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

Optimal. Leaf size=551 \[ -\frac{8 b d^2 \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right ),\frac{2 e}{c d+e}\right )}{c^2 e^3 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}-\frac{4 b \sqrt{1-c^2 x^2} \left (2 c^2 d^2+e^2\right ) \sqrt{\frac{c (d+e x)}{c d+e}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right ),\frac{2 e}{c d+e}\right )}{15 c^4 e^3 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}+\frac{2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt{d+e x}}+\frac{6 d^2 \sqrt{d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}-\frac{2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac{2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}-\frac{64 b d^3 \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{5 c e^4 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}-\frac{4 b \left (1-c^2 x^2\right ) \sqrt{d+e x}}{15 c^3 e^2 x \sqrt{1-\frac{1}{c^2 x^2}}}+\frac{32 b d \sqrt{1-c^2 x^2} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{15 c^2 e^3 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{c (d+e x)}{c d+e}}} \]

[Out]

(-4*b*Sqrt[d + e*x]*(1 - c^2*x^2))/(15*c^3*e^2*Sqrt[1 - 1/(c^2*x^2)]*x) + (2*d^3*(a + b*ArcCsc[c*x]))/(e^4*Sqr
t[d + e*x]) + (6*d^2*Sqrt[d + e*x]*(a + b*ArcCsc[c*x]))/e^4 - (2*d*(d + e*x)^(3/2)*(a + b*ArcCsc[c*x]))/e^4 +
(2*(d + e*x)^(5/2)*(a + b*ArcCsc[c*x]))/(5*e^4) + (32*b*d*Sqrt[d + e*x]*Sqrt[1 - c^2*x^2]*EllipticE[ArcSin[Sqr
t[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(15*c^2*e^3*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[(c*(d + e*x))/(c*d + e)]) - (8
*b*d^2*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^3*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x]) - (4*b*(2*c^2*d^2 + e^2)*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqr
t[1 - c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(15*c^4*e^3*Sqrt[1 - 1/(c^2*x^2)]*x*
Sqrt[d + e*x]) - (64*b*d^3*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*e^4*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x])

________________________________________________________________________________________

Rubi [A]  time = 2.47404, antiderivative size = 551, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 15, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.714, Rules used = {43, 5247, 12, 6721, 6742, 719, 419, 932, 168, 538, 537, 844, 424, 931, 1584} \[ \frac{2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt{d+e x}}+\frac{6 d^2 \sqrt{d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}-\frac{2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac{2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}-\frac{8 b d^2 \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{c^2 e^3 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}-\frac{4 b \sqrt{1-c^2 x^2} \left (2 c^2 d^2+e^2\right ) \sqrt{\frac{c (d+e x)}{c d+e}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{15 c^4 e^3 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}-\frac{64 b d^3 \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{5 c e^4 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}-\frac{4 b \left (1-c^2 x^2\right ) \sqrt{d+e x}}{15 c^3 e^2 x \sqrt{1-\frac{1}{c^2 x^2}}}+\frac{32 b d \sqrt{1-c^2 x^2} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{15 c^2 e^3 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{c (d+e x)}{c d+e}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*b*Sqrt[d + e*x]*(1 - c^2*x^2))/(15*c^3*e^2*Sqrt[1 - 1/(c^2*x^2)]*x) + (2*d^3*(a + b*ArcCsc[c*x]))/(e^4*Sqr
t[d + e*x]) + (6*d^2*Sqrt[d + e*x]*(a + b*ArcCsc[c*x]))/e^4 - (2*d*(d + e*x)^(3/2)*(a + b*ArcCsc[c*x]))/e^4 +
(2*(d + e*x)^(5/2)*(a + b*ArcCsc[c*x]))/(5*e^4) + (32*b*d*Sqrt[d + e*x]*Sqrt[1 - c^2*x^2]*EllipticE[ArcSin[Sqr
t[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(15*c^2*e^3*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[(c*(d + e*x))/(c*d + e)]) - (8
*b*d^2*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^3*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x]) - (4*b*(2*c^2*d^2 + e^2)*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqr
t[1 - c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(15*c^4*e^3*Sqrt[1 - 1/(c^2*x^2)]*x*
Sqrt[d + e*x]) - (64*b*d^3*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*e^4*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 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 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 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 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 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 931

