Optimal. Leaf size=116 \[ \frac {\sqrt {\frac {a}{x}} x (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\left (1+\sqrt {3}\right ) x\right )^2}} F\left (\cos ^{-1}\left (\frac {1+\left (1-\sqrt {3}\right ) x}{1+\left (1+\sqrt {3}\right ) x}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{\sqrt [4]{3} \sqrt {\frac {x (1+x)}{\left (1+\left (1+\sqrt {3}\right ) x\right )^2}} \sqrt {1+x^3}} \]
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Rubi [A]
time = 0.05, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {15, 335, 231}
\begin {gather*} \frac {x (x+1) \sqrt {\frac {x^2-x+1}{\left (\left (1+\sqrt {3}\right ) x+1\right )^2}} \sqrt {\frac {a}{x}} F\left (\text {ArcCos}\left (\frac {\left (1-\sqrt {3}\right ) x+1}{\left (1+\sqrt {3}\right ) x+1}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{\sqrt [4]{3} \sqrt {\frac {x (x+1)}{\left (\left (1+\sqrt {3}\right ) x+1\right )^2}} \sqrt {x^3+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 231
Rule 335
Rubi steps
\begin {align*} \int \frac {\sqrt {\frac {a}{x}}}{\sqrt {1+x^3}} \, dx &=\left (\sqrt {\frac {a}{x}} \sqrt {x}\right ) \int \frac {1}{\sqrt {x} \sqrt {1+x^3}} \, dx\\ &=\left (2 \sqrt {\frac {a}{x}} \sqrt {x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+x^6}} \, dx,x,\sqrt {x}\right )\\ &=\frac {\sqrt {\frac {a}{x}} x (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\left (1+\sqrt {3}\right ) x\right )^2}} F\left (\cos ^{-1}\left (\frac {1+\left (1-\sqrt {3}\right ) x}{1+\left (1+\sqrt {3}\right ) x}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{\sqrt [4]{3} \sqrt {\frac {x (1+x)}{\left (1+\left (1+\sqrt {3}\right ) 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 = 0.23 \begin {gather*} 2 \sqrt {\frac {a}{x}} x \, _2F_1\left (\frac {1}{6},\frac {1}{2};\frac {7}{6};-x^3\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.37, size = 232, normalized size = 2.00
method | result | size |
meijerg | \(2 \sqrt {\frac {a}{x}}\, x \hypergeom \left (\left [\frac {1}{6}, \frac {1}{2}\right ], \left [\frac {7}{6}\right ], -x^{3}\right )\) | \(22\) |
default | \(\frac {4 \sqrt {\frac {a}{x}}\, x \sqrt {x^{3}+1}\, \left (1+i \sqrt {3}\right ) \sqrt {\frac {\left (i \sqrt {3}+3\right ) x}{\left (1+i \sqrt {3}\right ) \left (1+x \right )}}\, \left (1+x \right )^{2} \sqrt {\frac {i \sqrt {3}+2 x -1}{\left (-1+i \sqrt {3}\right ) \left (1+x \right )}}\, \sqrt {\frac {i \sqrt {3}-2 x +1}{\left (1+i \sqrt {3}\right ) \left (1+x \right )}}\, \EllipticF \left (\sqrt {\frac {\left (i \sqrt {3}+3\right ) x}{\left (1+i \sqrt {3}\right ) \left (1+x \right )}}, \sqrt {\frac {\left (-3+i \sqrt {3}\right ) \left (1+i \sqrt {3}\right )}{\left (-1+i \sqrt {3}\right ) \left (i \sqrt {3}+3\right )}}\right )}{\sqrt {x \left (x^{3}+1\right )}\, \left (i \sqrt {3}+3\right ) \sqrt {-x \left (1+x \right ) \left (i \sqrt {3}+2 x -1\right ) \left (i \sqrt {3}-2 x +1\right )}}\) | \(232\) |
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 = 11, normalized size = 0.09 \begin {gather*} -2 \, \sqrt {a} {\rm weierstrassPInverse}\left (0, -4, \frac {1}{x}\right ) \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}}}{\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 = -0.01 \begin {gather*} \int \frac {\sqrt {\frac {a}{x}}}{\sqrt {x^3+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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