3.166 \(\int \frac{e+f x}{(c+d x) \sqrt{-1+x^3}} \, dx\)

Optimal. Leaf size=477 \[ -\frac{(1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} (d e-c f) \tanh ^{-1}\left (\frac{\sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{c^2-c d+d^2}}{\sqrt{d} \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{c+d}}\right )}{\sqrt{d} \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{x^3-1} \sqrt{c+d} \sqrt{c^2-c d+d^2}}+\frac{4 \sqrt [4]{3} \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} (d e-c f) \Pi \left (\frac{\left (c+\sqrt{3} d+d\right )^2}{\left (c-\sqrt{3} d+d\right )^2};-\sin ^{-1}\left (\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{x^3-1} \left (c^2+2 c d-2 d^2\right )}-\frac{2 \sqrt{2-\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x-\sqrt{3}+1\right )^2}} \left (e+\sqrt{3} f+f\right ) F\left (\sin ^{-1}\left (\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{-\frac{1-x}{\left (-x-\sqrt{3}+1\right )^2}} \sqrt{x^3-1} \left (c+\sqrt{3} d+d\right )} \]

[Out]

-(((d*e - c*f)*(1 - x)*Sqrt[(1 + x + x^2)/(1 + Sqrt[3] - x)^2]*ArcTanh[(Sqrt[c^2 - c*d + d^2]*Sqrt[(1 - x)/(1
+ Sqrt[3] - x)^2])/(Sqrt[d]*Sqrt[c + d]*Sqrt[(1 + x + x^2)/(1 + Sqrt[3] - x)^2])])/(Sqrt[d]*Sqrt[c + d]*Sqrt[c
^2 - c*d + d^2]*Sqrt[(1 - x)/(1 + Sqrt[3] - x)^2]*Sqrt[-1 + x^3])) - (2*Sqrt[2 - Sqrt[3]]*(e + f + Sqrt[3]*f)*
(1 - x)*Sqrt[(1 + x + x^2)/(1 - Sqrt[3] - x)^2]*EllipticF[ArcSin[(1 + Sqrt[3] - x)/(1 - Sqrt[3] - x)], -7 + 4*
Sqrt[3]])/(3^(1/4)*(c + d + Sqrt[3]*d)*Sqrt[-((1 - x)/(1 - Sqrt[3] - x)^2)]*Sqrt[-1 + x^3]) + (4*3^(1/4)*Sqrt[
2 + Sqrt[3]]*(d*e - c*f)*(1 - x)*Sqrt[(1 + x + x^2)/(1 + Sqrt[3] - x)^2]*EllipticPi[(c + d + Sqrt[3]*d)^2/(c +
d - Sqrt[3]*d)^2, -ArcSin[(1 - Sqrt[3] - x)/(1 + Sqrt[3] - x)], -7 - 4*Sqrt[3]])/((c^2 + 2*c*d - 2*d^2)*Sqrt[
(1 - x)/(1 + Sqrt[3] - x)^2]*Sqrt[-1 + x^3])

________________________________________________________________________________________

Rubi [A]  time = 0.924345, antiderivative size = 477, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {2144, 219, 2142, 2113, 537, 571, 93, 208} \[ -\frac{(1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} (d e-c f) \tanh ^{-1}\left (\frac{\sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{c^2-c d+d^2}}{\sqrt{d} \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{c+d}}\right )}{\sqrt{d} \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{x^3-1} \sqrt{c+d} \sqrt{c^2-c d+d^2}}+\frac{4 \sqrt [4]{3} \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} (d e-c f) \Pi \left (\frac{\left (c+\sqrt{3} d+d\right )^2}{\left (c-\sqrt{3} d+d\right )^2};-\sin ^{-1}\left (\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{x^3-1} \left (c^2+2 c d-2 d^2\right )}-\frac{2 \sqrt{2-\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x-\sqrt{3}+1\right )^2}} \left (e+\sqrt{3} f+f\right ) F\left (\sin ^{-1}\left (\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{-\frac{1-x}{\left (-x-\sqrt{3}+1\right )^2}} \sqrt{x^3-1} \left (c+\sqrt{3} d+d\right )} \]

Antiderivative was successfully verified.

