Optimal. Leaf size=36 \[ \frac{\log (a+b x)}{b d-a e}-\frac{\log (d+e x)}{b d-a e} \]
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Rubi [A] time = 0.0078569, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {27, 36, 31} \[ \frac{\log (a+b x)}{b d-a e}-\frac{\log (d+e x)}{b d-a e} \]
Antiderivative was successfully verified.
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Rule 27
Rule 36
Rule 31
Rubi steps
\begin{align*} \int \frac{a+b x}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )} \, dx &=\int \frac{1}{(a+b x) (d+e x)} \, dx\\ &=\frac{b \int \frac{1}{a+b x} \, dx}{b d-a e}-\frac{e \int \frac{1}{d+e x} \, dx}{b d-a e}\\ &=\frac{\log (a+b x)}{b d-a e}-\frac{\log (d+e x)}{b d-a e}\\ \end{align*}
Mathematica [A] time = 0.0122795, size = 26, normalized size = 0.72 \[ \frac{\log (a+b x)-\log (d+e x)}{b d-a e} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 37, normalized size = 1. \begin{align*}{\frac{\ln \left ( ex+d \right ) }{ae-bd}}-{\frac{\ln \left ( bx+a \right ) }{ae-bd}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.959161, size = 49, normalized size = 1.36 \begin{align*} \frac{\log \left (b x + a\right )}{b d - a e} - \frac{\log \left (e x + d\right )}{b d - a e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.50342, size = 58, normalized size = 1.61 \begin{align*} \frac{\log \left (b x + a\right ) - \log \left (e x + d\right )}{b d - a e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.323191, size = 128, normalized size = 3.56 \begin{align*} \frac{\log{\left (x + \frac{- \frac{a^{2} e^{2}}{a e - b d} + \frac{2 a b d e}{a e - b d} + a e - \frac{b^{2} d^{2}}{a e - b d} + b d}{2 b e} \right )}}{a e - b d} - \frac{\log{\left (x + \frac{\frac{a^{2} e^{2}}{a e - b d} - \frac{2 a b d e}{a e - b d} + a e + \frac{b^{2} d^{2}}{a e - b d} + b d}{2 b e} \right )}}{a e - b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1129, size = 93, normalized size = 2.58 \begin{align*} \frac{\log \left (\frac{{\left | 2 \, b x e + b d + a e -{\left | b d - a e \right |} \right |}}{{\left | 2 \, b x e + b d + a e +{\left | b d - a e \right |} \right |}}\right )}{{\left | b d - a e \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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