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 = {1979, 2008, 206} \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {\frac {a}{x}+b x^2}}\right )}{3 \sqrt {b}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 1979
Rule 2008
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} \operatorname {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 = 0.03, size = 63, normalized size = 1.97 \[ \frac {2 \sqrt {a+b x^3} \tanh ^{-1}\left (\frac {\sqrt {b} x^{3/2}}{\sqrt {a+b x^3}}\right )}{3 \sqrt {b} \sqrt {x} \sqrt {\frac {a+b x^3}{x}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 102, normalized size = 3.19 \[ \left [\frac {\log \left (-8 \, b^{2} x^{6} - 8 \, a b x^{3} - a^{2} - 4 \, {\left (2 \, b x^{5} + a x^{2}\right )} \sqrt {b} \sqrt {\frac {b x^{3} + a}{x}}\right )}{6 \, \sqrt {b}}, -\frac {\sqrt {-b} \arctan \left (\frac {2 \, \sqrt {-b} x^{2} \sqrt {\frac {b x^{3} + a}{x}}}{2 \, b x^{3} + a}\right )}{3 \, b}\right ] \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.57, size = 477, normalized size = 14.91 \[ -\frac {4 \left (b \,x^{3}+a \right ) \left (i \sqrt {3}-1\right ) \sqrt {-\frac {\left (i \sqrt {3}-3\right ) b x}{\left (i \sqrt {3}-1\right ) \left (-b x +\left (-a \,b^{2}\right )^{\frac {1}{3}}\right )}}\, \left (-b x +\left (-a \,b^{2}\right )^{\frac {1}{3}}\right )^{2} \sqrt {\frac {2 b x +i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}+\left (-a \,b^{2}\right )^{\frac {1}{3}}}{\left (1+i \sqrt {3}\right ) \left (-b x +\left (-a \,b^{2}\right )^{\frac {1}{3}}\right )}}\, \sqrt {\frac {-2 b x +i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}-\left (-a \,b^{2}\right )^{\frac {1}{3}}}{\left (i \sqrt {3}-1\right ) \left (-b x +\left (-a \,b^{2}\right )^{\frac {1}{3}}\right )}}\, \left (\EllipticF \left (\sqrt {-\frac {\left (i \sqrt {3}-3\right ) b x}{\left (i \sqrt {3}-1\right ) \left (-b x +\left (-a \,b^{2}\right )^{\frac {1}{3}}\right )}}, \sqrt {\frac {\left (i \sqrt {3}+3\right ) \left (i \sqrt {3}-1\right )}{\left (1+i \sqrt {3}\right ) \left (i \sqrt {3}-3\right )}}\right )-\EllipticPi \left (\sqrt {-\frac {\left (i \sqrt {3}-3\right ) b x}{\left (i \sqrt {3}-1\right ) \left (-b x +\left (-a \,b^{2}\right )^{\frac {1}{3}}\right )}}, \frac {i \sqrt {3}-1}{i \sqrt {3}-3}, \sqrt {\frac {\left (i \sqrt {3}+3\right ) \left (i \sqrt {3}-1\right )}{\left (1+i \sqrt {3}\right ) \left (i \sqrt {3}-3\right )}}\right )\right )}{\sqrt {\frac {b \,x^{3}+a}{x}}\, \sqrt {\left (b \,x^{3}+a \right ) x}\, \left (i \sqrt {3}-3\right ) \sqrt {\frac {\left (-b x +\left (-a \,b^{2}\right )^{\frac {1}{3}}\right ) \left (2 b x +i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}+\left (-a \,b^{2}\right )^{\frac {1}{3}}\right ) \left (-2 b x +i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}-\left (-a \,b^{2}\right )^{\frac {1}{3}}\right ) x}{b^{2}}}\, b^{2}} \]
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 \[ \int \frac {1}{\sqrt {\frac {b x^{3} + a}{x}}}\,{d x} \]
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 \[ \int \frac {1}{\sqrt {\frac {b\,x^3+a}{x}}} \,d x \]
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 \[ \int \frac {1}{\sqrt {\frac {a + b x^{3}}{x}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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