\(\int \frac {a+b \csc ^{-1}(c x)}{(d+e x)^{7/2}} \, dx\) [75]

Optimal result

Integrand size = 18, antiderivative size = 540 \[ \int \frac {a+b \csc ^{-1}(c x)}{(d+e x)^{7/2}} \, dx=\frac {4 b e \left (1-c^2 x^2\right )}{15 c d \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x (d+e x)^{3/2}}+\frac {16 b c e \left (1-c^2 x^2\right )}{15 \left (c^2 d^2-e^2\right )^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {4 b e \left (1-c^2 x^2\right )}{5 c d^2 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}-\frac {4 b \left (7 c^2 d^2-3 e^2\right ) \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 \left (c^2 d^3-d e^2\right )^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}+\frac {4 b \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{15 d \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {4 b \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{5 c d^2 e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \]

[Out]

-2/5*(a+b*arccsc(c*x))/e/(e*x+d)^(5/2)+4/15*b*e*(-c^2*x^2+1)/c/d/(c^2*d^2-e^2)/x/(e*x+d)^(3/2)/(1-1/c^2/x^2)^(
1/2)+16/15*b*c*e*(-c^2*x^2+1)/(c^2*d^2-e^2)^2/x/(1-1/c^2/x^2)^(1/2)/(e*x+d)^(1/2)+4/5*b*e*(-c^2*x^2+1)/c/d^2/(
c^2*d^2-e^2)/x/(1-1/c^2/x^2)^(1/2)/(e*x+d)^(1/2)-4/15*b*(7*c^2*d^2-3*e^2)*EllipticE(1/2*(-c*x+1)^(1/2)*2^(1/2)
,2^(1/2)*(e/(c*d+e))^(1/2))*(e*x+d)^(1/2)*(-c^2*x^2+1)^(1/2)/(c^2*d^3-d*e^2)^2/x/(1-1/c^2/x^2)^(1/2)/(c*(e*x+d
)/(c*d+e))^(1/2)+4/15*b*EllipticF(1/2*(-c*x+1)^(1/2)*2^(1/2),2^(1/2)*(e/(c*d+e))^(1/2))*(c*(e*x+d)/(c*d+e))^(1
/2)*(-c^2*x^2+1)^(1/2)/d/(c^2*d^2-e^2)/x/(1-1/c^2/x^2)^(1/2)/(e*x+d)^(1/2)+4/5*b*EllipticPi(1/2*(-c*x+1)^(1/2)
*2^(1/2),2,2^(1/2)*(e/(c*d+e))^(1/2))*(c*(e*x+d)/(c*d+e))^(1/2)*(-c^2*x^2+1)^(1/2)/c/d^2/e/x/(1-1/c^2/x^2)^(1/
2)/(e*x+d)^(1/2)

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 637, normalized size of antiderivative = 1.18, number of steps used = 19, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.778, Rules used = {5335, 1588, 972, 759, 849, 858, 733, 435, 430, 21, 947, 174, 552, 551} \[ \int \frac {a+b \csc ^{-1}(c x)}{(d+e x)^{7/2}} \, dx=-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}+\frac {4 b \sqrt {1-c^2 x^2} \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{15 d x \sqrt {1-\frac {1}{c^2 x^2}} \left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {16 b c^2 \sqrt {1-c^2 x^2} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 x \sqrt {1-\frac {1}{c^2 x^2}} \left (c^2 d^2-e^2\right )^2 \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {4 b \sqrt {1-c^2 x^2} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{5 d^2 x \sqrt {1-\frac {1}{c^2 x^2}} \left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}+\frac {4 b \sqrt {1-c^2 x^2} \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{5 c d^2 e x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {d+e x}}+\frac {16 b c e \left (1-c^2 x^2\right )}{15 x \sqrt {1-\frac {1}{c^2 x^2}} \left (c^2 d^2-e^2\right )^2 \sqrt {d+e x}}+\frac {4 b e \left (1-c^2 x^2\right )}{5 c d^2 x \sqrt {1-\frac {1}{c^2 x^2}} \left (c^2 d^2-e^2\right ) \sqrt {d+e x}}+\frac {4 b e \left (1-c^2 x^2\right )}{15 c d x \sqrt {1-\frac {1}{c^2 x^2}} \left (c^2 d^2-e^2\right ) (d+e x)^{3/2}} \]

