^2-159 c^2 d e-37 e^2\right ) \sqrt{-1-c^2 x^2} \sqrt{d+e x^2}}{3675 d x^2 \sqrt{-c^2 x^2}}-\frac{b c \left (30 c^2 d-11 e\right ) \sqrt{-1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{1225 d x^4 \sqrt{-c^2 x^2}}+\frac{b c \sqrt{-1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{49 d x^6 \sqrt{-c^2 x^2}}-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{7 d x^7}+\frac{2 e \left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{35 d^2 x^5}+\frac{(b c x) \int \frac{-d \left (240 c^6 d^3-528 c^4 d^2 e+193 c^2 d e^2+247 e^3\right )-e \left (120 c^6 d^3-249 c^4 d^2 e+71 c^2 d e^2+210 e^3\right ) x^2}{x^2 \sqrt{-1-c^2 x^2} \sqrt{d+e x^2}} \, dx}{3675 d^2 \sqrt{-c^2 x^2}}\\ &=-\frac{b c \left (240 c^6 d^3-528 c^4 d^2 e+193 c^2 d e^2+247 e^3\right ) \sqrt{-1-c^2 x^2} \sqrt{d+e x^2}}{3675 d^2 \sqrt{-c^2 x^2}}+\frac{b c \left (120 c^4 d^2-159 c^2 d e-37 e^2\right ) \sqrt{-1-c^2 x^2} \sqrt{d+e x^2}}{3675 d x^2 \sqrt{-c^2 x^2}}-\frac{b c \left (30 c^2 d-11 e\right ) \sqrt{-1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{1225 d x^4 \sqrt{-c^2 x^2}}+\frac{b c \sqrt{-1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{49 d x^6 \sqrt{-c^2 x^2}}-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{7 d x^7}+\frac{2 e \left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{35 d^2 x^5}+\frac{(b c x) \int \frac{-d e \left (120 c^6 d^3-249 c^4 d^2 e+71 c^2 d e^2+210 e^3\right )-c^2 d e \left (240 c^6 d^3-528 c^4 d^2 e+193 c^2 d e^2+247 e^3\right ) x^2}{\sqrt{-1-c^2 x^2} \sqrt{d+e x^2}} \, dx}{3675 d^3 \sqrt{-c^2 x^2}}\\ &=-\frac{b c \left (240 c^6 d^3-528 c^4 d^2 e+193 c^2 d e^2+247 e^3\right ) \sqrt{-1-c^2 x^2} \sqrt{d+e x^2}}{3675 d^2 \sqrt{-c^2 x^2}}+\frac{b c \left (120 c^4 d^2-159 c^2 d e-37 e^2\right ) \sqrt{-1-c^2 x^2} \sqrt{d+e x^2}}{3675 d x^2 \sqrt{-c^2 x^2}}-\frac{b c \left (30 c^2 d-11 e\right ) \sqrt{-1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{1225 d x^4 \sqrt{-c^2 x^2}}+\frac{b c \sqrt{-1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{49 d x^6 \sqrt{-c^2 x^2}}-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{7 d x^7}+\frac{2 e \left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{35 d^2 x^5}-\frac{\left (b c e \left (120 c^6 d^3-249 c^4 d^2 e+71 c^2 d e^2+210 e^3\right ) x\right ) \int \frac{1}{\sqrt{-1-c^2 x^2} \sqrt{d+e x^2}} \, dx}{3675 d^2 \sqrt{-c^2 x^2}}-\frac{\left (b c^3 e \left (240 c^6 d^3-528 c^4 d^2 e+193 c^2 d e^2+247 e^3\right ) x\right ) \int \frac{x^2}{\sqrt{-1-c^2 x^2} \sqrt{d+e x^2}} \, dx}{3675 d^2 \sqrt{-c^2 x^2}}\\ &=-\frac{b c^3 \left (240 c^6 d^3-528 c^4 d^2 e+193 c^2 d e^2+247 e^3\right ) x^2 \sqrt{d+e x^2}}{3675 d^2 \sqrt{-c^2 x^2} \sqrt{-1-c^2 x^2}}-\frac{b c \left (240 c^6 d^3-528 c^4 d^2 e+193 c^2 d e^2+247 e^3\right ) \sqrt{-1-c^2 x^2} \sqrt{d+e x^2}}{3675 d^2 \sqrt{-c^2 x^2}}+\frac{b c \left (120 c^4 d^2-159 c^2 d e-37 e^2\right ) \sqrt{-1-c^2 x^2} \sqrt{d+e x^2}}{3675 d x^2 \sqrt{-c^2 x^2}}-\frac{b c \left (30 c^2 d-11 e\right ) \sqrt{-1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{1225 d x^4 \sqrt{-c^2 x^2}}+\frac{b c \sqrt{-1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{49 d x^6 \sqrt{-c^2 x^2}}-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{7 d x^7}+\frac{2 e \left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{35 d^2 x^5}-\frac{b e \left (120 c^6 d^3-249 c^4 d^2 e+71 c^2 d e^2+210 e^3\right ) x \sqrt{d+e x^2} F\left (\tan ^{-1}(c x)|1-\frac{e}{c^2 d}\right )}{3675 d^3 \sqrt{-c^2 x^2} \sqrt{-1-c^2 x^2} \sqrt{\frac{d+e x^2}{d \left (1+c^2 x^2\right )}}}-\frac{\left (b c^3 \left (240 c^6 d^3-528 c^4 d^2 e+193 c^2 d e^2+247 e^3\right ) x\right ) \int \frac{\sqrt{d+e x^2}}{\left (-1-c^2 x^2\right )^{3/2}} \, dx}{3675 d^2 \sqrt{-c^2 x^2}}\\ &=-\frac{b c^3 \left (240 c^6 d^3-528 c^4 d^2 e+193 c^2 d e^2+247 e^3\right ) x^2 \sqrt{d+e x^2}}{3675 d^2 \sqrt{-c^2 x^2} \sqrt{-1-c^2 x^2}}-\frac{b c \left (240 c^6 d^3-528 c^4 d^2 e+193 c^2 d e^2+247 e^3\right ) \sqrt{-1-c^2 x^2} \sqrt{d+e x^2}}{3675 d^2 \sqrt{-c^2 x^2}}+\frac{b c \left (120 c^4 d^2-159 c^2 d e-37 e^2\right ) \sqrt{-1-c^2 x^2} \sqrt{d+e x^2}}{3675 d x^2 \sqrt{-c^2 x^2}}-\frac{b c \left (30 c^2 d-11 e\right ) \sqrt{-1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{1225 d x^4 \sqrt{-c^2 x^2}}+\frac{b c \sqrt{-1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{49 d x^6 \sqrt{-c^2 x^2}}-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{7 d x^7}+\frac{2 e \left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{35 d^2 x^5}+\frac{b c^2 \left (240 c^6 d^3-528 c^4 d^2 e+193 c^2 d e^2+247 e^3\right ) x \sqrt{d+e x^2} E\left (\tan ^{-1}(c x)|1-\frac{e}{c^2 d}\right )}{3675 d^2 \sqrt{-c^2 x^2} \sqrt{-1-c^2 x^2} \sqrt{\frac{d+e x^2}{d \left (1+c^2 x^2\right )}}}-\frac{b e \left (120 c^6 d^3-249 c^4 d^2 e+71 c^2 d e^2+210 e^3\right ) x \sqrt{d+e x^2} F\left (\tan ^{-1}(c x)|1-\frac{e}{c^2 d}\right )}{3675 d^3 \sqrt{-c^2 x^2} \sqrt{-1-c^2 x^2} \sqrt{\frac{d+e x^2}{d \left (1+c^2 x^2\right )}}}\\ \end{align*}

