Optimal. Leaf size=131 \[ \frac{2}{3} e \tan ^{-1}\left (\sqrt{-x^3-1}\right )+\frac{2 \sqrt{2-\sqrt{3}} f (x+1) \sqrt{\frac{x^2-x+1}{\left (x-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{-x^3-1}} \]
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Rubi [A] time = 0.126055, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ \frac{2}{3} e \tan ^{-1}\left (\sqrt{-x^3-1}\right )+\frac{2 \sqrt{2-\sqrt{3}} f (x+1) \sqrt{\frac{x^2-x+1}{\left (x-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{-x^3-1}} \]
Antiderivative was successfully verified.
[In] Int[(e + f*x)/(x*Sqrt[-1 - x^3]),x]
[Out]
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Rubi in Sympy [A] time = 12.6552, size = 116, normalized size = 0.89 \[ \frac{2 e \operatorname{atan}{\left (\sqrt{- x^{3} - 1} \right )}}{3} + \frac{2 \cdot 3^{\frac{3}{4}} f \sqrt{\frac{x^{2} - x + 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (x + 1\right ) F\left (\operatorname{asin}{\left (\frac{x + 1 + \sqrt{3}}{x - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{3 \sqrt{\frac{- x - 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \sqrt{- x^{3} - 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((f*x+e)/x/(-x**3-1)**(1/2),x)
[Out]
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Mathematica [A] time = 0.623961, size = 138, normalized size = 1.05 \[ \frac{2}{3} e \tan ^{-1}\left (\sqrt{-x^3-1}\right )-\frac{2 f \left (\sqrt [3]{-1}-x\right ) \sqrt{\frac{x+1}{1+\sqrt [3]{-1}}} \sqrt{-\frac{(-1)^{2/3} \left (x+(-1)^{2/3}\right )}{1+\sqrt [3]{-1}}} F\left (\sin ^{-1}\left (\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}} \sqrt{-x^3-1}} \]
Antiderivative was successfully verified.
[In] Integrate[(e + f*x)/(x*Sqrt[-1 - x^3]),x]
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Maple [A] time = 0.008, size = 122, normalized size = 0.9 \[{-{\frac{2\,i}{3}}f\sqrt{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{1+x}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticF} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}-1}}}}+{\frac{2\,e}{3}\arctan \left ( \sqrt{-{x}^{3}-1} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((f*x+e)/x/(-x^3-1)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{f x + e}{\sqrt{-x^{3} - 1} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x + e)/(sqrt(-x^3 - 1)*x),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{f x + e}{\sqrt{-x^{3} - 1} x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x + e)/(sqrt(-x^3 - 1)*x),x, algorithm="fricas")
[Out]
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Sympy [A] time = 5.39559, size = 46, normalized size = 0.35 \[ \frac{2 i e \operatorname{asinh}{\left (\frac{1}{x^{\frac{3}{2}}} \right )}}{3} - \frac{i f x \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{1}{2} \\ \frac{4}{3} \end{matrix}\middle |{x^{3} e^{i \pi }} \right )}}{3 \Gamma \left (\frac{4}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x+e)/x/(-x**3-1)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{f x + e}{\sqrt{-x^{3} - 1} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x + e)/(sqrt(-x^3 - 1)*x),x, algorithm="giac")
[Out]