[In]

Int[(a + b*ArcCsc[c*x])/(d + e*x)^(7/2),x]

[Out]

(4*b*e*(1 - c^2*x^2))/(15*c*d*(c^2*d^2 - e^2)*Sqrt[1 - 1/(c^2*x^2)]*x*(d + e*x)^(3/2)) + (16*b*c*e*(1 - c^2*x^
2))/(15*(c^2*d^2 - e^2)^2*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x]) + (4*b*e*(1 - c^2*x^2))/(5*c*d^2*(c^2*d^2 - e
^2)*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x]) - (2*(a + b*ArcCsc[c*x]))/(5*e*(d + e*x)^(5/2)) - (16*b*c^2*Sqrt[d
+ e*x]*Sqrt[1 - c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(15*(c^2*d^2 - e^2)^2*Sqrt
[1 - 1/(c^2*x^2)]*x*Sqrt[(c*(d + e*x))/(c*d + e)]) - (4*b*Sqrt[d + e*x]*Sqrt[1 - c^2*x^2]*EllipticE[ArcSin[Sqr
t[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(5*d^2*(c^2*d^2 - e^2)*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[(c*(d + e*x))/(c*d
+ e)]) + (4*b*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(
c*d + e)])/(15*d*(c^2*d^2 - e^2)*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x]) + (4*b*Sqrt[(c*(d + e*x))/(c*d + e)]*S
qrt[1 - c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(5*c*d^2*e*Sqrt[1 - 1/(c^2*x^2
)]*x*Sqrt[d + e*x])

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 174

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 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 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 551

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1/(a*Sqr
t[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b*(c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], 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 552

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

Rule 733

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 759

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m + 1)*((a + c*x^2)^(p
+ 1)/((m + 1)*(c*d^2 + a*e^2))), x] + Dist[c/((m + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^(m + 1)*Simp[d*(m + 1)
- e*(m + 2*p + 3)*x, x]*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[
m, -1] && ((LtQ[m, -1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]) || (SumSimplerQ[m, 1] && IntegerQ[p]) || ILtQ
[Simplify[m + 2*p + 3], 0])

Rule 849

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*f - d*g)*
(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/((m + 1)*(c*d^2 + a*e^2))), x] + Dist[1/((m + 1)*(c*d^2 + a*e^2)), Int[
(d + e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; Fr
eeQ[{a, c, d, e, f, g, p}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || Integer
sQ[2*m, 2*p])

Rule 858

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 947

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 972

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 1588

Int[(x_)^(m_.)*((a_.) + (c_.)*(x_)^(mn2_.))^(p_)*((d_) + (e_.)*(x_)^(n_.))^(q_.), x_Symbol] :> Dist[x^(2*n*Fra
cPart[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 5335

Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*((a + b
*ArcCsc[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]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}-\frac {(2 b) \int \frac {1}{\sqrt {1-\frac {1}{c^2 x^2}} x^2 (d+e x)^{5/2}} \, dx}{5 c e} \\ & = -\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}-\frac {\left (2 b \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \frac {1}{x (d+e x)^{5/2} \sqrt {-\frac {1}{c^2}+x^2}} \, dx}{5 c e \sqrt {1-\frac {1}{c^2 x^2}} x} \\ & = -\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}-\frac {\left (2 b \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \left (-\frac {e}{d (d+e x)^{5/2} \sqrt {-\frac {1}{c^2}+x^2}}-\frac {e}{d^2 (d+e x)^{3/2} \sqrt {-\frac {1}{c^2}+x^2}}+\frac {1}{d^2 x \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 \left (a+b \csc ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}+\frac {\left (2 b \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \frac {1}{(d+e x)^{3/2} \sqrt {-\frac {1}{c^2}+x^2}} \, dx}{5 c d^2 \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (2 b \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \frac {1}{(d+e x)^{5/2} \sqrt {-\frac {1}{c^2}+x^2}} \, dx}{5 c d \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (2 b \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 d^2 e \sqrt {1-\frac {1}{c^2 x^2}} x} \\ & = \frac {4 b e \left (1-c^2 x^2\right )}{15 c d \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x (d+e x)^{3/2}}+\frac {4 b e \left (1-c^2 x^2\right )}{5 c d^2 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}-\frac {\left (4 b \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \frac {-\frac {d}{2}-\frac {e x}{2}}{\sqrt {d+e x} \sqrt {-\frac {1}{c^2}+x^2}} \, dx}{5 c d^2 \left (d^2-\frac {e^2}{c^2}\right ) \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (4 b \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \frac {-\frac {3 d}{2}+\frac {e x}{2}}{(d+e x)^{3/2} \sqrt {-\frac {1}{c^2}+x^2}} \, dx}{15 c d \left (d^2-\frac {e^2}{c^2}\right ) \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \int \frac {1}{x \sqrt {1-c x} \sqrt {1+c x} \sqrt {d+e x}} \, dx}{5 c d^2 e \sqrt {1-\frac {1}{c^2 x^2}} x} \\ & = \frac {4 b e \left (1-c^2 x^2\right )}{15 c d \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x (d+e x)^{3/2}}+\frac {16 b c e \left (1-c^2 x^2\right )}{15 \left (c^2 d^2-e^2\right )^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {4 b e \left (1-c^2 x^2\right )}{5 c d^2 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}+\frac {\left (8 b \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \frac {\frac {1}{4} \left (3 d^2+\frac {e^2}{c^2}\right )+d e x}{\sqrt {d+e x} \sqrt {-\frac {1}{c^2}+x^2}} \, dx}{15 c d \left (d^2-\frac {e^2}{c^2}\right )^2 \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (2 b \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {-\frac {1}{c^2}+x^2}} \, dx}{5 c d^2 \left (d^2-\frac {e^2}{c^2}\right ) \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (4 b \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {2-x^2} \sqrt {d+\frac {e}{c}-\frac {e x^2}{c}}} \, dx,x,\sqrt {1-c x}\right )}{5 c d^2 e \sqrt {1-\frac {1}{c^2 x^2}} x} \\ & = \frac {4 b e \left (1-c^2 x^2\right )}{15 c d \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x (d+e x)^{3/2}}+\frac {16 b c e \left (1-c^2 x^2\right )}{15 \left (c^2 d^2-e^2\right )^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {4 b e \left (1-c^2 x^2\right )}{5 c d^2 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}+\frac {\left (8 b \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {-\frac {1}{c^2}+x^2}} \, dx}{15 c \left (d^2-\frac {e^2}{c^2}\right )^2 \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (2 b \left (-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 d \left (d^2-\frac {e^2}{c^2}\right )^2 \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (4 b \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {2-x^2} \sqrt {1-\frac {e x^2}{c \left (d+\frac {e}{c}\right )}}} \, dx,x,\sqrt {1-c x}\right )}{5 c d^2 e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {\left (4 b \sqrt {d+e x} \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\sqrt {1-\frac {2 e x^2}{c \left (d+\frac {e}{c}\right )}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {2}}\right )}{5 c^2 d^2 \left (d^2-\frac {e^2}{c^2}\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {d+e