Mathematica [C]  time = 0.823573, size = 372, normalized size = 0.58 \[ -\frac{\sqrt{d+e x^2} \left (105 a \left (5 d-2 e x^2\right ) \left (d+e x^2\right )^2+b c x \sqrt{\frac{1}{c^2 x^2}+1} \left (-3 d^2 e x^2 \left (176 c^4 x^4-83 c^2 x^2+61\right )+15 d^3 \left (16 c^6 x^6-8 c^4 x^4+6 c^2 x^2-5\right )+d e^2 x^4 \left (193 c^2 x^2-71\right )+247 e^3 x^6\right )+105 b \text{csch}^{-1}(c x) \left (5 d-2 e x^2\right ) \left (d+e x^2\right )^2\right )}{3675 d^2 x^7}-\frac{i b c x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{\frac{e x^2}{d}+1} \left (c^2 d \left (-528 c^4 d^2 e+240 c^6 d^3+193 c^2 d e^2+247 e^3\right ) E\left (i \sinh ^{-1}\left (\sqrt{c^2} x\right )|\frac{e}{c^2 d}\right )-2 \left (221 c^4 d^2 e^2-324 c^6 d^3 e+120 c^8 d^4+88 c^2 d e^3-105 e^4\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{c^2} x\right ),\frac{e}{c^2 d}\right )\right )}{3675 \sqrt{c^2} d^2 \sqrt{c^2 x^2+1} \sqrt{d+e x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x^2)^(3/2)*(a + b*ArcCsch[c*x]))/x^8,x]

