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

Optimal. Leaf size=540 \[ \frac{4 b \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right ),\frac{2 e}{c d+e}\right )}{15 d x \sqrt{1-\frac{1}{c^2 x^2}} \left (c^2 d^2-e^2\right ) \sqrt{d+e x}}-\frac{2 \left (a+b \csc ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}+\frac{4 b e \left (1-c^2 x^2\right )}{5 c d^2 x \sqrt{1-\frac{1}{c^2 x^2}} \left (c^2 d^2-e^2\right ) \sqrt{d+e x}}+\frac{16 b c e \left (1-c^2 x^2\right )}{15 x \sqrt{1-\frac{1}{c^2 x^2}} \left (c^2 d^2-e^2\right )^2 \sqrt{d+e x}}+\frac{4 b e \left (1-c^2 x^2\right )}{15 c d x \sqrt{1-\frac{1}{c^2 x^2}} \left (c^2 d^2-e^2\right ) (d+e x)^{3/2}}-\frac{4 b \sqrt{1-c^2 x^2} \left (7 c^2 d^2-3 e^2\right ) \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{15 x \sqrt{1-\frac{1}{c^2 x^2}} \left (c^2 d^3-d e^2\right )^2 \sqrt{\frac{c (d+e x)}{c d+e}}}+\frac{4 b \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{5 c d^2 e x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}} \]

[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)) - (4*b*(7*c^2*d^2 -
3*e^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^
3 - d*e^2)^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)]*Sqrt[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])

________________________________________________________________________________________

Rubi [A]  time = 0.751693, antiderivative size = 637, normalized size of antiderivative = 1.18, number of steps used = 19, number of rules used = 14, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.778, Rules used = {5227, 1574, 958, 745, 835, 844, 719, 424, 419, 21, 933, 168, 538, 537} \[ -\frac{2 \left (a+b \csc ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}+\frac{16 b c e \left (1-c^2 x^2\right )}{15 x \sqrt{1-\frac{1}{c^2 x^2}} \left (c^2 d^2-e^2\right )^2 \sqrt{d+e x}}+\frac{4 b e \left (1-c^2 x^2\right )}{5 c d^2 x \sqrt{1-\frac{1}{c^2 x^2}} \left (c^2 d^2-e^2\right ) \sqrt{d+e x}}+\frac{4 b e \left (1-c^2 x^2\right )}{15 c d x \sqrt{1-\frac{1}{c^2 x^2}} \left (c^2 d^2-e^2\right ) (d+e x)^{3/2}}+\frac{4 b \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{15 d x \sqrt{1-\frac{1}{c^2 x^2}} \left (c^2 d^2-e^2\right ) \sqrt{d+e x}}-\frac{16 b c^2 \sqrt{1-c^2 x^2} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{15 x \sqrt{1-\frac{1}{c^2 x^2}} \left (c^2 d^2-e^2\right )^2 \sqrt{\frac{c (d+e x)}{c d+e}}}-\frac{4 b \sqrt{1-c^2 x^2} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{5 d^2 x \sqrt{1-\frac{1}{c^2 x^2}} \left (c^2 d^2-e^2\right ) \sqrt{\frac{c (d+e x)}{c d+e}}}+\frac{4 b \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{5 c d^2 e x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[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 5227

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]

Rule 1574

Int[(x_)^(m_.)*((a_.) + (c_.)*(x_)^(mn2_.))^(p_)*((d_) + (e_.)*(x_)^(n_.))^(q_.), x_Symbol] :> Dist[(x^(2*n*Fr
acPart[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 958

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 745

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 835

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 844

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 719

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 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 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)])

