Optimal. Leaf size=21 \[ -\frac {2 \sqrt {a+b x}}{a \sqrt {c x}} \]
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Rubi [A]
time = 0.00, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {16, 37}
\begin {gather*} -\frac {2 \sqrt {a+b x}}{a \sqrt {c x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 16
Rule 37
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt {c x} \sqrt {a+b x}} \, dx &=c \int \frac {1}{(c x)^{3/2} \sqrt {a+b x}} \, dx\\ &=-\frac {2 \sqrt {a+b x}}{a \sqrt {c x}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 23, normalized size = 1.10 \begin {gather*} -\frac {2 c x \sqrt {a+b x}}{a (c x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 18, normalized size = 0.86
method | result | size |
gosper | \(-\frac {2 \sqrt {b x +a}}{a \sqrt {c x}}\) | \(18\) |
default | \(-\frac {2 \sqrt {b x +a}}{a \sqrt {c x}}\) | \(18\) |
risch | \(-\frac {2 \sqrt {b x +a}}{a \sqrt {c x}}\) | \(18\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 24, normalized size = 1.14 \begin {gather*} -\frac {2 \, \sqrt {b c x^{2} + a c x}}{a c x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.88, size = 23, normalized size = 1.10 \begin {gather*} -\frac {2 \, \sqrt {b x + a} \sqrt {c x}}{a c x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.57, size = 24, normalized size = 1.14 \begin {gather*} - \frac {2 \sqrt {b} \sqrt {\frac {a}{b x} + 1}}{a \sqrt {c}} \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.
\(2 (17) = 34\).
time = 0.74, size = 35, normalized size = 1.67 \begin {gather*} -\frac {2 \, \sqrt {b x + a} b^{2}}{\sqrt {{\left (b x + a\right )} b c - a b c} a {\left | b \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.18, size = 17, normalized size = 0.81 \begin {gather*} -\frac {2\,\sqrt {a+b\,x}}{a\,\sqrt {c\,x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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