Optimal. Leaf size=20 \[ \frac {2 \sqrt {x}}{a \sqrt {a-b x}} \]
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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 {x}}{a \sqrt {a-b x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {x} (a-b x)^{3/2}} \, dx &=\frac {2 \sqrt {x}}{a \sqrt {a-b x}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 20, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {x}}{a \sqrt {a-b x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 17, normalized size = 0.85
method | result | size |
gosper | \(\frac {2 \sqrt {x}}{a \sqrt {-b x +a}}\) | \(17\) |
default | \(\frac {2 \sqrt {x}}{a \sqrt {-b x +a}}\) | \(17\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 16, normalized size = 0.80 \begin {gather*} \frac {2 \, \sqrt {x}}{\sqrt {-b x + a} a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.95, size = 25, normalized size = 1.25 \begin {gather*} -\frac {2 \, \sqrt {-b x + a} \sqrt {x}}{a b x - a^{2}} \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 = 0.46, size = 44, normalized size = 2.20 \begin {gather*} \begin {cases} \frac {2}{a \sqrt {b} \sqrt {\frac {a}{b x} - 1}} & \text {for}\: \left |{\frac {a}{b x}}\right | > 1 \\- \frac {2 i}{a \sqrt {b} \sqrt {- \frac {a}{b x} + 1}} & \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. 53 vs.
\(2 (16) = 32\).
time = 0.89, size = 53, normalized size = 2.65 \begin {gather*} -\frac {4 \, \sqrt {-b} b}{{\left ({\left (\sqrt {-b x + a} \sqrt {-b} - \sqrt {{\left (b x - a\right )} b + a b}\right )}^{2} - a b\right )} {\left | b \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.34, size = 24, normalized size = 1.20 \begin {gather*} \frac {2\,\sqrt {x}\,\sqrt {a-b\,x}}{a^2-a\,b\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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