3.70 \(\int \frac{x^2 (a+b \csc ^{-1}(c x))}{(d+e x)^{5/2}} \, dx\)

Optimal. Leaf size=440 \[ -\frac{4 b \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right ),\frac{2 e}{c d+e}\right )}{c^2 e^2 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}-\frac{2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 (d+e x)^{3/2}}+\frac{4 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac{4 b d \left (1-c^2 x^2\right )}{3 c e x \sqrt{1-\frac{1}{c^2 x^2}} \left (c^2 d^2-e^2\right ) \sqrt{d+e x}}-\frac{4 b d \sqrt{1-c^2 x^2} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{3 e^2 x \sqrt{1-\frac{1}{c^2 x^2}} \left (c^2 d^2-e^2\right ) \sqrt{\frac{c (d+e x)}{c d+e}}}-\frac{32 b d \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{3 c e^3 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}} \]

[Out]

(4*b*d*(1 - c^2*x^2))/(3*c*e*(c^2*d^2 - e^2)*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x]) - (2*d^2*(a + b*ArcCsc[c*x
]))/(3*e^3*(d + e*x)^(3/2)) + (4*d*(a + b*ArcCsc[c*x]))/(e^3*Sqrt[d + e*x]) + (2*Sqrt[d + e*x]*(a + b*ArcCsc[c
*x]))/e^3 - (4*b*d*Sqrt[d + e*x]*Sqrt[1 - c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/
(3*e^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)])/(c^2*e^2*Sqrt[1 - 1/(c^2*x^2
)]*x*Sqrt[d + e*x]) - (32*b*d*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)])/(3*c*e^3*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x])

________________________________________________________________________________________

Rubi [A]  time = 2.18477, antiderivative size = 440, normalized size of antiderivative = 1., number of steps used = 25, number of rules used = 17, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.81, Rules used = {43, 5247, 12, 6721, 6742, 745, 21, 719, 424, 958, 932, 168, 538, 537, 835, 844, 419} \[ -\frac{2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 (d+e x)^{3/2}}+\frac{4 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac{4 b d \left (1-c^2 x^2\right )}{3 c e x \sqrt{1-\frac{1}{c^2 x^2}} \left (c^2 d^2-e^2\right ) \sqrt{d+e x}}-\frac{4 b d \sqrt{1-c^2 x^2} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{3 e^2 x \sqrt{1-\frac{1}{c^2 x^2}} \left (c^2 d^2-e^2\right ) \sqrt{\frac{c (d+e x)}{c d+e}}}-\frac{4 b \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{c^2 e^2 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}-\frac{32 b d \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{3 c e^3 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

Int[(x^2*(a + b*ArcCsc[c*x]))/(d + e*x)^(5/2),x]

[Out]

(4*b*d*(1 - c^2*x^2))/(3*c*e*(c^2*d^2 - e^2)*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x]) - (2*d^2*(a + b*ArcCsc[c*x
]))/(3*e^3*(d + e*x)^(3/2)) + (4*d*(a + b*ArcCsc[c*x]))/(e^3*Sqrt[d + e*x]) + (2*Sqrt[d + e*x]*(a + b*ArcCsc[c
*x]))/e^3 - (4*b*d*Sqrt[d + e*x]*Sqrt[1 - c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/
(3*e^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)])/(c^2*e^2*Sqrt[1 - 1/(c^2*x^2
)]*x*Sqrt[d + e*x]) - (32*b*d*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)])/(3*c*e^3*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x])

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 5247

Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))*(u_), x_Symbol] :> With[{v = IntHide[u, x]}, Dist[a + b*ArcCsc[c*x], v,
x] + Dist[b/c, Int[SimplifyIntegrand[v/(x^2*Sqrt[1 - 1/(c^2*x^2)]), x], x], x] /; InverseFunctionFreeQ[v, x]]
/; FreeQ[{a, b, c}, x]

