Optimal. Leaf size=24 \[ -\frac {2}{3} \sqrt {\frac {a}{x^2}} x \tanh ^{-1}\left (\sqrt {1+x^3}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {15, 272, 65,
213} \begin {gather*} -\frac {2}{3} x \sqrt {\frac {a}{x^2}} \tanh ^{-1}\left (\sqrt {x^3+1}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 65
Rule 213
Rule 272
Rubi steps
\begin {align*} \int \frac {\sqrt {\frac {a}{x^2}}}{\sqrt {1+x^3}} \, dx &=\left (\sqrt {\frac {a}{x^2}} x\right ) \int \frac {1}{x \sqrt {1+x^3}} \, dx\\ &=\frac {1}{3} \left (\sqrt {\frac {a}{x^2}} x\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,x^3\right )\\ &=\frac {1}{3} \left (2 \sqrt {\frac {a}{x^2}} x\right ) \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+x^3}\right )\\ &=-\frac {2}{3} \sqrt {\frac {a}{x^2}} x \tanh ^{-1}\left (\sqrt {1+x^3}\right )\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 24, normalized size = 1.00 \begin {gather*} -\frac {2}{3} \sqrt {\frac {a}{x^2}} x \tanh ^{-1}\left (\sqrt {1+x^3}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 19, normalized size = 0.79
method | result | size |
default | \(-\frac {2 x \arctanh \left (\sqrt {x^{3}+1}\right ) \sqrt {\frac {a}{x^{2}}}}{3}\) | \(19\) |
meijerg | \(\frac {\sqrt {\frac {a}{x^{2}}}\, x \left (-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {x^{3}+1}}{2}\right )+\left (-2 \ln \left (2\right )+3 \ln \left (x \right )\right ) \sqrt {\pi }\right )}{3 \sqrt {\pi }}\) | \(45\) |
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 [A] Leaf count of result is larger than twice the leaf count of optimal. 38 vs.
\(2 (18) = 36\).
time = 0.33, size = 68, normalized size = 2.83 \begin {gather*} \left [\frac {1}{3} \, x \sqrt {\frac {a}{x^{2}}} \log \left (\frac {x^{3} - 2 \, \sqrt {x^{3} + 1} + 2}{x^{3}}\right ), \frac {2}{3} \, \sqrt {-a} \arctan \left (\frac {\sqrt {x^{3} + 1} \sqrt {-a} x \sqrt {\frac {a}{x^{2}}}}{a x^{3} + a}\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^{2}}}}{\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 [A]
time = 3.45, size = 31, normalized size = 1.29 \begin {gather*} -\frac {1}{3} \, \sqrt {a} {\left (\log \left (\sqrt {x^{3} + 1} + 1\right ) - \log \left ({\left | \sqrt {x^{3} + 1} - 1 \right |}\right )\right )} \mathrm {sgn}\left (x\right ) \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 = -0.04 \begin {gather*} \int \frac {\sqrt {\frac {a}{x^2}}}{\sqrt {x^3+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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