Int[((d_.) + (e_.)*(x_))^(m_)/(Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> Simp[(2*e^2*(
d + e*x)^(m - 2)*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(c*g*(2*m - 1)), x] - Dist[1/(c*g*(2*m - 1)), Int[((d + e*x)^(
m - 3)*Simp[a*e^2*(d*g + 2*e*f*(m - 2)) - c*d^3*g*(2*m - 1) + e*(e*(a*e*g*(2*m - 3)) + c*d*(2*e*f - 3*d*g*(2*m
- 1)))*x + 2*e^2*(c*e*f - 3*c*d*g)*(m - 1)*x^2, x])/(Sqrt[f + g*x]*Sqrt[a + c*x^2]), 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[2*m] && GeQ[m, 2]

Rule 1584

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rubi steps

\begin{align*} \int \frac{x^3 \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{3/2}} \, dx &=\frac{2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt{d+e x}}+\frac{6 d^2 \sqrt{d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}-\frac{2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac{2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}+\frac{b \int \frac{2 \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )}{5 e^4 \sqrt{1-\frac{1}{c^2 x^2}} x^2 \sqrt{d+e x}} \, dx}{c}\\ &=\frac{2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt{d+e x}}+\frac{6 d^2 \sqrt{d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}-\frac{2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac{2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}+\frac{(2 b) \int \frac{16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3}{\sqrt{1-\frac{1}{c^2 x^2}} x^2 \sqrt{d+e x}} \, dx}{5 c e^4}\\ &=\frac{2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt{d+e x}}+\frac{6 d^2 \sqrt{d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}-\frac{2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac{2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}+\frac{\left (2 b \sqrt{1-c^2 x^2}\right ) \int \frac{16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3}{x \sqrt{d+e x} \sqrt{1-c^2 x^2}} \, dx}{5 c e^4 \sqrt{1-\frac{1}{c^2 x^2}} x}\\ &=\frac{2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt{d+e x}}+\frac{6 d^2 \sqrt{d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}-\frac{2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac{2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}+\frac{\left (2 b \sqrt{1-c^2 x^2}\right ) \int \left (\frac{8 d^2 e}{\sqrt{d+e x} \sqrt{1-c^2 x^2}}+\frac{16 d^3}{x \sqrt{d+e x} \sqrt{1-c^2 x^2}}-\frac{2 d e^2 x}{\sqrt{d+e x} \sqrt{1-c^2 x^2}}+\frac{e^3 x^2}{\sqrt{d+e x} \sqrt{1-c^2 x^2}}\right ) \, dx}{5 c e^4 \sqrt{1-\frac{1}{c^2 x^2}} x}\\ &=\frac{2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt{d+e x}}+\frac{6 d^2 \sqrt{d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}-\frac{2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac{2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}+\frac{\left (32 b d^3 \sqrt{1-c^2 x^2}\right ) \int \frac{1}{x \sqrt{d+e x} \sqrt{1-c^2 x^2}} \, dx}{5 c e^4 \sqrt{1-\frac{1}{c^2 x^2}} x}+\frac{\left (16 b d^2 \sqrt{1-c^2 x^2}\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{1-c^2 x^2}} \, dx}{5 c e^3 \sqrt{1-\frac{1}{c^2 x^2}} x}-\frac{\left (4 b d \sqrt{1-c^2 x^2}\right ) \int \frac{x}{\sqrt{d+e x} \sqrt{1-c^2 x^2}} \, dx}{5 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^2}{\sqrt{d+e x} \sqrt{1-c^2 x^2}} \, dx}{5 c e \sqrt{1-\frac{1}{c^2 x^2}} x}\\ &=-\frac{4 b \sqrt{d+e x} \left (1-c^2 x^2\right )}{15 c^3 e^2 \sqrt{1-\frac{1}{c^2 x^2}} x}+\frac{2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt{d+e x}}+\frac{6 d^2 \sqrt{d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}-\frac{2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac{2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}+\frac{\left (32 b d^3 \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 e^4 \sqrt{1-\frac{1}{c^2 x^2}} x}-\frac{\left (4 b d \sqrt{1-c^2 x^2}\right ) \int \frac{\sqrt{d+e x}}{\sqrt{1-c^2 x^2}} \, dx}{5 c e^3 \sqrt{1-\frac{1}{c^2 x^2}} x}+\frac{\left (4 b d^2 \sqrt{1-c^2 x^2}\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{1-c^2 x^2}} \, dx}{5 c e^3 \sqrt{1-\frac{1}{c^2 x^2}} x}+\frac{\left (2 b \sqrt{1-c^2 x^2}\right ) \int \frac{e x-2 c^2 d x^2}{x \sqrt{d+e x} \sqrt{1-c^2 x^2}} \, dx}{15 c^3 e^2 \sqrt{1-\frac{1}{c^2 x^2}} x}-\frac{\left (32 b d^2 \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 )}{5 c^2 e^3 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}\\ &=-\frac{4 b \sqrt{d+e x} \left (1-c^2 x^2\right )}{15 c^3 e^2 \sqrt{1-\frac{1}{c^2 x^2}} x}+\frac{2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt{d+e x}}+\frac{6 d^2 \sqrt{d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}-\frac{2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac{2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}-\frac{32 b d^2 \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 )}{5 c^2 e^3 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}-\frac{\left (64 b d^3 \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 e^4 \sqrt{1-\frac{1}{c^2 x^2}} x}+\frac{\left (2 b \sqrt{1-c^2 x^2}\right ) \int \frac{e-2 c^2 d x}{\sqrt{d+e x} \sqrt{1-c^2 x^2}} \, dx}{15 c^3 e^2 \sqrt{1-\frac{1}{c^2 x^2}} x}+\frac{\left (8 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 )}{5 c^2 e^3 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{-\frac{c^2 (d+e x)}{-c^2 d-c e}}}-\frac{\left (8 b d^2 \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 )}{5 c^2 e^3 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}\\ &=-\frac{4 b \sqrt{d+e x} \left (1-c^2 x^2\right )}{15 c^3 e^2 \sqrt{1-\frac{1}{c^2 x^2}} x}+\frac{2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt{d+e x}}+\frac{6 d^2 \sqrt{d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}-\frac{2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac{2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}+\frac{8 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 )}{5 c^2 e^3 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{\frac{c (d+e x)}{c d+e}}}-\frac{8 b d^2 \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^3 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}-\frac{\left (4 b d \sqrt{1-c^2 x^2}\right ) \int \frac{\sqrt{d+e x}}{\sqrt{1-c^2 x^2}} \, dx}{15 c e^3 \sqrt{1-\frac{1}{c^2 x^2}} x}+\frac{\left (2 b \left (2 c^2 d^2+e^2\right ) \sqrt{1-c^2 x^2}\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{1-c^2 x^2}} \, dx}{15 c^3 e^3 \sqrt{1-\frac{1}{c^2 x^2}} x}-\frac{\left (64 b d^3 \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 e^4 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}\\ &=-\frac{4 b \sqrt{d+e x} \left (1-c^2 x^2\right )}{15 c^3 e^2 \sqrt{1-\frac{1}{c^2 x^2}} x}+\frac{2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt{d+e x}}+\frac{6 d^2 \sqrt{d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}-\frac{2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac{2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}+\frac{8 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 )}{5 c^2 e^3 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{\frac{c (d+e x)}{c d+e}}}-\frac{8 b d^2 \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^3 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}-\frac{64 b d^3 \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 e^4 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}+\frac{\left (8 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 )}{15 c^2 e^3 \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 \left (2 c^2 d^2+e^2\right ) \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 )}{15 c^4 e^3 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}\\ &=-\frac{4 b \sqrt{d+e x} \left (1-c^2 x^2\right )}{15 c^3 e^2 \sqrt{1-\frac{1}{c^2 x^2}} x}+\frac{2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt{d+e x}}+\frac{6 d^2 \sqrt{d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}-\frac{2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac{2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}+\frac{32 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 )}{15 c^2 e^3 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{\frac{c (d+e x)}{c d+e}}}-\frac{8 b d^2 \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^3 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}-\frac{4 b \left (2 c^2 d^2+e^2\right ) \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 c^4 e^3 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}-\frac{64 b d^3 \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 e^4 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}\\ \end{align*}

