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 = 0.01, antiderivative size = 36, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {\log (a+b x)}{b d-a e}-\frac {\log (d+e x)}{b d-a e} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 31
Rule 36
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*}
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Mathematica [A] time = 0.01, size = 26, normalized size = 0.72 \begin {gather*} \frac {\log (a+b x)-\log (d+e x)}{b d-a e} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a+b x}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.40, size = 26, normalized size = 0.72 \begin {gather*} \frac {\log \left (b x + a\right ) - \log \left (e x + d\right )}{b d - a e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 49, normalized size = 1.36 \begin {gather*} \frac {b \log \left ({\left | b x + a \right |}\right )}{b^{2} d - a b e} - \frac {e \log \left ({\left | x e + d \right |}\right )}{b d e - a e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 37, normalized size = 1.03 \begin {gather*} -\frac {\ln \left (b x +a \right )}{a e -b d}+\frac {\ln \left (e x +d \right )}{a e -b d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.62, size = 36, normalized size = 1.00 \begin {gather*} \frac {\log \left (b x + a\right )}{b d - a e} - \frac {\log \left (e x + d\right )}{b d - a e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 40, normalized size = 1.11 \begin {gather*} \frac {\mathrm {atan}\left (\frac {b\,d\,2{}\mathrm {i}+b\,e\,x\,2{}\mathrm {i}}{a\,e-b\,d}+1{}\mathrm {i}\right )\,2{}\mathrm {i}}{a\,e-b\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.38, size = 128, normalized size = 3.56 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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