x}{d+\frac {e}{c}}}} \\ & = \frac {4 b e \left (1-c^2 x^2\right )}{15 c d \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x (d+e x)^{3/2}}+\frac {16 b c e \left (1-c^2 x^2\right )}{15 \left (c^2 d^2-e^2\right )^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {4 b e \left (1-c^2 x^2\right )}{5 c d^2 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}-\frac {4 b \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{5 d^2 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}+\frac {4 b \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{5 c d^2 e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {\left (16 b \sqrt {d+e x} \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\sqrt {1-\frac {2 e x^2}{c \left (d+\frac {e}{c}\right )}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {2}}\right )}{15 c^2 \left (d^2-\frac {e^2}{c^2}\right )^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {d+e x}{d+\frac {e}{c}}}}-\frac {\left (4 b \left (-d^2+\frac {e^2}{c^2}\right ) \sqrt {\frac {d+e x}{d+\frac {e}{c}}} \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1-\frac {2 e x^2}{c \left (d+\frac {e}{c}\right )}}} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {2}}\right )}{15 c^2 d \left (d^2-\frac {e^2}{c^2}\right )^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \\ & = \frac {4 b e \left (1-c^2 x^2\right )}{15 c d \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x (d+e x)^{3/2}}+\frac {16 b c e \left (1-c^2 x^2\right )}{15 \left (c^2 d^2-e^2\right )^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {4 b e \left (1-c^2 x^2\right )}{5 c d^2 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}-\frac {16 b c^2 \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 \left (c^2 d^2-e^2\right )^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {4 b \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{5 d^2 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}+\frac {4 b \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{15 d \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {4 b \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{5 c d^2 e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 34.05 (sec) , antiderivative size = 1002, normalized size of antiderivative = 1.86 \[ \int \frac {a+b \csc ^{-1}(c x)}{(d+e x)^{7/2}} \, dx=-\frac {2 a}{5 e (d+e x)^{5/2}}+\frac {b \left (-\frac {c^4 \left (e+\frac {d}{x}\right )^4 x^4 \left (\frac {4 \left (-7 c^2 d^2+3 e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}}}{15 c^2 d^2 \left (-c^2 d^2+e^2\right )^2}+\frac {2 \csc ^{-1}(c x)}{5 c^3 d^3 e}-\frac {2 e^2 \csc ^{-1}(c x)}{5 c^3 d^3 \left (e+\frac {d}{x}\right )^3}-\frac {2 \left (2 c d e^2 \sqrt {1-\frac {1}{c^2 x^2}}-9 c^2 d^2 e \csc ^{-1}(c x)+9 e^3 \csc ^{-1}(c x)\right )}{15 c^3 d^3 \left (c^2 d^2-e^2\right ) \left (e+\frac {d}{x}\right )^2}-\frac {2 \left (-16 c^3 d^3 e \sqrt {1-\frac {1}{c^2 x^2}}+8 c d e^3 \sqrt {1-\frac {1}{c^2 x^2}}+9 c^4 d^4 \csc ^{-1}(c x)-18 c^2 d^2 e^2 \csc ^{-1}(c x)+9 e^4 \csc ^{-1}(c x)\right )}{15 c^3 d^3 \left (c^2 d^2-e^2\right )^2 \left (e+\frac {d}{x}\right )}\right )}{(d+e x)^{7/2}}+\frac {2 \left (e+\frac {d}{x}\right )^{7/2} (c x)^{7/2} \left (\frac {2 \left (c^2 d^2 e-e^3\right ) \sqrt {\frac {c d+c e x}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{\sqrt {1-\frac {1}{c^2 x^2}} \sqrt {e+\frac {d}{x}} (c x)^{3/2}}+\frac {2 \left (3 c^3 d^3+c d e^2\right ) \sqrt {\frac {c d+c e x}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{\sqrt {1-\frac {1}{c^2 x^2}} \sqrt {e+\frac {d}{x}} (c x)^{3/2}}+\frac {2 \left (-7 c^2 d^2 e+3 e^3\right ) \cos \left (2 \csc ^{-1}(c x)\right ) \left ((c d+c e x) \left (-1+c^2 x^2\right )+c^2 d x \sqrt {\frac {c d+c e x}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )-\frac {c x (1+c x) \sqrt {\frac {e-c e x}{c d+e}} \sqrt {\frac {c d+c e x}{c d-e}} \left ((c d+e) E\left (\arcsin \left (\sqrt {\frac {c d+c e x}{c d-e}}\right )|\frac {c d-e}{c d+e}\right )-e \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {c d+c e x}{c d-e}}\right ),\frac {c d-e}{c d+e}\right )\right )}{\sqrt {\frac {e (1+c x)}{-c d+e}}}+c e x \sqrt {\frac {c d+c e x}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )\right )}{c d \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {e+\frac {d}{x}} \sqrt {c x} \left (-2+c^2 x^2\right )}\right )}{15 c d (c d-e)^2 e (c d+e)^2 (d+e x)^{7/2}}\right )}{c} \]

