Optimal. Leaf size=30 \[ \frac {c \log (x)}{a}-\frac {(b c-a d) \log (a+b x)}{a b} \]
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Rubi [A]
time = 0.01, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {78}
\begin {gather*} \frac {c \log (x)}{a}-\frac {(b c-a d) \log (a+b x)}{a b} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rubi steps
\begin {align*} \int \frac {c+d x}{x (a+b x)} \, dx &=\int \left (\frac {c}{a x}+\frac {-b c+a d}{a (a+b x)}\right ) \, dx\\ &=\frac {c \log (x)}{a}-\frac {(b c-a d) \log (a+b x)}{a b}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 29, normalized size = 0.97 \begin {gather*} \frac {c \log (x)}{a}+\frac {(-b c+a d) \log (a+b x)}{a b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 30, normalized size = 1.00
method | result | size |
default | \(\frac {\left (a d -b c \right ) \ln \left (b x +a \right )}{a b}+\frac {c \ln \left (x \right )}{a}\) | \(30\) |
norman | \(\frac {\left (a d -b c \right ) \ln \left (b x +a \right )}{a b}+\frac {c \ln \left (x \right )}{a}\) | \(30\) |
risch | \(\frac {c \ln \left (x \right )}{a}+\frac {\ln \left (-b x -a \right ) d}{b}-\frac {\ln \left (-b x -a \right ) c}{a}\) | \(38\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.33, size = 30, normalized size = 1.00 \begin {gather*} \frac {c \log \left (x\right )}{a} - \frac {{\left (b c - a d\right )} \log \left (b x + a\right )}{a b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.50, size = 29, normalized size = 0.97 \begin {gather*} \frac {b c \log \left (x\right ) - {\left (b c - a d\right )} \log \left (b x + a\right )}{a b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.25, size = 41, normalized size = 1.37 \begin {gather*} \frac {c \log {\left (x \right )}}{a} + \frac {\left (a d - b c\right ) \log {\left (x + \frac {- a c + \frac {a \left (a d - b c\right )}{b}}{a d - 2 b c} \right )}}{a b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.32, size = 32, normalized size = 1.07 \begin {gather*} \frac {c \log \left ({\left | x \right |}\right )}{a} - \frac {{\left (b c - a d\right )} \log \left ({\left | b x + a \right |}\right )}{a b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.35, size = 28, normalized size = 0.93 \begin {gather*} \frac {c\,\ln \left (x\right )}{a}-\ln \left (a+b\,x\right )\,\left (\frac {c}{a}-\frac {d}{b}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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