Optimal. Leaf size=294 \[ -\frac {80 \sqrt {-1+x^3}}{91 \left (1-\sqrt {3}-x\right )}+\frac {20}{91} x^2 \sqrt {-1+x^3}+\frac {2}{13} x^5 \sqrt {-1+x^3}+\frac {40 \sqrt [4]{3} \sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} E\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right )|-7+4 \sqrt {3}\right )}{91 \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}-\frac {80 \sqrt {2} (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right )|-7+4 \sqrt {3}\right )}{91 \sqrt [4]{3} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}} \]
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Rubi [A]
time = 0.07, antiderivative size = 294, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {327, 310, 225,
1893} \begin {gather*} -\frac {80 \sqrt {2} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x-\sqrt {3}+1\right )^2}} F\left (\text {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 (\text {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 \end {gather*}
Antiderivative was successfully verified.
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Rule 225
Rule 310
Rule 327
Rule 1893
Rubi steps
\begin {align*} \int \frac {x^7}{\sqrt {-1+x^3}} \, dx &=\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 \sqrt {2 \left (2+\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*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.02, size = 54, normalized size = 0.18 \begin {gather*} \frac {2 x^2 \left (-10+3 x^3+7 x^6+10 \sqrt {1-x^3} \, _2F_1\left (\frac {1}{2},\frac {2}{3};\frac {5}{3};x^3\right )\right )}{91 \sqrt {-1+x^3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 198, normalized size = 0.67
method | result | size |
meijerg | \(\frac {\sqrt {-\mathrm {signum}\left (x^{3}-1\right )}\, x^{8} \hypergeom \left (\left [\frac {1}{2}, \frac {8}{3}\right ], \left [\frac {11}{3}\right ], x^{3}\right )}{8 \sqrt {\mathrm {signum}\left (x^{3}-1\right )}}\) | \(33\) |
risch | \(\frac {2 x^{2} \left (7 x^{3}+10\right ) \sqrt {x^{3}-1}}{91}+\frac {80 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \left (\left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \EllipticE \left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )+\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \EllipticF \left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )\right )}{91 \sqrt {x^{3}-1}}\) | \(193\) |
default | \(\frac {2 x^{5} \sqrt {x^{3}-1}}{13}+\frac {20 x^{2} \sqrt {x^{3}-1}}{91}+\frac {80 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \left (\left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \EllipticE \left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )+\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \EllipticF \left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )\right )}{91 \sqrt {x^{3}-1}}\) | \(198\) |
elliptic | \(\frac {2 x^{5} \sqrt {x^{3}-1}}{13}+\frac {20 x^{2} \sqrt {x^{3}-1}}{91}+\frac {80 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \left (\left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \EllipticE \left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )+\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \EllipticF \left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )\right )}{91 \sqrt {x^{3}-1}}\) | \(198\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.07, size = 30, normalized size = 0.10 \begin {gather*} \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.40, size = 27, normalized size = 0.09 \begin {gather*} - \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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.97, size = 240, normalized size = 0.82 \begin {gather*} \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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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