Integrand size = 13, antiderivative size = 294 \[ \int \frac {x^7}{\sqrt {-1+x^3}} \, dx=-\frac {80 \sqrt {-1+x^3}}{91 \left (1-\sqrt {3}-x\right )}+\frac {20}{91} x^2 \sqrt {-1+x^3}+\frac {2}{13} x^5 \sqrt {-1+x^3}+\frac {40 \sqrt [4]{3} \sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} E\left (\arcsin \left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right )|-7+4 \sqrt {3}\right )}{91 \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}-\frac {80 \sqrt {2} (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right ),-7+4 \sqrt {3}\right )}{91 \sqrt [4]{3} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}} \]
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Time = 0.08 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {327, 310, 225, 1893} \[ \int \frac {x^7}{\sqrt {-1+x^3}} \, dx=-\frac {80 \sqrt {2} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-x+\sqrt {3}+1}{-x-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{91 \sqrt [4]{3} \sqrt {-\frac {1-x}{\left (-x-\sqrt {3}+1\right )^2}} \sqrt {x^3-1}}+\frac {40 \sqrt [4]{3} \sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x-\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {-x+\sqrt {3}+1}{-x-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{91 \sqrt {-\frac {1-x}{\left (-x-\sqrt {3}+1\right )^2}} \sqrt {x^3-1}}-\frac {80 \sqrt {x^3-1}}{91 \left (-x-\sqrt {3}+1\right )}+\frac {2}{13} \sqrt {x^3-1} x^5+\frac {20}{91} \sqrt {x^3-1} x^2 \]
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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 {80 \sqrt {-1+x^3}}{91 \left (1-\sqrt {3}-x\right )}+\frac {20}{91} x^2 \sqrt {-1+x^3}+\frac {2}{13} x^5 \sqrt {-1+x^3}+\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 = 54, normalized size of antiderivative = 0.18 \[ \int \frac {x^7}{\sqrt {-1+x^3}} \, dx=\frac {2 x^2 \left (-10+3 x^3+7 x^6+10 \sqrt {1-x^3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2}{3},\frac {5}{3},x^3\right )\right )}{91 \sqrt {-1+x^3}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 4.07 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.11
method | result | size |
meijerg | \(\frac {\sqrt {-\operatorname {signum}\left (x^{3}-1\right )}\, x^{8} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{2},\frac {8}{3};\frac {11}{3};x^{3}\right )}{8 \sqrt {\operatorname {signum}\left (x^{3}-1\right )}}\) | \(33\) |
risch | \(\frac {2 x^{2} \left (7 x^{3}+10\right ) \sqrt {x^{3}-1}}{91}+\frac {80 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \left (\left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) E\left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )+\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) F\left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )\right )}{91 \sqrt {x^{3}-1}}\) | \(193\) |
default | \(\frac {2 x^{5} \sqrt {x^{3}-1}}{13}+\frac {20 x^{2} \sqrt {x^{3}-1}}{91}+\frac {80 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \left (\left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) E\left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )+\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) F\left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )\right )}{91 \sqrt {x^{3}-1}}\) | \(198\) |
elliptic | \(\frac {2 x^{5} \sqrt {x^{3}-1}}{13}+\frac {20 x^{2} \sqrt {x^{3}-1}}{91}+\frac {80 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \left (\left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) E\left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )+\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) F\left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )\right )}{91 \sqrt {x^{3}-1}}\) | \(198\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.10 \[ \int \frac {x^7}{\sqrt {-1+x^3}} \, dx=\frac {2}{91} \, {\left (7 \, x^{5} + 10 \, x^{2}\right )} \sqrt {x^{3} - 1} - \frac {80}{91} \, {\rm weierstrassZeta}\left (0, 4, {\rm weierstrassPInverse}\left (0, 4, x\right )\right ) \]
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Time = 0.44 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.09 \[ \int \frac {x^7}{\sqrt {-1+x^3}} \, dx=- \frac {i x^{8} \Gamma \left (\frac {8}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {8}{3} \\ \frac {11}{3} \end {matrix}\middle | {x^{3}} \right )}}{3 \Gamma \left (\frac {11}{3}\right )} \]
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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 = 5.42 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.82 \[ \int \frac {x^7}{\sqrt {-1+x^3}} \, dx=\frac {20\,x^2\,\sqrt {x^3-1}}{91}+\frac {2\,x^5\,\sqrt {x^3-1}}{13}-\frac {80\,\left (\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )-\left (-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\mathrm {E}\left (\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )\right )\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {-\frac {x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}}{91\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x+\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \]
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