3.52 \(\int x \sqrt{d+e x} (a+b \csc ^{-1}(c x)) \, dx\)

Optimal. Leaf size=404 \[ \frac{4 b \sqrt{1-c^2 x^2} \left (3 c^2 d^2-e^2\right ) \sqrt{\frac{c (d+e x)}{c d+e}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right ),\frac{2 e}{c d+e}\right )}{15 c^4 e x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}-\frac{2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}+\frac{2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}+\frac{8 b d^3 \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{15 c e^2 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}-\frac{4 b \left (1-c^2 x^2\right ) \sqrt{d+e x}}{15 c^3 x \sqrt{1-\frac{1}{c^2 x^2}}}-\frac{8 b d \sqrt{1-c^2 x^2} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{15 c^2 e x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{c (d+e x)}{c d+e}}} \]

[Out]

(-4*b*Sqrt[d + e*x]*(1 - c^2*x^2))/(15*c^3*Sqrt[1 - 1/(c^2*x^2)]*x) - (2*d*(d + e*x)^(3/2)*(a + b*ArcCsc[c*x])
)/(3*e^2) + (2*(d + e*x)^(5/2)*(a + b*ArcCsc[c*x]))/(5*e^2) - (8*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)])/(15*c^2*e*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[(c*(d + e*x))/(c*d +
e)]) + (4*b*(3*c^2*d^2 - e^2)*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - c*x]/S
qrt[2]], (2*e)/(c*d + e)])/(15*c^4*e*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x]) + (8*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)])/(15*c*e^2*Sqrt[1 - 1/(
c^2*x^2)]*x*Sqrt[d + e*x])

________________________________________________________________________________________

Rubi [A]  time = 2.19491, antiderivative size = 502, normalized size of antiderivative = 1.24, number of steps used = 24, number of rules used = 15, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.79, Rules used = {43, 5247, 12, 6721, 6742, 743, 844, 719, 424, 419, 958, 932, 168, 538, 537} \[ -\frac{2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}+\frac{2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}+\frac{8 b d^3 \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{15 c e^2 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}+\frac{8 b d^2 \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{15 c^2 e x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}-\frac{4 b \left (1-c^2 x^2\right ) \sqrt{d+e x}}{15 c^3 x \sqrt{1-\frac{1}{c^2 x^2}}}+\frac{4 b \sqrt{1-c^2 x^2} (c d-e) (c d+e) \sqrt{\frac{c (d+e x)}{c d+e}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{15 c^4 e x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}-\frac{8 b d \sqrt{1-c^2 x^2} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{15 c^2 e x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{c (d+e x)}{c d+e}}} \]

Antiderivative was successfully verified.

[In]

Int[x*Sqrt[d + e*x]*(a + b*ArcCsc[c*x]),x]

[Out]

(-4*b*Sqrt[d + e*x]*(1 - c^2*x^2))/(15*c^3*Sqrt[1 - 1/(c^2*x^2)]*x) - (2*d*(d + e*x)^(3/2)*(a + b*ArcCsc[c*x])
)/(3*e^2) + (2*(d + e*x)^(5/2)*(a + b*ArcCsc[c*x]))/(5*e^2) - (8*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)])/(15*c^2*e*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)])/(15*c^2*e*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x]) + (4*b*(c*d - e)*(c*d + e)*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*c^4*e*Sqrt[1 - 1/(c^
2*x^2)]*x*Sqrt[d + e*x]) + (8*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)])/(15*c*e^2*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 743

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))/(c*(m + 2*p + 1)), x] + Dist[1/(c*(m + 2*p + 1)), Int[(d + e*x)^(m - 2)*Simp[c*d^2*(m + 2*p + 1) - a*e^
2*(m - 1) + 2*c*d*e*(m + p)*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] && If[RationalQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadraticQ[a, 0, c, d, e, m
, p, x]

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 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)
])

Rubi steps

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

Mathematica [C]  time = 1.61413, size = 368, normalized size = 0.91 \[ \frac{1}{15} \left (-\frac{4 i b \sqrt{\frac{e (c x+1)}{e-c d}} \sqrt{\frac{e-c e x}{c d+e}} \left (\left (-c^2 d^2-2 c d e+e^2\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{-\frac{c}{c d+e}} \sqrt{d+e x}\right ),\frac{c d+e}{c d-e}\right )+2 c^2 d^2 \Pi \left (\frac{e}{c d}+1;i \sinh ^{-1}\left (\sqrt{-\frac{c}{c d+e}} \sqrt{d+e x}\right )|\frac{c d+e}{c d-e}\right )-2 c d (c d-e) E\left (i \sinh ^{-1}\left (\sqrt{-\frac{c}{c d+e}} \sqrt{d+e x}\right )|\frac{c d+e}{c d-e}\right )\right )}{c^3 e^2 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{-\frac{c}{c d+e}}}+\frac{2 a \sqrt{d+e x} \left (-2 d^2+d e x+3 e^2 x^2\right )}{e^2}+\frac{4 b x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}{c}+\frac{2 b \csc ^{-1}(c x) \sqrt{d+e x} \left (-2 d^2+d e x+3 e^2 x^2\right )}{e^2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x*Sqrt[d + e*x]*(a + b*ArcCsc[c*x]),x]

[Out]

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

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Maple [B]  time = 0.281, size = 836, normalized size = 2.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/e^2*(a*(1/5*(e*x+d)^(5/2)-1/3*(e*x+d)^(3/2)*d)+b*(1/5*arccsc(c*x)*(e*x+d)^(5/2)-1/3*arccsc(c*x)*(e*x+d)^(3/2
)*d+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-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^2-2*(-((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))*c^2*d^2+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)*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+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*d*e-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))*
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)))

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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*(a+b*arccsc(c*x))*(e*x+d)^(1/2),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(x*(a+b*arccsc(c*x))*(e*x+d)^(1/2),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(x*(a+b*acsc(c*x))*(e*x+d)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*arccsc(c*x))*(e*x+d)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(e*x + d)*(b*arccsc(c*x) + a)*x, x)