Optimal. Leaf size=183 \[ \frac{\left (e-\left (1+\sqrt{3}\right ) f\right ) \tanh ^{-1}\left (\frac{\sqrt{3+2 \sqrt{3}} (x+1)}{\sqrt{-x^3-1}}\right )}{\sqrt{3 \left (3+2 \sqrt{3}\right )}}+\frac{\sqrt{2-\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x-\sqrt{3}+1\right )^2}} \left (e-\left (1-\sqrt{3}\right ) f\right ) F\left (\sin ^{-1}\left (\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{3^{3/4} \sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{-x^3-1}} \]
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Rubi [A] time = 0.503347, antiderivative size = 183, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148 \[ \frac{\left (e-\left (1+\sqrt{3}\right ) f\right ) \tanh ^{-1}\left (\frac{\sqrt{3+2 \sqrt{3}} (x+1)}{\sqrt{-x^3-1}}\right )}{\sqrt{3 \left (3+2 \sqrt{3}\right )}}+\frac{\sqrt{2-\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x-\sqrt{3}+1\right )^2}} \left (e-\left (1-\sqrt{3}\right ) f\right ) F\left (\sin ^{-1}\left (\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{3^{3/4} \sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{-x^3-1}} \]
Antiderivative was successfully verified.
[In] Int[(e + f*x)/((1 + Sqrt[3] + x)*Sqrt[-1 - x^3]),x]
[Out]
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Rubi in Sympy [A] time = 44.4323, size = 252, normalized size = 1.38 \[ \frac{\sqrt [4]{3} \sqrt{\frac{x^{2} - x + 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \left (e - f \left (1 + \sqrt{3}\right )\right ) \left (x + 1\right ) \operatorname{atanh}{\left (\frac{\sqrt{1 - \frac{\left (x + 1 + \sqrt{3}\right )^{2}}{\left (- x - 1 + \sqrt{3}\right )^{2}}} \left (\sqrt{3} + 2\right )}{\sqrt{4 \sqrt{3} + 7 + \frac{\left (x + 1 + \sqrt{3}\right )^{2}}{\left (- x - 1 + \sqrt{3}\right )^{2}}}} \right )}}{3 \sqrt{\frac{- x - 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \sqrt{\sqrt{3} + 2} \sqrt{- x^{3} - 1}} + \frac{\sqrt [4]{3} \sqrt{\frac{x^{2} - x + 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (x + 1\right ) \left (e - f + \sqrt{3} f\right ) F\left (\operatorname{asin}{\left (\frac{x + 1 + \sqrt{3}}{x - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{3 \sqrt{\frac{- x - 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \sqrt{- x^{3} - 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((f*x+e)/(1+x+3**(1/2))/(-x**3-1)**(1/2),x)
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Mathematica [C] time = 0.727635, size = 293, normalized size = 1.6 \[ \frac{2 \sqrt{\frac{2}{3}} \sqrt{\frac{i (x+1)}{\sqrt{3}+3 i}} \left (2 \sqrt{-2 i x+\sqrt{3}+i} \sqrt{x^2-x+1} \left (\left (3+\sqrt{3}\right ) f-\sqrt{3} e\right ) \Pi \left (\frac{2 \sqrt{3}}{3 i+(1+2 i) \sqrt{3}};\sin ^{-1}\left (\frac{\sqrt{-2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right )+3 f \sqrt{2 i x+\sqrt{3}-i} \left (\left ((1+2 i)+i \sqrt{3}\right ) x-\sqrt{3}-(2+i)\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{-2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right )\right )}{\left (3 i+(1+2 i) \sqrt{3}\right ) \sqrt{-2 i x+\sqrt{3}+i} \sqrt{-x^3-1}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(e + f*x)/((1 + Sqrt[3] + x)*Sqrt[-1 - x^3]),x]
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Maple [A] time = 0.035, size = 258, normalized size = 1.4 \[{-{\frac{2\,i}{3}}f\sqrt{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{1+x}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticF} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}-1}}}}-{\frac{{\frac{2\,i}{3}} \left ( e-f-f\sqrt{3} \right ) \sqrt{3}}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}+\sqrt{3}}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{1+x}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},{\frac{i\sqrt{3}}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}+\sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}-1}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((f*x+e)/(1+x+3^(1/2))/(-x^3-1)^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{f x + e}{\sqrt{-x^{3} - 1}{\left (x + \sqrt{3} + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x + e)/(sqrt(-x^3 - 1)*(x + sqrt(3) + 1)),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{f x + e}{\sqrt{-x^{3} - 1}{\left (x + \sqrt{3} + 1\right )}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x + e)/(sqrt(-x^3 - 1)*(x + sqrt(3) + 1)),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{e + f x}{\sqrt{- \left (x + 1\right ) \left (x^{2} - x + 1\right )} \left (x + 1 + \sqrt{3}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x+e)/(1+x+3**(1/2))/(-x**3-1)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{f x + e}{\sqrt{-x^{3} - 1}{\left (x + \sqrt{3} + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x + e)/(sqrt(-x^3 - 1)*(x + sqrt(3) + 1)),x, algorithm="giac")
[Out]