Optimal. Leaf size=15 \[ \frac {1}{2} b x \sqrt {b x^2} \]
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Rubi [A]
time = 0.00, antiderivative size = 15, 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} b 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 {\left (b x^2\right )^{3/2}}{x^2} \, dx &=\frac {\left (b \sqrt {b x^2}\right ) \int x \, dx}{x}\\ &=\frac {1}{2} b x \sqrt {b x^2}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 15, normalized size = 1.00 \begin {gather*} \frac {1}{2} b x \sqrt {b x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 13, normalized size = 0.87
method | result | size |
risch | \(\frac {b x \sqrt {b \,x^{2}}}{2}\) | \(12\) |
gosper | \(\frac {\left (b \,x^{2}\right )^{\frac {3}{2}}}{2 x}\) | \(13\) |
default | \(\frac {\left (b \,x^{2}\right )^{\frac {3}{2}}}{2 x}\) | \(13\) |
trager | \(\frac {b \left (x -1\right ) \left (x +1\right ) \sqrt {b \,x^{2}}}{2 x}\) | \(20\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 11, normalized size = 0.73 \begin {gather*} \frac {1}{2} \, \sqrt {b x^{2}} b x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.11, size = 10, normalized size = 0.67 \begin {gather*} \frac {\left (b x^{2}\right )^{\frac {3}{2}}}{2 x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.07, size = 10, normalized size = 0.67 \begin {gather*} \frac {1}{2} \, b^{\frac {3}{2}} 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 = 8, normalized size = 0.53 \begin {gather*} \frac {b^{3/2}\,x\,\left |x\right |}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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