Optimal. Leaf size=32 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {\frac {a}{x^3}+b x^2}}\right )}{5 \sqrt {b}} \]
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Rubi [A] time = 0.02, 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} \begin {gather*} \frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {\frac {a}{x^3}+b x^2}}\right )}{5 \sqrt {b}} \end {gather*}
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^5}{x^3}}} \, dx &=\int \frac {1}{\sqrt {\frac {a}{x^3}+b x^2}} \, dx\\ &=\frac {2}{5} \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {\frac {a}{x^3}+b x^2}}\right )\\ &=\frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {\frac {a}{x^3}+b x^2}}\right )}{5 \sqrt {b}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 63, normalized size = 1.97 \begin {gather*} \frac {2 \sqrt {a+b x^5} \tanh ^{-1}\left (\frac {\sqrt {b} x^{5/2}}{\sqrt {a+b x^5}}\right )}{5 \sqrt {b} x^{3/2} \sqrt {\frac {a+b x^5}{x^3}}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 75.15, size = 64, normalized size = 2.00 \begin {gather*} \frac {2 x^{3/2} \sqrt {\frac {a+b x^5}{x^3}} \log \left (\sqrt {a+b x^5}+\sqrt {b} x^{5/2}\right )}{5 \sqrt {b} \sqrt {a+b x^5}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 102, normalized size = 3.19 \begin {gather*} \left [\frac {\log \left (-8 \, b^{2} x^{10} - 8 \, a b x^{5} - a^{2} - 4 \, {\left (2 \, b x^{9} + a x^{4}\right )} \sqrt {b} \sqrt {\frac {b x^{5} + a}{x^{3}}}\right )}{10 \, \sqrt {b}}, -\frac {\sqrt {-b} \arctan \left (\frac {2 \, \sqrt {-b} x^{4} \sqrt {\frac {b x^{5} + a}{x^{3}}}}{2 \, b x^{5} + a}\right )}{5 \, b}\right ] \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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maple [F] time = 0.12, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {\frac {b \,x^{5}+a}{x^{3}}}}\, dx \end {gather*}
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*} \int \frac {1}{\sqrt {\frac {b x^{5} + a}{x^{3}}}}\,{d x} \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^5+a}{x^3}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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