Optimal. Leaf size=21 \[ -\frac {2 \sqrt {b x+c x^2}}{b x} \]
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Rubi [A]
time = 0.01, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {664}
\begin {gather*} -\frac {2 \sqrt {b x+c x^2}}{b x} \end {gather*}
Antiderivative was successfully verified.
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Rule 664
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt {b x+c x^2}} \, dx &=-\frac {2 \sqrt {b x+c x^2}}{b x}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 21, normalized size = 1.00 \begin {gather*} -\frac {2 (b+c x)}{b \sqrt {x (b+c x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.41, size = 20, normalized size = 0.95
method | result | size |
default | \(-\frac {2 \sqrt {c \,x^{2}+b x}}{b x}\) | \(20\) |
trager | \(-\frac {2 \sqrt {c \,x^{2}+b x}}{b x}\) | \(20\) |
risch | \(-\frac {2 \left (c x +b \right )}{b \sqrt {x \left (c x +b \right )}}\) | \(20\) |
gosper | \(-\frac {2 \left (c x +b \right )}{b \sqrt {c \,x^{2}+b x}}\) | \(22\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 19, normalized size = 0.90 \begin {gather*} -\frac {2 \, \sqrt {c x^{2} + b x}}{b x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.53, size = 19, normalized size = 0.90 \begin {gather*} -\frac {2 \, \sqrt {c x^{2} + b x}}{b x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \sqrt {x \left (b + c x\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.06, size = 23, normalized size = 1.10 \begin {gather*} \frac {2}{\sqrt {c} x - \sqrt {c x^{2} + b x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.17, size = 19, normalized size = 0.90 \begin {gather*} -\frac {2\,\sqrt {c\,x^2+b\,x}}{b\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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