Optimal. Leaf size=32 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {\frac {a}{x}+b x^2}}\right )}{3 \sqrt {b}} \]
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Rubi [A]
time = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2004, 2033,
212} \begin {gather*} \frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {\frac {a}{x}+b x^2}}\right )}{3 \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 2004
Rule 2033
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {\frac {a+b x^3}{x}}} \, dx &=\int \frac {1}{\sqrt {\frac {a}{x}+b x^2}} \, dx\\ &=\frac {2}{3} \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {\frac {a}{x}+b x^2}}\right )\\ &=\frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {\frac {a}{x}+b x^2}}\right )}{3 \sqrt {b}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 63, normalized size = 1.97 \begin {gather*} \frac {2 \sqrt {a+b x^3} \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {b} x^{3/2}}\right )}{3 \sqrt {b} \sqrt {x} \sqrt {\frac {a+b x^3}{x}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.41, size = 477, normalized size = 14.91
method | result | size |
default | \(-\frac {4 \left (b \,x^{3}+a \right ) \left (-1+i \sqrt {3}\right ) \sqrt {-\frac {\left (i \sqrt {3}-3\right ) x b}{\left (-1+i \sqrt {3}\right ) \left (-b x +\left (-a \,b^{2}\right )^{\frac {1}{3}}\right )}}\, \left (-b x +\left (-a \,b^{2}\right )^{\frac {1}{3}}\right )^{2} \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}+2 b x +\left (-a \,b^{2}\right )^{\frac {1}{3}}}{\left (1+i \sqrt {3}\right ) \left (-b x +\left (-a \,b^{2}\right )^{\frac {1}{3}}\right )}}\, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}-2 b x -\left (-a \,b^{2}\right )^{\frac {1}{3}}}{\left (-1+i \sqrt {3}\right ) \left (-b x +\left (-a \,b^{2}\right )^{\frac {1}{3}}\right )}}\, \left (\EllipticF \left (\sqrt {-\frac {\left (i \sqrt {3}-3\right ) x b}{\left (-1+i \sqrt {3}\right ) \left (-b x +\left (-a \,b^{2}\right )^{\frac {1}{3}}\right )}}, \sqrt {\frac {\left (i \sqrt {3}+3\right ) \left (-1+i \sqrt {3}\right )}{\left (1+i \sqrt {3}\right ) \left (i \sqrt {3}-3\right )}}\right )-\EllipticPi \left (\sqrt {-\frac {\left (i \sqrt {3}-3\right ) x b}{\left (-1+i \sqrt {3}\right ) \left (-b x +\left (-a \,b^{2}\right )^{\frac {1}{3}}\right )}}, \frac {-1+i \sqrt {3}}{i \sqrt {3}-3}, \sqrt {\frac {\left (i \sqrt {3}+3\right ) \left (-1+i \sqrt {3}\right )}{\left (1+i \sqrt {3}\right ) \left (i \sqrt {3}-3\right )}}\right )\right )}{b^{2} \sqrt {\frac {b \,x^{3}+a}{x}}\, \sqrt {x \left (b \,x^{3}+a \right )}\, \left (i \sqrt {3}-3\right ) \sqrt {\frac {x \left (-b x +\left (-a \,b^{2}\right )^{\frac {1}{3}}\right ) \left (i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}+2 b x +\left (-a \,b^{2}\right )^{\frac {1}{3}}\right ) \left (i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}-2 b x -\left (-a \,b^{2}\right )^{\frac {1}{3}}\right )}{b^{2}}}}\) | \(477\) |
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]
time = 1.71, size = 102, normalized size = 3.19 \begin {gather*} \left [\frac {\log \left (-8 \, b^{2} x^{6} - 8 \, a b x^{3} - a^{2} - 4 \, {\left (2 \, b x^{5} + a x^{2}\right )} \sqrt {b} \sqrt {\frac {b x^{3} + a}{x}}\right )}{6 \, \sqrt {b}}, -\frac {\sqrt {-b} \arctan \left (\frac {2 \, \sqrt {-b} x^{2} \sqrt {\frac {b x^{3} + a}{x}}}{2 \, b x^{3} + a}\right )}{3 \, b}\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 {1}{\sqrt {\frac {a + b x^{3}}{x}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 = -0.03 \begin {gather*} \int \frac {1}{\sqrt {\frac {b\,x^3+a}{x}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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