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

Optimal. Leaf size=275 \[ -\frac{2 \sqrt{2-\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x-\sqrt{3}+1\right )^2}} \left (c+\sqrt{3} d+d\right ) F\left (\sin ^{-1}\left (\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{-\frac{1-x}{\left (-x-\sqrt{3}+1\right )^2}} \sqrt{x^3-1}}-\frac{2 d \sqrt{x^3-1}}{-x-\sqrt{3}+1}+\frac{\sqrt [4]{3} \sqrt{2+\sqrt{3}} d (1-x) \sqrt{\frac{x^2+x+1}{\left (-x-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{\sqrt{-\frac{1-x}{\left (-x-\sqrt{3}+1\right )^2}} \sqrt{x^3-1}} \]

[Out]

(-2*d*Sqrt[-1 + x^3])/(1 - Sqrt[3] - x) + (3^(1/4)*Sqrt[2 + Sqrt[3]]*d*(1 - x)*S
qrt[(1 + x + x^2)/(1 - Sqrt[3] - x)^2]*EllipticE[ArcSin[(1 + Sqrt[3] - x)/(1 - S
qrt[3] - x)], -7 + 4*Sqrt[3]])/(Sqrt[-((1 - x)/(1 - Sqrt[3] - x)^2)]*Sqrt[-1 + x
^3]) - (2*Sqrt[2 - Sqrt[3]]*(c + d + Sqrt[3]*d)*(1 - x)*Sqrt[(1 + x + x^2)/(1 -
Sqrt[3] - x)^2]*EllipticF[ArcSin[(1 + Sqrt[3] - x)/(1 - Sqrt[3] - x)], -7 + 4*Sq
rt[3]])/(3^(1/4)*Sqrt[-((1 - x)/(1 - Sqrt[3] - x)^2)]*Sqrt[-1 + x^3])

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Rubi [A]  time = 0.208419, antiderivative size = 275, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{2 \sqrt{2-\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x-\sqrt{3}+1\right )^2}} \left (c+\sqrt{3} d+d\right ) F\left (\sin ^{-1}\left (\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{-\frac{1-x}{\left (-x-\sqrt{3}+1\right )^2}} \sqrt{x^3-1}}-\frac{2 d \sqrt{x^3-1}}{-x-\sqrt{3}+1}+\frac{\sqrt [4]{3} \sqrt{2+\sqrt{3}} d (1-x) \sqrt{\frac{x^2+x+1}{\left (-x-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{\sqrt{-\frac{1-x}{\left (-x-\sqrt{3}+1\right )^2}} \sqrt{x^3-1}} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)/Sqrt[-1 + x^3],x]

[Out]

(-2*d*Sqrt[-1 + x^3])/(1 - Sqrt[3] - x) + (3^(1/4)*Sqrt[2 + Sqrt[3]]*d*(1 - x)*S
qrt[(1 + x + x^2)/(1 - Sqrt[3] - x)^2]*EllipticE[ArcSin[(1 + Sqrt[3] - x)/(1 - S
qrt[3] - x)], -7 + 4*Sqrt[3]])/(Sqrt[-((1 - x)/(1 - Sqrt[3] - x)^2)]*Sqrt[-1 + x
^3]) - (2*Sqrt[2 - Sqrt[3]]*(c + d + Sqrt[3]*d)*(1 - x)*Sqrt[(1 + x + x^2)/(1 -
Sqrt[3] - x)^2]*EllipticF[ArcSin[(1 + Sqrt[3] - x)/(1 - Sqrt[3] - x)], -7 + 4*Sq
rt[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 = 13.1052, size = 218, normalized size = 0.79 \[ - \frac{2 d \sqrt{x^{3} - 1}}{- x - \sqrt{3} + 1} + \frac{\sqrt [4]{3} d \sqrt{\frac{x^{2} + x + 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (- x + 1\right ) E\left (\operatorname{asin}{\left (\frac{- x + 1 + \sqrt{3}}{- x - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{\sqrt{\frac{x - 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \sqrt{x^{3} - 1}} - \frac{2 \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} + x + 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (- x + 1\right ) \left (c + d + \sqrt{3} d\right ) F\left (\operatorname{asin}{\left (\frac{- x + 1 + \sqrt{3}}{- x - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{3 \sqrt{\frac{x - 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \sqrt{x^{3} - 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)/(x**3-1)**(1/2),x)

[Out]

