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

Optimal. Leaf size=183 \[ \frac{\left (e-\left (1+\sqrt{3}\right ) f\right ) \tanh ^{-1}\left (\frac{\sqrt{3+2 \sqrt{3}} (x+1)}{\sqrt{-x^3-1}}\right )}{\sqrt{3 \left (3+2 \sqrt{3}\right )}}+\frac{\sqrt{2-\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x-\sqrt{3}+1\right )^2}} \left (e-\left (1-\sqrt{3}\right ) f\right ) F\left (\sin ^{-1}\left (\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{3^{3/4} \sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{-x^3-1}} \]

[Out]

((e - (1 + Sqrt[3])*f)*ArcTanh[(Sqrt[3 + 2*Sqrt[3]]*(1 + x))/Sqrt[-1 - x^3]])/Sqrt[3*(3 + 2*Sqrt[3])] + (Sqrt[
2 - Sqrt[3]]*(e - (1 - 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^(3/4)*Sqrt[-((1 + x)/(1 - Sqrt[3] + x)^2)]*Sqrt[-1 - x^3])

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Rubi [A]  time = 0.23645, antiderivative size = 183, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {2141, 219, 2140, 206} \[ \frac{\left (e-\left (1+\sqrt{3}\right ) f\right ) \tanh ^{-1}\left (\frac{\sqrt{3+2 \sqrt{3}} (x+1)}{\sqrt{-x^3-1}}\right )}{\sqrt{3 \left (3+2 \sqrt{3}\right )}}+\frac{\sqrt{2-\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x-\sqrt{3}+1\right )^2}} \left (e-\left (1-\sqrt{3}\right ) f\right ) F\left (\sin ^{-1}\left (\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{3^{3/4} \sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{-x^3-1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((e - (1 + Sqrt[3])*f)*ArcTanh[(Sqrt[3 + 2*Sqrt[3]]*(1 + x))/Sqrt[-1 - x^3]])/Sqrt[3*(3 + 2*Sqrt[3])] + (Sqrt[
2 - Sqrt[3]]*(e - (1 - 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^(3/4)*Sqrt[-((1 + x)/(1 - Sqrt[3] + x)^2)]*Sqrt[-1 - x^3])

Rule 2141

Int[((e_.) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_Symbol] :> -Dist[(6*a*d^4*e - c*f*
(b*c^3 - 22*a*d^3))/(c*d*(b*c^3 - 28*a*d^3)), Int[1/Sqrt[a + b*x^3], x], x] + Dist[(d*e - c*f)/(c*d*(b*c^3 - 2
8*a*d^3)), Int[(c*(b*c^3 - 22*a*d^3) + 6*a*d^4*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] && EqQ[b^2*c^6 - 20*a*b*c^3*d^3 - 8*a^2*d^6, 0] && NeQ[6*a*d^4*e - c*f*(b*c^3 - 2
2*a*d^3), 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 2140

Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_Symbol] :> With[{k = Simplify[(d*e
+ 2*c*f)/(c*f)]}, Dist[((1 + k)*e)/d, Subst[Int[1/(1 + (3 + 2*k)*a*x^2), x], x, (1 + ((1 + k)*d*x)/c)/Sqrt[a +
b*x^3]], x]] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && EqQ[b^2*c^6 - 20*a*b*c^3*d^3 - 8*a^2*d^6
, 0] && EqQ[6*a*d^4*e - c*f*(b*c^3 - 22*a*d^3), 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{e+f x}{\left (1+\sqrt{3}+x\right ) \sqrt{-1-x^3}} \, dx &=\frac{\left (e-\left (1-\sqrt{3}\right ) f\right ) \int \frac{1}{\sqrt{-1-x^3}} \, dx}{2 \sqrt{3}}+\frac{\left (e-\left (1+\sqrt{3}\right ) f\right ) \int \frac{\left (1+\sqrt{3}\right ) \left (22-\left (1+\sqrt{3}\right )^3\right )-6 x}{\left (1+\sqrt{3}+x\right ) \sqrt{-1-x^3}} \, dx}{12 \sqrt{3}}\\ &=\frac{\sqrt{2-\sqrt{3}} \left (e-\left (1-\sqrt{3}\right ) 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 )}{3^{3/4} \sqrt{-\frac{1+x}{\left (1-\sqrt{3}+x\right )^2}} \sqrt{-1-x^3}}+\frac{\left (e-\left (1+\sqrt{3}\right ) f\right ) \operatorname{Subst}\left (\int \frac{1}{1-\left (3+2 \sqrt{3}\right ) x^2} \, dx,x,\frac{1+x}{\sqrt{-1-x^3}}\right )}{\sqrt{3}}\\ &=\frac{\left (e-\left (1+\sqrt{3}\right ) f\right ) \tanh ^{-1}\left (\frac{\sqrt{3+2 \sqrt{3}} (1+x)}{\sqrt{-1-x^3}}\right )}{\sqrt{3 \left (3+2 \sqrt{3}\right )}}+\frac{\sqrt{2-\sqrt{3}} \left (e-\left (1-\sqrt{3}\right ) 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 )}{3^{3/4} \sqrt{-\frac{1+x}{\left (1-\sqrt{3}+x\right )^2}} \sqrt{-1-x^3}}\\ \end{align*}

Mathematica [C]  time = 0.457237, size = 293, normalized size = 1.6 \[ \frac{2 \sqrt{\frac{2}{3}} \sqrt{\frac{i (x+1)}{\sqrt{3}+3 i}} \left (2 \sqrt{-2 i x+\sqrt{3}+i} \sqrt{x^2-x+1} \left (\left (3+\sqrt{3}\right ) f-\sqrt{3} e\right ) \Pi \left (\frac{2 \sqrt{3}}{3 i+(1+2 i) \sqrt{3}};\sin ^{-1}\left (\frac{\sqrt{-2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right )+3 f \sqrt{2 i x+\sqrt{3}-i} \left (\left ((1+2 i)+i \sqrt{3}\right ) x-\sqrt{3}-(2+i)\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{-2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right )\right )}{\left (3 i+(1+2 i) \sqrt{3}\right ) \sqrt{-2 i x+\sqrt{3}+i} \sqrt{-x^3-1}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [A]  time = 0.023, size = 258, normalized size = 1.4 \begin{align*}{-{\frac{2\,i}{3}}f\sqrt{3}\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 ( e-f-f\sqrt{3} \right ) \sqrt{3}}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}+\sqrt{3}}\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}}},{\frac{i\sqrt{3}}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}+\sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}-1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*I*f*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*(e-f-f*3^(1/2))*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)/(3/2+1/2*I*3^(1/2)+3^(1/2))*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)/(3/2+1/2*I*3^(1/2)+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., size = 0, normalized size = 0. \begin{align*} \int \frac{f x + e}{\sqrt{-x^{3} - 1}{\left (x + \sqrt{3} + 1\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (f x^{2} +{\left (e + f\right )} x - \sqrt{3}{\left (f x + e\right )} + e\right )} \sqrt{-x^{3} - 1}}{x^{5} + 2 \, x^{4} - 2 \, x^{3} + x^{2} + 2 \, x - 2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(f*x^2 + (e + f)*x - sqrt(3)*(f*x + e) + e)*sqrt(-x^3 - 1)/(x^5 + 2*x^4 - 2*x^3 + x^2 + 2*x - 2), x)

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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 (x + 1 + \sqrt{3}\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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