Optimal. Leaf size=14 \[ \frac {1}{2} x \sqrt {b x^2} \]
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Rubi [A]
time = 0.00, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, 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 \sqrt {b x^2} \, dx &=\frac {\sqrt {b x^2} \int x \, dx}{x}\\ &=\frac {1}{2} x \sqrt {b x^2}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 14, normalized size = 1.00 \begin {gather*} \frac {1}{2} x \sqrt {b x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 11, normalized size = 0.79
method | result | size |
gosper | \(\frac {x \sqrt {b \,x^{2}}}{2}\) | \(11\) |
default | \(\frac {x \sqrt {b \,x^{2}}}{2}\) | \(11\) |
risch | \(\frac {x \sqrt {b \,x^{2}}}{2}\) | \(11\) |
trager | \(\frac {\left (x -1\right ) \left (x +1\right ) \sqrt {b \,x^{2}}}{2 x}\) | \(19\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 10, normalized size = 0.71 \begin {gather*} \frac {1}{2} \, \sqrt {b x^{2}} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 10, normalized size = 0.71 \begin {gather*} \frac {1}{2} \, \sqrt {b x^{2}} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 10, normalized size = 0.71 \begin {gather*} \frac {x \sqrt {b x^{2}}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.74, size = 10, normalized size = 0.71 \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 = 1.24, size = 8, normalized size = 0.57 \begin {gather*} \frac {\sqrt {b}\,x\,\left |x\right |}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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