[In]

Integrate[(a + b*ArcCsc[c*x])/(d + e*x)^(7/2),x]

[Out]

(-2*a)/(5*e*(d + e*x)^(5/2)) + (b*(-((c^4*(e + d/x)^4*x^4*((4*(-7*c^2*d^2 + 3*e^2)*Sqrt[1 - 1/(c^2*x^2)])/(15*
c^2*d^2*(-(c^2*d^2) + e^2)^2) + (2*ArcCsc[c*x])/(5*c^3*d^3*e) - (2*e^2*ArcCsc[c*x])/(5*c^3*d^3*(e + d/x)^3) -
(2*(2*c*d*e^2*Sqrt[1 - 1/(c^2*x^2)] - 9*c^2*d^2*e*ArcCsc[c*x] + 9*e^3*ArcCsc[c*x]))/(15*c^3*d^3*(c^2*d^2 - e^2
)*(e + d/x)^2) - (2*(-16*c^3*d^3*e*Sqrt[1 - 1/(c^2*x^2)] + 8*c*d*e^3*Sqrt[1 - 1/(c^2*x^2)] + 9*c^4*d^4*ArcCsc[
c*x] - 18*c^2*d^2*e^2*ArcCsc[c*x] + 9*e^4*ArcCsc[c*x]))/(15*c^3*d^3*(c^2*d^2 - e^2)^2*(e + d/x))))/(d + e*x)^(
7/2)) + (2*(e + d/x)^(7/2)*(c*x)^(7/2)*((2*(c^2*d^2*e - e^3)*Sqrt[(c*d + c*e*x)/(c*d + e)]*Sqrt[1 - c^2*x^2]*E
llipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(Sqrt[1 - 1/(c^2*x^2)]*Sqrt[e + d/x]*(c*x)^(3/2)) +
(2*(3*c^3*d^3 + c*d*e^2)*Sqrt[(c*d + c*e*x)/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sq
rt[2]], (2*e)/(c*d + e)])/(Sqrt[1 - 1/(c^2*x^2)]*Sqrt[e + d/x]*(c*x)^(3/2)) + (2*(-7*c^2*d^2*e + 3*e^3)*Cos[2*
ArcCsc[c*x]]*((c*d + c*e*x)*(-1 + c^2*x^2) + c^2*d*x*Sqrt[(c*d + c*e*x)/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticF
[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)] - (c*x*(1 + c*x)*Sqrt[(e - c*e*x)/(c*d + e)]*Sqrt[(c*d + c*e*
x)/(c*d - e)]*((c*d + e)*EllipticE[ArcSin[Sqrt[(c*d + c*e*x)/(c*d - e)]], (c*d - e)/(c*d + e)] - e*EllipticF[A
rcSin[Sqrt[(c*d + c*e*x)/(c*d - e)]], (c*d - e)/(c*d + e)]))/Sqrt[(e*(1 + c*x))/(-(c*d) + e)] + c*e*x*Sqrt[(c*
d + c*e*x)/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)]))/(c*d*S
qrt[1 - 1/(c^2*x^2)]*Sqrt[e + d/x]*Sqrt[c*x]*(-2 + c^2*x^2))))/(15*c*d*(c*d - e)^2*e*(c*d + e)^2*(d + e*x)^(7/
2))))/c

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1617\) vs. \(2(491)=982\).

