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

Optimal. Leaf size=344 \[ \frac{8 b d \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right ),\frac{2 e}{c d+e}\right )}{3 c^2 e x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}-\frac{2 d \sqrt{d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^2}+\frac{2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}+\frac{8 b d^2 \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{3 c e^2 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}-\frac{4 b \sqrt{1-c^2 x^2} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{3 c^2 e x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{c (d+e x)}{c d+e}}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.60028, antiderivative size = 344, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 13, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.684, Rules used = {43, 5247, 12, 6721, 6742, 719, 424, 944, 419, 932, 168, 538, 537} \[ -\frac{2 d \sqrt{d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^2}+\frac{2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}+\frac{8 b d^2 \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{3 c e^2 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}+\frac{8 b d \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{3 c^2 e x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}-\frac{4 b \sqrt{1-c^2 x^2} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{3 c^2 e x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{c (d+e x)}{c d+e}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*d*Sqrt[d + e*x]*(a + b*ArcCsc[c*x]))/e^2 + (2*(d + e*x)^(3/2)*(a + b*ArcCsc[c*x]))/(3*e^2) - (4*b*Sqrt[d +
e*x]*Sqrt[1 - c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(3*c^2*e*Sqrt[1 - 1/(c^2*x^
2)]*x*Sqrt[(c*(d + e*x))/(c*d + e)]) + (8*b*d*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)])/(3*c^2*e*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x]) + (8*b*d^2*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^
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 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 944

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

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)
])

Rubi steps

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

Mathematica [C]  time = 1.42907, size = 289, normalized size = 0.84 \[ \frac{2 \left (\frac{2 i b \sqrt{\frac{e (c x+1)}{e-c d}} \sqrt{\frac{e-c e x}{c d+e}} \left ((c d+e) \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{-\frac{c}{c d+e}} \sqrt{d+e x}\right ),\frac{c d+e}{c d-e}\right )+(c d-e) E\left (i \sinh ^{-1}\left (\sqrt{-\frac{c}{c d+e}} \sqrt{d+e x}\right )|\frac{c d+e}{c d-e}\right )-2 c d \Pi \left (\frac{e}{c d}+1;i \sinh ^{-1}\left (\sqrt{-\frac{c}{c d+e}} \sqrt{d+e x}\right )|\frac{c d+e}{c d-e}\right )\right )}{c^2 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{-\frac{c}{c d+e}}}+a \sqrt{d+e x} (e x-2 d)+b \csc ^{-1}(c x) \sqrt{d+e x} (e x-2 d)\right )}{3 e^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.278, size = 412, normalized size = 1.2 \begin{align*} 2\,{\frac{1}{{e}^{2}} \left ( a \left ( 1/3\, \left ( ex+d \right ) ^{3/2}-d\sqrt{ex+d} \right ) +b \left ( 1/3\, \left ( ex+d \right ) ^{3/2}{\rm arccsc} \left (cx\right )-{\rm arccsc} \left (cx\right )d\sqrt{ex+d}-2/3\,{\frac{1}{{c}^{2}x} \left ( d{\it EllipticF} \left ( \sqrt{ex+d}\sqrt{{\frac{c}{dc-e}}},\sqrt{{\frac{dc-e}{dc+e}}} \right ) c+{\it EllipticE} \left ( \sqrt{ex+d}\sqrt{{\frac{c}{dc-e}}},\sqrt{{\frac{dc-e}{dc+e}}} \right ) cd-2\,d{\it EllipticPi} \left ( \sqrt{ex+d}\sqrt{{\frac{c}{dc-e}}},{\frac{dc-e}{dc}},{\sqrt{{\frac{c}{dc+e}}}{\frac{1}{\sqrt{{\frac{c}{dc-e}}}}}} \right ) c-{\it EllipticF} \left ( \sqrt{ex+d}\sqrt{{\frac{c}{dc-e}}},\sqrt{{\frac{dc-e}{dc+e}}} \right ) e+{\it EllipticE} \left ( \sqrt{ex+d}\sqrt{{\frac{c}{dc-e}}},\sqrt{{\frac{dc-e}{dc+e}}} \right ) e \right ) \sqrt{-{\frac{ \left ( ex+d \right ) c-dc-e}{dc+e}}}\sqrt{-{\frac{ \left ( ex+d \right ) c-dc+e}{dc-e}}}{\frac{1}{\sqrt{{\frac{c}{dc-e}}}}}{\frac{1}{\sqrt{{\frac{{c}^{2} \left ( ex+d \right ) ^{2}-2\,d{c}^{2} \left ( ex+d \right ) +{c}^{2}{d}^{2}-{e}^{2}}{{c}^{2}{e}^{2}{x}^{2}}}}}}} \right ) \right ) } \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*(1/3*(e*x+d)^(3/2)-d*(e*x+d)^(1/2))+b*(1/3*(e*x+d)^(3/2)*arccsc(c*x)-arccsc(c*x)*d*(e*x+d)^(1/2)-2/3/
c^2*(d*EllipticF((e*x+d)^(1/2)*(c/(c*d-e))^(1/2),((c*d-e)/(c*d+e))^(1/2))*c+EllipticE((e*x+d)^(1/2)*(c/(c*d-e)
)^(1/2),((c*d-e)/(c*d+e))^(1/2))*c*d-2*d*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-EllipticF((e*x+d)^(1/2)*(c/(c*d-e))^(1/2),((c*d-e)/(c*d+e))^(1/2))*e+EllipticE((e*x
+d)^(1/2)*(c/(c*d-e))^(1/2),((c*d-e)/(c*d+e))^(1/2))*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)/(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]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \left (a + b \operatorname{acsc}{\left (c x \right )}\right )}{\sqrt{d + e x}}\, dx \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]

Integral(x*(a + b*acsc(c*x))/sqrt(d + e*x), x)

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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}{\sqrt{e x + d}}\,{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((b*arccsc(c*x) + a)*x/sqrt(e*x + d), x)