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

Optimal. Leaf size=364 \[ \frac{4 \sqrt [4]{3} \sqrt{2-\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x-\sqrt{3}+1\right )^2}} \Pi \left (\frac{\left (c-\left (1-\sqrt{3}\right ) d\right )^2}{\left (c-\left (1+\sqrt{3}\right ) d\right )^2};-\sin ^{-1}\left (\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{\sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{-x^3-1} \left (c-\sqrt{3} d-d\right )}-\frac{(x+1) \sqrt{\frac{x^2-x+1}{\left (x-\sqrt{3}+1\right )^2}} \left (c-\left (1-\sqrt{3}\right ) d\right ) \tanh ^{-1}\left (\frac{2 \sqrt{2+\sqrt{3}} \sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{c^2+c d+d^2}}{\sqrt{d} \sqrt{\frac{\left (x+\sqrt{3}+1\right )^2}{\left (x-\sqrt{3}+1\right )^2}+4 \sqrt{3}+7} \sqrt{c-d}}\right )}{\sqrt{d} \sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{-x^3-1} \sqrt{c-d} \sqrt{c^2+c d+d^2}} \]

[Out]

-(((c - (1 - Sqrt[3])*d)*(1 + x)*Sqrt[(1 - x + x^2)/(1 - Sqrt[3] + x)^2]*ArcTanh[(2*Sqrt[2 + Sqrt[3]]*Sqrt[c^2
+ c*d + d^2]*Sqrt[-((1 + x)/(1 - Sqrt[3] + x)^2)])/(Sqrt[c - d]*Sqrt[d]*Sqrt[7 + 4*Sqrt[3] + (1 + Sqrt[3] + 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)]*Sq
rt[-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 -
(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
- d - Sqrt[3]*d)*Sqrt[-((1 + x)/(1 - Sqrt[3] + x)^2)]*Sqrt[-1 - x^3])

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Rubi [A]  time = 0.792385, antiderivative size = 364, normalized size of antiderivative = 1., 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 = {2143, 2113, 537, 571, 93, 208} \[ \frac{4 \sqrt [4]{3} \sqrt{2-\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x-\sqrt{3}+1\right )^2}} \Pi \left (\frac{\left (c-\left (1-\sqrt{3}\right ) d\right )^2}{\left (c-\left (1+\sqrt{3}\right ) d\right )^2};-\sin ^{-1}\left (\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{\sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{-x^3-1} \left (c-\sqrt{3} d-d\right )}-\frac{(x+1) \sqrt{\frac{x^2-x+1}{\left (x-\sqrt{3}+1\right )^2}} \left (c-\left (1-\sqrt{3}\right ) d\right ) \tanh ^{-1}\left (\frac{2 \sqrt{2+\sqrt{3}} \sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{c^2+c d+d^2}}{\sqrt{d} \sqrt{\frac{\left (x+\sqrt{3}+1\right )^2}{\left (x-\sqrt{3}+1\right )^2}+4 \sqrt{3}+7} \sqrt{c-d}}\right )}{\sqrt{d} \sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{-x^3-1} \sqrt{c-d} \sqrt{c^2+c d+d^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((c - (1 - Sqrt[3])*d)*(1 + x)*Sqrt[(1 - x + x^2)/(1 - Sqrt[3] + x)^2]*ArcTanh[(2*Sqrt[2 + Sqrt[3]]*Sqrt[c^2
+ c*d + d^2]*Sqrt[-((1 + x)/(1 - Sqrt[3] + x)^2)])/(Sqrt[c - d]*Sqrt[d]*Sqrt[7 + 4*Sqrt[3] + (1 + Sqrt[3] + 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)]*Sq
rt[-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 -
(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
- d - Sqrt[3]*d)*Sqrt[-((1 + x)/(1 - Sqrt[3] + x)^2)]*Sqrt[-1 - x^3])

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]

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{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{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{\left (4 \sqrt [4]{3} \sqrt{2+\sqrt{3}} \left (-c+\left (1-\sqrt{3}\right ) 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{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-\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-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+\left (1-\sqrt{3}\right ) 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-\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-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+\left (1-\sqrt{3}\right ) 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{2 \sqrt [4]{3} \sqrt{-\frac{1+x}{\left (1-\sqrt{3}+x\right )^2}}}{\sqrt{7+4 \sqrt{3}+\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{\left (c-\left (1-\sqrt{3}\right ) d\right ) (1+x) \sqrt{\frac{1-x+x^2}{\left (1-\sqrt{3}+x\right )^2}} \tanh ^{-1}\left (\frac{2 \sqrt{2+\sqrt{3}} \sqrt{c^2+c d+d^2} \sqrt{-\frac{1+x}{\left (1-\sqrt{3}+x\right )^2}}}{\sqrt{c-d} \sqrt{d} \sqrt{7+4 \sqrt{3}+\frac{\left (1+\sqrt{3}+x\right )^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{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-\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-d-\sqrt{3} d\right ) \sqrt{-\frac{1+x}{\left (1-\sqrt{3}+x\right )^2}} \sqrt{-1-x^3}}\\ \end{align*}

Mathematica [C]  time = 0.200449, size = 215, normalized size = 0.59 \[ \frac{2 \sqrt{\frac{x+1}{1+\sqrt [3]{-1}}} \left (-\frac{\left (\sqrt [3]{-1}-x\right ) \sqrt{\frac{\sqrt [3]{-1}-(-1)^{2/3} x}{1+\sqrt [3]{-1}}} F\left (\sin ^{-1}\left (\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}}+\frac{i \sqrt{x^2-x+1} \left (c+\left (\sqrt{3}-1\right ) d\right ) \Pi \left (\frac{i \sqrt{3} d}{c+\sqrt [3]{-1} d};\sin ^{-1}\left (\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{c+\sqrt [3]{-1} d}\right )}{d \sqrt{-x^3-1}} \]

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))]*(-((((-1)^(1/3) - x)*Sqrt[((-1)^(1/3) - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]*Elli
pticF[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*(c + (-1 + 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)))/(d*Sqrt[-1 - x^3])

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Maple [A]  time = 0.019, size = 266, normalized size = 0.7 \begin{align*}{\frac{-{\frac{2\,i}{3}}\sqrt{3}}{d}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{1+x}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticF} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}-1}}}}+{\frac{{\frac{2\,i}{3}} \left ( d\sqrt{3}+c-d \right ) \sqrt{3}}{{d}^{2}}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{1+x}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},{i\sqrt{3} \left ({\frac{1}{2}}+{\frac{i}{2}}\sqrt{3}+{\frac{c}{d}} \right ) ^{-1}},\sqrt{{\frac{i\sqrt{3}}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}-1}}} \left ({\frac{1}{2}}+{\frac{i}{2}}\sqrt{3}+{\frac{c}{d}} \right ) ^{-1}} \end{align*}

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)*((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*(d*3^(1/2)+c-d)/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)*Elli
pticPi(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., size = 0, normalized size = 0. \begin{align*} \int \frac{x - \sqrt{3} + 1}{\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((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)]  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((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., size = 0, normalized size = 0. \begin{align*} \int \frac{x - \sqrt{3} + 1}{\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((1+x-3**(1/2))/(d*x+c)/(-x**3-1)**(1/2),x)

[Out]

Integral((x - sqrt(3) + 1)/(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{x - \sqrt{3} + 1}{\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((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)