Time = 9.34 (sec) , antiderivative size = 1618, normalized size of antiderivative = 3.00

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1618\)
default	\(\text {Expression too large to display}\)	\(1618\)
parts	\(\text {Expression too large to display}\)	\(1642\)

[In]

int((a+b*arccsc(c*x))/(e*x+d)^(7/2),x,method=_RETURNVERBOSE)

[Out]

2/e*(-1/5*a/(e*x+d)^(5/2)+b*(-1/5/(e*x+d)^(5/2)*arccsc(c*x)-2/15/c*(7*(c/(c*d-e))^(1/2)*c^4*d^3*(e*x+d)^3-6*((
-c*(e*x+d)+c*d-e)/(c*d-e))^(1/2)*((-c*(e*x+d)+c*d+e)/(c*d+e))^(1/2)*EllipticF((e*x+d)^(1/2)*(c/(c*d-e))^(1/2),
((c*d-e)/(c*d+e))^(1/2))*c^4*d^4*(e*x+d)^(3/2)+7*((-c*(e*x+d)+c*d-e)/(c*d-e))^(1/2)*((-c*(e*x+d)+c*d+e)/(c*d+e
))^(1/2)*EllipticE((e*x+d)^(1/2)*(c/(c*d-e))^(1/2),((c*d-e)/(c*d+e))^(1/2))*c^4*d^4*(e*x+d)^(3/2)-3*((-c*(e*x+
d)+c*d-e)/(c*d-e))^(1/2)*((-c*(e*x+d)+c*d+e)/(c*d+e))^(1/2)*EllipticPi((e*x+d)^(1/2)*(c/(c*d-e))^(1/2),1/c*(c*
d-e)/d,(c/(c*d+e))^(1/2)/(c/(c*d-e))^(1/2))*c^4*d^4*(e*x+d)^(3/2)-13*(c/(c*d-e))^(1/2)*c^4*d^4*(e*x+d)^2-7*((-
c*(e*x+d)+c*d-e)/(c*d-e))^(1/2)*((-c*(e*x+d)+c*d+e)/(c*d+e))^(1/2)*EllipticF((e*x+d)^(1/2)*(c/(c*d-e))^(1/2),(
(c*d-e)/(c*d+e))^(1/2))*c^3*d^3*e*(e*x+d)^(3/2)+7*((-c*(e*x+d)+c*d-e)/(c*d-e))^(1/2)*((-c*(e*x+d)+c*d+e)/(c*d+
e))^(1/2)*EllipticE((e*x+d)^(1/2)*(c/(c*d-e))^(1/2),((c*d-e)/(c*d+e))^(1/2))*c^3*d^3*e*(e*x+d)^(3/2)+5*(c/(c*d
-e))^(1/2)*c^4*d^5*(e*x+d)-3*(c/(c*d-e))^(1/2)*c^2*d*e^2*(e*x+d)^3+2*((-c*(e*x+d)+c*d-e)/(c*d-e))^(1/2)*((-c*(
e*x+d)+c*d+e)/(c*d+e))^(1/2)*EllipticF((e*x+d)^(1/2)*(c/(c*d-e))^(1/2),((c*d-e)/(c*d+e))^(1/2))*c^2*d^2*e^2*(e
*x+d)^(3/2)-3*((-c*(e*x+d)+c*d-e)/(c*d-e))^(1/2)*((-c*(e*x+d)+c*d+e)/(c*d+e))^(1/2)*EllipticE((e*x+d)^(1/2)*(c
/(c*d-e))^(1/2),((c*d-e)/(c*d+e))^(1/2))*c^2*d^2*e^2*(e*x+d)^(3/2)+6*((-c*(e*x+d)+c*d-e)/(c*d-e))^(1/2)*((-c*(
e*x+d)+c*d+e)/(c*d+e))^(1/2)*EllipticPi((e*x+d)^(1/2)*(c/(c*d-e))^(1/2),1/c*(c*d-e)/d,(c/(c*d+e))^(1/2)/(c/(c*
d-e))^(1/2))*c^2*d^2*e^2*(e*x+d)^(3/2)+(c/(c*d-e))^(1/2)*c^4*d^6+5*(c/(c*d-e))^(1/2)*c^2*d^2*e^2*(e*x+d)^2+3*(
(-c*(e*x+d)+c*d-e)/(c*d-e))^(1/2)*((-c*(e*x+d)+c*d+e)/(c*d+e))^(1/2)*EllipticF((e*x+d)^(1/2)*(c/(c*d-e))^(1/2)
,((c*d-e)/(c*d+e))^(1/2))*c*d*e^3*(e*x+d)^(3/2)-3*((-c*(e*x+d)+c*d-e)/(c*d-e))^(1/2)*((-c*(e*x+d)+c*d+e)/(c*d+
e))^(1/2)*EllipticE((e*x+d)^(1/2)*(c/(c*d-e))^(1/2),((c*d-e)/(c*d+e))^(1/2))*c*d*e^3*(e*x+d)^(3/2)-8*(c/(c*d-e
))^(1/2)*c^2*d^3*e^2*(e*x+d)-3*((-c*(e*x+d)+c*d-e)/(c*d-e))^(1/2)*((-c*(e*x+d)+c*d+e)/(c*d+e))^(1/2)*EllipticP
i((e*x+d)^(1/2)*(c/(c*d-e))^(1/2),1/c*(c*d-e)/d,(c/(c*d+e))^(1/2)/(c/(c*d-e))^(1/2))*e^4*(e*x+d)^(3/2)-2*(c/(c
*d-e))^(1/2)*c^2*d^4*e^2+3*(c/(c*d-e))^(1/2)*d*e^4*(e*x+d)+(c/(c*d-e))^(1/2)*d^2*e^4)/(c*d-e)/(c/(c*d-e))^(1/2
)/(e*x+d)^(3/2)/(c*d+e)/(c^2*d^2-e^2)/d^3/x/((c^2*(e*x+d)^2-2*c^2*d*(e*x+d)+c^2*d^2-e^2)/c^2/e^2/x^2)^(1/2)))