Mathematica [C]  time = 13.8299, size = 814, normalized size = 1.48 \[ \frac{a \left (\frac{e x}{d}+1\right )^{3/2} B_{-\frac{e x}{d}}\left (4,-\frac{1}{2}\right ) d^4}{e^4 (d+e x)^{3/2}}+\frac{b \left (-\frac{c^2 \left (\frac{d}{x}+e\right )^2 \left (\frac{2 c^2 \csc ^{-1}(c x) d^2}{e^3 \left (\frac{d}{x}+e\right )}-\frac{32 c^2 \csc ^{-1}(c x) d^2}{5 e^4}+\frac{32 c \sqrt{1-\frac{1}{c^2 x^2}} d}{15 e^3}-\frac{2 c^2 x^2 \csc ^{-1}(c x)}{5 e^2}-\frac{2 c x \left (2 e \sqrt{1-\frac{1}{c^2 x^2}}-9 c d \csc ^{-1}(c x)\right )}{15 e^3}\right ) x^2}{(d+e x)^{3/2}}-\frac{2 \left (\frac{d}{x}+e\right )^{3/2} (c x)^{3/2} \left (\frac{2 \left (e^3+32 c^2 d^2 e\right ) \sqrt{\frac{c d+c e x}{c d+e}} \sqrt{1-c^2 x^2} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right ),\frac{2 e}{c d+e}\right )}{\sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} (c x)^{3/2}}+\frac{2 \left (48 c^3 d^3+8 c e^2 d\right ) \sqrt{\frac{c d+c e x}{c d+e}} \sqrt{1-c^2 x^2} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{\sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} (c x)^{3/2}}-\frac{16 c d e \cos \left (2 \csc ^{-1}(c x)\right ) \left (d x \sqrt{\frac{c d+c e x}{c d+e}} \sqrt{1-c^2 x^2} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right ),\frac{2 e}{c d+e}\right ) c^2-\frac{x (c x+1) \sqrt{\frac{e-c e x}{c d+e}} \sqrt{\frac{c d+c e x}{c d-e}} \left ((c d+e) E\left (\sin ^{-1}\left (\sqrt{\frac{c d+c e x}{c d-e}}\right )|\frac{c d-e}{c d+e}\right )-e \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{c d+c e x}{c d-e}}\right ),\frac{c d-e}{c d+e}\right )\right ) c}{\sqrt{\frac{e (c x+1)}{e-c d}}}+e x \sqrt{\frac{c d+c e x}{c d+e}} \sqrt{1-c^2 x^2} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right ) c+(c d+c e x) \left (c^2 x^2-1\right )\right )}{\sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} \sqrt{c x} \left (c^2 x^2-2\right )}\right )}{15 e^4 (d+e x)^{3/2}}\right )}{c^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*d^4*(1 + (e*x)/d)^(3/2)*Beta[-((e*x)/d), 4, -1/2])/(e^4*(d + e*x)^(3/2)) + (b*(-((c^2*(e + d/x)^2*x^2*((32*
c*d*Sqrt[1 - 1/(c^2*x^2)])/(15*e^3) - (32*c^2*d^2*ArcCsc[c*x])/(5*e^4) + (2*c^2*d^2*ArcCsc[c*x])/(e^3*(e + d/x
)) - (2*c^2*x^2*ArcCsc[c*x])/(5*e^2) - (2*c*x*(2*e*Sqrt[1 - 1/(c^2*x^2)] - 9*c*d*ArcCsc[c*x]))/(15*e^3)))/(d +
e*x)^(3/2)) - (2*(e + d/x)^(3/2)*(c*x)^(3/2)*((2*(32*c^2*d^2*e + 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*(48*c^3*d^3 + 8*c*d*e^2)*Sqrt[(c*d + c*e*x)/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticPi[2, ArcSin[Sqr
t[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)) - (16*c*d*e*Cos[2*Arc
Csc[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[Ar
cSin[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[ArcS
in[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))))/(15*e^4*(d + e*x)^(3/2))))/c^4

