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 = 0.0535521, 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, Rules used = {1832, 266, 63, 204, 12, 219} \[ \frac{2}{3} e \tan ^{-1}\left (\sqrt{-x^3-1}\right )+\frac{2 \sqrt{2-\sqrt{3}} f (x+1) \sqrt{\frac{x^2-x+1}{\left (x-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{-x^3-1}} \]
Antiderivative was successfully verified.
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Rule 1832
Rule 266
Rule 63
Rule 204
Rule 12
Rule 219
Rubi steps
\begin{align*} \int \frac{e+f x}{x \sqrt{-1-x^3}} \, dx &=e \int \frac{1}{x \sqrt{-1-x^3}} \, dx+\int \frac{f}{\sqrt{-1-x^3}} \, dx\\ &=\frac{1}{3} e \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1-x} x} \, dx,x,x^3\right )+f \int \frac{1}{\sqrt{-1-x^3}} \, dx\\ &=\frac{2 \sqrt{2-\sqrt{3}} f (1+x) \sqrt{\frac{1-x+x^2}{\left (1-\sqrt{3}+x\right )^2}} F\left (\sin ^{-1}\left (\frac{1+\sqrt{3}+x}{1-\sqrt{3}+x}\right )|-7+4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{-\frac{1+x}{\left (1-\sqrt{3}+x\right )^2}} \sqrt{-1-x^3}}-\frac{1}{3} (2 e) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,\sqrt{-1-x^3}\right )\\ &=\frac{2}{3} e \tan ^{-1}\left (\sqrt{-1-x^3}\right )+\frac{2 \sqrt{2-\sqrt{3}} f (1+x) \sqrt{\frac{1-x+x^2}{\left (1-\sqrt{3}+x\right )^2}} F\left (\sin ^{-1}\left (\frac{1+\sqrt{3}+x}{1-\sqrt{3}+x}\right )|-7+4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{-\frac{1+x}{\left (1-\sqrt{3}+x\right )^2}} \sqrt{-1-x^3}}\\ \end{align*}
Mathematica [C] time = 0.0319325, size = 56, normalized size = 0.43 \[ \frac{2}{3} e \tan ^{-1}\left (\sqrt{-x^3-1}\right )+\frac{f x \sqrt{x^3+1} \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{4}{3};-x^3\right )}{\sqrt{-x^3-1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 122, normalized size = 0.9 \begin{align*}{-{\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 ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f x + e}{\sqrt{-x^{3} - 1} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-x^{3} - 1}{\left (f x + e\right )}}{x^{4} + x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.55529, size = 46, normalized size = 0.35 \begin{align*} \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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f x + e}{\sqrt{-x^{3} - 1} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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