Rule 12

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

Rule 6721

Int[(u_.)*((a_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(b^IntPart[p]*(a + b*x^n)^FracPart[p])/(x^(n*FracP
art[p])*(1 + a/(x^n*b))^FracPart[p]), Int[u*x^(n*p)*(1 + a/(x^n*b))^p, x], x] /; FreeQ[{a, b, p}, x] &&  !Inte
gerQ[p] && ILtQ[n, 0] &&  !RationalFunctionQ[u, x] && IntegerQ[p + 1/2]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

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 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 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 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 932

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[-(c
/a), 2]}, Dist[1/Sqrt[a], 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)
])

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 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{x^2 \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{5/2}} \, dx &=-\frac{2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 (d+e x)^{3/2}}+\frac{4 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac{b \int \frac{2 \left (8 d^2+12 d e x+3 e^2 x^2\right )}{3 e^3 \sqrt{1-\frac{1}{c^2 x^2}} x^2 (d+e x)^{3/2}} \, dx}{c}\\ &=-\frac{2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 (d+e x)^{3/2}}+\frac{4 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac{(2 b) \int \frac{8 d^2+12 d e x+3 e^2 x^2}{\sqrt{1-\frac{1}{c^2 x^2}} x^2 (d+e x)^{3/2}} \, dx}{3 c e^3}\\ &=-\frac{2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 (d+e x)^{3/2}}+\frac{4 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac{\left (2 b \sqrt{1-c^2 x^2}\right ) \int \frac{8 d^2+12 d e x+3 e^2 x^2}{x (d+e x)^{3/2} \sqrt{1-c^2 x^2}} \, dx}{3 c e^3 \sqrt{1-\frac{1}{c^2 x^2}} x}\\ &=-\frac{2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 (d+e x)^{3/2}}+\frac{4 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac{\left (2 b \sqrt{1-c^2 x^2}\right ) \int \left (\frac{12 d e}{(d+e x)^{3/2} \sqrt{1-c^2 x^2}}+\frac{8 d^2}{x (d+e x)^{3/2} \sqrt{1-c^2 x^2}}+\frac{3 e^2 x}{(d+e x)^{3/2} \sqrt{1-c^2 x^2}}\right ) \, dx}{3 c e^3 \sqrt{1-\frac{1}{c^2 x^2}} x}\\ &=-\frac{2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 (d+e x)^{3/2}}+\frac{4 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac{\left (16 b d^2 \sqrt{1-c^2 x^2}\right ) \int \frac{1}{x (d+e x)^{3/2} \sqrt{1-c^2 x^2}} \, dx}{3 c e^3 \sqrt{1-\frac{1}{c^2 x^2}} x}+\frac{\left (8 b d \sqrt{1-c^2 x^2}\right ) \int \frac{1}{(d+e x)^{3/2} \sqrt{1-c^2 x^2}} \, dx}{c e^2 \sqrt{1-\frac{1}{c^2 x^2}} x}+\frac{\left (2 b \sqrt{1-c^2 x^2}\right ) \int \frac{x}{(d+e x)^{3/2} \sqrt{1-c^2 x^2}} \, dx}{c e \sqrt{1-\frac{1}{c^2 x^2}} x}\\ &=\frac{12 b d \left (1-c^2 x^2\right )}{c e \left (c^2 d^2-e^2\right ) \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}-\frac{2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 (d+e x)^{3/2}}+\frac{4 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac{\left (16 b d^2 \sqrt{1-c^2 x^2}\right ) \int \left (-\frac{e}{d (d+e x)^{3/2} \sqrt{1-c^2 x^2}}+\frac{1}{d x \sqrt{d+e x} \sqrt{1-c^2 x^2}}\right ) \, dx}{3 c e^3 \sqrt{1-\frac{1}{c^2 x^2}} x}-\frac{\left (16 b c d \sqrt{1-c^2 x^2}\right ) \int \frac{-\frac{d}{2}-\frac{e x}{2}}{\sqrt{d+e x} \sqrt{1-c^2 x^2}} \, dx}{e^2 \left (c^2 d^2-e^2\right ) \sqrt{1-\frac{1}{c^2 x^2}} x}+\frac{\left (4 b \sqrt{1-c^2 x^2}\right ) \int \frac{-\frac{e}{2}-\frac{1}{2} c^2 d x}{\sqrt{d+e x} \sqrt{1-c^2 x^2}} \, dx}{c e \left (c^2 d^2-e^2\right ) \sqrt{1-\frac{1}{c^2 x^2}} x}\\ &=\frac{12 b d \left (1-c^2 x^2\right )}{c e \left (c^2 d^2-e^2\right ) \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}-\frac{2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 (d+e x)^{3/2}}+\frac{4 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac{\left (16 b d \sqrt{1-c^2 x^2}\right ) \int \frac{1}{x \sqrt{d+e x} \sqrt{1-c^2 x^2}} \, dx}{3 c e^3 \sqrt{1-\frac{1}{c^2 x^2}} x}-\frac{\left (16 b d \sqrt{1-c^2 x^2}\right ) \int \frac{1}{(d+e x)^{3/2} \sqrt{1-c^2 x^2}} \, dx}{3 c e^2 \sqrt{1-\frac{1}{c^2 x^2}} x}-\frac{\left (2 b c d \sqrt{1-c^2 x^2}\right ) \int \frac{\sqrt{d+e x}}{\sqrt{1-c^2 