Optimal. Leaf size=43 \[ \frac {2}{3} \sqrt {a+b x^3}-\frac {2}{3} \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 43, 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 = {266, 50, 63, 208} \[ \frac {2}{3} \sqrt {a+b x^3}-\frac {2}{3} \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right ) \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 208
Rule 266
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x^3}}{x} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x} \, dx,x,x^3\right )\\ &=\frac {2}{3} \sqrt {a+b x^3}+\frac {1}{3} a \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^3\right )\\ &=\frac {2}{3} \sqrt {a+b x^3}+\frac {(2 a) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^3}\right )}{3 b}\\ &=\frac {2}{3} \sqrt {a+b x^3}-\frac {2}{3} \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 43, normalized size = 1.00 \[ \frac {2}{3} \sqrt {a+b x^3}-\frac {2}{3} \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 84, normalized size = 1.95 \[ \left [\frac {1}{3} \, \sqrt {a} \log \left (\frac {b x^{3} - 2 \, \sqrt {b x^{3} + a} \sqrt {a} + 2 \, a}{x^{3}}\right ) + \frac {2}{3} \, \sqrt {b x^{3} + a}, \frac {2}{3} \, \sqrt {-a} \arctan \left (\frac {\sqrt {b x^{3} + a} \sqrt {-a}}{a}\right ) + \frac {2}{3} \, \sqrt {b x^{3} + a}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 36, normalized size = 0.84 \[ \frac {2 \, a \arctan \left (\frac {\sqrt {b x^{3} + a}}{\sqrt {-a}}\right )}{3 \, \sqrt {-a}} + \frac {2}{3} \, \sqrt {b x^{3} + a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.28, size = 32, normalized size = 0.74 \[ -\frac {2 \sqrt {a}\, \arctanh \left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )}{3}+\frac {2 \sqrt {b \,x^{3}+a}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.95, size = 49, normalized size = 1.14 \[ \frac {1}{3} \, \sqrt {a} \log \left (\frac {\sqrt {b x^{3} + a} - \sqrt {a}}{\sqrt {b x^{3} + a} + \sqrt {a}}\right ) + \frac {2}{3} \, \sqrt {b x^{3} + a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.21, size = 52, normalized size = 1.21 \[ \frac {\sqrt {a}\,\ln \left (\frac {{\left (\sqrt {b\,x^3+a}-\sqrt {a}\right )}^3\,\left (\sqrt {b\,x^3+a}+\sqrt {a}\right )}{x^6}\right )}{3}+\frac {2\,\sqrt {b\,x^3+a}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.07, size = 76, normalized size = 1.77 \[ - \frac {2 \sqrt {a} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{\frac {3}{2}}} \right )}}{3} + \frac {2 a}{3 \sqrt {b} x^{\frac {3}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} + \frac {2 \sqrt {b} x^{\frac {3}{2}}}{3 \sqrt {\frac {a}{b x^{3}} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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