Integrand size = 13, antiderivative size = 262 \[ \int \frac {x^7}{\sqrt {1+x^3}} \, dx=-\frac {20}{91} x^2 \sqrt {1+x^3}+\frac {2}{13} x^5 \sqrt {1+x^3}+\frac {80 \sqrt {1+x^3}}{91 \left (1+\sqrt {3}+x\right )}-\frac {40 \sqrt [4]{3} \sqrt {2-\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} E\left (\arcsin \left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right )|-7-4 \sqrt {3}\right )}{91 \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}+\frac {80 \sqrt {2} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right ),-7-4 \sqrt {3}\right )}{91 \sqrt [4]{3} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}} \]
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Time = 0.06 (sec) , antiderivative size = 262, 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, 309, 224, 1891} \[ \int \frac {x^7}{\sqrt {1+x^3}} \, dx=\frac {80 \sqrt {2} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{91 \sqrt [4]{3} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}-\frac {40 \sqrt [4]{3} \sqrt {2-\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{91 \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}+\frac {80 \sqrt {x^3+1}}{91 \left (x+\sqrt {3}+1\right )}+\frac {2}{13} \sqrt {x^3+1} x^5-\frac {20}{91} \sqrt {x^3+1} x^2 \]
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Rule 224
Rule 309
Rule 327
Rule 1891
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 = 42, normalized size of antiderivative = 0.16 \[ \int \frac {x^7}{\sqrt {1+x^3}} \, dx=\frac {2}{91} x^2 \left (\sqrt {1+x^3} \left (-10+7 x^3\right )+10 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2}{3},\frac {5}{3},-x^3\right )\right ) \]
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Result contains higher order function than in optimal. Order 5 vs. order 4.
Time = 3.99 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.06
method | result | size |
meijerg | \(\frac {x^{8} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{2},\frac {8}{3};\frac {11}{3};-x^{3}\right )}{8}\) | \(17\) |
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 = 0.07 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.11 \[ \int \frac {x^7}{\sqrt {1+x^3}} \, dx=\frac {2}{91} \, {\left (7 \, x^{5} - 10 \, x^{2}\right )} \sqrt {x^{3} + 1} - \frac {80}{91} \, {\rm weierstrassZeta}\left (0, -4, {\rm weierstrassPInverse}\left (0, -4, x\right )\right ) \]
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Time = 0.43 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.11 \[ \int \frac {x^7}{\sqrt {1+x^3}} \, dx=\frac {x^{8} \Gamma \left (\frac {8}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {8}{3} \\ \frac {11}{3} \end {matrix}\middle | {x^{3} e^{i \pi }} \right )}}{3 \Gamma \left (\frac {11}{3}\right )} \]
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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 = 5.44 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.91 \[ \int \frac {x^7}{\sqrt {1+x^3}} \, dx=\frac {2\,x^5\,\sqrt {x^3+1}}{13}-\frac {20\,x^2\,\sqrt {x^3+1}}{91}-\frac {80\,\left (\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )-\left (-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\mathrm {E}\left (\mathrm {asin}\left (\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )\right )\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {\frac {x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {\frac {1}{2}-x+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}}{91\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \]
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