x^2}} \, dx}{e^2 \left (c^2 d^2-e^2\right ) \sqrt{1-\frac{1}{c^2 x^2}} x}+\frac{\left (8 b c d \sqrt{1-c^2 x^2}\right ) \int \frac{\sqrt{d+e x}}{\sqrt{1-c^2 x^2}} \, dx}{e^2 \left (c^2 d^2-e^2\right ) \sqrt{1-\frac{1}{c^2 x^2}} x}+\frac{\left (2 b (c d-e) (c d+e) \sqrt{1-c^2 x^2}\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{1-c^2 x^2}} \, dx}{c e^2 \left (c^2 d^2-e^2\right ) \sqrt{1-\frac{1}{c^2 x^2}} x}\\ &=\frac{4 b d \left (1-c^2 x^2\right )}{3 c e \left (c^2 d^2-e^2\right ) \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}-\frac{2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 (d+e x)^{3/2}}+\frac{4 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac{\left (16 b d \sqrt{1-c^2 x^2}\right ) \int \frac{1}{x \sqrt{1-c x} \sqrt{1+c x} \sqrt{d+e x}} \, dx}{3 c e^3 \sqrt{1-\frac{1}{c^2 x^2}} x}+\frac{\left (32 b c d \sqrt{1-c^2 x^2}\right ) \int \frac{-\frac{d}{2}-\frac{e x}{2}}{\sqrt{d+e x} \sqrt{1-c^2 x^2}} \, dx}{3 e^2 \left (c^2 d^2-e^2\right ) \sqrt{1-\frac{1}{c^2 x^2}} x}+\frac{\left (4 b d \sqrt{d+e x} \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{2 c e x^2}{-c^2 d-c e}}}{\sqrt{1-x^2}} \, dx,x,\frac{\sqrt{1-c x}}{\sqrt{2}}\right )}{e^2 \left (c^2 d^2-e^2\right ) \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{-\frac{c^2 (d+e x)}{-c^2 d-c e}}}-\frac{\left (16 b d \sqrt{d+e x} \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{2 c e x^2}{-c^2 d-c e}}}{\sqrt{1-x^2}} \, dx,x,\frac{\sqrt{1-c x}}{\sqrt{2}}\right )}{e^2 \left (c^2 d^2-e^2\right ) \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{-\frac{c^2 (d+e x)}{-c^2 d-c e}}}-\frac{\left (4 b (c d-e) (c d+e) \sqrt{-\frac{c^2 (d+e x)}{-c^2 d-c e}} \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1+\frac{2 c e x^2}{-c^2 d-c e}}} \, dx,x,\frac{\sqrt{1-c x}}{\sqrt{2}}\right )}{c^2 e^2 \left (c^2 d^2-e^2\right ) \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}\\ &=\frac{4 b d \left (1-c^2 x^2\right )}{3 c e \left (c^2 d^2-e^2\right ) \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}-\frac{2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 (d+e x)^{3/2}}+\frac{4 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}-\frac{12 b d \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 )}{e^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 )}{c^2 e^2 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}-\frac{\left (32 b d \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 )}{3 c e^3 \sqrt{1-\frac{1}{c^2 x^2}} x}-\frac{\left (16 b c d \sqrt{1-c^2 x^2}\right ) \int \frac{\sqrt{d+e x}}{\sqrt{1-c^2 x^2}} \, dx}{3 e^2 \left (c^2 d^2-e^2\right ) \sqrt{1-\frac{1}{c^2 x^2}} x}\\ &=\frac{4 b d \left (1-c^2 x^2\right )}{3 c e \left (c^2 d^2-e^2\right ) \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}-\frac{2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 (d+e x)^{3/2}}+\frac{4 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}-\frac{12 b d \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 )}{e^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 )}{c^2 e^2 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}-\frac{\left (32 b d \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 )}{3 c e^3 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}+\frac{\left (32 b d \sqrt{d+e x} \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{2 c e x^2}{-c^2 d-c e}}}{\sqrt{1-x^2}} \, dx,x,\frac{\sqrt{1-c x}}{\sqrt{2}}\right )}{3 e^2 \left (c^2 d^2-e^2\right ) \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{-\frac{c^2 (d+e x)}{-c^2 d-c e}}}\\ &=\frac{4 b d \left (1-c^2 x^2\right )}{3 c e \left (c^2 d^2-e^2\right ) \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}-\frac{2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 (d+e x)^{3/2}}+\frac{4 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}-\frac{4 b d \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 )}{3 e^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 )}{c^2 e^2 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}-\frac{32 b d \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 )}{3 c e^3 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}\\ \end{align*}

