3.2.67 \(\int \frac {e+f x}{(c+d x) \sqrt {-1-x^3}} \, dx\) [167]

Optimal. Leaf size=463 \[ \frac {(d e-c f) (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \tan ^{-1}\left (\frac {\sqrt {c^2+c d+d^2} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}}}{\sqrt {c-d} \sqrt {d} \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}}}\right )}{\sqrt {c-d} \sqrt {d} \sqrt {c^2+c d+d^2} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {-1-x^3}}+\frac {2 \sqrt {2-\sqrt {3}} \left (e-f-\sqrt {3} f\right ) (1+x) \sqrt {\frac {1-x+x^2}{\left (1-\sqrt {3}+x\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}+x}{1-\sqrt {3}+x}\right )|-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \left (c-d-\sqrt {3} d\right ) \sqrt {-\frac {1+x}{\left (1-\sqrt {3}+x\right )^2}} \sqrt {-1-x^3}}-\frac {4 \sqrt [4]{3} \sqrt {2+\sqrt {3}} (d e-c f) (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \Pi \left (\frac {\left (c-\left (1+\sqrt {3}\right ) d\right )^2}{\left (c-\left (1-\sqrt {3}\right ) d\right )^2};\sin ^{-1}\left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right )|-7-4 \sqrt {3}\right )}{\left (c^2-2 c d-2 d^2\right ) \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {-1-x^3}} \]

[Out]

2/3*(1+x)*EllipticF((1+x+3^(1/2))/(1+x-3^(1/2)),2*I-I*3^(1/2))*(e-f-f*3^(1/2))*(1/2*6^(1/2)-1/2*2^(1/2))*((x^2
-x+1)/(1+x-3^(1/2))^2)^(1/2)*3^(3/4)/(-d*3^(1/2)+c-d)/(-x^3-1)^(1/2)/((-1-x)/(1+x-3^(1/2))^2)^(1/2)+(-c*f+d*e)
*(1+x)*arctan((c^2+c*d+d^2)^(1/2)*((1+x)/(1+x+3^(1/2))^2)^(1/2)/(c-d)^(1/2)/d^(1/2)/((x^2-x+1)/(1+x+3^(1/2))^2
)^(1/2))*((x^2-x+1)/(1+x+3^(1/2))^2)^(1/2)/(c-d)^(1/2)/d^(1/2)/(c^2+c*d+d^2)^(1/2)/(-x^3-1)^(1/2)/((1+x)/(1+x+
3^(1/2))^2)^(1/2)+4*3^(1/4)*(-c*f+d*e)*(1+x)*EllipticPi((-1-x+3^(1/2))/(1+x+3^(1/2)),(c-d*(1+3^(1/2)))^2/(c-d*
(1-3^(1/2)))^2,I*3^(1/2)+2*I)*(1/2*6^(1/2)+1/2*2^(1/2))*((x^2-x+1)/(1+x+3^(1/2))^2)^(1/2)/(c^2-2*c*d-2*d^2)/(-
x^3-1)^(1/2)/((1+x)/(1+x+3^(1/2))^2)^(1/2)

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Rubi [A]
time = 0.92, antiderivative size = 463, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2169, 225, 2167, 2138, 551, 585, 95, 211} \begin {gather*} -\frac {4 \sqrt [4]{3} \sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} (d e-c f) \Pi \left (\frac {\left (c-\left (1+\sqrt {3}\right ) d\right )^2}{\left (c-\left (1-\sqrt {3}\right ) d\right )^2};\text {ArcSin}\left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{\sqrt {\frac {x+1}{\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}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x-\sqrt {3}+1\right )^2}} \left (e-\sqrt {3} f-f\right ) F\left (\text {ArcSin}\left (\frac {x+\sqrt {3}+1}{x-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-\frac {x+1}{\left (x-\sqrt {3}+1\right )^2}} \sqrt {-x^3-1} \left (c-\sqrt {3} d-d\right )}+\frac {(x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \text {ArcTan}\left (\frac {\sqrt {\frac {x+1}{\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 ) (d e-c f)}{\sqrt {d} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {-x^3-1} \sqrt {c-d} \sqrt {c^2+c d+d^2}} \end {gather*}

Warning: Unable to verify antiderivative.

[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]*ArcTan[(Sqrt[c^2 + c*d + d^2]*Sqrt[(1 + x)/(1 + S
qrt[3] + x)^2])/(Sqrt[c - d]*Sqrt[d]*Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2])])/(Sqrt[c - d]*Sqrt[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 - (1 + Sqrt[3])*d)^2/(c - (
1 - 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 95

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 211

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

Rule 225

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

Rule 551

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1/(a*Sqr
t[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b*(c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], 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 585

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 2138

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 2167

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 2169

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]

