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

Optimal. Leaf size=261 \[ \frac{2 \sqrt{2-\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x-\sqrt{3}+1\right )^2}} \left (c-\left (1+\sqrt{3}\right ) d\right ) F\left (\sin ^{-1}\left (\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{-x^3-1}}-\frac{2 d \sqrt{-x^3-1}}{x-\sqrt{3}+1}+\frac{\sqrt [4]{3} \sqrt{2+\sqrt{3}} d (x+1) \sqrt{\frac{x^2-x+1}{\left (x-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{\sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{-x^3-1}} \]

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

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Rubi [A]  time = 0.200216, antiderivative size = 261, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{2 \sqrt{2-\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x-\sqrt{3}+1\right )^2}} \left (c-\left (1+\sqrt{3}\right ) d\right ) F\left (\sin ^{-1}\left (\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{-x^3-1}}-\frac{2 d \sqrt{-x^3-1}}{x-\sqrt{3}+1}+\frac{\sqrt [4]{3} \sqrt{2+\sqrt{3}} d (x+1) \sqrt{\frac{x^2-x+1}{\left (x-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{\sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{-x^3-1}} \]

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 - (1 + 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*
Sqrt[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 = 12.9329, size = 226, normalized size = 0.87 \[ - \frac{2 d \sqrt{- x^{3} - 1}}{x - \sqrt{3} + 1} + \frac{\sqrt [4]{3} d \sqrt{\frac{x^{2} - x + 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (x + 1\right ) E\left (\operatorname{asin}{\left (\frac{x + 1 + \sqrt{3}}{x - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{\sqrt{\frac{- x - 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \sqrt{- x^{3} - 1}} + \frac{2 \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} - x + 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (c - d \left (1 + \sqrt{3}\right )\right ) \left (x + 1\right ) F\left (\operatorname{asin}{\left (\frac{x + 1 + \sqrt{3}}{x - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{3 \sqrt{\frac{- x - 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \sqrt{- x^{3} - 1}} \]

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 - sqr
t(3) + 1)**2)*sqrt(sqrt(3) + 2)*(x + 1)*elliptic_e(asin((x + 1 + sqrt(3))/(x - s
qrt(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)*(c
- d*(1 + sqrt(3)))*(x + 1)*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 = 0.181432, size = 152, normalized size = 0.58 \[ \frac{2 \sqrt [6]{-1} \sqrt{-(-1)^{5/6}+i x} \sqrt{-\sqrt [3]{-1} x^2-(-1)^{2/3} x+1} \left (\sqrt{3} \left ((-1)^{2/3} c-d\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{-\sqrt [6]{-1} \left (x+(-1)^{2/3}\right )}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )+3 \sqrt [6]{-1} d E\left (\sin ^{-1}\left (\frac{\sqrt{-\sqrt [6]{-1} \left (x+(-1)^{2/3}\right )}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )\right )}{3^{3/4} \sqrt{-x^3-1}} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [A]  time = 0.007, size = 269, normalized size = 1. \[{-{\frac{2\,i}{3}}c\sqrt{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{1+x}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticF} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}-1}}}}-{{\frac{2\,i}{3}}d\sqrt{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{1+x}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}} \left ( \left ({\frac{3}{2}}+{\frac{i}{2}}\sqrt{3} \right ){\it EllipticE} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ) -{\it EllipticF} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ) \right ){\frac{1}{\sqrt{-{x}^{3}-1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)/(-x^3-1)^(1/2),x)

[Out]

-2/3*I*c*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*d*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))*EllipticE(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))-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. \[ \int \frac{d x + c}{\sqrt{-x^{3} - 1}}\,{d x} \]

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 = 1.75819, size = 66, normalized size = 0.25 \[ - \frac{i c x \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{1}{2} \\ \frac{4}{3} \end{matrix}\middle |{x^{3} e^{i \pi }} \right )}}{3 \Gamma \left (\frac{4}{3}\right )} - \frac{i d x^{2} \Gamma \left (\frac{2}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle |{x^{3} e^{i \pi }} \right )}}{3 \Gamma \left (\frac{5}{3}\right )} \]

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*exp_polar(I*pi))/(3*gamma(4/3))
- I*d*x**2*gamma(2/3)*hyper((1/2, 2/3), (5/3,), x**3*exp_polar(I*pi))/(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)