Integrand size = 18, antiderivative size = 120 \[ \int \frac {e+f x}{x \sqrt {1+x^3}} \, dx=-\frac {2}{3} e \text {arctanh}\left (\sqrt {1+x^3}\right )+\frac {2 \sqrt {2+\sqrt {3}} f (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}} \]
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Time = 0.04 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1846, 272, 65, 213, 12, 224} \[ \int \frac {e+f x}{x \sqrt {1+x^3}} \, dx=\frac {2 \sqrt {2+\sqrt {3}} f (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}-\frac {2}{3} e \text {arctanh}\left (\sqrt {x^3+1}\right ) \]
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Rule 12
Rule 65
Rule 213
Rule 224
Rule 272
Rule 1846
Rubi steps \begin{align*} \text {integral}& = e \int \frac {1}{x \sqrt {1+x^3}} \, dx+\int \frac {f}{\sqrt {1+x^3}} \, dx \\ & = \frac {1}{3} e \text {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) \text {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*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.04 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.28 \[ \int \frac {e+f x}{x \sqrt {1+x^3}} \, dx=-\frac {2}{3} e \text {arctanh}\left (\sqrt {1+x^3}\right )+f x \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},-x^3\right ) \]
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Result contains higher order function than in optimal. Order 5 vs. order 4.
Time = 0.98 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.44
method | result | size |
meijerg | \(f x {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{3},\frac {1}{2};\frac {4}{3};-x^{3}\right )+\frac {e \left (-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {x^{3}+1}}{2}\right )+\left (-2 \ln \left (2\right )+3 \ln \left (x \right )\right ) \sqrt {\pi }\right )}{3 \sqrt {\pi }}\) | \(53\) |
default | \(\frac {2 f \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, F\left (\sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}-\frac {2 e \,\operatorname {arctanh}\left (\sqrt {x^{3}+1}\right )}{3}\) | \(129\) |
elliptic | \(\frac {2 f \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, F\left (\sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}-\frac {2 e \,\operatorname {arctanh}\left (\sqrt {x^{3}+1}\right )}{3}\) | \(129\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.25 \[ \int \frac {e+f x}{x \sqrt {1+x^3}} \, dx=\frac {1}{3} \, e \log \left (\frac {x^{3} - 2 \, \sqrt {x^{3} + 1} + 2}{x^{3}}\right ) + 2 \, f {\rm weierstrassPInverse}\left (0, -4, x\right ) \]
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Time = 1.26 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.35 \[ \int \frac {e+f x}{x \sqrt {1+x^3}} \, dx=- \frac {2 e \operatorname {asinh}{\left (\frac {1}{x^{\frac {3}{2}}} \right )}}{3} + \frac {f x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{2} \\ \frac {4}{3} \end {matrix}\middle | {x^{3} e^{i \pi }} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} \]
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\[ \int \frac {e+f x}{x \sqrt {1+x^3}} \, dx=\int { \frac {f x + e}{\sqrt {x^{3} + 1} x} \,d x } \]
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\[ \int \frac {e+f x}{x \sqrt {1+x^3}} \, dx=\int { \frac {f x + e}{\sqrt {x^{3} + 1} x} \,d x } \]
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Time = 20.50 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.72 \[ \int \frac {e+f x}{x \sqrt {1+x^3}} \, dx=\frac {\left (3+\sqrt {3}\,1{}\mathrm {i}\right )\,\sqrt {\frac {x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\left (f\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )-e\,\Pi \left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2};\mathrm {asin}\left (\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )\right )\,\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {\frac {1}{2}-x+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}}{\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \]
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