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

Optimal. Leaf size=554 \[ -\frac{2 b x \sqrt{1-c^2 x^2} \left (c^2 d+e\right ) \left (204 c^4 d^2 e+120 c^6 d^3+17 c^2 d e^2-105 e^3\right ) \sqrt{\frac{e x^2}{d}+1} \text{EllipticF}\left (\sin ^{-1}(c x),-\frac{e}{c^2 d}\right )}{3675 d^2 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}+\frac{2 e \left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{35 d^2 x^5}-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{7 d x^7}-\frac{b c \sqrt{c^2 x^2-1} \left (528 c^4 d^2 e+240 c^6 d^3+193 c^2 d e^2-247 e^3\right ) \sqrt{d+e x^2}}{3675 d^2 \sqrt{c^2 x^2}}-\frac{b c \sqrt{c^2 x^2-1} \left (120 c^4 d^2+159 c^2 d e-37 e^2\right ) \sqrt{d+e x^2}}{3675 d x^2 \sqrt{c^2 x^2}}+\frac{b c^2 x \sqrt{1-c^2 x^2} \left (528 c^4 d^2 e+240 c^6 d^3+193 c^2 d e^2-247 e^3\right ) \sqrt{d+e x^2} E\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{3675 d^2 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{\frac{e x^2}{d}+1}}-\frac{b c \sqrt{c^2 x^2-1} \left (d+e x^2\right )^{5/2}}{49 d x^6 \sqrt{c^2 x^2}}-\frac{b c \sqrt{c^2 x^2-1} \left (30 c^2 d+11 e\right ) \left (d+e x^2\right )^{3/2}}{1225 d x^4 \sqrt{c^2 x^2}} \]

[Out]

-(b*c*(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*Sq
rt[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*Sqr
t[-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*ArcCsc[c*x]))/(7*d*x^7
) + (2*e*(d + e*x^2)^(5/2)*(a + b*ArcCsc[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[1 - c^2*x^2]*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/(3675*d^2*Sqrt[c^2*x
^2]*Sqrt[-1 + c^2*x^2]*Sqrt[1 + (e*x^2)/d]) - (2*b*(c^2*d + e)*(120*c^6*d^3 + 204*c^4*d^2*e + 17*c^2*d*e^2 - 1
05*e^3)*x*Sqrt[1 - c^2*x^2]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(3675*d^2*Sqrt[c^2*x^2]*
Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.891112, antiderivative size = 554, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 12, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.522, Rules used = {271, 264, 5239, 12, 580, 583, 524, 427, 426, 424, 421, 419} \[ \frac{2 e \left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{35 d^2 x^5}-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{7 d x^7}-\frac{b c \sqrt{c^2 x^2-1} \left (528 c^4 d^2 e+240 c^6 d^3+193 c^2 d e^2-247 e^3\right ) \sqrt{d+e x^2}}{3675 d^2 \sqrt{c^2 x^2}}-\frac{b c \sqrt{c^2 x^2-1} \left (120 c^4 d^2+159 c^2 d e-37 e^2\right ) \sqrt{d+e x^2}}{3675 d x^2 \sqrt{c^2 x^2}}-\frac{2 b x \sqrt{1-c^2 x^2} \left (c^2 d+e\right ) \left (204 c^4 d^2 e+120 c^6 d^3+17 c^2 d e^2-105 e^3\right ) \sqrt{\frac{e x^2}{d}+1} F\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{3675 d^2 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}+\frac{b c^2 x \sqrt{1-c^2 x^2} \left (528 c^4 d^2 e+240 c^6 d^3+193 c^2 d e^2-247 e^3\right ) \sqrt{d+e x^2} E\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{3675 d^2 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{\frac{e x^2}{d}+1}}-\frac{b c \sqrt{c^2 x^2-1} \left (d+e x^2\right )^{5/2}}{49 d x^6 \sqrt{c^2 x^2}}-\frac{b c \sqrt{c^2 x^2-1} \left (30 c^2 d+11 e\right ) \left (d+e x^2\right )^{3/2}}{1225 d x^4 \sqrt{c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(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*Sq
rt[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*Sqr
t[-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*ArcCsc[c*x]))/(7*d*x^7
) + (2*e*(d + e*x^2)^(5/2)*(a + b*ArcCsc[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[1 - c^2*x^2]*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/(3675*d^2*Sqrt[c^2*x
^2]*Sqrt[-1 + c^2*x^2]*Sqrt[1 + (e*x^2)/d]) - (2*b*(c^2*d + e)*(120*c^6*d^3 + 204*c^4*d^2*e + 17*c^2*d*e^2 - 1
05*e^3)*x*Sqrt[1 - c^2*x^2]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(3675*d^2*Sqrt[c^2*x^2]*
Sqrt[-1 + c^2*x^2]*Sqrt[d + e*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 5239

Int[((a_.) + ArcCsc[(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*ArcCsc[c*x], u, x] + Dist[(b*c*x)/Sqrt[c^2*x^2], Int[SimplifyI
ntegrand[u/(x*Sqrt[c^2*x^2 - 1]), 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])) || (I
LtQ[(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 524

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

Rule 427

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

Rule 426

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[a + b*x^2]/Sqrt[1 + (b*x^2)/a]
, Int[Sqrt[1 + (b*x^2)/a]/Sqrt[c + d*x^2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &&  !GtQ
[a, 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 421