Rubi steps

\begin {align*} \int \frac {e+f x}{(c+d x) \sqrt {-1-x^3}} \, dx &=\frac {\left (e-\left (1+\sqrt {3}\right ) f\right ) \int \frac {1}{\sqrt {-1-x^3}} \, dx}{c-\left (1+\sqrt {3}\right ) d}-\frac {(d e-c f) \int \frac {1+\sqrt {3}+x}{(c+d x) \sqrt {-1-x^3}} \, dx}{c-\left (1+\sqrt {3}\right ) 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 ) \text {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-\left (1+\sqrt {3}\right ) 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}} \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 ) \text {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 )}{\left (c-\left (1+\sqrt {3}\right ) 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}} \left (-c+\left (1-\sqrt {3}\right ) d\right ) (d e-c f) (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}}\right ) \text {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-\left (1+\sqrt {3}\right ) 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} \sqrt {2+\sqrt {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-\left (1+\sqrt {3}\right ) d\right )^2}{\left (c-\left (1-\sqrt {3}\right ) d\right )^2};-\sin ^{-1}\left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right )|-7-4 \sqrt {3}\right )}{\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}} \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 ) \text {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 )}{\left (c-\left (1+\sqrt {3}\right ) 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} \sqrt {2+\sqrt {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-\left (1+\sqrt {3}\right ) d\right )^2}{\left (c-\left (1-\sqrt {3}\right ) d\right )^2};-\sin ^{-1}\left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right )|-7-4 \sqrt {3}\right )}{\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}} \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 ) \text {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 )}{\left (c-\left (1+\sqrt {3}\right ) d\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}} \tan ^{-1}\left (\frac {\sqrt {c^2+c d+d^2} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}}}{\sqrt {c-d} \sqrt {d} \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}}}\right )}{\sqrt {c-d} \sqrt {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} \sqrt {2+\sqrt {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-\left (1+\sqrt {3}\right ) d\right )^2}{\left (c-\left (1-\sqrt {3}\right ) d\right )^2};-\sin ^{-1}\left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right )|-7-4 \sqrt {3}\right )}{\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*}

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Mathematica [C] Result contains complex when optimal does not.
time = 10.29, size = 213, normalized size = 0.46 \begin {gather*} \frac {2 \sqrt {\frac {1+x}{1+\sqrt [3]{-1}}} \left (-\frac {f \left (\sqrt [3]{-1}-x\right ) \sqrt {\frac {\sqrt [3]{-1}-(-1)^{2/3} x}{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 {i (-d e+c f) \sqrt {1-x+x^2} \Pi \left (\frac {i \sqrt {3} d}{c+\sqrt [3]{-1} d};\sin ^{-1}\left (\sqrt {\frac {1+(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{c+\sqrt [3]{-1} d}\right )}{d \sqrt {-1-x^3}} \end {gather*}

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))]*(-((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
))]) + (I*(-(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)))/(d*Sqrt[-1 - x^3])

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Maple [A]
time = 0.25, size = 265, normalized size = 0.57

method	result	size
default	\(-\frac {2 i f \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {1+x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 d \sqrt {-x^{3}-1}}-\frac {2 i \left (-c f +d e \right ) \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {1+x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}+\frac {c}{d}}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 d^{2} \sqrt {-x^{3}-1}\, \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}+\frac {c}{d}\right )}\)	\(265\)
elliptic	\(-\frac {2 i f \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {1+x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 d \sqrt {-x^{3}-1}}+\frac {2 i \left (c f -d e \right ) \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {1+x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}+\frac {c}{d}}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 d^{2} \sqrt {-x^{3}-1}\, \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}+\frac {c}{d}\right )}\)	\(265\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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]

integral(-sqrt(-x^3 - 1)*(f*x + e)/(d*x^4 + c*x^3 + d*x + c), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e + f x}{\sqrt {- \left (x + 1\right ) \left (x^{2} - x + 1\right )} \left (c + d x\right )}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [B]
time = 0.10, size = 388, normalized size = 0.84 \begin {gather*} \frac {2\,f\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {x^3+1}\,\sqrt {\frac {x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {\frac {1}{2}-x+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{d\,\sqrt {-x^3-1}\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}}-\frac {2\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {x^3+1}\,\sqrt {\frac {x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\left (c\,f-d\,e\right )\,\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {\frac {1}{2}-x+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\Pi \left (-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {c}{d}-1};\mathrm {asin}\left (\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{d^2\,\sqrt {-x^3-1}\,\left (\frac {c}{d}-1\right )\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*f*((3^(1/2)*1i)/2 + 3/2)*(x^3 + 1)^(1/2)*((x + (3^(1/2)*1i)/2 - 1/2)/((3^(1/2)*1i)/2 - 3/2))^(1/2)*((x + 1)
/((3^(1/2)*1i)/2 + 3/2))^(1/2)*(((3^(1/2)*1i)/2 - x + 1/2)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*ellipticF(asin(((x +
1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)), -((3^(1/2)*1i)/2 + 3/2)/((3^(1/2)*1i)/2 - 3/2)))/(d*(- x^3 - 1)^(1/2)*(x^3
- x*(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) + 1) - ((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2))^(1/2)
) - (2*((3^(1/2)*1i)/2 + 3/2)*(x^3 + 1)^(1/2)*((x + (3^(1/2)*1i)/2 - 1/2)/((3^(1/2)*1i)/2 - 3/2))^(1/2)*(c*f -
d*e)*((x + 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*(((3^(1/2)*1i)/2 - x + 1/2)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*ellipti
cPi(-((3^(1/2)*1i)/2 + 3/2)/(c/d - 1), asin(((x + 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)), -((3^(1/2)*1i)/2 + 3/2)/(
(3^(1/2)*1i)/2 - 3/2)))/(d^2*(- x^3 - 1)^(1/2)*(c/d - 1)*(x^3 - x*(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/
2) + 1) - ((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2))^(1/2))

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