Optimal. Leaf size=19 \[ \frac {x^3}{4 b \sqrt {\frac {b}{x^2}}} \]
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Rubi [A]
time = 0.00, antiderivative size = 19, 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 {x^3}{4 b \sqrt {\frac {b}{x^2}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 30
Rubi steps
\begin {align*} \int \frac {1}{\left (\frac {b}{x^2}\right )^{3/2}} \, dx &=\frac {\int x^3 \, dx}{b \sqrt {\frac {b}{x^2}} x}\\ &=\frac {x^3}{4 b \sqrt {\frac {b}{x^2}}}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 14, normalized size = 0.74 \begin {gather*} \frac {x}{4 \left (\frac {b}{x^2}\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 11, normalized size = 0.58
method | result | size |
gosper | \(\frac {x}{4 \left (\frac {b}{x^{2}}\right )^{\frac {3}{2}}}\) | \(11\) |
default | \(\frac {x}{4 \left (\frac {b}{x^{2}}\right )^{\frac {3}{2}}}\) | \(11\) |
risch | \(\frac {x^{3}}{4 b \sqrt {\frac {b}{x^{2}}}}\) | \(16\) |
trager | \(\frac {x \left (x^{3}+x^{2}+x +1\right ) \left (x -1\right ) \sqrt {\frac {b}{x^{2}}}}{4 b^{2}}\) | \(26\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 10, normalized size = 0.53 \begin {gather*} \frac {x}{4 \, \left (\frac {b}{x^{2}}\right )^{\frac {3}{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 = 0.79 \begin {gather*} \frac {x^{5} \sqrt {\frac {b}{x^{2}}}}{4 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.20, size = 10, normalized size = 0.53 \begin {gather*} \frac {x}{4 \left (\frac {b}{x^{2}}\right )^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.55, size = 12, normalized size = 0.63 \begin {gather*} \frac {x^{4}}{4 \, b^{\frac {3}{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 = 0.93, size = 13, normalized size = 0.68 \begin {gather*} \frac {x^5\,\sqrt {\frac {1}{x^2}}}{4\,b^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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