Optimal. Leaf size=25 \[ \frac {d x}{b}+\frac {(b c-a d) \log (a+b x)}{b^2} \]
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Rubi [A]
time = 0.02, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {45}
\begin {gather*} \frac {(b c-a d) \log (a+b x)}{b^2}+\frac {d x}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int \frac {c+d x}{a+b x} \, dx &=\int \left (\frac {d}{b}+\frac {b c-a d}{b (a+b x)}\right ) \, dx\\ &=\frac {d x}{b}+\frac {(b c-a d) \log (a+b x)}{b^2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 25, normalized size = 1.00 \begin {gather*} \frac {d x}{b}+\frac {(b c-a d) \log (a+b x)}{b^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 26, normalized size = 1.04
method | result | size |
default | \(\frac {d x}{b}+\frac {\left (-a d +b c \right ) \ln \left (b x +a \right )}{b^{2}}\) | \(26\) |
norman | \(\frac {d x}{b}-\frac {\left (a d -b c \right ) \ln \left (b x +a \right )}{b^{2}}\) | \(27\) |
risch | \(\frac {d x}{b}-\frac {\ln \left (b x +a \right ) a d}{b^{2}}+\frac {c \ln \left (b x +a \right )}{b}\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 25, normalized size = 1.00 \begin {gather*} \frac {d x}{b} + \frac {{\left (b c - a d\right )} \log \left (b x + a\right )}{b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.51, size = 24, normalized size = 0.96 \begin {gather*} \frac {b d x + {\left (b c - a d\right )} \log \left (b x + a\right )}{b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 20, normalized size = 0.80 \begin {gather*} \frac {d x}{b} - \frac {\left (a d - b c\right ) \log {\left (a + b x \right )}}{b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.33, size = 26, normalized size = 1.04 \begin {gather*} \frac {d x}{b} + \frac {{\left (b c - a d\right )} \log \left ({\left | b x + a \right |}\right )}{b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 26, normalized size = 1.04 \begin {gather*} \frac {d\,x}{b}-\frac {\ln \left (a+b\,x\right )\,\left (a\,d-b\,c\right )}{b^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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