Fricas [F(-1)]

Timed out. \[ \int \frac {a+b \csc ^{-1}(c x)}{(d+e x)^{7/2}} \, dx=\text {Timed out} \]

[In]

integrate((a+b*arccsc(c*x))/(e*x+d)^(7/2),x, algorithm="fricas")

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int \frac {a+b \csc ^{-1}(c x)}{(d+e x)^{7/2}} \, dx=\text {Timed out} \]

[In]

integrate((a+b*acsc(c*x))/(e*x+d)**(7/2),x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {a+b \csc ^{-1}(c x)}{(d+e x)^{7/2}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((a+b*arccsc(c*x))/(e*x+d)^(7/2),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+c*d>0)', see `assume?` for m
ore details)

Giac [F]

\[ \int \frac {a+b \csc ^{-1}(c x)}{(d+e x)^{7/2}} \, dx=\int { \frac {b \operatorname {arccsc}\left (c x\right ) + a}{{\left (e x + d\right )}^{\frac {7}{2}}} \,d x } \]

[In]

integrate((a+b*arccsc(c*x))/(e*x+d)^(7/2),x, algorithm="giac")

[Out]

integrate((b*arccsc(c*x) + a)/(e*x + d)^(7/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \csc ^{-1}(c x)}{(d+e x)^{7/2}} \, dx=\int \frac {a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )}{{\left (d+e\,x\right )}^{7/2}} \,d x \]

[In]

int((a + b*asin(1/(c*x)))/(d + e*x)^(7/2),x)

[Out]

int((a + b*asin(1/(c*x)))/(d + e*x)^(7/2), x)