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 933

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 168

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 538

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

Rule 537

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

Rubi steps

\begin{align*} \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{(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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 (\sin ^{-1}\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} \Pi \left (2;\sin ^{-1}\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 ) \operatorname{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 ) \operatorname{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 (\sin ^{-1}\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 (\sin ^{-1}\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} F\left (\sin ^{-1}\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} \Pi \left (2;\sin ^{-1}\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]  time = 14.053, size = 1002, normalized size = 1.86 \[ \frac{b \left (\frac{2 \left (\frac{d}{x}+e\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} \text{EllipticF}\left (\sin ^{-1}\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{\frac{d}{x}+e} (c x)^{3/2}}+\frac{2 \left (3 c^3 d^3+c e^2 d\right ) \sqrt{\frac{c d+c e x}{c d+e}} \sqrt{1-c^2 x^2} \Pi \left (2;\sin ^{-1}\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{\frac{d}{x}+e} (c x)^{3/2}}+\frac{2 \left (3 e^3-7 c^2 d^2 e\right ) \cos \left (2 \csc ^{-1}(c x)\right ) \left (d x \sqrt{\frac{c d+c e x}{c d+e}} \sqrt{1-c^2 x^2} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right ),\frac{2 e}{c d+e}\right ) c^2-\frac{x (c x+1) \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 (\sin ^{-1}\left (\sqrt{\frac{c d+c e x}{c d-e}}\right )|\frac{c d-e}{c d+e}\right )-e \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{c d+c e x}{c d-e}}\right ),\frac{c d-e}{c d+e}\right )\right ) c}{\sqrt{\frac{e (c x+1)}{e-c d}}}+e x \sqrt{\frac{c d+c e x}{c d+e}} \sqrt{1-c^2 x^2} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right ) c+(c d+c e x) \left (c^2 x^2-1\right )\right )}{c d \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} \sqrt{c x} \left (c^2 x^2-2\right )}\right )}{15 c d (c d-e)^2 e (c d+e)^2 (d+e x)^{7/2}}-\frac{c^4 \left (\frac{d}{x}+e\right )^4 x^4 \left (-\frac{2 \csc ^{-1}(c x) e^2}{5 c^3 d^3 \left (\frac{d}{x}+e\right )^3}-\frac{2 \left (9 \csc ^{-1}(c x) e^3+2 c d \sqrt{1-\frac{1}{c^2 x^2}} e^2-9 c^2 d^2 \csc ^{-1}(c x) e\right )}{15 c^3 d^3 \left (c^2 d^2-e^2\right ) \left (\frac{d}{x}+e\right )^2}-\frac{2 \left (9 c^4 \csc ^{-1}(c x) d^4-16 c^3 e \sqrt{1-\frac{1}{c^2 x^2}} d^3-18 c^2 e^2 \csc ^{-1}(c x) d^2+8 c e^3 \sqrt{1-\frac{1}{c^2 x^2}} d+9 e^4 \csc ^{-1}(c x)\right )}{15 c^3 d^3 \left (c^2 d^2-e^2\right )^2 \left (\frac{d}{x}+e\right )}+\frac{4 \left (3 e^2-7 c^2 d^2\right ) \sqrt{1-\frac{1}{c^2 x^2}}}{15 c^2 d^2 \left (e^2-c^2 d^2\right )^2}+\frac{2 \csc ^{-1}(c x)}{5 c^3 d^3 e}\right )}{(d+e x)^{7/2}}\right )}{c}-\frac{2 a}{5 e (d+e x)^{5/2}} \]

Warning: Unable to verify antiderivative.

[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

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Maple [B]  time = 0.296, size = 1640, normalized size = 3. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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)*(e*x+d)^3*c^4*d^3-3*(-
((e*x+d)*c-d*c+e)/(c*d-e))^(1/2)*(-((e*x+d)*c-d*c-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))*(e*x+d)^(3/2)*c^4*d^4-6*(-((e*x+d)*c-d*c+e)/(c*d-e))^(1/2)
*(-((e*x+d)*c-d*c-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))*(e*x+d)
^(3/2)*c^4*d^4+7*(-((e*x+d)*c-d*c+e)/(c*d-e))^(1/2)*(-((e*x+d)*c-d*c-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))*(e*x+d)^(3/2)*c^4*d^4-13*(c/(c*d-e))^(1/2)*(e*x+d)^2*c^4*d^4-7*(-(
(e*x+d)*c-d*c+e)/(c*d-e))^(1/2)*(-((e*x+d)*c-d*c-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))*(e*x+d)^(3/2)*c^3*d^3*e+7*(-((e*x+d)*c-d*c+e)/(c*d-e))^(1/2)*(-((e*x+d)*c-d*c-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))*(e*x+d)^(3/2)*c^3*d^3*e-3*(c/(c*d
-e))^(1/2)*(e*x+d)^3*c^2*d*e^2+5*(c/(c*d-e))^(1/2)*(e*x+d)*c^4*d^5+6*(-((e*x+d)*c-d*c+e)/(c*d-e))^(1/2)*(-((e*
x+d)*c-d*c-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))*(e*x+d)^(3/2)*c^2*d^2*e^2+2*(-((e*x+d)*c-d*c+e)/(c*d-e))^(1/2)*(-((e*x+d)*c-d*c-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))*(e*x+d)^(3/2)*c^2*d^2*e^2-3*(-((e*x+d)*c-d
*c+e)/(c*d-e))^(1/2)*(-((e*x+d)*c-d*c-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))*(e*x+d)^(3/2)*c^2*d^2*e^2+5*(c/(c*d-e))^(1/2)*(e*x+d)^2*c^2*d^2*e^2+(c/(c*d-e))^(1/2)*c^4*d^6+3*(
-((e*x+d)*c-d*c+e)/(c*d-e))^(1/2)*(-((e*x+d)*c-d*c-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))*(e*x+d)^(3/2)*c*d*e^3-3*(-((e*x+d)*c-d*c+e)/(c*d-e))^(1/2)*(-((e*x+d)*c-d*c-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))*(e*x+d)^(3/2)*c*d*e^3-8*(c/(c*d-e
))^(1/2)*(e*x+d)*c^2*d^3*e^2-3*(-((e*x+d)*c-d*c+e)/(c*d-e))^(1/2)*(-((e*x+d)*c-d*c-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*x+d)^(3/2)*e^4-2*(c/(c
*d-e))^(1/2)*c^2*d^4*e^2+3*(c/(c*d-e))^(1/2)*(e*x+d)*d*e^4+(c/(c*d-e))^(1/2)*d^2*e^4)/(c*d-e)/(e*x+d)^(3/2)/(c
*d+e)/(c^2*d^2-e^2)/(c/(c*d-e))^(1/2)/d^3/x/((c^2*(e*x+d)^2-2*d*c^2*(e*x+d)+c^2*d^2-e^2)/c^2/e^2/x^2)^(1/2)))

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

[Out]

Exception raised: ValueError

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

[Out]

Timed out

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \operatorname{arccsc}\left (c x\right ) + a}{{\left (e x + d\right )}^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)