Optimal. Leaf size=32 \[ \frac {d \log (a+b x)}{b^2}-\frac {b c-a d}{b^2 (a+b x)} \]
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Rubi [A] time = 0.02, antiderivative size = 32, 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 = {43} \[ \frac {d \log (a+b x)}{b^2}-\frac {b c-a d}{b^2 (a+b x)} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int \frac {c+d x}{(a+b x)^2} \, dx &=\int \left (\frac {b c-a d}{b (a+b x)^2}+\frac {d}{b (a+b x)}\right ) \, dx\\ &=-\frac {b c-a d}{b^2 (a+b x)}+\frac {d \log (a+b x)}{b^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 31, normalized size = 0.97 \[ \frac {a d-b c}{b^2 (a+b x)}+\frac {d \log (a+b x)}{b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 39, normalized size = 1.22 \[ -\frac {b c - a d - {\left (b d x + a d\right )} \log \left (b x + a\right )}{b^{3} x + a b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.02, size = 57, normalized size = 1.78 \[ -\frac {d {\left (\frac {\log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b} - \frac {a}{{\left (b x + a\right )} b}\right )}}{b} - \frac {c}{{\left (b x + a\right )} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 39, normalized size = 1.22 \[ \frac {a d}{\left (b x +a \right ) b^{2}}-\frac {c}{\left (b x +a \right ) b}+\frac {d \ln \left (b x +a \right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 35, normalized size = 1.09 \[ -\frac {b c - a d}{b^{3} x + a b^{2}} + \frac {d \log \left (b x + a\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.17, size = 31, normalized size = 0.97 \[ \frac {a\,d-b\,c}{b^2\,\left (a+b\,x\right )}+\frac {d\,\ln \left (a+b\,x\right )}{b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.19, size = 27, normalized size = 0.84 \[ \frac {a d - b c}{a b^{2} + b^{3} x} + \frac {d \log {\left (a + b x \right )}}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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