Optimal. Leaf size=51 \[ -\frac{(b d-a e)^2}{b^3 (a+b x)}+\frac{2 e (b d-a e) \log (a+b x)}{b^3}+\frac{e^2 x}{b^2} \]
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Rubi [A] time = 0.0393365, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {27, 43} \[ -\frac{(b d-a e)^2}{b^3 (a+b x)}+\frac{2 e (b d-a e) \log (a+b x)}{b^3}+\frac{e^2 x}{b^2} \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin{align*} \int \frac{(d+e x)^2}{a^2+2 a b x+b^2 x^2} \, dx &=\int \frac{(d+e x)^2}{(a+b x)^2} \, dx\\ &=\int \left (\frac{e^2}{b^2}+\frac{(b d-a e)^2}{b^2 (a+b x)^2}+\frac{2 e (b d-a e)}{b^2 (a+b x)}\right ) \, dx\\ &=\frac{e^2 x}{b^2}-\frac{(b d-a e)^2}{b^3 (a+b x)}+\frac{2 e (b d-a e) \log (a+b x)}{b^3}\\ \end{align*}
Mathematica [A] time = 0.0371345, size = 47, normalized size = 0.92 \[ \frac{-\frac{(b d-a e)^2}{a+b x}+2 e (b d-a e) \log (a+b x)+b e^2 x}{b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 86, normalized size = 1.7 \begin{align*}{\frac{{e}^{2}x}{{b}^{2}}}-2\,{\frac{{e}^{2}\ln \left ( bx+a \right ) a}{{b}^{3}}}+2\,{\frac{e\ln \left ( bx+a \right ) d}{{b}^{2}}}-{\frac{{a}^{2}{e}^{2}}{{b}^{3} \left ( bx+a \right ) }}+2\,{\frac{ade}{{b}^{2} \left ( bx+a \right ) }}-{\frac{{d}^{2}}{b \left ( bx+a \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.13578, size = 90, normalized size = 1.76 \begin{align*} \frac{e^{2} x}{b^{2}} - \frac{b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}}{b^{4} x + a b^{3}} + \frac{2 \,{\left (b d e - a e^{2}\right )} \log \left (b x + a\right )}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.67671, size = 184, normalized size = 3.61 \begin{align*} \frac{b^{2} e^{2} x^{2} + a b e^{2} x - b^{2} d^{2} + 2 \, a b d e - a^{2} e^{2} + 2 \,{\left (a b d e - a^{2} e^{2} +{\left (b^{2} d e - a b e^{2}\right )} x\right )} \log \left (b x + a\right )}{b^{4} x + a b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.592445, size = 60, normalized size = 1.18 \begin{align*} - \frac{a^{2} e^{2} - 2 a b d e + b^{2} d^{2}}{a b^{3} + b^{4} x} + \frac{e^{2} x}{b^{2}} - \frac{2 e \left (a e - b d\right ) \log{\left (a + b x \right )}}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24279, size = 86, normalized size = 1.69 \begin{align*} \frac{x e^{2}}{b^{2}} + \frac{2 \,{\left (b d e - a e^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{3}} - \frac{b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}}{{\left (b x + a\right )} b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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