-2*d*sqrt(x**3 - 1)/(-x - sqrt(3) + 1) + 3**(1/4)*d*sqrt((x**2 + x + 1)/(-x - sq
rt(3) + 1)**2)*sqrt(sqrt(3) + 2)*(-x + 1)*elliptic_e(asin((-x + 1 + sqrt(3))/(-x
- sqrt(3) + 1)), -7 + 4*sqrt(3))/(sqrt((x - 1)/(-x - sqrt(3) + 1)**2)*sqrt(x**3
- 1)) - 2*3**(3/4)*sqrt((x**2 + x + 1)/(-x - sqrt(3) + 1)**2)*sqrt(-sqrt(3) + 2
)*(-x + 1)*(c + d + sqrt(3)*d)*elliptic_f(asin((-x + 1 + sqrt(3))/(-x - sqrt(3)
+ 1)), -7 + 4*sqrt(3))/(3*sqrt((x - 1)/(-x - sqrt(3) + 1)**2)*sqrt(x**3 - 1))

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Mathematica [C]  time = 0.133674, size = 119, normalized size = 0.43 \[ \frac{2 i \sqrt{(-1)^{5/6} (x-1)} \sqrt{x^2+x+1} \left (\sqrt{3} (c+d) F\left (\sin ^{-1}\left (\frac{\sqrt{-i x-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )-3 \sqrt [6]{-1} d E\left (\sin ^{-1}\left (\frac{\sqrt{-i x-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )\right )}{3^{3/4} \sqrt{x^3-1}} \]

Warning: Unable to verify antiderivative.

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

[Out]

((2*I)*Sqrt[(-1)^(5/6)*(-1 + x)]*Sqrt[1 + x + x^2]*(-3*(-1)^(1/6)*d*EllipticE[Ar
cSin[Sqrt[-(-1)^(5/6) - I*x]/3^(1/4)], (-1)^(1/3)] + Sqrt[3]*(c + d)*EllipticF[A
rcSin[Sqrt[-(-1)^(5/6) - I*x]/3^(1/4)], (-1)^(1/3)]))/(3^(3/4)*Sqrt[-1 + x^3])

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Maple [A]  time = 0.006, size = 291, normalized size = 1.1 \[ 2\,{\frac{c \left ( -3/2-i/2\sqrt{3} \right ) }{\sqrt{{x}^{3}-1}}\sqrt{{\frac{-1+x}{-3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2-i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2+i/2\sqrt{3}}{3/2+i/2\sqrt{3}}}}{\it EllipticF} \left ( \sqrt{{\frac{-1+x}{-3/2-i/2\sqrt{3}}}},\sqrt{{\frac{3/2+i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}} \right ) }+2\,{\frac{d \left ( -3/2-i/2\sqrt{3} \right ) }{\sqrt{{x}^{3}-1}}\sqrt{{\frac{-1+x}{-3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2-i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2+i/2\sqrt{3}}{3/2+i/2\sqrt{3}}}} \left ( \left ( 3/2-i/2\sqrt{3} \right ){\it EllipticE} \left ( \sqrt{{\frac{-1+x}{-3/2-i/2\sqrt{3}}}},\sqrt{{\frac{3/2+i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}} \right ) + \left ( -1/2+i/2\sqrt{3} \right ){\it EllipticF} \left ( \sqrt{{\frac{-1+x}{-3/2-i/2\sqrt{3}}}},\sqrt{{\frac{3/2+i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}} \right ) \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)/(x^3-1)^(1/2),x)

[Out]

2*c*(-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*d*(-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)*((3/2-1/2*I*3^(1/2))*EllipticE(((-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))
+(-1/2+1/2*I*3^(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)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{d x + c}{\sqrt{x^{3} - 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((d*x + c)/sqrt(x^3 - 1), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{d x + c}{\sqrt{x^{3} - 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((d*x + c)/sqrt(x^3 - 1), x)

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Sympy [A]  time = 1.8178, size = 56, normalized size = 0.2 \[ - \frac{i c x \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{1}{2} \\ \frac{4}{3} \end{matrix}\middle |{x^{3}} \right )}}{3 \Gamma \left (\frac{4}{3}\right )} - \frac{i d x^{2} \Gamma \left (\frac{2}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle |{x^{3}} \right )}}{3 \Gamma \left (\frac{5}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x+c)/(x**3-1)**(1/2),x)

[Out]

-I*c*x*gamma(1/3)*hyper((1/3, 1/2), (4/3,), x**3)/(3*gamma(4/3)) - I*d*x**2*gamm
a(2/3)*hyper((1/2, 2/3), (5/3,), x**3)/(3*gamma(5/3))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{d x + c}{\sqrt{x^{3} - 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((d*x + c)/sqrt(x^3 - 1), x)