[In]

Int[(e + f*x)/((c + d*x)*Sqrt[-1 + x^3]),x]

[Out]

-(((d*e - c*f)*(1 - x)*Sqrt[(1 + x + x^2)/(1 + Sqrt[3] - x)^2]*ArcTanh[(Sqrt[c^2 - c*d + d^2]*Sqrt[(1 - x)/(1
+ Sqrt[3] - x)^2])/(Sqrt[d]*Sqrt[c + d]*Sqrt[(1 + x + x^2)/(1 + Sqrt[3] - x)^2])])/(Sqrt[d]*Sqrt[c + d]*Sqrt[c
^2 - c*d + d^2]*Sqrt[(1 - x)/(1 + Sqrt[3] - x)^2]*Sqrt[-1 + x^3])) - (2*Sqrt[2 - Sqrt[3]]*(e + f + Sqrt[3]*f)*
(1 - x)*Sqrt[(1 + x + x^2)/(1 - Sqrt[3] - x)^2]*EllipticF[ArcSin[(1 + Sqrt[3] - x)/(1 - Sqrt[3] - x)], -7 + 4*
Sqrt[3]])/(3^(1/4)*(c + d + Sqrt[3]*d)*Sqrt[-((1 - x)/(1 - Sqrt[3] - x)^2)]*Sqrt[-1 + x^3]) + (4*3^(1/4)*Sqrt[
2 + Sqrt[3]]*(d*e - c*f)*(1 - x)*Sqrt[(1 + x + x^2)/(1 + Sqrt[3] - x)^2]*EllipticPi[(c + d + Sqrt[3]*d)^2/(c +
d - Sqrt[3]*d)^2, -ArcSin[(1 - Sqrt[3] - x)/(1 + Sqrt[3] - x)], -7 - 4*Sqrt[3]])/((c^2 + 2*c*d - 2*d^2)*Sqrt[
(1 - x)/(1 + Sqrt[3] - x)^2]*Sqrt[-1 + x^3])

Rule 2144

Int[((e_.) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_Symbol] :> With[{q = Rt[b/a, 3]},
Dist[((1 + Sqrt[3])*f - e*q)/((1 + Sqrt[3])*d - c*q), Int[1/Sqrt[a + b*x^3], x], x] + Dist[(d*e - c*f)/((1 + S
qrt[3])*d - c*q), Int[(1 + Sqrt[3] + q*x)/((c + d*x)*Sqrt[a + b*x^3]), x], x]] /; FreeQ[{a, b, c, d, e, f}, x]
&& NeQ[d*e - c*f, 0] && NeQ[b^2*c^6 - 20*a*b*c^3*d^3 - 8*a^2*d^6, 0] && NeQ[b^2*e^6 - 20*a*b*e^3*f^3 - 8*a^2*
f^6, 0]

Rule 219

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(2*Sqr
t[2 - Sqrt[3]]*(s + r*x)*Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 - Sqrt[3])*s + r*x)^2]*EllipticF[ArcSin[((1 + Sqrt[3
])*s + r*x)/((1 - Sqrt[3])*s + r*x)], -7 + 4*Sqrt[3]])/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[-((s*(s + r*x))/((1 - S
qrt[3])*s + r*x)^2)]), x]] /; FreeQ[{a, b}, x] && NegQ[a]

Rule 2142

Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_Symbol] :> With[{q = Simplify[((1 +
Sqrt[3])*f)/e]}, Dist[(4*3^(1/4)*Sqrt[2 - Sqrt[3]]*f*(1 + q*x)*Sqrt[(1 - q*x + q^2*x^2)/(1 + Sqrt[3] + q*x)^2
])/(q*Sqrt[a + b*x^3]*Sqrt[(1 + q*x)/(1 + Sqrt[3] + q*x)^2]), Subst[Int[1/(((1 - Sqrt[3])*d - c*q + ((1 + Sqrt
[3])*d - c*q)*x)*Sqrt[1 - x^2]*Sqrt[7 - 4*Sqrt[3] + x^2]), x], x, (-1 + Sqrt[3] - q*x)/(1 + Sqrt[3] + q*x)], x
]] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && EqQ[b*e^3 - 2*(5 + 3*Sqrt[3])*a*f^3, 0] && NeQ[b*c^
3 - 2*(5 - 3*Sqrt[3])*a*d^3, 0]

