Optimal. Leaf size=16 \[ -\frac {1}{2 x \sqrt {b x^2}} \]
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Rubi [A]
time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {15, 30}
\begin {gather*} -\frac {1}{2 x \sqrt {b x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 30
Rubi steps
\begin {align*} \int \frac {1}{x^2 \sqrt {b x^2}} \, dx &=\frac {x \int \frac {1}{x^3} \, dx}{\sqrt {b x^2}}\\ &=-\frac {1}{2 x \sqrt {b x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 15, normalized size = 0.94 \begin {gather*} -\frac {b x}{2 \left (b x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 13, normalized size = 0.81
method | result | size |
gosper | \(-\frac {1}{2 x \sqrt {b \,x^{2}}}\) | \(13\) |
default | \(-\frac {1}{2 x \sqrt {b \,x^{2}}}\) | \(13\) |
risch | \(-\frac {1}{2 x \sqrt {b \,x^{2}}}\) | \(13\) |
trager | \(\frac {\left (x -1\right ) \left (x +1\right ) \sqrt {b \,x^{2}}}{2 b \,x^{3}}\) | \(22\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 8, normalized size = 0.50 \begin {gather*} -\frac {1}{2 \, \sqrt {b} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 15, normalized size = 0.94 \begin {gather*} -\frac {\sqrt {b x^{2}}}{2 \, b x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.19, size = 14, normalized size = 0.88 \begin {gather*} - \frac {1}{2 x \sqrt {b x^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.87, size = 12, normalized size = 0.75 \begin {gather*} -\frac {1}{2 \, \sqrt {b} x^{2} \mathrm {sgn}\left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.97, size = 13, normalized size = 0.81 \begin {gather*} -\frac {1}{2\,\sqrt {b}\,x\,\sqrt {x^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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