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

Optimal. Leaf size=187 \[ -\frac {\left (e+\sqrt {3} f+f\right ) \tan ^{-1}\left (\frac {\sqrt {3+2 \sqrt {3}} (1-x)}{\sqrt {1-x^3}}\right )}{\sqrt {3 \left (3+2 \sqrt {3}\right )}}-\frac {\sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x+\sqrt {3}+1\right )^2}} \left (e+\left (1-\sqrt {3}\right ) f\right ) F\left (\sin ^{-1}\left (\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{3^{3/4} \sqrt {\frac {1-x}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {1-x^3}} \]

[Out]

-1/3*(1-x)*EllipticF((1-x-3^(1/2))/(1-x+3^(1/2)),I*3^(1/2)+2*I)*(e+f*(1-3^(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^(1/4)/(-x^3+1)^(1/2)/((1-x)/(1-x+3^(1/2))^2)^(1/2)-arctan((1-x)*(3+2*3^(1/2)
)^(1/2)/(-x^3+1)^(1/2))*(e+f+f*3^(1/2))/(9+6*3^(1/2))^(1/2)

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Rubi [A]  time = 0.28, antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2141, 218, 2140, 203} \[ -\frac {\left (e+\sqrt {3} f+f\right ) \tan ^{-1}\left (\frac {\sqrt {3+2 \sqrt {3}} (1-x)}{\sqrt {1-x^3}}\right )}{\sqrt {3 \left (3+2 \sqrt {3}\right )}}-\frac {\sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x+\sqrt {3}+1\right )^2}} \left (e+\left (1-\sqrt {3}\right ) f\right ) F\left (\sin ^{-1}\left (\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{3^{3/4} \sqrt {\frac {1-x}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {1-x^3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((e + f + Sqrt[3]*f)*ArcTan[(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 203

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

Rule 218

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 + Sqr
t[3])*s + r*x)^2]), x]] /; FreeQ[{a, b}, x] && PosQ[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 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]

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 {\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}{\left (1+\sqrt {3}\right ) \left (28-\left (1+\sqrt {3}\right )^3\right )}+\frac {\left (6 e-\left (1+\sqrt {3}\right ) \left (22-\left (1+\sqrt {3}\right )^3\right ) f\right ) \int \frac {1}{\sqrt {1-x^3}} \, dx}{\left (1+\sqrt {3}\right ) \left (28-\left (1+\sqrt {3}\right )^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}}+\frac {\left (12 \left (-e-\left (1+\sqrt {3}\right ) f\right )\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 )}{\left (1+\sqrt {3}\right ) \left (28-\left (1+\sqrt {3}\right )^3\right )}\\ &=-\frac {\left (e+f+\sqrt {3} f\right ) \tan ^{-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*}

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Mathematica [C]  time = 0.58, size = 291, normalized size = 1.56 \[ \frac {2 \sqrt {\frac {2}{3}} \sqrt {-\frac {i (x-1)}{\sqrt {3}+3 i}} \left (2 \sqrt {2 i x+\sqrt {3}+i} \sqrt {x^2+x+1} \left (\sqrt {3} e+\left (3+\sqrt {3}\right ) f\right ) \Pi \left (\frac {2 \sqrt {3}}{3 i+(1+2 i) \sqrt {3}};\sin ^{-1}\left (\frac {\sqrt {2 i x+\sqrt {3}+i}}{\sqrt {2} \sqrt [4]{3}}\right )|\frac {2 \sqrt {3}}{3 i+\sqrt {3}}\right )-3 i f \sqrt {-2 i x+\sqrt {3}-i} \left (\left (\sqrt {3}+(2-i)\right ) x-i \left (\sqrt {3}+(2+i)\right )\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {2 i x+\sqrt {3}+i}}{\sqrt {2} \sqrt [4]{3}}\right )|\frac {2 \sqrt {3}}{3 i+\sqrt {3}}\right )\right )}{\left (3 i+(1+2 i) \sqrt {3}\right ) \sqrt {2 i x+\sqrt {3}+i} \sqrt {1-x^3}} \]

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*I)*f*Sqrt[-I + Sqrt[3] - (2*I)*x]*((-I)*((2 + I) + Sqr
t[3]) + ((2 - 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*S
qrt[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 + S
qrt[3])]))/((3*I + (1 + 2*I)*Sqrt[3])*Sqrt[I + Sqrt[3] + (2*I)*x]*Sqrt[1 - x^3])

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fricas [F]  time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (f x^{2} + {\left (e - f\right )} x + \sqrt {3} {\left (f x + e\right )} - e\right )} \sqrt {-x^{3} + 1}}{x^{5} - 2 \, x^{4} - 2 \, x^{3} - x^{2} + 2 \, x + 2}, x\right ) \]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Unab
le to divide, perhaps due to rounding error%%%{1,[2]%%%} / %%%{%%{[2,4]:[1,0,-3]%%},[2]%%%} Error: Bad Argumen
t Value

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maple [A]  time = 0.04, size = 264, normalized size = 1.41 \[ \frac {2 i \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x -1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, f \EllipticF \left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}+1}}-\frac {2 i \left (-e -f -\sqrt {3}\, f \right ) \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x -1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}-\sqrt {3}}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}+1}\, \left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}-\sqrt {3}\right )} \]

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)*((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*(-e-f-3^(1/2)*f)*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)/(-3/2+1/2*I*3^(1/2)-3^(1/2))*
EllipticPi(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 = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {f x + e}{\sqrt {-x^{3} + 1} {\left (x - \sqrt {3} - 1\right )}}\,{d x} \]

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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mupad [F(-1)]  time = 0.00, size = -1, normalized size = -0.01 \[ \text {Hanged} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

\text{Hanged}

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {e}{x \sqrt {1 - x^{3}} - \sqrt {3} \sqrt {1 - x^{3}} - \sqrt {1 - x^{3}}}\, dx - \int \frac {f x}{x \sqrt {1 - x^{3}} - \sqrt {3} \sqrt {1 - x^{3}} - \sqrt {1 - x^{3}}}\, dx \]

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/(x*sqrt(1 - x**3) - sqrt(3)*sqrt(1 - x**3) - sqrt(1 - x**3)), x) - Integral(f*x/(x*sqrt(1 - x**3)
- sqrt(3)*sqrt(1 - x**3) - sqrt(1 - x**3)), x)

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