Mathematica [C]  time = 13.7893, size = 856, normalized size = 1.95 \[ \frac{b \left (-\frac{c^3 \left (\frac{d}{x}+e\right )^3 \left (\frac{4 c \sqrt{1-\frac{1}{c^2 x^2}} d}{3 e^2 \left (e^2-c^2 d^2\right )}+\frac{2 \csc ^{-1}(c x)}{3 e \left (\frac{d}{x}+e\right )^2}-\frac{16 \csc ^{-1}(c x)}{3 e^3}+\frac{4 \left (-2 c^2 \csc ^{-1}(c x) d^2-c e \sqrt{1-\frac{1}{c^2 x^2}} d+2 e^2 \csc ^{-1}(c x)\right )}{3 e^2 \left (e^2-c^2 d^2\right ) \left (\frac{d}{x}+e\right )}\right ) x^3}{(d+e x)^{5/2}}-\frac{2 \left (\frac{d}{x}+e\right )^{5/2} (c x)^{5/2} \left (\frac{2 \left (3 c^2 d^2 e-3 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 (8 c^3 d^3-9 c d e^2\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 c d e \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 )}{\sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} \sqrt{c x} \left (c^2 x^2-2\right )}\right )}{3 (c d-e) e^3 (c d+e) (d+e x)^{5/2}}\right )}{c^3}-\frac{a d^3 \left (\frac{e x}{d}+1\right )^{5/2} B_{-\frac{e x}{d}}\left (3,-\frac{3}{2}\right )}{e^3 (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^2*(a + b*ArcCsc[c*x]))/(d + e*x)^(5/2),x]

[Out]

