Integrand size = 13, antiderivative size = 153 \[ \int \frac {x^6}{\sqrt {-1+x^3}} \, dx=\frac {16}{55} x \sqrt {-1+x^3}+\frac {2}{11} x^4 \sqrt {-1+x^3}-\frac {32 \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right ),-7+4 \sqrt {3}\right )}{55 \sqrt [4]{3} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}} \]
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Time = 0.03 (sec) , antiderivative size = 153, 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 = {327, 225} \[ \int \frac {x^6}{\sqrt {-1+x^3}} \, dx=-\frac {32 \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-x+\sqrt {3}+1}{-x-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{55 \sqrt [4]{3} \sqrt {-\frac {1-x}{\left (-x-\sqrt {3}+1\right )^2}} \sqrt {x^3-1}}+\frac {16}{55} \sqrt {x^3-1} x+\frac {2}{11} \sqrt {x^3-1} x^4 \]
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Rule 225
Rule 327
Rubi steps \begin{align*} \text {integral}& = \frac {2}{11} x^4 \sqrt {-1+x^3}+\frac {8}{11} \int \frac {x^3}{\sqrt {-1+x^3}} \, dx \\ & = \frac {16}{55} x \sqrt {-1+x^3}+\frac {2}{11} x^4 \sqrt {-1+x^3}+\frac {16}{55} \int \frac {1}{\sqrt {-1+x^3}} \, dx \\ & = \frac {16}{55} x \sqrt {-1+x^3}+\frac {2}{11} x^4 \sqrt {-1+x^3}-\frac {32 \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 )}{55 \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 = 52, normalized size of antiderivative = 0.34 \[ \int \frac {x^6}{\sqrt {-1+x^3}} \, dx=\frac {2 x \left (-8+3 x^3+5 x^6+8 \sqrt {1-x^3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},x^3\right )\right )}{55 \sqrt {-1+x^3}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 4.03 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.22
method | result | size |
meijerg | \(\frac {\sqrt {-\operatorname {signum}\left (x^{3}-1\right )}\, x^{7} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{2},\frac {7}{3};\frac {10}{3};x^{3}\right )}{7 \sqrt {\operatorname {signum}\left (x^{3}-1\right )}}\) | \(33\) |
risch | \(\frac {2 x \left (5 x^{3}+8\right ) \sqrt {x^{3}-1}}{55}+\frac {32 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, F\left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{55 \sqrt {x^{3}-1}}\) | \(134\) |
default | \(\frac {2 x^{4} \sqrt {x^{3}-1}}{11}+\frac {16 x \sqrt {x^{3}-1}}{55}+\frac {32 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, F\left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{55 \sqrt {x^{3}-1}}\) | \(139\) |
elliptic | \(\frac {2 x^{4} \sqrt {x^{3}-1}}{11}+\frac {16 x \sqrt {x^{3}-1}}{55}+\frac {32 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, F\left (\sqrt {\frac {-1+x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{55 \sqrt {x^{3}-1}}\) | \(139\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.16 \[ \int \frac {x^6}{\sqrt {-1+x^3}} \, dx=\frac {2}{55} \, {\left (5 \, x^{4} + 8 \, x\right )} \sqrt {x^{3} - 1} + \frac {32}{55} \, {\rm weierstrassPInverse}\left (0, 4, x\right ) \]
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Time = 0.42 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.18 \[ \int \frac {x^6}{\sqrt {-1+x^3}} \, dx=- \frac {i x^{7} \Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {7}{3} \\ \frac {10}{3} \end {matrix}\middle | {x^{3}} \right )}}{3 \Gamma \left (\frac {10}{3}\right )} \]
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\[ \int \frac {x^6}{\sqrt {-1+x^3}} \, dx=\int { \frac {x^{6}}{\sqrt {x^{3} - 1}} \,d x } \]
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\[ \int \frac {x^6}{\sqrt {-1+x^3}} \, dx=\int { \frac {x^{6}}{\sqrt {x^{3} - 1}} \,d x } \]
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Time = 0.05 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.17 \[ \int \frac {x^6}{\sqrt {-1+x^3}} \, dx=\frac {16\,x\,\sqrt {x^3-1}}{55}+\frac {2\,x^4\,\sqrt {x^3-1}}{11}-\frac {32\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {-\frac {x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{55\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x+\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \]
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