Rule 2113

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

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 571

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_))^(r_.), x
_Symbol] :> Dist[1/n, Subst[Int[(a + b*x)^p*(c + d*x)^q*(e + f*x)^r, x], x, x^n], x] /; FreeQ[{a, b, c, d, e,
f, m, n, p, q, r}, x] && EqQ[m - n + 1, 0]

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \int \frac{e+f x}{(c+d x) \sqrt{-1+x^3}} \, dx &=\frac{\left (e+f+\sqrt{3} f\right ) \int \frac{1}{\sqrt{-1+x^3}} \, dx}{c+d+\sqrt{3} d}+\frac{(d e-c f) \int \frac{1+\sqrt{3}-x}{(c+d x) \sqrt{-1+x^3}} \, dx}{c+d+\sqrt{3} d}\\ &=-\frac{2 \sqrt{2-\sqrt{3}} \left (e+f+\sqrt{3} f\right ) (1-x) \sqrt{\frac{1+x+x^2}{\left (1-\sqrt{3}-x\right )^2}} F\left (\sin ^{-1}\left (\frac{1+\sqrt{3}-x}{1-\sqrt{3}-x}\right )|-7+4 \sqrt{3}\right )}{\sqrt [4]{3} \left (c+d+\sqrt{3} d\right ) \sqrt{-\frac{1-x}{\left (1-\sqrt{3}-x\right )^2}} \sqrt{-1+x^3}}+\frac{\left (4 \sqrt [4]{3} \sqrt{2-\sqrt{3}} (d e-c f) (1-x) \sqrt{\frac{1+x+x^2}{\left (1+\sqrt{3}-x\right )^2}}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (c+\left (1-\sqrt{3}\right ) d+\left (c+\left (1+\sqrt{3}\right ) d\right ) x\right ) \sqrt{1-x^2} \sqrt{7-4 \sqrt{3}+x^2}} \, dx,x,\frac{-1+\sqrt{3}+x}{1+\sqrt{3}-x}\right )}{\left (c+d+\sqrt{3} d\right ) \sqrt{\frac{1-x}{\left (1+\sqrt{3}-x\right )^2}} \sqrt{-1+x^3}}\\ &=-\frac{2 \sqrt{2-\sqrt{3}} \left (e+f+\sqrt{3} f\right ) (1-x) \sqrt{\frac{1+x+x^2}{\left (1-\sqrt{3}-x\right )^2}} F\left (\sin ^{-1}\left (\frac{1+\sqrt{3}-x}{1-\sqrt{3}-x}\right )|-7+4 \sqrt{3}\right )}{\sqrt [4]{3} \left (c+d+\sqrt{3} d\right ) \sqrt{-\frac{1-x}{\left (1-\sqrt{3}-x\right )^2}} \sqrt{-1+x^3}}-\frac{\left (4 \sqrt [4]{3} \sqrt{2-\sqrt{3}} (d e-c f) (1-x) \sqrt{\frac{1+x+x^2}{\left (1+\sqrt{3}-x\right )^2}}\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{1-x^2} \sqrt{7-4 \sqrt{3}+x^2} \left (\left (c+\left (1-\sqrt{3}\right ) d\right )^2-\left (c+\left (1+\sqrt{3}\right ) d\right )^2 x^2\right )} \, dx,x,\frac{-1+\sqrt{3}+x}{1+\sqrt{3}-x}\right )}{\sqrt{\frac{1-x}{\left (1+\sqrt{3}-x\right )^2}} \sqrt{-1+x^3}}+\frac{\left (4 \sqrt [4]{3} \sqrt{2-\sqrt{3}} \left (c+d-\sqrt{3} d\right ) (d e-c f) (1-x) \sqrt{\frac{1+x+x^2}{\left (1+\sqrt{3}-x\right )^2}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{7-4 \sqrt{3}+x^2} \left (\left (c+\left (1-\sqrt{3}\right ) d\right )^2-\left (c+\left (1+\sqrt{3}\right ) d\right )^2 x^2\right )} \, dx,x,\frac{-1+\sqrt{3}+x}{1+\sqrt{3}-x}\right )}{\left (c+d+\sqrt{3} d\right ) \sqrt{\frac{1-x}{\left (1+\sqrt{3}-x\right )^2}} \sqrt{-1+x^3}}\\ &=-\frac{2 \sqrt{2-\sqrt{3}} \left (e+f+\sqrt{3} f\right ) (1-x) \sqrt{\frac{1+x+x^2}{\left (1-\sqrt{3}-x\right )^2}} F\left (\sin ^{-1}\left (\frac{1+\sqrt{3}-x}{1-\sqrt{3}-x}\right )|-7+4 \sqrt{3}\right )}{\sqrt [4]{3} \left (c+d+\sqrt{3} d\right ) \sqrt{-\frac{1-x}{\left (1-\sqrt{3}-x\right )^2}} \sqrt{-1+x^3}}+\frac{4 \sqrt [4]{3} (d e-c f) (1-x) \sqrt{\frac{1+x+x^2}{\left (1+\sqrt{3}-x\right )^2}} \Pi \left (\frac{\left (c+d+\sqrt{3} d\right )^2}{\left (c+d-\sqrt{3} d\right )^2};-\sin ^{-1}\left (\frac{1-\sqrt{3}-x}{1+\sqrt{3}-x}\right )|-7-4 \sqrt{3}\right )}{\sqrt{2-\sqrt{3}} \left (c^2+2 c d-2 d^2\right ) \sqrt{\frac{1-x}{\left (1+\sqrt{3}-x\right )^2}} \sqrt{-1+x^3}}-\frac{\left (2 \sqrt [4]{3} \sqrt{2-\sqrt{3}} (d e-c f) (1-x) \sqrt{\frac{1+x+x^2}{\left (1+\sqrt{3}-x\right )^2}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x} \sqrt{7-4 \sqrt{3}+x} \left (\left (c+\left (1-\sqrt{3}\right ) d\right )^2-\left (c+\left (1+\sqrt{3}\right ) d\right )^2 x\right )} \, dx,x,\frac{\left (-1+\sqrt{3}+x\right )^2}{\left (1+\sqrt{3}-x\right )^2}\right )}{\sqrt{\frac{1-x}{\left (1+\sqrt{3}-x\right )^2}} \sqrt{-1+x^3}}\\ &=-\frac{2 \sqrt{2-\sqrt{3}} \left (e+f+\sqrt{3} f\right ) (1-x) \sqrt{\frac{1+x+x^2}{\left (1-\sqrt{3}-x\right )^2}} F\left (\sin ^{-1}\left (\frac{1+\sqrt{3}-x}{1-\sqrt{3}-x}\right )|-7+4 \sqrt{3}\right )}{\sqrt [4]{3} \left (c+d+\sqrt{3} d\right ) \sqrt{-\frac{1-x}{\left (1-\sqrt{3}-x\right )^2}} \sqrt{-1+x^3}}+\frac{4 \sqrt [4]{3} (d e-c