Optimal. Leaf size=19 \[ \frac {2 \sqrt {x}}{a \sqrt {a+b x}} \]
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Rubi [A]
time = 0.00, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, 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 = 19, 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 = 16, normalized size = 0.84
method | result | size |
gosper | \(\frac {2 \sqrt {x}}{a \sqrt {b x +a}}\) | \(16\) |
default | \(\frac {2 \sqrt {x}}{a \sqrt {b x +a}}\) | \(16\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 15, normalized size = 0.79 \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 = 0.81, size = 22, normalized size = 1.16 \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 [A]
time = 0.44, size = 17, normalized size = 0.89 \begin {gather*} \frac {2}{a \sqrt {b} \sqrt {\frac {a}{b x} + 1}} \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. 45 vs.
\(2 (15) = 30\).
time = 0.97, size = 45, normalized size = 2.37 \begin {gather*} \frac {4 \, b^{\frac {3}{2}}}{{\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 = 0.33, size = 22, normalized size = 1.16 \begin {gather*} \frac {2\,\sqrt {x}\,\sqrt {a+b\,x}}{a^2+b\,x\,a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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