[Out]

-(Sqrt[d + e*x^2]*(105*a*(5*d - 2*e*x^2)*(d + e*x^2)^2 + b*c*Sqrt[1 + 1/(c^2*x^2)]*x*(247*e^3*x^6 + d*e^2*x^4*
(-71 + 193*c^2*x^2) - 3*d^2*e*x^2*(61 - 83*c^2*x^2 + 176*c^4*x^4) + 15*d^3*(-5 + 6*c^2*x^2 - 8*c^4*x^4 + 16*c^
6*x^6)) + 105*b*(5*d - 2*e*x^2)*(d + e*x^2)^2*ArcCsch[c*x]))/(3675*d^2*x^7) - ((I/3675)*b*c*Sqrt[1 + 1/(c^2*x^
2)]*x*Sqrt[1 + (e*x^2)/d]*(c^2*d*(240*c^6*d^3 - 528*c^4*d^2*e + 193*c^2*d*e^2 + 247*e^3)*EllipticE[I*ArcSinh[S
qrt[c^2]*x], e/(c^2*d)] - 2*(120*c^8*d^4 - 324*c^6*d^3*e + 221*c^4*d^2*e^2 + 88*c^2*d*e^3 - 105*e^4)*EllipticF
[I*ArcSinh[Sqrt[c^2]*x], e/(c^2*d)]))/(Sqrt[c^2]*d^2*Sqrt[1 + c^2*x^2]*Sqrt[d + e*x^2])

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Maple [F]  time = 0.453, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b{\rm arccsch} \left (cx\right )}{{x}^{8}} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d)^(3/2)*(a+b*arccsch(c*x))/x^8,x)

[Out]

int((e*x^2+d)^(3/2)*(a+b*arccsch(c*x))/x^8,x)

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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^2+d)^(3/2)*(a+b*arccsch(c*x))/x^8,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 (\frac{{\left (a e x^{2} + a d +{\left (b e x^{2} + b d\right )} \operatorname{arcsch}\left (c x\right )\right )} \sqrt{e x^{2} + d}}{x^{8}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^(3/2)*(a+b*arccsch(c*x))/x^8,x, algorithm="fricas")

[Out]

integral((a*e*x^2 + a*d + (b*e*x^2 + b*d)*arccsch(c*x))*sqrt(e*x^2 + d)/x^8, 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**2+d)**(3/2)*(a+b*acsch(c*x))/x**8,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x^{2} + d\right )}^{\frac{3}{2}}{\left (b \operatorname{arcsch}\left (c x\right ) + a\right )}}{x^{8}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^(3/2)*(a+b*arccsch(c*x))/x^8,x, algorithm="giac")

[Out]

integrate((e*x^2 + d)^(3/2)*(b*arccsch(c*x) + a)/x^8, x)