Optimal. Leaf size=8 \[ \log (x (b+c x)) \]
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Rubi [A]
time = 0.00, antiderivative size = 9, normalized size of antiderivative = 1.12, number of steps
used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {78}
\begin {gather*} \log (b+c x)+\log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rubi steps
\begin {align*} \int \frac {b+2 c x}{x (b+c x)} \, dx &=\int \left (\frac {1}{x}+\frac {c}{b+c x}\right ) \, dx\\ &=\log (x)+\log (b+c x)\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 9, normalized size = 1.12 \begin {gather*} \log (x)+\log (b+c x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.31, size = 9, normalized size = 1.12
method | result | size |
default | \(\ln \left (x \left (c x +b \right )\right )\) | \(9\) |
norman | \(\ln \left (x \right )+\ln \left (c x +b \right )\) | \(10\) |
risch | \(\ln \left (c \,x^{2}+b x \right )\) | \(11\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 9, normalized size = 1.12 \begin {gather*} \log \left (c x + b\right ) + \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.98, size = 10, normalized size = 1.25 \begin {gather*} \log \left (c x^{2} + b x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 8, normalized size = 1.00 \begin {gather*} \log {\left (b x + c x^{2} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.58, size = 11, normalized size = 1.38 \begin {gather*} \log \left ({\left | c x + b \right |}\right ) + \log \left ({\left | x \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.66, size = 8, normalized size = 1.00 \begin {gather*} \ln \left (x\,\left (b+c\,x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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