Optimal. Leaf size=120 \[ \frac{2 \sqrt{2+\sqrt{3}} f (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^3+1}}-\frac{2}{3} e \tanh ^{-1}\left (\sqrt{x^3+1}\right ) \]
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Rubi [A] time = 0.112841, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{2 \sqrt{2+\sqrt{3}} f (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^3+1}}-\frac{2}{3} e \tanh ^{-1}\left (\sqrt{x^3+1}\right ) \]
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 = 11.0902, size = 112, normalized size = 0.93 \[ - \frac{2 e \operatorname{atanh}{\left (\sqrt{x^{3} + 1} \right )}}{3} + \frac{2 \cdot 3^{\frac{3}{4}} f \sqrt{\frac{x^{2} - x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (x + 1\right ) F\left (\operatorname{asin}{\left (\frac{x - \sqrt{3} + 1}{x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{3 \sqrt{\frac{x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{x^{3} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((f*x+e)/x/(x**3+1)**(1/2),x)
[Out]
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Mathematica [A] time = 0.608211, size = 134, normalized size = 1.12 \[ -\frac{2}{3} e \tanh ^{-1}\left (\sqrt{x^3+1}\right )-\frac{2 f \left (\sqrt [3]{-1}-x\right ) \sqrt{\frac{x+1}{1+\sqrt [3]{-1}}} \sqrt{-\frac{(-1)^{2/3} \left (x+(-1)^{2/3}\right )}{1+\sqrt [3]{-1}}} F\left (\sin ^{-1}\left (\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}} \sqrt{x^3+1}} \]
Antiderivative was successfully verified.
[In] Integrate[(e + f*x)/(x*Sqrt[1 + x^3]),x]
[Out]
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Maple [A] time = 0.009, size = 129, normalized size = 1.1 \[ 2\,{\frac{f \left ( 3/2-i/2\sqrt{3} \right ) }{\sqrt{{x}^{3}+1}}\sqrt{{\frac{1+x}{3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x-1/2-i/2\sqrt{3}}{-3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x-1/2+i/2\sqrt{3}}{-3/2+i/2\sqrt{3}}}}{\it EllipticF} \left ( \sqrt{{\frac{1+x}{3/2-i/2\sqrt{3}}}},\sqrt{{\frac{-3/2+i/2\sqrt{3}}{-3/2-i/2\sqrt{3}}}} \right ) }-{\frac{2\,e}{3}{\it Artanh} \left ( \sqrt{{x}^{3}+1} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((f*x+e)/x/(x^3+1)^(1/2),x)
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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")
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Sympy [A] time = 5.34849, size = 42, normalized size = 0.35 \[ - \frac{2 e \operatorname{asinh}{\left (\frac{1}{x^{\frac{3}{2}}} \right )}}{3} + \frac{f x \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{1}{2} \\ \frac{4}{3} \end{matrix}\middle |{x^{3} e^{i \pi }} \right )}}{3 \Gamma \left (\frac{4}{3}\right )} \]
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]