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

Optimal. Leaf size=159 \[ \frac{2 \left (e-2^{2/3} f\right ) \tan ^{-1}\left (\frac{\sqrt{3} \left (\sqrt [3]{2} x+1\right )}{\sqrt{x^3+1}}\right )}{3 \sqrt{3}}+\frac{2 \sqrt{2+\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} \left (\sqrt [3]{2} e+f\right ) F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{3 \sqrt [4]{3} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^3+1}} \]

[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 = 0.375366, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{2 \left (e-2^{2/3} f\right ) \tan ^{-1}\left (\frac{\sqrt{3} \left (\sqrt [3]{2} x+1\right )}{\sqrt{x^3+1}}\right )}{3 \sqrt{3}}+\frac{2 \sqrt{2+\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} \left (\sqrt [3]{2} e+f\right ) F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{3 \sqrt [4]{3} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^3+1}} \]

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 = 144.083, size = 483, normalized size = 3.04 \[ \frac{2 \sqrt{\frac{x^{2} - x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (e - 2^{\frac{2}{3}} f\right ) \left (x + 1\right ) \operatorname{atan}{\left (\frac{3^{\frac{3}{4}} \sqrt{1 + \sqrt [3]{2}} \sqrt{- 4 \sqrt{3} + 8} \sqrt{- \frac{\left (- x - 1 + \sqrt{3}\right )^{2}}{\left (x + 1 + \sqrt{3}\right )^{2}} + 1}}{6 \sqrt{-1 + \sqrt [3]{2}} \sqrt{\frac{\left (- x - 1 + \sqrt{3}\right )^{2}}{\left (x + 1 + \sqrt{3}\right )^{2}} - 4 \sqrt{3} + 7}} \right )}}{\sqrt{\frac{x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{-1 + \sqrt [3]{2}} \left (1 + \sqrt [3]{2}\right )^{\frac{3}{2}} \sqrt{- 4 \sqrt{3} + 8} \sqrt{x^{3} + 1}} + \frac{4 \sqrt [4]{3} \sqrt{\frac{x^{2} - x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (e - 2^{\frac{2}{3}} f\right ) \left (x + 1\right ) \Pi \left (\frac{\left (- 2^{\frac{2}{3}} + 1 + \sqrt{3}\right )^{2}}{\left (-1 + 2^{\frac{2}{3}} + \sqrt{3}\right )^{2}}; \operatorname{asin}{\left (\frac{- x - 1 + \sqrt{3}}{x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{\sqrt{\frac{x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- 4 \sqrt{3} + 7} \sqrt{x^{3} + 1} \left (- 2^{\frac{2}{3}} + 1 + \sqrt{3}\right ) \left (- \sqrt{3} - 2^{\frac{2}{3}} + 1\right )} - \frac{2 \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} - x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (e - f \left (1 + \sqrt{3}\right )\right ) \left (x + 1\right ) F\left (\operatorname{asin}{\left (\frac{x - \sqrt{3} + 1}{x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{3 \sqrt{\frac{x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{x^{3} + 1} \left (- 2^{\frac{2}{3}} + 1 + \sqrt{3}\right )} \]

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(-4*sqrt(3) + 8)*sqrt(-(-x - 1 + sq
rt(3))**2/(x + 1 + sqrt(3))**2 + 1)/(6*sqrt(-1 + 2**(1/3))*sqrt((-x - 1 + sqrt(3
))**2/(x + 1 + sqrt(3))**2 - 4*sqrt(3) + 7)))/(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) + sq
rt(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 + sq
rt(3))**2)*sqrt(sqrt(3) + 2)*(e - f*(1 + sqrt(3)))*(x + 1)*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 = 0.710208, size = 340, normalized size = 2.14 \[ \frac{2 \sqrt [6]{2} \sqrt{\frac{i (x+1)}{\sqrt{3}+3 i}} \left (f \sqrt{2 i x+\sqrt{3}-i} \left (\left (3 \sqrt [3]{2}+4 i \sqrt{3}+i \sqrt [3]{2} \sqrt{3}\right ) x+i \sqrt [3]{2} \sqrt{3}-2 i \sqrt{3}-3 \sqrt [3]{2}-6\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{-2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right )-2 \sqrt{3} \sqrt{-2 i x+\sqrt{3}+i} \sqrt{x^2-x+1} \left (\sqrt [3]{2} e-2 f\right ) \Pi \left (\frac{2 \sqrt{3}}{i+2 i 2^{2/3}+\sqrt{3}};\sin ^{-1}\left (\frac{\sqrt{-2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right )\right )}{\sqrt{3} \left (i+2 i 2^{2/3}+\sqrt{3}\right ) \sqrt{-2 i x+\sqrt{3}+i} \sqrt{x^3+1}} \]

