Optimal. Leaf size=31 \[ \frac {b c-a d}{d^2 (c+d x)}+\frac {b \log (c+d x)}{d^2} \]
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Rubi [A]
time = 0.01, antiderivative size = 31, 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}{d^2 (c+d x)}+\frac {b \log (c+d x)}{d^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int \frac {a+b x}{(c+d x)^2} \, dx &=\int \left (\frac {-b c+a d}{d (c+d x)^2}+\frac {b}{d (c+d x)}\right ) \, dx\\ &=\frac {b c-a d}{d^2 (c+d x)}+\frac {b \log (c+d x)}{d^2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 31, normalized size = 1.00 \begin {gather*} \frac {b c-a d}{d^2 (c+d x)}+\frac {b \log (c+d x)}{d^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 33, normalized size = 1.06
method | result | size |
default | \(-\frac {a d -b c}{d^{2} \left (d x +c \right )}+\frac {b \ln \left (d x +c \right )}{d^{2}}\) | \(33\) |
norman | \(-\frac {a d -b c}{d^{2} \left (d x +c \right )}+\frac {b \ln \left (d x +c \right )}{d^{2}}\) | \(33\) |
risch | \(-\frac {a}{d \left (d x +c \right )}+\frac {b c}{d^{2} \left (d x +c \right )}+\frac {b \ln \left (d x +c \right )}{d^{2}}\) | \(39\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 34, normalized size = 1.10 \begin {gather*} \frac {b c - a d}{d^{3} x + c d^{2}} + \frac {b \log \left (d x + c\right )}{d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.83, size = 37, normalized size = 1.19 \begin {gather*} \frac {b c - a d + {\left (b d x + b c\right )} \log \left (d x + c\right )}{d^{3} x + c d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.08, size = 27, normalized size = 0.87 \begin {gather*} \frac {b \log {\left (c + d x \right )}}{d^{2}} + \frac {- a d + b c}{c d^{2} + d^{3} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.78, size = 57, normalized size = 1.84 \begin {gather*} -\frac {b {\left (\frac {\log \left (\frac {{\left | d x + c \right |}}{{\left (d x + c\right )}^{2} {\left | d \right |}}\right )}{d} - \frac {c}{{\left (d x + c\right )} d}\right )}}{d} - \frac {a}{{\left (d x + c\right )} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 32, normalized size = 1.03 \begin {gather*} \frac {b\,\ln \left (c+d\,x\right )}{d^2}-\frac {a\,d-b\,c}{d^2\,\left (c+d\,x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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