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

Rubi steps

\begin{align*} \int \frac{\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{x^8} \, dx &=-\frac{\left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{7 d x^7}+\frac{2 e \left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-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 \csc ^{-1}(c x)\right )}{7 d x^7}+\frac{2 e \left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-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 \csc ^{-1}(c x)\right )}{7 d x^7}+\frac{2 e \left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-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 )+\left (5 c^2 d-14 e\right ) e 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 \csc ^{-1}(c x)\right )}{7 d x^7}+\frac{2 e \left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-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 \csc ^{-1}(c x)\right )}{7 d x^7}+\frac{2 e \left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-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 \csc ^{-1}(c x)\right )}{7 d x^7}+\frac{2 e \left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-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 \csc ^{-1}(c x)\right )}{7 d x^7}+\frac{2 e \left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{35 d^2 x^5}+\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}}{\sqrt{-1+c^2 x^2}} \, dx}{3675 d^2 \sqrt{c^2 x^2}}-\frac{\left (2 b c \left (c^2 d+e\right ) \left (120 c^6 d^3+204 c^4 d^2 e+17 c^2 d e^2-105 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{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 \csc ^{-1}(c x)\right )}{7 d x^7}+\frac{2 e \left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{35 d^2 x^5}+\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 \sqrt{1-c^2 x^2}\right ) \int \frac{\sqrt{d+e x^2}}{\sqrt{1-c^2 x^2}} \, dx}{3675 d^2 \sqrt{c^2 x^2} \sqrt{-1+c^2 x^2}}-\frac{\left (2 b c \left (c^2 d+e\right ) \left (120 c^6 d^3+204 c^4 d^2 e+17 c^2 d e^2-105 e^3\right ) x \sqrt{1+\frac{e x^2}{d}}\right ) \int \frac{1}{\sqrt{-1+c^2 x^2} \sqrt{1+\frac{e x^2}{d}}} \, dx}{3675 d^2 \sqrt{c^2 x^2} \sqrt{d+e 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 \csc ^{-1}(c x)\right )}{7 d x^7}+\frac{2 e \left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{35 d^2 x^5}+\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 \sqrt{1-c^2 x^2} \sqrt{d+e x^2}\right ) \int \frac{\sqrt{1+\frac{e x^2}{d}}}{\sqrt{1-c^2 x^2}} \, dx}{3675 d^2 \sqrt{c^2 x^2} \sqrt{-1+c^2 x^2} \sqrt{1+\frac{e x^2}{d}}}-\frac{\left (2 b c \left (c^2 d+e\right ) \left (120 c^6 d^3+204 c^4 d^2 e+17 c^2 d e^2-105 e^3\right ) x \sqrt{1-c^2 x^2} \sqrt{1+\frac{e x^2}{d}}\right ) \int \frac{1}{\sqrt{1-c^2 x^2} \sqrt{1+\frac{e x^2}{d}}} \, dx}{3675 d^2 \sqrt{c^2 x^2} \sqrt{-1+c^2 x^2} \sqrt{d+e 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 \csc ^{-1}(c x)\right )}{7 d x^7}+\frac{2 e \left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-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{1-c^2 x^2} \sqrt{d+e x^2} E\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{3675 d^2 \sqrt{c^2 x^2} \sqrt{-1+c^2 x^2} \sqrt{1+\frac{e x^2}{d}}}-\frac{2 b \left (c^2 d+e\right ) \left (120 c^6 d^3+204 c^4 d^2 e+17 c^2 d e^2-105 e^3\right ) x \sqrt{1-c^2 x^2} \sqrt{1+\frac{e x^2}{d}} F\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{3675 d^2 \sqrt{c^2 x^2} \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}\\ \end{align*}

Mathematica [C]  time = 0.851825, size = 383, normalized size = 0.69 \[ -\frac{\sqrt{d+e x^2} \left (105 a \left (5 d-2 e x^2\right ) \left (d+e x^2\right )^2+b c x \sqrt{1-\frac{1}{c^2 x^2}} \left (3 d^2 e x^2 \left (176 c^4 x^4+83 c^2 x^2+61\right )+15 d^3 \left (16 c^6 x^6+8 c^4 x^4+6 c^2 x^2+5\right )+d e^2 x^4 \left (193 c^2 x^2+71\right )-247 e^3 x^6\right )+105 b \csc ^{-1}(c x) \left (5 d-2 e x^2\right ) \left (d+e x^2\right )^2\right )}{3675 d^2 x^7}+\frac{i b c x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{e x^2}{d}+1} \left (c^2 d \left (528 c^4 d^2 e+240 c^6 d^3+193 c^2 d e^2-247 e^3\right ) E\left (i \sinh ^{-1}\left (\sqrt{-c^2} x\right )|-\frac{e}{c^2 d}\right )-2 \left (221 c^4 d^2 e^2+324 c^6 d^3 e+120 c^8 d^4-88 c^2 d e^3-105 e^4\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{-c^2} x\right ),-\frac{e}{c^2 d}\right )\right )}{3675 \sqrt{-c^2} d^2 \sqrt{1-c^2 x^2} \sqrt{d+e x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x^2)^(3/2)*(a + b*ArcCsc[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*ArcCsc[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[Sqr
t[-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 = 5.132, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b{\rm arccsc} \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*arccsc(c*x))/x^8,x)

[Out]

int((e*x^2+d)^(3/2)*(a+b*arccsc(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*arccsc(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{arccsc}\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*arccsc(c*x))/x^8,x, algorithm="fricas")

[Out]

integral((a*e*x^2 + a*d + (b*e*x^2 + b*d)*arccsc(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*acsc(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{arccsc}\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*arccsc(c*x))/x^8,x, algorithm="giac")

[Out]

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