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])]*(f*Sqrt[-I + Sqrt[3] + (2*I)*x]*(-6
- 3*2^(1/3) - (2*I)*Sqrt[3] + I*2^(1/3)*Sqrt[3] + (3*2^(1/3) + (4*I)*Sqrt[3] +
I*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]*S
qrt[1 + x^3])

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Maple [B]  time = 0.038, size = 264, normalized size = 1.7 \[ 2\,{\frac{f \left ( 3/2-i/2\sqrt{3} \right ) }{\sqrt{{x}^{3}+1}}\sqrt{{\frac{1+x}{3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x-1/2-i/2\sqrt{3}}{-3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x-1/2+i/2\sqrt{3}}{-3/2+i/2\sqrt{3}}}}{\it EllipticF} \left ( \sqrt{{\frac{1+x}{3/2-i/2\sqrt{3}}}},\sqrt{{\frac{-3/2+i/2\sqrt{3}}{-3/2-i/2\sqrt{3}}}} \right ) }+2\,{\frac{ \left ( e-{2}^{2/3}f \right ) \left ( 3/2-i/2\sqrt{3} \right ) }{\sqrt{{x}^{3}+1} \left ({2}^{2/3}-1 \right ) }\sqrt{{\frac{1+x}{3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x-1/2-i/2\sqrt{3}}{-3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x-1/2+i/2\sqrt{3}}{-3/2+i/2\sqrt{3}}}}{\it EllipticPi} \left ( \sqrt{{\frac{1+x}{3/2-i/2\sqrt{3}}}},{\frac{-3/2+i/2\sqrt{3}}{{2}^{2/3}-1}},\sqrt{{\frac{-3/2+i/2\sqrt{3}}{-3/2-i/2\sqrt{3}}}} \right ) } \]

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*f*(3/2-1/2*I*3^(1/2))*((1+x)/(3/2-1/2*I*3^(1/2)))^(1/2)*((x-1/2-1/2*I*3^(1/2))
/(-3/2-1/2*I*3^(1/2)))^(1/2)*((x-1/2+1/2*I*3^(1/2))/(-3/2+1/2*I*3^(1/2)))^(1/2)/
(x^3+1)^(1/2)*EllipticF(((1+x)/(3/2-1/2*I*3^(1/2)))^(1/2),((-3/2+1/2*I*3^(1/2))/
(-3/2-1/2*I*3^(1/2)))^(1/2))+2*(e-2^(2/3)*f)*(3/2-1/2*I*3^(1/2))*((1+x)/(3/2-1/2
*I*3^(1/2)))^(1/2)*((x-1/2-1/2*I*3^(1/2))/(-3/2-1/2*I*3^(1/2)))^(1/2)*((x-1/2+1/
2*I*3^(1/2))/(-3/2+1/2*I*3^(1/2)))^(1/2)/(x^3+1)^(1/2)/(2^(2/3)-1)*EllipticPi(((
1+x)/(3/2-1/2*I*3^(1/2)))^(1/2),(-3/2+1/2*I*3^(1/2))/(2^(2/3)-1),((-3/2+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. \[ \int \frac{e + f x}{\sqrt{\left (x + 1\right ) \left (x^{2} - x + 1\right )} \left (x + 2^{\frac{2}{3}}\right )}\, dx \]

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