Integrand size = 15, antiderivative size = 282 \[ \int \frac {x^7}{\sqrt {-1-x^3}} \, dx=\frac {20}{91} x^2 \sqrt {-1-x^3}-\frac {2}{13} x^5 \sqrt {-1-x^3}-\frac {80 \sqrt {-1-x^3}}{91 \left (1-\sqrt {3}+x\right )}+\frac {40 \sqrt [4]{3} \sqrt {2+\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1-\sqrt {3}+x\right )^2}} E\left (\arcsin \left (\frac {1+\sqrt {3}+x}{1-\sqrt {3}+x}\right )|-7+4 \sqrt {3}\right )}{91 \sqrt {-\frac {1+x}{\left (1-\sqrt {3}+x\right )^2}} \sqrt {-1-x^3}}-\frac {80 \sqrt {2} (1+x) \sqrt {\frac {1-x+x^2}{\left (1-\sqrt {3}+x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}+x}{1-\sqrt {3}+x}\right ),-7+4 \sqrt {3}\right )}{91 \sqrt [4]{3} \sqrt {-\frac {1+x}{\left (1-\sqrt {3}+x\right )^2}} \sqrt {-1-x^3}} \]
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Time = 0.07 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {327, 310, 225, 1893} \[ \int \frac {x^7}{\sqrt {-1-x^3}} \, dx=-\frac {80 \sqrt {2} (x+1) \sqrt {\frac {x^2-x+1}{\left (x-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x+\sqrt {3}+1}{x-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{91 \sqrt [4]{3} \sqrt {-\frac {x+1}{\left (x-\sqrt {3}+1\right )^2}} \sqrt {-x^3-1}}+\frac {40 \sqrt [4]{3} \sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x-\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {x+\sqrt {3}+1}{x-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{91 \sqrt {-\frac {x+1}{\left (x-\sqrt {3}+1\right )^2}} \sqrt {-x^3-1}}-\frac {80 \sqrt {-x^3-1}}{91 \left (x-\sqrt {3}+1\right )}-\frac {2}{13} \sqrt {-x^3-1} x^5+\frac {20}{91} \sqrt {-x^3-1} x^2 \]
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Rule 225
Rule 310
Rule 327
Rule 1893
Rubi steps \begin{align*} \text {integral}& = -\frac {2}{13} x^5 \sqrt {-1-x^3}-\frac {10}{13} \int \frac {x^4}{\sqrt {-1-x^3}} \, dx \\ & = \frac {20}{91} x^2 \sqrt {-1-x^3}-\frac {2}{13} x^5 \sqrt {-1-x^3}+\frac {40}{91} \int \frac {x}{\sqrt {-1-x^3}} \, dx \\ & = \frac {20}{91} x^2 \sqrt {-1-x^3}-\frac {2}{13} x^5 \sqrt {-1-x^3}+\frac {40}{91} \int \frac {1+\sqrt {3}+x}{\sqrt {-1-x^3}} \, dx-\frac {1}{91} \left (40 \left (1+\sqrt {3}\right )\right ) \int \frac {1}{\sqrt {-1-x^3}} \, dx \\ & = \frac {20}{91} x^2 \sqrt {-1-x^3}-\frac {2}{13} x^5 \sqrt {-1-x^3}-\frac {80 \sqrt {-1-x^3}}{91 \left (1-\sqrt {3}+x\right )}+\frac {40 \sqrt [4]{3} \sqrt {2+\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1-\sqrt {3}+x\right )^2}} E\left (\sin ^{-1}\left (\frac {1+\sqrt {3}+x}{1-\sqrt {3}+x}\right )|-7+4 \sqrt {3}\right )}{91 \sqrt {-\frac {1+x}{\left (1-\sqrt {3}+x\right )^2}} \sqrt {-1-x^3}}-\frac {80 \sqrt {2} (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 )}{91 \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 = 10.02 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.20 \[ \int \frac {x^7}{\sqrt {-1-x^3}} \, dx=\frac {2 x^2 \left (-10-3 x^3+7 x^6+10 \sqrt {1+x^3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2}{3},\frac {5}{3},-x^3\right )\right )}{91 \sqrt {-1-x^3}} \]
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Result contains higher order function than in optimal. Order 5 vs. order 4.
Time = 3.92 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.06
method | result | size |
meijerg | \(-\frac {i x^{8} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{2},\frac {8}{3};\frac {11}{3};-x^{3}\right )}{8}\) | \(18\) |
risch | \(\frac {2 x^{2} \left (7 x^{3}-10\right ) \left (x^{3}+1\right )}{91 \sqrt {-x^{3}-1}}-\frac {80 i \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {1+x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \left (\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) E\left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )-F\left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )\right )}{273 \sqrt {-x^{3}-1}}\) | \(187\) |
default | \(-\frac {2 x^{5} \sqrt {-x^{3}-1}}{13}+\frac {20 x^{2} \sqrt {-x^{3}-1}}{91}-\frac {80 i \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {1+x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \left (\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) E\left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )-F\left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )\right )}{273 \sqrt {-x^{3}-1}}\) | \(189\) |
elliptic | \(-\frac {2 x^{5} \sqrt {-x^{3}-1}}{13}+\frac {20 x^{2} \sqrt {-x^{3}-1}}{91}-\frac {80 i \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {1+x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \left (\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) E\left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )-F\left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )\right )}{273 \sqrt {-x^{3}-1}}\) | \(189\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.11 \[ \int \frac {x^7}{\sqrt {-1-x^3}} \, dx=-\frac {2}{91} \, {\left (7 \, x^{5} - 10 \, x^{2}\right )} \sqrt {-x^{3} - 1} + \frac {80}{91} i \, {\rm weierstrassZeta}\left (0, -4, {\rm weierstrassPInverse}\left (0, -4, x\right )\right ) \]
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Time = 0.43 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.11 \[ \int \frac {x^7}{\sqrt {-1-x^3}} \, dx=- \frac {i x^{8} \Gamma \left (\frac {8}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {8}{3} \\ \frac {11}{3} \end {matrix}\middle | {x^{3} e^{i \pi }} \right )}}{3 \Gamma \left (\frac {11}{3}\right )} \]
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\[ \int \frac {x^7}{\sqrt {-1-x^3}} \, dx=\int { \frac {x^{7}}{\sqrt {-x^{3} - 1}} \,d x } \]
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\[ \int \frac {x^7}{\sqrt {-1-x^3}} \, dx=\int { \frac {x^{7}}{\sqrt {-x^{3} - 1}} \,d x } \]
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Time = 0.05 (sec) , antiderivative size = 259, normalized size of antiderivative = 0.92 \[ \int \frac {x^7}{\sqrt {-1-x^3}} \, dx=\frac {20\,x^2\,\sqrt {-x^3-1}}{91}-\frac {2\,x^5\,\sqrt {-x^3-1}}{13}-\frac {80\,\left (\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )-\left (-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\mathrm {E}\left (\mathrm {asin}\left (\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )\right )\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {x^3+1}\,\sqrt {\frac {x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {\frac {1}{2}-x+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}}{91\,\sqrt {-x^3-1}\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \]
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