Optimal. Leaf size=312 \[ -2 \sqrt {\frac {a}{x^3}} x \sqrt {1+x^3}+\frac {2 \left (1+\sqrt {3}\right ) \sqrt {\frac {a}{x^3}} x^2 \sqrt {1+x^3}}{1+\left (1+\sqrt {3}\right ) x}-\frac {2 \sqrt [4]{3} \sqrt {\frac {a}{x^3}} x^2 (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\left (1+\sqrt {3}\right ) x\right )^2}} E\left (\cos ^{-1}\left (\frac {1+\left (1-\sqrt {3}\right ) x}{1+\left (1+\sqrt {3}\right ) x}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{\sqrt {\frac {x (1+x)}{\left (1+\left (1+\sqrt {3}\right ) x\right )^2}} \sqrt {1+x^3}}-\frac {\left (1-\sqrt {3}\right ) \sqrt {\frac {a}{x^3}} x^2 (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\left (1+\sqrt {3}\right ) x\right )^2}} F\left (\cos ^{-1}\left (\frac {1+\left (1-\sqrt {3}\right ) x}{1+\left (1+\sqrt {3}\right ) x}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{\sqrt [4]{3} \sqrt {\frac {x (1+x)}{\left (1+\left (1+\sqrt {3}\right ) x\right )^2}} \sqrt {1+x^3}} \]
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Rubi [A]
time = 0.19, antiderivative size = 312, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {15, 331, 335,
314, 231, 1895} \begin {gather*} -\frac {\left (1-\sqrt {3}\right ) (x+1) \sqrt {\frac {x^2-x+1}{\left (\left (1+\sqrt {3}\right ) x+1\right )^2}} x^2 \sqrt {\frac {a}{x^3}} 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}}-\frac {2 \sqrt [4]{3} (x+1) \sqrt {\frac {x^2-x+1}{\left (\left (1+\sqrt {3}\right ) x+1\right )^2}} x^2 \sqrt {\frac {a}{x^3}} E\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 {\frac {x (x+1)}{\left (\left (1+\sqrt {3}\right ) x+1\right )^2}} \sqrt {x^3+1}}-2 \sqrt {x^3+1} x \sqrt {\frac {a}{x^3}}+\frac {2 \left (1+\sqrt {3}\right ) \sqrt {x^3+1} x^2 \sqrt {\frac {a}{x^3}}}{\left (1+\sqrt {3}\right ) x+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 231
Rule 314
Rule 331
Rule 335
Rule 1895
Rubi steps
\begin {align*} \int \frac {\sqrt {\frac {a}{x^3}}}{\sqrt {1+x^3}} \, dx &=\left (\sqrt {\frac {a}{x^3}} x^{3/2}\right ) \int \frac {1}{x^{3/2} \sqrt {1+x^3}} \, dx\\ &=-2 \sqrt {\frac {a}{x^3}} x \sqrt {1+x^3}+\left (2 \sqrt {\frac {a}{x^3}} x^{3/2}\right ) \int \frac {x^{3/2}}{\sqrt {1+x^3}} \, dx\\ &=-2 \sqrt {\frac {a}{x^3}} x \sqrt {1+x^3}+\left (4 \sqrt {\frac {a}{x^3}} x^{3/2}\right ) \text {Subst}\left (\int \frac {x^4}{\sqrt {1+x^6}} \, dx,x,\sqrt {x}\right )\\ &=-2 \sqrt {\frac {a}{x^3}} x \sqrt {1+x^3}-\left (2 \sqrt {\frac {a}{x^3}} x^{3/2}\right ) \text {Subst}\left (\int \frac {-1+\sqrt {3}-2 x^4}{\sqrt {1+x^6}} \, dx,x,\sqrt {x}\right )+\left (2 \left (-1+\sqrt {3}\right ) \sqrt {\frac {a}{x^3}} x^{3/2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+x^6}} \, dx,x,\sqrt {x}\right )\\ &=-2 \sqrt {\frac {a}{x^3}} x \sqrt {1+x^3}+\frac {2 \left (1+\sqrt {3}\right ) \sqrt {\frac {a}{x^3}} x^2 \sqrt {1+x^3}}{1+\left (1+\sqrt {3}\right ) x}-\frac {2 \sqrt [4]{3} \sqrt {\frac {a}{x^3}} x^2 (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\left (1+\sqrt {3}\right ) x\right )^2}} E\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 {\frac {x (1+x)}{\left (1+\left (1+\sqrt {3}\right ) x\right )^2}} \sqrt {1+x^3}}-\frac {\left (1-\sqrt {3}\right ) \sqrt {\frac {a}{x^3}} x^2 (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 = 10.01, size = 27, normalized size = 0.09 \begin {gather*} -2 \sqrt {\frac {a}{x^3}} x \, _2F_1\left (-\frac {1}{6},\frac {1}{2};\frac {5}{6};-x^3\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.39, size = 1784, normalized size = 5.72
method | result | size |
meijerg | \(-2 \sqrt {\frac {a}{x^{3}}}\, x \hypergeom \left (\left [-\frac {1}{6}, \frac {1}{2}\right ], \left [\frac {5}{6}\right ], -x^{3}\right )\) | \(22\) |
risch | \(-2 x \sqrt {\frac {a}{x^{3}}}\, \sqrt {x^{3}+1}+\frac {2 \left (x \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )+\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (1+x \right )^{2} \sqrt {-\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \sqrt {-\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (-\frac {\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \EllipticF \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \EllipticE \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )\right ) \sqrt {\frac {a}{x^{3}}}\, x \sqrt {x \left (x^{3}+1\right ) a}}{\sqrt {a x \left (1+x \right ) \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}\, \sqrt {x^{3}+1}}\) | \(355\) |
default | \(\text {Expression too large to display}\) | \(1784\) |
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 = 14, normalized size = 0.04 \begin {gather*} 2 \, \sqrt {a} {\rm weierstrassZeta}\left (0, -4, {\rm weierstrassPInverse}\left (0, -4, \frac {1}{x}\right )\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^{3}}}}{\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^3}}}{\sqrt {x^3+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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