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

Optimal result

Integrand size = 23, antiderivative size = 554 \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{x^8} \, dx=-\frac {b c \left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{3675 d^2 \sqrt {c^2 x^2}}-\frac {b c \left (120 c^4 d^2+159 c^2 d e-37 e^2\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{3675 d x^2 \sqrt {c^2 x^2}}-\frac {b c \left (30 c^2 d+11 e\right ) \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{1225 d x^4 \sqrt {c^2 x^2}}-\frac {b c \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{5/2}}{49 d x^6 \sqrt {c^2 x^2}}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{7 d x^7}+\frac {2 e \left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{35 d^2 x^5}+\frac {b c^2 \left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right ) x \sqrt {1-c^2 x^2} \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{3675 d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}-\frac {2 b \left (c^2 d+e\right ) \left (120 c^6 d^3+204 c^4 d^2 e+17 c^2 d e^2-105 e^3\right ) x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{3675 d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \]

[Out]

-1/7*(e*x^2+d)^(5/2)*(a+b*arccsc(c*x))/d/x^7+2/35*e*(e*x^2+d)^(5/2)*(a+b*arccsc(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*(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*EllipticE(c*x,(-e/c^2/d)^(1/
2))*(-c^2*x^2+1)^(1/2)*(e*x^2+d)^(1/2)/d^2/(c^2*x^2)^(1/2)/(c^2*x^2-1)^(1/2)/(1+e*x^2/d)^(1/2)-2/3675*b*(c^2*d
+e)*(120*c^6*d^3+204*c^4*d^2*e+17*c^2*d*e^2-105*e^3)*x*EllipticF(c*x,(-e/c^2/d)^(1/2))*(-c^2*x^2+1)^(1/2)*(1+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)^(1/2)

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 554, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {277, 270, 5347, 12, 594, 597, 538, 438, 437, 435, 432, 430} \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{x^8} \, dx=\frac {2 e \left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{35 d^2 x^5}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{7 d x^7}-\frac {2 b x \sqrt {1-c^2 x^2} \left (c^2 d+e\right ) \left (120 c^6 d^3+204 c^4 d^2 e+17 c^2 d e^2-105 e^3\right ) \sqrt {\frac {e x^2}{d}+1} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{3675 d^2 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}+\frac {b c^2 x \sqrt {1-c^2 x^2} \left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right ) \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{3675 d^2 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {\frac {e x^2}{d}+1}}-\frac {b c \sqrt {c^2 x^2-1} \left (d+e x^2\right )^{5/2}}{49 d x^6 \sqrt {c^2 x^2}}-\frac {b c \sqrt {c^2 x^2-1} \left (30 c^2 d+11 e\right ) \left (d+e x^2\right )^{3/2}}{1225 d x^4 \sqrt {c^2 x^2}}-\frac {b c \sqrt {c^2 x^2-1} \left (120 c^4 d^2+159 c^2 d e-37 e^2\right ) \sqrt {d+e x^2}}{3675 d x^2 \sqrt {c^2 x^2}}-\frac {b c \sqrt {c^2 x^2-1} \left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right ) \sqrt {d+e x^2}}{3675 d^2 \sqrt {c^2 x^2}} \]

[In]

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

[Out]

-1/3675*(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])/(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*Sqr
t[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*S
qrt[-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 -
105*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 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 430

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

Rule 432

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

Rule 435

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*Ell
ipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0
]

Rule 437

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

Rule 438

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

Rule 538

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 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 5347

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]))

Rubi steps \begin{align*} \text {integral}& = -\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 (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\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}} \operatorname {EllipticF}\left (\arcsin (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] (verified)

Result contains complex when optimal does not.

Time = 11.82 (sec) , antiderivative size = 383, normalized size of antiderivative = 0.69 \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{x^8} \, dx=-\frac {\sqrt {d+e x^2} \left (105 a \left (5 d-2 e x^2\right ) \left (d+e x^2\right )^2+b c \sqrt {1-\frac {1}{c^2 x^2}} x \left (-247 e^3 x^6+d e^2 x^4 \left (71+193 c^2 x^2\right )+3 d^2 e x^2 \left (61+83 c^2 x^2+176 c^4 x^4\right )+15 d^3 \left (5+6 c^2 x^2+8 c^4 x^4+16 c^6 x^6\right )\right )+105 b \left (5 d-2 e x^2\right ) \left (d+e x^2\right )^2 \csc ^{-1}(c x)\right )}{3675 d^2 x^7}+\frac {i b c \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {1+\frac {e x^2}{d}} \left (c^2 d \left (240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right ) E\left (i \text {arcsinh}\left (\sqrt {-c^2} x\right )|-\frac {e}{c^2 d}\right )-2 \left (120 c^8 d^4+324 c^6 d^3 e+221 c^4 d^2 e^2-88 c^2 d e^3-105 e^4\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-c^2} x\right ),-\frac {e}{c^2 d}\right )\right )}{3675 \sqrt {-c^2} d^2 \sqrt {1-c^2 x^2} \sqrt {d+e x^2}} \]

