Optimal. Leaf size=51 \[ -\frac{(b d-a e)^2}{e^3 (d+e x)}-\frac{2 b (b d-a e) \log (d+e x)}{e^3}+\frac{b^2 x}{e^2} \]
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Rubi [A] time = 0.0477378, antiderivative size = 51, normalized size of antiderivative = 1., 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} \[ -\frac{(b d-a e)^2}{e^3 (d+e x)}-\frac{2 b (b d-a e) \log (d+e x)}{e^3}+\frac{b^2 x}{e^2} \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin{align*} \int \frac{a^2+2 a b x+b^2 x^2}{(d+e x)^2} \, dx &=\int \frac{(a+b x)^2}{(d+e x)^2} \, dx\\ &=\int \left (\frac{b^2}{e^2}+\frac{(-b d+a e)^2}{e^2 (d+e x)^2}-\frac{2 b (b d-a e)}{e^2 (d+e x)}\right ) \, dx\\ &=\frac{b^2 x}{e^2}-\frac{(b d-a e)^2}{e^3 (d+e x)}-\frac{2 b (b d-a e) \log (d+e x)}{e^3}\\ \end{align*}
Mathematica [A] time = 0.0369414, size = 47, normalized size = 0.92 \[ \frac{-\frac{(b d-a e)^2}{d+e x}+2 b (a e-b d) \log (d+e x)+b^2 e x}{e^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 86, normalized size = 1.7 \begin{align*}{\frac{{b}^{2}x}{{e}^{2}}}+2\,{\frac{b\ln \left ( ex+d \right ) a}{{e}^{2}}}-2\,{\frac{{b}^{2}\ln \left ( ex+d \right ) d}{{e}^{3}}}-{\frac{{a}^{2}}{e \left ( ex+d \right ) }}+2\,{\frac{abd}{{e}^{2} \left ( ex+d \right ) }}-{\frac{{b}^{2}{d}^{2}}{{e}^{3} \left ( ex+d \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.18164, size = 90, normalized size = 1.76 \begin{align*} \frac{b^{2} x}{e^{2}} - \frac{b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}}{e^{4} x + d e^{3}} - \frac{2 \,{\left (b^{2} d - a b e\right )} \log \left (e x + d\right )}{e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.74924, size = 184, normalized size = 3.61 \begin{align*} \frac{b^{2} e^{2} x^{2} + b^{2} d e x - b^{2} d^{2} + 2 \, a b d e - a^{2} e^{2} - 2 \,{\left (b^{2} d^{2} - a b d e +{\left (b^{2} d e - a b e^{2}\right )} x\right )} \log \left (e x + d\right )}{e^{4} x + d e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.553262, size = 60, normalized size = 1.18 \begin{align*} \frac{b^{2} x}{e^{2}} + \frac{2 b \left (a e - b d\right ) \log{\left (d + e x \right )}}{e^{3}} - \frac{a^{2} e^{2} - 2 a b d e + b^{2} d^{2}}{d e^{3} + e^{4} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.14451, size = 150, normalized size = 2.94 \begin{align*} -2 \,{\left (e^{\left (-1\right )} \log \left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) - \frac{d e^{\left (-1\right )}}{x e + d}\right )} a b e^{\left (-1\right )} +{\left (2 \, d e^{\left (-3\right )} \log \left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) +{\left (x e + d\right )} e^{\left (-3\right )} - \frac{d^{2} e^{\left (-3\right )}}{x e + d}\right )} b^{2} - \frac{a^{2} e^{\left (-1\right )}}{x e + d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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