Integrand size = 13, antiderivative size = 155 \[ \int \frac {1}{x^6 \sqrt {-1+x^3}} \, dx=\frac {\sqrt {-1+x^3}}{5 x^5}+\frac {7 \sqrt {-1+x^3}}{20 x^2}-\frac {7 \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right ),-7+4 \sqrt {3}\right )}{20 \sqrt [4]{3} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}} \]
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Time = 0.03 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {331, 225} \[ \int \frac {1}{x^6 \sqrt {-1+x^3}} \, dx=-\frac {7 \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-x+\sqrt {3}+1}{-x-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{20 \sqrt [4]{3} \sqrt {-\frac {1-x}{\left (-x-\sqrt {3}+1\right )^2}} \sqrt {x^3-1}}+\frac {\sqrt {x^3-1}}{5 x^5}+\frac {7 \sqrt {x^3-1}}{20 x^2} \]
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Rule 225
Rule 331
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {-1+x^3}}{5 x^5}+\frac {7}{10} \int \frac {1}{x^3 \sqrt {-1+x^3}} \, dx \\ & = \frac {\sqrt {-1+x^3}}{5 x^5}+\frac {7 \sqrt {-1+x^3}}{20 x^2}+\frac {7}{40} \int \frac {1}{\sqrt {-1+x^3}} \, dx \\ & = \frac {\sqrt {-1+x^3}}{5 x^5}+\frac {7 \sqrt {-1+x^3}}{20 x^2}-\frac {7 \sqrt {2-\sqrt {3}} (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 )}{20 \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 = 10.01 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.26 \[ \int \frac {1}{x^6 \sqrt {-1+x^3}} \, dx=-\frac {\sqrt {1-x^3} \operatorname {Hypergeometric2F1}\left (-\frac {5}{3},\frac {1}{2},-\frac {2}{3},x^3\right )}{5 x^5 \sqrt {-1+x^3}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 4.04 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.21
method | result | size |
meijerg | \(-\frac {\sqrt {-\operatorname {signum}\left (x^{3}-1\right )}\, {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (-\frac {5}{3},\frac {1}{2};-\frac {2}{3};x^{3}\right )}{5 \sqrt {\operatorname {signum}\left (x^{3}-1\right )}\, x^{5}}\) | \(33\) |
default | \(\frac {\sqrt {x^{3}-1}}{5 x^{5}}+\frac {7 \sqrt {x^{3}-1}}{20 x^{2}}+\frac {7 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, F\left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{20 \sqrt {x^{3}-1}}\) | \(141\) |
risch | \(\frac {7 x^{6}-3 x^{3}-4}{20 x^{5} \sqrt {x^{3}-1}}+\frac {7 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, F\left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{20 \sqrt {x^{3}-1}}\) | \(141\) |
elliptic | \(\frac {\sqrt {x^{3}-1}}{5 x^{5}}+\frac {7 \sqrt {x^{3}-1}}{20 x^{2}}+\frac {7 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, F\left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{20 \sqrt {x^{3}-1}}\) | \(141\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.19 \[ \int \frac {1}{x^6 \sqrt {-1+x^3}} \, dx=\frac {7 \, x^{5} {\rm weierstrassPInverse}\left (0, 4, x\right ) + {\left (7 \, x^{3} + 4\right )} \sqrt {x^{3} - 1}}{20 \, x^{5}} \]
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Time = 0.50 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.22 \[ \int \frac {1}{x^6 \sqrt {-1+x^3}} \, dx=- \frac {i \Gamma \left (- \frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{3}, \frac {1}{2} \\ - \frac {2}{3} \end {matrix}\middle | {x^{3}} \right )}}{3 x^{5} \Gamma \left (- \frac {2}{3}\right )} \]
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\[ \int \frac {1}{x^6 \sqrt {-1+x^3}} \, dx=\int { \frac {1}{\sqrt {x^{3} - 1} x^{6}} \,d x } \]
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\[ \int \frac {1}{x^6 \sqrt {-1+x^3}} \, dx=\int { \frac {1}{\sqrt {x^{3} - 1} x^{6}} \,d x } \]
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Time = 5.38 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.17 \[ \int \frac {1}{x^6 \sqrt {-1+x^3}} \, dx=\frac {7\,\sqrt {x^3-1}}{20\,x^2}+\frac {\sqrt {x^3-1}}{5\,x^5}-\frac {7\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {-\frac {x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{20\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x+\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \]
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