Optimal. Leaf size=20 \[ -\frac {2 \sqrt {a-b x}}{a \sqrt {x}} \]
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Rubi [A]
time = 0.00, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {37}
\begin {gather*} -\frac {2 \sqrt {a-b x}}{a \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rubi steps
\begin {align*} \int \frac {1}{x^{3/2} \sqrt {a-b x}} \, dx &=-\frac {2 \sqrt {a-b x}}{a \sqrt {x}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 20, normalized size = 1.00 \begin {gather*} -\frac {2 \sqrt {a-b x}}{a \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 17, normalized size = 0.85
method | result | size |
gosper | \(-\frac {2 \sqrt {-b x +a}}{a \sqrt {x}}\) | \(17\) |
default | \(-\frac {2 \sqrt {-b x +a}}{a \sqrt {x}}\) | \(17\) |
risch | \(-\frac {2 \sqrt {-b x +a}}{a \sqrt {x}}\) | \(17\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 16, normalized size = 0.80 \begin {gather*} -\frac {2 \, \sqrt {-b x + a}}{a \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.07, size = 16, normalized size = 0.80 \begin {gather*} -\frac {2 \, \sqrt {-b x + a}}{a \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.48, size = 46, normalized size = 2.30 \begin {gather*} \begin {cases} - \frac {2 \sqrt {b} \sqrt {\frac {a}{b x} - 1}}{a} & \text {for}\: \left |{\frac {a}{b x}}\right | > 1 \\- \frac {2 i \sqrt {b} \sqrt {- \frac {a}{b x} + 1}}{a} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 35 vs.
\(2 (16) = 32\).
time = 1.34, size = 35, normalized size = 1.75 \begin {gather*} -\frac {2 \, \sqrt {-b x + a} b^{2}}{\sqrt {{\left (b x - a\right )} b + a b} a {\left | b \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.40, size = 16, normalized size = 0.80 \begin {gather*} -\frac {2\,\sqrt {a-b\,x}}{a\,\sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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