Optimal. Leaf size=23 \[ \frac {2 \sqrt {a x^2+b x^3}}{b x} \]
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Rubi [A] time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {1588} \begin {gather*} \frac {2 \sqrt {a x^2+b x^3}}{b x} \end {gather*}
Antiderivative was successfully verified.
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Rule 1588
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {a x^2+b x^3}} \, dx &=\frac {2 \sqrt {a x^2+b x^3}}{b x}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 21, normalized size = 0.91 \begin {gather*} \frac {2 \sqrt {x^2 (a+b x)}}{b x} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.03, size = 23, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {a x^2+b x^3}}{b x} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 21, normalized size = 0.91 \begin {gather*} \frac {2 \, \sqrt {b x^{3} + a x^{2}}}{b x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 26, normalized size = 1.13 \begin {gather*} \frac {2}{\sqrt {\frac {b}{x} + \frac {a}{x^{2}}} - \frac {\sqrt {a}}{x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 25, normalized size = 1.09 \begin {gather*} \frac {2 \left (b x +a \right ) x}{\sqrt {b \,x^{3}+a \,x^{2}}\, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.41, size = 12, normalized size = 0.52 \begin {gather*} \frac {2 \, \sqrt {b x + a}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.14, size = 17, normalized size = 0.74 \begin {gather*} \frac {2\,\relax |x|\,\sqrt {a+b\,x}}{b\,x} \end {gather*}
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 \begin {gather*} \int \frac {x}{\sqrt {x^{2} \left (a + b x\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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