Optimal. Leaf size=10 \[ \frac {\log (c+d x)}{d} \]
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Rubi [A]
time = 0.00, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {640, 31}
\begin {gather*} \frac {\log (c+d x)}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 640
Rubi steps
\begin {align*} \int \frac {a+b x}{a c+(b c+a d) x+b d x^2} \, dx &=\int \frac {1}{c+d x} \, dx\\ &=\frac {\log (c+d x)}{d}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 10, normalized size = 1.00 \begin {gather*} \frac {\log (c+d x)}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.59, size = 11, normalized size = 1.10
method | result | size |
default | \(\frac {\ln \left (d x +c \right )}{d}\) | \(11\) |
norman | \(\frac {\ln \left (d x +c \right )}{d}\) | \(11\) |
risch | \(\frac {\ln \left (d x +c \right )}{d}\) | \(11\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 10, normalized size = 1.00 \begin {gather*} \frac {\log \left (d x + c\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.70, size = 10, normalized size = 1.00 \begin {gather*} \frac {\log \left (d x + c\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.01, size = 7, normalized size = 0.70 \begin {gather*} \frac {\log {\left (c + d x \right )}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.18, size = 11, normalized size = 1.10 \begin {gather*} \frac {\log \left ({\left | d x + c \right |}\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.02, size = 10, normalized size = 1.00 \begin {gather*} \frac {\ln \left (c+d\,x\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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