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

Optimal. Leaf size=346 \[ \frac {4 \sqrt [4]{3} \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x-\sqrt {3}+1\right )^2}} \Pi \left (\frac {\left (c-\sqrt {3} d+d\right )^2}{\left (c+\sqrt {3} d+d\right )^2};\sin ^{-1}\left (\frac {-x+\sqrt {3}+1}{-x-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{\sqrt {-\frac {1-x}{\left (-x-\sqrt {3}+1\right )^2}} \sqrt {1-x^3} \left (c+\sqrt {3} d+d\right )}-\frac {(1-x) \sqrt {\frac {x^2+x+1}{\left (-x-\sqrt {3}+1\right )^2}} \left (c-\sqrt {3} d+d\right ) \tan ^{-1}\left (\frac {\sqrt {-\frac {1-x}{\left (-x-\sqrt {3}+1\right )^2}} \sqrt {c^2-c d+d^2}}{\sqrt {d} \sqrt {\frac {x^2+x+1}{\left (-x-\sqrt {3}+1\right )^2}} \sqrt {c+d}}\right )}{\sqrt {d} \sqrt {-\frac {1-x}{\left (-x-\sqrt {3}+1\right )^2}} \sqrt {1-x^3} \sqrt {c+d} \sqrt {c^2-c d+d^2}} \]

[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 = 0.98, antiderivative size = 348, normalized size of antiderivative = 1.01, number of steps used = 6, number of rules used = 6, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {2143, 2113, 537, 571, 93, 205} \[ -\frac {(1-x) \sqrt {\frac {x^2+x+1}{\left (-x-\sqrt {3}+1\right )^2}} \left (c-\sqrt {3} d+d\right ) \tan ^{-1}\left (\frac {\sqrt {-\frac {1-x}{\left (-x-\sqrt {3}+1\right )^2}} \sqrt {c^2-c d+d^2}}{\sqrt {d} \sqrt {\frac {x^2+x+1}{\left (-x-\sqrt {3}+1\right )^2}} \sqrt {c+d}}\right )}{\sqrt {d} \sqrt {-\frac {1-x}{\left (-x-\sqrt {3}+1\right )^2}} \sqrt {1-x^3} \sqrt {c+d} \sqrt {c^2-c d+d^2}}-\frac {4 \sqrt [4]{3} \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x-\sqrt {3}+1\right )^2}} \Pi \left (\frac {\left (c-\sqrt {3} d+d\right )^2}{\left (c+\sqrt {3} d+d\right )^2};-\sin ^{-1}\left (\frac {-x+\sqrt {3}+1}{-x-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{\sqrt {-\frac {1-x}{\left (-x-\sqrt {3}+1\right )^2}} \sqrt {1-x^3} \left (c+\sqrt {3} d+d\right )} \]

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 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 205

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

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 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 2143

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]

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 ) \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 )}{\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 ) \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 ) (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 )}{\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 ) \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 {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 ) \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 (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]  time = 0.70, size = 235, normalized size = 0.68 \[ \frac {2 \sqrt {\frac {1-x}{1+\sqrt [3]{-1}}} \left (-\frac {3 \left (x+\sqrt [3]{-1}\right ) \sqrt {\frac {(-1)^{2/3} x+\sqrt [3]{-1}}{1+\sqrt [3]{-1}}} F\left (\sin ^{-1}\left (\sqrt {\frac {1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{\sqrt {\frac {1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}}+\frac {\sqrt [3]{-1} \left (1+\sqrt [3]{-1}\right ) \sqrt {x^2+x+1} \left (\sqrt {3} c+\left (\sqrt {3}-3\right ) d\right ) \Pi \left (\frac {i \sqrt {3} d}{\sqrt [3]{-1} d-c};\sin ^{-1}\left (\sqrt {\frac {1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{c-\sqrt [3]{-1} d}\right )}{3 d \sqrt {1-x^3}} \]

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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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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]

Exception raised: TypeError >>  Error detected within library code:   catdef: division by zero

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

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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maple [A]  time = 0.03, size = 268, normalized size = 0.77 \[ \frac {2 i \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x -1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}+1}\, d}+\frac {2 i \left (-c +\sqrt {3}\, d -d \right ) \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x -1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{\frac {c}{d}-\frac {1}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}+1}\, \left (\frac {c}{d}-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) d^{2}} \]

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)

[Out]

2/3*I/d*3^(1/2)*(I*(x+1/2-1/2*I*3^(1/2))*3^(1/2))^(1/2)*((x-1)/(-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*(3^(1/2)*d-c-d)/d^2*3^(1/2)*(I*(x+1/2-1/2*I*3^(1/2))*3^(1/2))^(1/2)*((x-1)/
(-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)/(c/d-1/2+1/2*I*3^(1/2))*El
lipticPi(1/3*3^(1/2)*(I*(x+1/2-1/2*I*3^(1/2))*3^(1/2))^(1/2),I*3^(1/2)/(c/d-1/2+1/2*I*3^(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 \[ -\int \frac {x + \sqrt {3} - 1}{\sqrt {-x^{3} + 1} {\left (d x + c\right )}}\,{d x} \]

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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mupad [F(-1)]  time = 0.00, size = -1, normalized size = -0.00 \[ \text {Hanged} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

\text{Hanged}

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

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

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