3.2.50 \(\int \frac {1-\sqrt {3}-x}{(c+d x) \sqrt {-1+x^3}} \, dx\) [150]

Optimal. Leaf size=342 \[ -\frac {\left (c+d-\sqrt {3} d\right ) (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} \tan ^{-1}\left (\frac {\sqrt {c^2-c d+d^2} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}}}{\sqrt {d} \sqrt {c+d} \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}}}\right )}{\sqrt {d} \sqrt {c+d} \sqrt {c^2-c d+d^2} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}+\frac {4 \sqrt [4]{3} \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} \Pi \left (\frac {\left (c+d-\sqrt {3} d\right )^2}{\left (c+d+\sqrt {3} d\right )^2};\sin ^{-1}\left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right )|-7+4 \sqrt {3}\right )}{\left (c+d+\sqrt {3} d\right ) \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}} \]

[Out]

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

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Rubi [A]
time = 0.63, antiderivative size = 342, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2168, 2138, 551, 585, 95, 211} \begin {gather*} \frac {4 \sqrt [4]{3} \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x-\sqrt {3}+1\right )^2}} \Pi \left (\frac {\left (c-\sqrt {3} d+d\right )^2}{\left (c+\sqrt {3} d+d\right )^2};\text {ArcSin}\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+\sqrt {3} d+d\right )}-\frac {(1-x) \sqrt {\frac {x^2+x+1}{\left (-x-\sqrt {3}+1\right )^2}} \left (c-\sqrt {3} d+d\right ) \text {ArcTan}\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}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(((c + d - Sqrt[3]*d)*(1 - x)*Sqrt[(1 + x + x^2)/(1 - Sqrt[3] - x)^2]*ArcTan[(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])) + (4*3^(1/4)*Sqrt[2 - Sqrt[3]]
*(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, ArcSi
n[(1 + Sqrt[3] - x)/(1 - Sqrt[3] - x)], -7 + 4*Sqrt[3]])/((c + d + Sqrt[3]*d)*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 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 2168

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 - Sq
rt[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]

Rubi steps

\begin {align*} \int \frac {1-\sqrt {3}-x}{(c+d x) \sqrt {-1+x^3}} \, dx &=-\frac {\left (4 \sqrt [4]{3} \sqrt {2+\sqrt {3}} (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 )}{\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 ) (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 )}{\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 ) (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 )}{\sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}\\ &=-\frac {4 \sqrt [4]{3} \sqrt {2-\sqrt {3}} (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 )}{\left (c+d+\sqrt {3} d\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 ) (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 )}{\sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}\\ &=-\frac {4 \sqrt [4]{3} \sqrt {2-\sqrt {3}} (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 )}{\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 ) (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 (7+4 \sqrt {3}\right ) \left (c+\left (1-\sqrt {3}\right ) d\right )^2+\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 {2+\sqrt {3}} \sqrt {\frac {1+x+x^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 {\left (c+d-\sqrt {3} d\right ) (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 {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 {4 \sqrt [4]{3} \sqrt {2-\sqrt {3}} (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 )}{\left (c+d+\sqrt {3} d\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 = 10.12, size = 233, normalized size = 0.68 \begin {gather*} \frac {2 \sqrt {\frac {1-x}{1+\sqrt [3]{-1}}} \left (-\frac {3 \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 {\sqrt [3]{-1} \left (1+\sqrt [3]{-1}\right ) \left (\sqrt {3} c+\left (-3+\sqrt {3}\right ) d\right ) \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 )}{3 d \sqrt {-1+x^3}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [A]
time = 0.26, size = 277, normalized size = 0.81

method	result	size
default	\(-\frac {2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \EllipticF \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{d \sqrt {x^{3}-1}}-\frac {2 \left (d \sqrt {3}-c -d \right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{1+\frac {c}{d}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{d^{2} \sqrt {x^{3}-1}\, \left (1+\frac {c}{d}\right )}\)	\(277\)
elliptic	\(-\frac {2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \EllipticF \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{d \sqrt {x^{3}-1}}-\frac {2 \left (d \sqrt {3}-c -d \right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{1+\frac {c}{d}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{d^{2} \sqrt {x^{3}-1}\, \left (1+\frac {c}{d}\right )}\)	\(277\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/d*(-3/2-1/2*I*3^(1/2))*((-1+x)/(-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(((-1+x)/(-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*(d*3^(1/2)-c-d)/d^2*(-3/2-1/2*I*3^(1/2))*((-1+x)/(-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(((-1+x)/(-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 = 0.00, 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((1-x-3^(1/2))/(d*x+c)/(x^3-1)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {\sqrt {3}}{c \sqrt {x^{3} - 1} + d x \sqrt {x^{3} - 1}}\, dx - \int \frac {x}{c \sqrt {x^{3} - 1} + d x \sqrt {x^{3} - 1}}\, dx - \int \left (- \frac {1}{c \sqrt {x^{3} - 1} + d x \sqrt {x^{3} - 1}}\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(sqrt(3)/(c*sqrt(x**3 - 1) + d*x*sqrt(x**3 - 1)), x) - Integral(x/(c*sqrt(x**3 - 1) + d*x*sqrt(x**3 -
1)), x) - Integral(-1/(c*sqrt(x**3 - 1) + d*x*sqrt(x**3 - 1)), 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((1-x-3^(1/2))/(d*x+c)/(x^3-1)^(1/2),x, algorithm="giac")

[Out]

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

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Mupad [F(-1)]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \text {Hanged} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

\text{Hanged}

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