3.2.37 \(\int \frac {(d+e x^2)^{3/2} (a+b \text {csch}^{-1}(c x))}{x^8} \, dx\) [137]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.55, antiderivative size = 643, normalized size of antiderivative = 1.00, 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 = {277, 270, 6437, 12, 594, 597, 545, 429, 506, 422} \begin {gather*} \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 e x \left (120 c^6 d^3-249 c^4 d^2 e+71 c^2 d e^2+210 e^3\right ) \sqrt {d+e x^2} F\left (\text {ArcTan}(c x)\left |1-\frac {e}{c^2 d}\right .\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 (240 c^6 d^3-528 c^4 d^2 e+193 c^2 d e^2+247 e^3\right ) \sqrt {d+e x^2} E\left (\text {ArcTan}(c x)\left |1-\frac {e}{c^2 d}\right .\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}}+\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 \sqrt {-c^2 x^2-1} \left (240 c^6 d^3-528 c^4 d^2 e+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^3 x^2 \left (240 c^6 d^3-528 c^4 d^2 e+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}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3675*(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])/(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*(
240*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 12

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

Rule 270

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 277

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 422

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

Rule 429

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

Rule 506

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt
[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 545

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 594

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 597

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 6437

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]))

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*}

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Mathematica [C] Result contains complex when optimal does not.
time = 10.53, size = 372, normalized size = 0.58 \begin {gather*} -\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 \sqrt {1+\frac {1}{c^2 x^2}} x \left (247 e^3 x^6+d e^2 x^4 \left (-71+193 c^2 x^2\right )-3 d^2 e x^2 \left (61-83 c^2 x^2+176 c^4 x^4\right )+15 d^3 \left (-5+6 c^2 x^2-8 c^4 x^4+16 c^6 x^6\right )\right )+105 b \left (5 d-2 e x^2\right ) \left (d+e x^2\right )^2 \text {csch}^{-1}(c x)\right )}{3675 d^2 x^7}-\frac {i b c \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {1+\frac {e x^2}{d}} \left (c^2 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 (i \sinh ^{-1}\left (\sqrt {c^2} x\right )|\frac {e}{c^2 d}\right )-2 \left (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\right ) F\left (i \sinh ^{-1}\left (\sqrt {c^2} x\right )|\frac {e}{c^2 d}\right )\right )}{3675 \sqrt {c^2} d^2 \sqrt {1+c^2 x^2} \sqrt {d+e x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3675*(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]))/(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
[Sqrt[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)*Ellipti
cF[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 = 0.13, size = 0, normalized size = 0.00 \[\int \frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\mathrm {arccsch}\left (c x \right )\right )}{x^{8}}\, dx\]

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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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]

1/35*a*(2*(x^2*e + d)^(5/2)*e/(d^2*x^5) - 5*(x^2*e + d)^(5/2)/(d*x^7)) + 1/35*b*((2*x^6*e^3 - d*x^4*e^2 - 8*d^
2*x^2*e - 5*d^3)*sqrt(x^2*e + d)*log(sqrt(c^2*x^2 + 1) + 1)/(d^2*x^7) - 35*integrate(1/35*(2*c^2*x^8*e^3 - c^2
*d*x^6*e^2 + (35*d^2*log(c) - 8*d^2)*c^2*x^4*e + 35*d^3*log(c) + 5*(7*d^2*e*log(c) + (7*d^3*log(c) - d^3)*c^2)
*x^2 + 35*(c^2*d^2*x^4*e + d^3 + (c^2*d^3 + d^2*e)*x^2)*log(x))*sqrt(x^2*e + d)/(c^2*d^2*x^10 + d^2*x^8), x) +
35*integrate(1/35*(2*c^2*x^6*e^3 - c^2*d*x^4*e^2 - 8*c^2*d^2*x^2*e - 5*c^2*d^3)*sqrt(x^2*e + d)/(c^2*d^2*x^8
+ d^2*x^6 + (c^2*d^2*x^8 + d^2*x^6)*sqrt(c^2*x^2 + 1)), x))

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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]

Exception raised: TypeError >> Symbolic function elliptic_ec takes exactly 1 arguments (2 given)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

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]

Exception raised: SystemError >> excessive stack use: stack is 3061 deep

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (e\,x^2+d\right )}^{3/2}\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{x^8} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((d + e*x^2)^(3/2)*(a + b*asinh(1/(c*x))))/x^8,x)

[Out]

int(((d + e*x^2)^(3/2)*(a + b*asinh(1/(c*x))))/x^8, x)

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