Optimal. Leaf size=46 \[ \frac {2}{3 a \sqrt {a+b x^3}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{3 a^{3/2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {272, 53, 65,
214} \begin {gather*} \frac {2}{3 a \sqrt {a+b x^3}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{3 a^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 53
Rule 65
Rule 214
Rule 272
Rubi steps
\begin {align*} \int \frac {1}{x \left (a+b x^3\right )^{3/2}} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {1}{x (a+b x)^{3/2}} \, dx,x,x^3\right )\\ &=\frac {2}{3 a \sqrt {a+b x^3}}+\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^3\right )}{3 a}\\ &=\frac {2}{3 a \sqrt {a+b x^3}}+\frac {2 \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^3}\right )}{3 a b}\\ &=\frac {2}{3 a \sqrt {a+b x^3}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{3 a^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 46, normalized size = 1.00 \begin {gather*} \frac {2}{3 a \sqrt {a+b x^3}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{3 a^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 39, normalized size = 0.85
method | result | size |
default | \(\frac {2}{3 a \sqrt {\left (x^{3}+\frac {a}{b}\right ) b}}-\frac {2 \arctanh \left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )}{3 a^{\frac {3}{2}}}\) | \(39\) |
elliptic | \(\frac {2}{3 a \sqrt {\left (x^{3}+\frac {a}{b}\right ) b}}-\frac {2 \arctanh \left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )}{3 a^{\frac {3}{2}}}\) | \(39\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 52, normalized size = 1.13 \begin {gather*} \frac {\log \left (\frac {\sqrt {b x^{3} + a} - \sqrt {a}}{\sqrt {b x^{3} + a} + \sqrt {a}}\right )}{3 \, a^{\frac {3}{2}}} + \frac {2}{3 \, \sqrt {b x^{3} + a} a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 129, normalized size = 2.80 \begin {gather*} \left [\frac {{\left (b x^{3} + a\right )} \sqrt {a} \log \left (\frac {b x^{3} - 2 \, \sqrt {b x^{3} + a} \sqrt {a} + 2 \, a}{x^{3}}\right ) + 2 \, \sqrt {b x^{3} + a} a}{3 \, {\left (a^{2} b x^{3} + a^{3}\right )}}, \frac {2 \, {\left ({\left (b x^{3} + a\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b x^{3} + a} \sqrt {-a}}{a}\right ) + \sqrt {b x^{3} + a} a\right )}}{3 \, {\left (a^{2} b x^{3} + a^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 184 vs.
\(2 (39) = 78\).
time = 0.85, size = 184, normalized size = 4.00 \begin {gather*} \frac {2 a^{3} \sqrt {1 + \frac {b x^{3}}{a}}}{3 a^{\frac {9}{2}} + 3 a^{\frac {7}{2}} b x^{3}} + \frac {a^{3} \log {\left (\frac {b x^{3}}{a} \right )}}{3 a^{\frac {9}{2}} + 3 a^{\frac {7}{2}} b x^{3}} - \frac {2 a^{3} \log {\left (\sqrt {1 + \frac {b x^{3}}{a}} + 1 \right )}}{3 a^{\frac {9}{2}} + 3 a^{\frac {7}{2}} b x^{3}} + \frac {a^{2} b x^{3} \log {\left (\frac {b x^{3}}{a} \right )}}{3 a^{\frac {9}{2}} + 3 a^{\frac {7}{2}} b x^{3}} - \frac {2 a^{2} b x^{3} \log {\left (\sqrt {1 + \frac {b x^{3}}{a}} + 1 \right )}}{3 a^{\frac {9}{2}} + 3 a^{\frac {7}{2}} b x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.04, size = 41, normalized size = 0.89 \begin {gather*} \frac {2 \, \arctan \left (\frac {\sqrt {b x^{3} + a}}{\sqrt {-a}}\right )}{3 \, \sqrt {-a} a} + \frac {2}{3 \, \sqrt {b x^{3} + a} a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.23, size = 55, normalized size = 1.20 \begin {gather*} \frac {\ln \left (\frac {{\left (\sqrt {b\,x^3+a}-\sqrt {a}\right )}^3\,\left (\sqrt {b\,x^3+a}+\sqrt {a}\right )}{x^6}\right )}{3\,a^{3/2}}+\frac {2}{3\,a\,\sqrt {b\,x^3+a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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