Optimal. Leaf size=137 \[ \frac{2}{3} e \tan ^{-1}\left (\sqrt{x^3-1}\right )-\frac{2 \sqrt{2-\sqrt{3}} f (1-x) \sqrt{\frac{x^2+x+1}{\left (-x-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{-\frac{1-x}{\left (-x-\sqrt{3}+1\right )^2}} \sqrt{x^3-1}} \]
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Rubi [A] time = 0.0512827, antiderivative size = 137, 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, Rules used = {1832, 266, 63, 203, 12, 219} \[ \frac{2}{3} e \tan ^{-1}\left (\sqrt{x^3-1}\right )-\frac{2 \sqrt{2-\sqrt{3}} f (1-x) \sqrt{\frac{x^2+x+1}{\left (-x-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{-\frac{1-x}{\left (-x-\sqrt{3}+1\right )^2}} \sqrt{x^3-1}} \]
Antiderivative was successfully verified.
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Rule 1832
Rule 266
Rule 63
Rule 203
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 = 0.0269654, size = 52, normalized size = 0.38 \[ \frac{2}{3} e \tan ^{-1}\left (\sqrt{x^3-1}\right )+\frac{f x \sqrt{1-x^3} \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{4}{3};x^3\right )}{\sqrt{x^3-1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 129, normalized size = 0.9 \begin{align*} 2\,{\frac{f \left ( -3/2-i/2\sqrt{3} \right ) }{\sqrt{{x}^{3}-1}}\sqrt{{\frac{x-1}{-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{x-1}{-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}\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 = 2.41762, size = 60, normalized size = 0.44 \begin{align*} e \left (\begin{cases} \frac{2 i \operatorname{acosh}{\left (\frac{1}{x^{\frac{3}{2}}} \right )}}{3} & \text{for}\: \frac{1}{\left |{x^{3}}\right |} > 1 \\- \frac{2 \operatorname{asin}{\left (\frac{1}{x^{\frac{3}{2}}} \right )}}{3} & \text{otherwise} \end{cases}\right ) - \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}} \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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