3.34 \(\int \frac{e+f x}{\left (2^{2/3}-x\right ) \sqrt{1-x^3}} \, dx\)

Optimal. Leaf size=175 \[ -\frac{2 \left (e+2^{2/3} f\right ) \tan ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} x\right )}{\sqrt{1-x^3}}\right )}{3 \sqrt{3}}-\frac{2 \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} \left (\sqrt [3]{2} e-f\right ) F\left (\sin ^{-1}\left (\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{3 \sqrt [4]{3} \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{1-x^3}} \]

[Out]

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

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Rubi [A]  time = 0.424549, antiderivative size = 175, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ -\frac{2 \left (e+2^{2/3} f\right ) \tan ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} x\right )}{\sqrt{1-x^3}}\right )}{3 \sqrt{3}}-\frac{2 \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} \left (\sqrt [3]{2} e-f\right ) F\left (\sin ^{-1}\left (\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{3 \sqrt [4]{3} \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{1-x^3}} \]

Antiderivative was successfully verified.

[In]  Int[(e + f*x)/((2^(2/3) - x)*Sqrt[1 - x^3]),x]

[Out]

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

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Rubi in Sympy [A]  time = 148.262, size = 483, normalized size = 2.76 \[ - \frac{2 \sqrt{\frac{x^{2} + x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (e + 2^{\frac{2}{3}} f\right ) \left (- x + 1\right ) \operatorname{atan}{\left (\frac{3^{\frac{3}{4}} \sqrt{1 + \sqrt [3]{2}} \sqrt{1 - \frac{\left (x - 1 + \sqrt{3}\right )^{2}}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- 4 \sqrt{3} + 8}}{6 \sqrt{-1 + \sqrt [3]{2}} \sqrt{- 4 \sqrt{3} + 7 + \frac{\left (x - 1 + \sqrt{3}\right )^{2}}{\left (- x + 1 + \sqrt{3}\right )^{2}}}} \right )}}{\sqrt{\frac{- x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \sqrt{-1 + \sqrt [3]{2}} \left (1 + \sqrt [3]{2}\right )^{\frac{3}{2}} \sqrt{- 4 \sqrt{3} + 8} \sqrt{- x^{3} + 1}} - \frac{4 \sqrt [4]{3} \sqrt{\frac{x^{2} + x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (e + 2^{\frac{2}{3}} f\right ) \left (- x + 1\right ) \Pi \left (\frac{\left (- 2^{\frac{2}{3}} + 1 + \sqrt{3}\right )^{2}}{\left (-1 + 2^{\frac{2}{3}} + \sqrt{3}\right )^{2}}; \operatorname{asin}{\left (\frac{x - 1 + \sqrt{3}}{- x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{\sqrt{\frac{- x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- 4 \sqrt{3} + 7} \sqrt{- x^{3} + 1} \left (- 2^{\frac{2}{3}} + 1 + \sqrt{3}\right ) \left (- \sqrt{3} - 2^{\frac{2}{3}} + 1\right )} + \frac{2 \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} + x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (- x + 1\right ) \left (e + f + \sqrt{3} f\right ) F\left (\operatorname{asin}{\left (\frac{- x - \sqrt{3} + 1}{- x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{3 \sqrt{\frac{- x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- x^{3} + 1} \left (- 2^{\frac{2}{3}} + 1 + \sqrt{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((f*x+e)/(2**(2/3)-x)/(-x**3+1)**(1/2),x)

[Out]

-2*sqrt((x**2 + x + 1)/(-x + 1 + sqrt(3))**2)*sqrt(-sqrt(3) + 2)*(e + 2**(2/3)*f
)*(-x + 1)*atan(3**(3/4)*sqrt(1 + 2**(1/3))*sqrt(1 - (x - 1 + sqrt(3))**2/(-x +
1 + sqrt(3))**2)*sqrt(-4*sqrt(3) + 8)/(6*sqrt(-1 + 2**(1/3))*sqrt(-4*sqrt(3) + 7
+ (x - 1 + sqrt(3))**2/(-x + 1 + sqrt(3))**2)))/(sqrt((-x + 1)/(-x + 1 + sqrt(3
))**2)*sqrt(-1 + 2**(1/3))*(1 + 2**(1/3))**(3/2)*sqrt(-4*sqrt(3) + 8)*sqrt(-x**3
+ 1)) - 4*3**(1/4)*sqrt((x**2 + x + 1)/(-x + 1 + sqrt(3))**2)*sqrt(-sqrt(3) + 2
)*(e + 2**(2/3)*f)*(-x + 1)*elliptic_pi((-2**(2/3) + 1 + sqrt(3))**2/(-1 + 2**(2
/3) + sqrt(3))**2, asin((x - 1 + sqrt(3))/(-x + 1 + sqrt(3))), -7 - 4*sqrt(3))/(
sqrt((-x + 1)/(-x + 1 + sqrt(3))**2)*sqrt(-4*sqrt(3) + 7)*sqrt(-x**3 + 1)*(-2**(
2/3) + 1 + sqrt(3))*(-sqrt(3) - 2**(2/3) + 1)) + 2*3**(3/4)*sqrt((x**2 + x + 1)/
(-x + 1 + sqrt(3))**2)*sqrt(sqrt(3) + 2)*(-x + 1)*(e + f + sqrt(3)*f)*elliptic_f
(asin((-x - sqrt(3) + 1)/(-x + 1 + sqrt(3))), -7 - 4*sqrt(3))/(3*sqrt((-x + 1)/(
-x + 1 + sqrt(3))**2)*sqrt(-x**3 + 1)*(-2**(2/3) + 1 + sqrt(3)))

