Optimal. Leaf size=32 \[ \frac {e \log (a+b x)}{b^2}-\frac {b d-a e}{b^2 (a+b x)} \]
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Rubi [A] time = 0.02, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {27, 43} \[ \frac {e \log (a+b x)}{b^2}-\frac {b d-a e}{b^2 (a+b x)} \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin {align*} \int \frac {d+e x}{a^2+2 a b x+b^2 x^2} \, dx &=\int \frac {d+e x}{(a+b x)^2} \, dx\\ &=\int \left (\frac {b d-a e}{b (a+b x)^2}+\frac {e}{b (a+b x)}\right ) \, dx\\ &=-\frac {b d-a e}{b^2 (a+b x)}+\frac {e \log (a+b x)}{b^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 31, normalized size = 0.97 \[ \frac {a e-b d}{b^2 (a+b x)}+\frac {e \log (a+b x)}{b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.95, size = 39, normalized size = 1.22 \[ -\frac {b d - a e - {\left (b e x + a e\right )} \log \left (b x + a\right )}{b^{3} x + a b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 35, normalized size = 1.09 \[ \frac {e \log \left ({\left | b x + a \right |}\right )}{b^{2}} - \frac {b d - a e}{{\left (b x + a\right )} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 39, normalized size = 1.22 \[ \frac {a e}{\left (b x +a \right ) b^{2}}-\frac {d}{\left (b x +a \right ) b}+\frac {e \ln \left (b x +a \right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.26, size = 35, normalized size = 1.09 \[ -\frac {b d - a e}{b^{3} x + a b^{2}} + \frac {e \log \left (b x + a\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.50, size = 31, normalized size = 0.97 \[ \frac {a\,e-b\,d}{b^2\,\left (a+b\,x\right )}+\frac {e\,\ln \left (a+b\,x\right )}{b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 27, normalized size = 0.84 \[ \frac {a e - b d}{a b^{2} + b^{3} x} + \frac {e \log {\left (a + b x \right )}}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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