-((a*d^3*(1 + (e*x)/d)^(5/2)*Beta[-((e*x)/d), 3, -3/2])/(e^3*(d + e*x)^(5/2))) + (b*(-((c^3*(e + d/x)^3*x^3*((
4*c*d*Sqrt[1 - 1/(c^2*x^2)])/(3*e^2*(-(c^2*d^2) + e^2)) - (16*ArcCsc[c*x])/(3*e^3) + (2*ArcCsc[c*x])/(3*e*(e +
d/x)^2) + (4*(-(c*d*e*Sqrt[1 - 1/(c^2*x^2)]) - 2*c^2*d^2*ArcCsc[c*x] + 2*e^2*ArcCsc[c*x]))/(3*e^2*(-(c^2*d^2)
+ e^2)*(e + d/x))))/(d + e*x)^(5/2)) - (2*(e + d/x)^(5/2)*(c*x)^(5/2)*((2*(3*c^2*d^2*e - 3*e^3)*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)])/(Sqrt[1 - 1/(c^2
*x^2)]*Sqrt[e + d/x]*(c*x)^(3/2)) + (2*(8*c^3*d^3 - 9*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]/Sqrt[2]], (2*e)/(c*d + e)])/(Sqrt[1 - 1/(c^2*x^2)]*Sqrt[e + d/x]*(c*x)^(3/
2)) + (2*c*d*e*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[ArcSin[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)]))/(Sqrt[1 - 1/(c^2*x^2)]*Sqrt[e + d/x]*Sqrt[c*x]*(-2 + c^2*x^2))))/(3*(c*d - e)*e^3*(c*d + e)*(d
+ e*x)^(5/2))))/c^3

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Maple [B]  time = 0.289, size = 1038, normalized size = 2.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a+b*arccsc(c*x))/(e*x+d)^(5/2),x)

[Out]

2/e^3*(a*((e*x+d)^(1/2)+2*d/(e*x+d)^(1/2)-1/3*d^2/(e*x+d)^(3/2))+b*((e*x+d)^(1/2)*arccsc(c*x)+2*arccsc(c*x)*d/
(e*x+d)^(1/2)-1/3*arccsc(c*x)*d^2/(e*x+d)^(3/2)-2/3/c*((c/(c*d-e))^(1/2)*(e*x+d)^2*c^2*d-4*(-((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)^(1/2)*c^2*d^2+8*(-((e*x+d)*c-d*c+e)/(c*d-e))^(1/2)*(-((e*x+d)*c-d*c-e)/(c*d+e))^(1/2)*Elliptic
Pi((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)^(1/2)*c^2*d^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)*EllipticE((e*x+d)^(1/2)*(c/(c*d-e))^(1/2),
((c*d-e)/(c*d+e))^(1/2))*(e*x+d)^(1/2)*c^2*d^2-2*(c/(c*d-e))^(1/2)*(e*x+d)*c^2*d^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)^(1/2)*c*d*e+(-((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)^(1/2)*c*d*e+(c/(c*d-e))^(1/2)*c^2*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)*EllipticF((e*x+d)^(1/2)*(c/(c*d-e))^(1/2),((c*d-e)/(c*d
+e))^(1/2))*(e*x+d)^(1/2)*e^2-8*(-((e*x+d)*c-d*c+e)/(c*d-e))^(1/2)*(-((e*x+d)*c-d*c-e)/(c*d+e))^(1/2)*Elliptic
Pi((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)^(1/2)*e^2-(c/(c*
d-e))^(1/2)*d*e^2)/(c*d-e)/(c/(c*d-e))^(1/2)/(e*x+d)^(1/2)/(c*d+e)/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(x^2*(a+b*arccsc(c*x))/(e*x+d)^(5/2),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 (b x^{2} \operatorname{arccsc}\left (c x\right ) + a x^{2}\right )} \sqrt{e x + d}}{e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*arccsc(c*x))/(e*x+d)^(5/2),x, algorithm="fricas")

[Out]

integral((b*x^2*arccsc(c*x) + a*x^2)*sqrt(e*x + d)/(e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x + d^3), 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(x**2*(a+b*acsc(c*x))/(e*x+d)**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arccsc}\left (c x\right ) + a\right )} x^{2}}{{\left (e x + d\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*arccsc(c*x))/(e*x+d)^(5/2),x, algorithm="giac")

[Out]

integrate((b*arccsc(c*x) + a)*x^2/(e*x + d)^(5/2), x)