[In]

Integrate[((d + e*x^2)^(3/2)*(a + b*ArcCsc[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*ArcCsc[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[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)*Ellip
ticF[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])

Maple [F]

\[\int \frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arccsc}\left (c x \right )\right )}{x^{8}}d x\]

[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)

Fricas [A] (verification not implemented)

none

Time = 0.14 (sec) , antiderivative size = 397, normalized size of antiderivative = 0.72 \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{x^8} \, dx=\frac {{\left (210 \, a c d e^{3} x^{6} - 105 \, a c d^{2} e^{2} x^{4} - 840 \, a c d^{3} e x^{2} - 525 \, a c d^{4} + 105 \, {\left (2 \, b c d e^{3} x^{6} - b c d^{2} e^{2} x^{4} - 8 \, b c d^{3} e x^{2} - 5 \, b c d^{4}\right )} \operatorname {arccsc}\left (c x\right ) - {\left ({\left (240 \, b c^{7} d^{4} + 528 \, b c^{5} d^{3} e + 193 \, b c^{3} d^{2} e^{2} - 247 \, b c d e^{3}\right )} x^{6} + 75 \, b c d^{4} + {\left (120 \, b c^{5} d^{4} + 249 \, b c^{3} d^{3} e + 71 \, b c d^{2} e^{2}\right )} x^{4} + 3 \, {\left (30 \, b c^{3} d^{4} + 61 \, b c d^{3} e\right )} x^{2}\right )} \sqrt {c^{2} x^{2} - 1}\right )} \sqrt {e x^{2} + d} - {\left ({\left (240 \, b c^{10} d^{4} + 528 \, b c^{8} d^{3} e + 193 \, b c^{6} d^{2} e^{2} - 247 \, b c^{4} d e^{3}\right )} x^{7} E(\arcsin \left (c x\right )\,|\,-\frac {e}{c^{2} d}) - {\left (240 \, b c^{10} d^{4} + 24 \, {\left (22 \, b c^{8} + 5 \, b c^{6}\right )} d^{3} e + {\left (193 \, b c^{6} + 249 \, b c^{4}\right )} d^{2} e^{2} - {\left (247 \, b c^{4} - 71 \, b c^{2}\right )} d e^{3} - 210 \, b e^{4}\right )} x^{7} F(\arcsin \left (c x\right )\,|\,-\frac {e}{c^{2} d})\right )} \sqrt {-d}}{3675 \, c d^{3} x^{7}} \]

[In]

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

[Out]

1/3675*((210*a*c*d*e^3*x^6 - 105*a*c*d^2*e^2*x^4 - 840*a*c*d^3*e*x^2 - 525*a*c*d^4 + 105*(2*b*c*d*e^3*x^6 - b*
c*d^2*e^2*x^4 - 8*b*c*d^3*e*x^2 - 5*b*c*d^4)*arccsc(c*x) - ((240*b*c^7*d^4 + 528*b*c^5*d^3*e + 193*b*c^3*d^2*e
^2 - 247*b*c*d*e^3)*x^6 + 75*b*c*d^4 + (120*b*c^5*d^4 + 249*b*c^3*d^3*e + 71*b*c*d^2*e^2)*x^4 + 3*(30*b*c^3*d^
4 + 61*b*c*d^3*e)*x^2)*sqrt(c^2*x^2 - 1))*sqrt(e*x^2 + d) - ((240*b*c^10*d^4 + 528*b*c^8*d^3*e + 193*b*c^6*d^2
*e^2 - 247*b*c^4*d*e^3)*x^7*elliptic_e(arcsin(c*x), -e/(c^2*d)) - (240*b*c^10*d^4 + 24*(22*b*c^8 + 5*b*c^6)*d^
3*e + (193*b*c^6 + 249*b*c^4)*d^2*e^2 - (247*b*c^4 - 71*b*c^2)*d*e^3 - 210*b*e^4)*x^7*elliptic_f(arcsin(c*x),
-e/(c^2*d)))*sqrt(-d))/(c*d^3*x^7)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{x^8} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{x^8} \, dx=\text {Exception raised: ValueError} \]

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [F]

\[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{x^8} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}}{x^{8}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{x^8} \, dx=\int \frac {{\left (e\,x^2+d\right )}^{3/2}\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{x^8} \,d x \]

[In]

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

[Out]

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