f) (1-x) \sqrt{\frac{1+x+x^2}{\left (1+\sqrt{3}-x\right )^2}} \Pi \left (\frac{\left (c+d+\sqrt{3} d\right )^2}{\left (c+d-\sqrt{3} d\right )^2};-\sin ^{-1}\left (\frac{1-\sqrt{3}-x}{1+\sqrt{3}-x}\right )|-7-4 \sqrt{3}\right )}{\sqrt{2-\sqrt{3}} \left (c^2+2 c d-2 d^2\right ) \sqrt{\frac{1-x}{\left (1+\sqrt{3}-x\right )^2}} \sqrt{-1+x^3}}-\frac{\left (4 \sqrt [4]{3} \sqrt{2-\sqrt{3}} (d e-c f) (1-x) \sqrt{\frac{1+x+x^2}{\left (1+\sqrt{3}-x\right )^2}}\right ) \operatorname{Subst}\left (\int \frac{1}{-\left (c+\left (1-\sqrt{3}\right ) d\right )^2+\left (c+\left (1+\sqrt{3}\right ) d\right )^2-\left (\left (c+\left (1-\sqrt{3}\right ) d\right )^2+\left (7-4 \sqrt{3}\right ) \left (c+\left (1+\sqrt{3}\right ) d\right )^2\right ) x^2} \, dx,x,\frac{\sqrt [4]{3} \sqrt{-\frac{-1+x}{\left (1+\sqrt{3}-x\right )^2}}}{\sqrt{-\frac{\left (-2+\sqrt{3}\right ) \left (1+x+x^2\right )}{\left (1+\sqrt{3}-x\right )^2}}}\right )}{\sqrt{\frac{1-x}{\left (1+\sqrt{3}-x\right )^2}} \sqrt{-1+x^3}}\\ &=-\frac{(d e-c f) (1-x) \sqrt{\frac{1+x+x^2}{\left (1+\sqrt{3}-x\right )^2}} \tanh ^{-1}\left (\frac{\sqrt{c^2-c d+d^2} \sqrt{\frac{1-x}{\left (1+\sqrt{3}-x\right )^2}}}{\sqrt{d} \sqrt{c+d} \sqrt{\frac{1+x+x^2}{\left (1+\sqrt{3}-x\right )^2}}}\right )}{\sqrt{d} \sqrt{c+d} \sqrt{c^2-c d+d^2} \sqrt{\frac{1-x}{\left (1+\sqrt{3}-x\right )^2}} \sqrt{-1+x^3}}-\frac{2 \sqrt{2-\sqrt{3}} \left (e+f+\sqrt{3} f\right ) (1-x) \sqrt{\frac{1+x+x^2}{\left (1-\sqrt{3}-x\right )^2}} F\left (\sin ^{-1}\left (\frac{1+\sqrt{3}-x}{1-\sqrt{3}-x}\right )|-7+4 \sqrt{3}\right )}{\sqrt [4]{3} \left (c+d+\sqrt{3} d\right ) \sqrt{-\frac{1-x}{\left (1-\sqrt{3}-x\right )^2}} \sqrt{-1+x^3}}+\frac{4 \sqrt [4]{3} (d e-c f) (1-x) \sqrt{\frac{1+x+x^2}{\left (1+\sqrt{3}-x\right )^2}} \Pi \left (\frac{\left (c+d+\sqrt{3} d\right )^2}{\left (c+d-\sqrt{3} d\right )^2};-\sin ^{-1}\left (\frac{1-\sqrt{3}-x}{1+\sqrt{3}-x}\right )|-7-4 \sqrt{3}\right )}{\sqrt{2-\sqrt{3}} \left (c^2+2 c d-2 d^2\right ) \sqrt{\frac{1-x}{\left (1+\sqrt{3}-x\right )^2}} \sqrt{-1+x^3}}\\ \end{align*}