________________________________________________________________________________________

Maple [A]  time = 0.281, size = 890, normalized size = 1.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/e^4*(a*(1/5*(e*x+d)^(5/2)-(e*x+d)^(3/2)*d+3*d^2*(e*x+d)^(1/2)+d^3/(e*x+d)^(1/2))+b*(1/5*arccsc(c*x)*(e*x+d)^
(5/2)-arccsc(c*x)*(e*x+d)^(3/2)*d+3*arccsc(c*x)*d^2*(e*x+d)^(1/2)+arccsc(c*x)*d^3/(e*x+d)^(1/2)+2/15/c^3*((c/(
c*d-e))^(1/2)*(e*x+d)^(5/2)*c^2-2*(c/(c*d-e))^(1/2)*(e*x+d)^(3/2)*c^2*d+24*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))*c^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)*EllipticE((e*x+d)^(1/2)*(c/(c*d-e))^(1
/2),((c*d-e)/(c*d+e))^(1/2))*c^2*d^2-48*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)*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+(c/(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)*Ellipt
icF((e*x+d)^(1/2)*(c/(c*d-e))^(1/2),((c*d-e)/(c*d+e))^(1/2))*c*d*e+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)*EllipticE((e*x+d)^(1/2)*(c/(c*d-e))^(1/2),((c*d-e)/(c*d+e))^(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)*EllipticF((e*x+d)^(1/2)*(c/(c*d-e))^(1/2),((c*d-
e)/(c*d+e))^(1/2))*e^2-(c/(c*d-e))^(1/2)*(e*x+d)^(1/2)*e^2)/(c/(c*d-e))^(1/2)/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)))

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arccsc}\left (c x\right ) + a\right )} x^{3}}{{\left (e x + d\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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