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 = 0.0492872, 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, Rules used = {1832, 266, 63, 207, 12, 218} \[ \frac{2 \sqrt{2+\sqrt{3}} f (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^3+1}}-\frac{2}{3} e \tanh ^{-1}\left (\sqrt{x^3+1}\right ) \]
Antiderivative was successfully verified.
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Rule 1832
Rule 266
Rule 63
Rule 207
Rule 12
Rule 218
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}{x \sqrt{1+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 \tanh ^{-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 = 0.0169898, size = 34, normalized size = 0.28 \[ f x \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{4}{3};-x^3\right )-\frac{2}{3} e \tanh ^{-1}\left (\sqrt{x^3+1}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 129, normalized size = 1.1 \begin{align*} 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 ) } \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 = 2.41032, size = 42, normalized size = 0.35 \begin{align*} - \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 )} \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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