Optimal. Leaf size=14 \[ -\frac {2}{b \sqrt {a+b x}} \]
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Rubi [A]
time = 0.00, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {32}
\begin {gather*} -\frac {2}{b \sqrt {a+b x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 32
Rubi steps
\begin {align*} \int \frac {1}{(a+b x)^{3/2}} \, dx &=-\frac {2}{b \sqrt {a+b x}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 14, normalized size = 1.00 \begin {gather*} -\frac {2}{b \sqrt {a+b x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 13, normalized size = 0.93
method | result | size |
gosper | \(-\frac {2}{b \sqrt {b x +a}}\) | \(13\) |
derivativedivides | \(-\frac {2}{b \sqrt {b x +a}}\) | \(13\) |
default | \(-\frac {2}{b \sqrt {b x +a}}\) | \(13\) |
trager | \(-\frac {2}{b \sqrt {b x +a}}\) | \(13\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 12, normalized size = 0.86 \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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Fricas [A]
time = 0.48, size = 20, normalized size = 1.43 \begin {gather*} -\frac {2 \, \sqrt {b x + a}}{b^{2} x + a b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.01, size = 12, normalized size = 0.86 \begin {gather*} - \frac {2}{b \sqrt {a + b x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.68, size = 12, normalized size = 0.86 \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 = 0.02, size = 12, normalized size = 0.86 \begin {gather*} -\frac {2}{b\,\sqrt {a+b\,x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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