Optimal. Leaf size=27 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^3}}}{\sqrt {a}}\right )}{3 \sqrt {a}} \]
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Rubi [A]
time = 0.01, antiderivative size = 27, 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 = {272, 65, 214}
\begin {gather*} \frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^3}}}{\sqrt {a}}\right )}{3 \sqrt {a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 214
Rule 272
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a+\frac {b}{x^3}} x} \, dx &=-\left (\frac {1}{3} \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x^3}\right )\right )\\ &=-\frac {2 \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x^3}}\right )}{3 b}\\ &=\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^3}}}{\sqrt {a}}\right )}{3 \sqrt {a}}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(59\) vs. \(2(27)=54\).
time = 0.14, size = 59, normalized size = 2.19 \begin {gather*} \frac {2 \sqrt {b+a x^3} \tanh ^{-1}\left (\frac {\sqrt {b+a x^3}}{\sqrt {a} x^{3/2}}\right )}{3 \sqrt {a} \sqrt {a+\frac {b}{x^3}} x^{3/2}} \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 = 0.03, size = 480, normalized size = 17.78
method | result | size |
default | \(-\frac {4 \left (a \,x^{3}+b \right ) \left (-1+i \sqrt {3}\right ) \sqrt {-\frac {\left (i \sqrt {3}-3\right ) x a}{\left (-1+i \sqrt {3}\right ) \left (-a x +\left (-a^{2} b \right )^{\frac {1}{3}}\right )}}\, \left (-a x +\left (-a^{2} b \right )^{\frac {1}{3}}\right )^{2} \sqrt {\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}+2 a x +\left (-a^{2} b \right )^{\frac {1}{3}}}{\left (1+i \sqrt {3}\right ) \left (-a x +\left (-a^{2} b \right )^{\frac {1}{3}}\right )}}\, \sqrt {\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}-2 a x -\left (-a^{2} b \right )^{\frac {1}{3}}}{\left (-1+i \sqrt {3}\right ) \left (-a x +\left (-a^{2} b \right )^{\frac {1}{3}}\right )}}\, \left (\EllipticF \left (\sqrt {-\frac {\left (i \sqrt {3}-3\right ) x a}{\left (-1+i \sqrt {3}\right ) \left (-a x +\left (-a^{2} b \right )^{\frac {1}{3}}\right )}}, \sqrt {\frac {\left (i \sqrt {3}+3\right ) \left (-1+i \sqrt {3}\right )}{\left (1+i \sqrt {3}\right ) \left (i \sqrt {3}-3\right )}}\right )-\EllipticPi \left (\sqrt {-\frac {\left (i \sqrt {3}-3\right ) x a}{\left (-1+i \sqrt {3}\right ) \left (-a x +\left (-a^{2} b \right )^{\frac {1}{3}}\right )}}, \frac {-1+i \sqrt {3}}{i \sqrt {3}-3}, \sqrt {\frac {\left (i \sqrt {3}+3\right ) \left (-1+i \sqrt {3}\right )}{\left (1+i \sqrt {3}\right ) \left (i \sqrt {3}-3\right )}}\right )\right )}{\sqrt {\frac {a \,x^{3}+b}{x^{3}}}\, x \,a^{2} \sqrt {x \left (a \,x^{3}+b \right )}\, \left (i \sqrt {3}-3\right ) \sqrt {\frac {x \left (-a x +\left (-a^{2} b \right )^{\frac {1}{3}}\right ) \left (i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}+2 a x +\left (-a^{2} b \right )^{\frac {1}{3}}\right ) \left (i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}-2 a x -\left (-a^{2} b \right )^{\frac {1}{3}}\right )}{a^{2}}}}\) | \(480\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 37, normalized size = 1.37 \begin {gather*} -\frac {\log \left (\frac {\sqrt {a + \frac {b}{x^{3}}} - \sqrt {a}}{\sqrt {a + \frac {b}{x^{3}}} + \sqrt {a}}\right )}{3 \, \sqrt {a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 44 vs.
\(2 (19) = 38\).
time = 0.47, size = 102, normalized size = 3.78 \begin {gather*} \left [\frac {\log \left (-8 \, a^{2} x^{6} - 8 \, a b x^{3} - b^{2} - 4 \, {\left (2 \, a x^{6} + b x^{3}\right )} \sqrt {a} \sqrt {\frac {a x^{3} + b}{x^{3}}}\right )}{6 \, \sqrt {a}}, -\frac {\sqrt {-a} \arctan \left (\frac {2 \, \sqrt {-a} x^{3} \sqrt {\frac {a x^{3} + b}{x^{3}}}}{2 \, a x^{3} + b}\right )}{3 \, a}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.54, size = 24, normalized size = 0.89 \begin {gather*} \frac {2 \operatorname {asinh}{\left (\frac {\sqrt {a} x^{\frac {3}{2}}}{\sqrt {b}} \right )}}{3 \sqrt {a}} \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 [B]
time = 1.31, size = 19, normalized size = 0.70 \begin {gather*} \frac {2\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x^3}}}{\sqrt {a}}\right )}{3\,\sqrt {a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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