Optimal. Leaf size=281 \[ -\sqrt {\frac {a}{x^4}} x \sqrt {1+x^3}+\frac {\sqrt {\frac {a}{x^4}} x^2 \sqrt {1+x^3}}{1+\sqrt {3}+x}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt {\frac {a}{x^4}} x^2 (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} E\left (\sin ^{-1}\left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right )|-7-4 \sqrt {3}\right )}{2 \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}+\frac {\sqrt {2} \sqrt {\frac {a}{x^4}} x^2 (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} F\left (\sin ^{-1}\left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right )|-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}} \]
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Rubi [A]
time = 0.06, antiderivative size = 281, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {15, 331, 309,
224, 1891} \begin {gather*} \frac {\sqrt {2} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} x^2 \sqrt {\frac {a}{x^4}} F\left (\text {ArcSin}\left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} x^2 \sqrt {\frac {a}{x^4}} E\left (\text {ArcSin}\left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{2 \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}-\sqrt {x^3+1} x \sqrt {\frac {a}{x^4}}+\frac {\sqrt {x^3+1} x^2 \sqrt {\frac {a}{x^4}}}{x+\sqrt {3}+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 224
Rule 309
Rule 331
Rule 1891
Rubi steps
\begin {align*} \int \frac {\sqrt {\frac {a}{x^4}}}{\sqrt {1+x^3}} \, dx &=\left (\sqrt {\frac {a}{x^4}} x^2\right ) \int \frac {1}{x^2 \sqrt {1+x^3}} \, dx\\ &=-\sqrt {\frac {a}{x^4}} x \sqrt {1+x^3}+\frac {1}{2} \left (\sqrt {\frac {a}{x^4}} x^2\right ) \int \frac {x}{\sqrt {1+x^3}} \, dx\\ &=-\sqrt {\frac {a}{x^4}} x \sqrt {1+x^3}+\frac {1}{2} \left (\sqrt {\frac {a}{x^4}} x^2\right ) \int \frac {1-\sqrt {3}+x}{\sqrt {1+x^3}} \, dx+\left (\sqrt {\frac {1}{2} \left (2-\sqrt {3}\right )} \sqrt {\frac {a}{x^4}} x^2\right ) \int \frac {1}{\sqrt {1+x^3}} \, dx\\ &=-\sqrt {\frac {a}{x^4}} x \sqrt {1+x^3}+\frac {\sqrt {\frac {a}{x^4}} x^2 \sqrt {1+x^3}}{1+\sqrt {3}+x}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt {\frac {a}{x^4}} x^2 (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 )}{2 \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}+\frac {\sqrt {2} \sqrt {\frac {a}{x^4}} x^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 )}{\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 = 27, normalized size = 0.10 \begin {gather*} -\sqrt {\frac {a}{x^4}} x \, _2F_1\left (-\frac {1}{3},\frac {1}{2};\frac {2}{3};-x^3\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.32, size = 353, normalized size = 1.26
method | result | size |
meijerg | \(-\sqrt {\frac {a}{x^{4}}}\, x \hypergeom \left (\left [-\frac {1}{3}, \frac {1}{2}\right ], \left [\frac {2}{3}\right ], -x^{3}\right )\) | \(22\) |
risch | \(-x \sqrt {\frac {a}{x^{4}}}\, \sqrt {x^{3}+1}-\frac {i \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {1+x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \left (\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \EllipticE \left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )-\EllipticF \left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )\right ) \sqrt {\frac {a}{x^{4}}}\, x^{2} \sqrt {a \left (x^{3}+1\right )}}{3 \sqrt {a \,x^{3}+a}\, \sqrt {x^{3}+1}}\) | \(204\) |
default | \(\frac {\sqrt {\frac {a}{x^{4}}}\, x \left (i \sqrt {3}\, \sqrt {-\frac {2 \left (1+x \right )}{-3+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}-2 x +1}{i \sqrt {3}+3}}\, \sqrt {\frac {i \sqrt {3}+2 x -1}{-3+i \sqrt {3}}}\, \EllipticF \left (\sqrt {-\frac {2 \left (1+x \right )}{-3+i \sqrt {3}}}, \sqrt {-\frac {-3+i \sqrt {3}}{i \sqrt {3}+3}}\right ) x -6 \sqrt {-\frac {2 \left (1+x \right )}{-3+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}-2 x +1}{i \sqrt {3}+3}}\, \sqrt {\frac {i \sqrt {3}+2 x -1}{-3+i \sqrt {3}}}\, \EllipticE \left (\sqrt {-\frac {2 \left (1+x \right )}{-3+i \sqrt {3}}}, \sqrt {-\frac {-3+i \sqrt {3}}{i \sqrt {3}+3}}\right ) x +3 \sqrt {-\frac {2 \left (1+x \right )}{-3+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}-2 x +1}{i \sqrt {3}+3}}\, \sqrt {\frac {i \sqrt {3}+2 x -1}{-3+i \sqrt {3}}}\, \EllipticF \left (\sqrt {-\frac {2 \left (1+x \right )}{-3+i \sqrt {3}}}, \sqrt {-\frac {-3+i \sqrt {3}}{i \sqrt {3}+3}}\right ) x -2 x^{3}-2\right )}{2 \sqrt {x^{3}+1}}\) | \(353\) |
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 = 0.08, size = 37, normalized size = 0.13 \begin {gather*} -x^{2} \sqrt {\frac {a}{x^{4}}} {\rm weierstrassZeta}\left (0, -4, {\rm weierstrassPInverse}\left (0, -4, x\right )\right ) - \sqrt {x^{3} + 1} x \sqrt {\frac {a}{x^{4}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {\frac {a}{x^{4}}}}{\sqrt {\left (x + 1\right ) \left (x^{2} - x + 1\right )}}\, dx \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 [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {\frac {a}{x^4}}}{\sqrt {x^3+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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