Optimal. Leaf size=173 \[ \frac{\left (e-\sqrt{3} f-f\right ) \tan ^{-1}\left (\frac{\sqrt{3+2 \sqrt{3}} (x+1)}{\sqrt{x^3+1}}\right )}{\sqrt{3 \left (3+2 \sqrt{3}\right )}}+\frac{\sqrt{2+\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} \left (e-\left (1-\sqrt{3}\right ) f\right ) F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{3^{3/4} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^3+1}} \]
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Rubi [A] time = 0.496004, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16 \[ \frac{\left (e-\left (1+\sqrt{3}\right ) f\right ) \tan ^{-1}\left (\frac{\sqrt{3+2 \sqrt{3}} (x+1)}{\sqrt{x^3+1}}\right )}{\sqrt{3 \left (3+2 \sqrt{3}\right )}}+\frac{\sqrt{2+\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} \left (e-\left (1-\sqrt{3}\right ) f\right ) F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{3^{3/4} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^3+1}} \]
Antiderivative was successfully verified.
[In] Int[(e + f*x)/((1 + Sqrt[3] + x)*Sqrt[1 + x^3]),x]
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Rubi in Sympy [A] time = 13.7666, size = 82, normalized size = 0.47 \[ \frac{2 \tilde{\infty } \sqrt{\frac{x^{2} - x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \left (e + f\right ) \left (x + 1\right ) F\left (\operatorname{asin}{\left (\frac{x - \sqrt{3} + 1}{x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{\sqrt{\frac{x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{x^{3} + 1}} \]
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)
[Out]
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Mathematica [C] time = 0.709146, size = 291, normalized size = 1.68 \[ \frac{2 \sqrt{\frac{2}{3}} \sqrt{\frac{i (x+1)}{\sqrt{3}+3 i}} \left (2 \sqrt{-2 i x+\sqrt{3}+i} \sqrt{x^2-x+1} \left (\left (3+\sqrt{3}\right ) f-\sqrt{3} e\right ) \Pi \left (\frac{2 \sqrt{3}}{3 i+(1+2 i) \sqrt{3}};\sin ^{-1}\left (\frac{\sqrt{-2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right )+3 f \sqrt{2 i x+\sqrt{3}-i} \left (\left ((1+2 i)+i \sqrt{3}\right ) x-\sqrt{3}-(2+i)\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{-2 i x+\sqrt{3}+i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{3 i+\sqrt{3}}\right )\right )}{\left (3 i+(1+2 i) \sqrt{3}\right ) \sqrt{-2 i x+\sqrt{3}+i} \sqrt{x^3+1}} \]
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 = 0.038, size = 260, normalized size = 1.5 \[ 2\,{\frac{f \left ( 3/2-i/2\sqrt{3} \right ) }{\sqrt{{x}^{3}+1}}\sqrt{{\frac{1+x}{3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x-1/2-i/2\sqrt{3}}{-3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x-1/2+i/2\sqrt{3}}{-3/2+i/2\sqrt{3}}}}{\it EllipticF} \left ( \sqrt{{\frac{1+x}{3/2-i/2\sqrt{3}}}},\sqrt{{\frac{-3/2+i/2\sqrt{3}}{-3/2-i/2\sqrt{3}}}} \right ) }+{\frac{ \left ( 2\,e-2\,f-2\,f\sqrt{3} \right ) \left ({\frac{3}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{3}\sqrt{{\frac{1+x}{{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}}\sqrt{{\frac{1}{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}} \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) }}\sqrt{{\frac{1}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}} \left ( x-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) }}{\it EllipticPi} \left ( \sqrt{{\frac{1+x}{{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}},{\frac{ \left ( -{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{3}},\sqrt{{\frac{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{{x}^{3}+1}}}} \]
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")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{e + f x}{\sqrt{\left (x + 1\right ) \left (x^{2} - x + 1\right )} \left (x + 1 + \sqrt{3}\right )}\, dx \]
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]