Mathematica [C]  time = 0.297834, size = 231, normalized size = 0.48 \[ \frac{2 \sqrt{\frac{1-x}{1+\sqrt [3]{-1}}} \left (\frac{3 f \left (x+\sqrt [3]{-1}\right ) \sqrt{\frac{(-1)^{2/3} x+\sqrt [3]{-1}}{1+\sqrt [3]{-1}}} F\left (\sin ^{-1}\left (\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}}+\frac{\sqrt [3]{-1} \sqrt{3} \left (1+\sqrt [3]{-1}\right ) \sqrt{x^2+x+1} (c f-d e) \Pi \left (\frac{i \sqrt{3} d}{\sqrt [3]{-1} d-c};\sin ^{-1}\left (\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{\sqrt [3]{-1} d-c}\right )}{3 d \sqrt{x^3-1}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(e + f*x)/((c + d*x)*Sqrt[-1 + x^3]),x]

[Out]

(2*Sqrt[(1 - x)/(1 + (-1)^(1/3))]*((3*f*((-1)^(1/3) + x)*Sqrt[((-1)^(1/3) + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]*El
lipticF[ArcSin[Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3
))] + ((-1)^(1/3)*Sqrt[3]*(1 + (-1)^(1/3))*(-(d*e) + c*f)*Sqrt[1 + x + x^2]*EllipticPi[(I*Sqrt[3]*d)/(-c + (-1
)^(1/3)*d), ArcSin[Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/(-c + (-1)^(1/3)*d)))/(3*d*Sqrt[-1
+ x^3])

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Maple [A]  time = 0.007, size = 274, normalized size = 0.6 \begin{align*} 2\,{\frac{f \left ( -3/2-i/2\sqrt{3} \right ) }{d\sqrt{{x}^{3}-1}}\sqrt{{\frac{x-1}{-3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2-i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2+i/2\sqrt{3}}{3/2+i/2\sqrt{3}}}}{\it EllipticF} \left ( \sqrt{{\frac{x-1}{-3/2-i/2\sqrt{3}}}},\sqrt{{\frac{3/2+i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}} \right ) }+2\,{\frac{ \left ( -cf+de \right ) \left ( -3/2-i/2\sqrt{3} \right ) }{{d}^{2}\sqrt{{x}^{3}-1}}\sqrt{{\frac{x-1}{-3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2-i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2+i/2\sqrt{3}}{3/2+i/2\sqrt{3}}}}{\it EllipticPi} \left ( \sqrt{{\frac{x-1}{-3/2-i/2\sqrt{3}}}},{(3/2+i/2\sqrt{3}) \left ( 1+{\frac{c}{d}} \right ) ^{-1}},\sqrt{{\frac{3/2+i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}} \right ) \left ( 1+{\frac{c}{d}} \right ) ^{-1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x+e)/(d*x+c)/(x^3-1)^(1/2),x)

[Out]

2*f/d*(-3/2-1/2*I*3^(1/2))*((x-1)/(-3/2-1/2*I*3^(1/2)))^(1/2)*((x+1/2-1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1/2
)*((x+1/2+1/2*I*3^(1/2))/(3/2+1/2*I*3^(1/2)))^(1/2)/(x^3-1)^(1/2)*EllipticF(((x-1)/(-3/2-1/2*I*3^(1/2)))^(1/2)
,((3/2+1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1/2))+2*(-c*f+d*e)/d^2*(-3/2-1/2*I*3^(1/2))*((x-1)/(-3/2-1/2*I*3^(
1/2)))^(1/2)*((x+1/2-1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1/2)*((x+1/2+1/2*I*3^(1/2))/(3/2+1/2*I*3^(1/2)))^(1/
2)/(x^3-1)^(1/2)/(1+c/d)*EllipticPi(((x-1)/(-3/2-1/2*I*3^(1/2)))^(1/2),(3/2+1/2*I*3^(1/2))/(1+c/d),((3/2+1/2*I
*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f x + e}{\sqrt{x^{3} - 1}{\left (d x + c\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)/(d*x+c)/(x^3-1)^(1/2),x, algorithm="maxima")

[Out]

integrate((f*x + e)/(sqrt(x^3 - 1)*(d*x + c)), x)

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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((f*x+e)/(d*x+c)/(x^3-1)^(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{e + f x}{\sqrt{\left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (c + d x\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)/(d*x+c)/(x**3-1)**(1/2),x)

[Out]

Integral((e + f*x)/(sqrt((x - 1)*(x**2 + x + 1))*(c + d*x)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f x + e}{\sqrt{x^{3} - 1}{\left (d x + c\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)/(d*x+c)/(x^3-1)^(1/2),x, algorithm="giac")

[Out]

integrate((f*x + e)/(sqrt(x^3 - 1)*(d*x + c)), x)