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Mathematica [C]  time = 0.697047, size = 340, normalized size = 1.94 \[ \frac{2 \sqrt [6]{2} \sqrt{-\frac{i (x-1)}{\sqrt{3}+3 i}} \left (2 \sqrt{3} \sqrt{2 i x+\sqrt{3}+i} \sqrt{x^2+x+1} \left (\sqrt [3]{2} e+2 f\right ) \Pi \left (\frac{2 \sqrt{3}}{i+2 i 2^{2/3}+\sqrt{3}};\sin ^{-1}\left (\frac{\sqrt{2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right )-i f \sqrt{-2 i x+\sqrt{3}-i} \left (\left (-3 i \sqrt [3]{2}+4 \sqrt{3}+\sqrt [3]{2} \sqrt{3}\right ) x-\sqrt [3]{2} \sqrt{3}+2 \sqrt{3}-3 i \sqrt [3]{2}-6 i\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right )\right )}{\sqrt{3} \left (i+2 i 2^{2/3}+\sqrt{3}\right ) \sqrt{2 i x+\sqrt{3}+i} \sqrt{1-x^3}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(e + f*x)/((2^(2/3) - x)*Sqrt[1 - x^3]),x]

[Out]

(2*2^(1/6)*Sqrt[((-I)*(-1 + x))/(3*I + Sqrt[3])]*((-I)*f*Sqrt[-I + Sqrt[3] - (2*
I)*x]*(-6*I - (3*I)*2^(1/3) + 2*Sqrt[3] - 2^(1/3)*Sqrt[3] + ((-3*I)*2^(1/3) + 4*
Sqrt[3] + 2^(1/3)*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]*(2^(1/3)*e + 2*f)*Sqrt[I
+ Sqrt[3] + (2*I)*x]*Sqrt[1 + x + x^2]*EllipticPi[(2*Sqrt[3])/(I + (2*I)*2^(2/3
) + Sqrt[3]), ArcSin[Sqrt[I + Sqrt[3] + (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])
/(3*I + Sqrt[3])]))/(Sqrt[3]*(I + (2*I)*2^(2/3) + Sqrt[3])*Sqrt[I + Sqrt[3] + (2
*I)*x]*Sqrt[1 - x^3])

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Maple [A]  time = 0.039, size = 261, normalized size = 1.5 \[{\frac{-{\frac{2\,i}{3}} \left ( -e-{2}^{{\frac{2}{3}}}f \right ) \sqrt{3}}{-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3}-{2}^{{\frac{2}{3}}}}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{-1+x}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x+{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},{\frac{i\sqrt{3}}{-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3}-{2}^{{\frac{2}{3}}}}},\sqrt{{\frac{i\sqrt{3}}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}+1}}}}+{{\frac{2\,i}{3}}f\sqrt{3}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{-1+x}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x+{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticF} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*I*(-e-2^(2/3)*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
)/(-1/2+1/2*I*3^(1/2)-2^(2/3))*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)/(-1/2+1/2*I*3^(1/2)-2^(2/3)),(I*3^(1/2)/(-3/2+1/2*I*3^(1
/2)))^(1/2))+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))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{f x + e}{\sqrt{-x^{3} + 1}{\left (x - 2^{\frac{2}{3}}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \frac{e}{x \sqrt{- x^{3} + 1} - 2^{\frac{2}{3}} \sqrt{- x^{3} + 1}}\, dx - \int \frac{f x}{x \sqrt{- x^{3} + 1} - 2^{\frac{2}{3}} \sqrt{- x^{3} + 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int -\frac{f x + e}{\sqrt{-x^{3} + 1}{\left (x - 2^{\frac{2}{3}}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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