Optimal. Leaf size=59 \[ \frac{2 b (b d-a e)}{e^3 (d+e x)}-\frac{(b d-a e)^2}{2 e^3 (d+e x)^2}+\frac{b^2 \log (d+e x)}{e^3} \]
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Rubi [A] time = 0.0379732, antiderivative size = 59, 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} \[ \frac{2 b (b d-a e)}{e^3 (d+e x)}-\frac{(b d-a e)^2}{2 e^3 (d+e x)^2}+\frac{b^2 \log (d+e x)}{e^3} \]
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)^3} \, dx &=\int \frac{(a+b x)^2}{(d+e x)^3} \, dx\\ &=\int \left (\frac{(-b d+a e)^2}{e^2 (d+e x)^3}-\frac{2 b (b d-a e)}{e^2 (d+e x)^2}+\frac{b^2}{e^2 (d+e x)}\right ) \, dx\\ &=-\frac{(b d-a e)^2}{2 e^3 (d+e x)^2}+\frac{2 b (b d-a e)}{e^3 (d+e x)}+\frac{b^2 \log (d+e x)}{e^3}\\ \end{align*}
Mathematica [A] time = 0.0239186, size = 48, normalized size = 0.81 \[ \frac{\frac{(b d-a e) (a e+3 b d+4 b e x)}{(d+e x)^2}+2 b^2 \log (d+e x)}{2 e^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 92, normalized size = 1.6 \begin{align*} -{\frac{{a}^{2}}{2\,e \left ( ex+d \right ) ^{2}}}+{\frac{abd}{{e}^{2} \left ( ex+d \right ) ^{2}}}-{\frac{{b}^{2}{d}^{2}}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}}+{\frac{{b}^{2}\ln \left ( ex+d \right ) }{{e}^{3}}}-2\,{\frac{ab}{{e}^{2} \left ( ex+d \right ) }}+2\,{\frac{{b}^{2}d}{{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 = 1.14444, size = 108, normalized size = 1.83 \begin{align*} \frac{3 \, b^{2} d^{2} - 2 \, a b d e - a^{2} e^{2} + 4 \,{\left (b^{2} d e - a b e^{2}\right )} x}{2 \,{\left (e^{5} x^{2} + 2 \, d e^{4} x + d^{2} e^{3}\right )}} + \frac{b^{2} \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 = 1.6027, size = 205, normalized size = 3.47 \begin{align*} \frac{3 \, b^{2} d^{2} - 2 \, a b d e - a^{2} e^{2} + 4 \,{\left (b^{2} d e - a b e^{2}\right )} x + 2 \,{\left (b^{2} e^{2} x^{2} + 2 \, b^{2} d e x + b^{2} d^{2}\right )} \log \left (e x + d\right )}{2 \,{\left (e^{5} x^{2} + 2 \, d e^{4} x + d^{2} e^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.795259, size = 80, normalized size = 1.36 \begin{align*} \frac{b^{2} \log{\left (d + e x \right )}}{e^{3}} - \frac{a^{2} e^{2} + 2 a b d e - 3 b^{2} d^{2} + x \left (4 a b e^{2} - 4 b^{2} d e\right )}{2 d^{2} e^{3} + 4 d e^{4} x + 2 e^{5} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19965, size = 93, normalized size = 1.58 \begin{align*} b^{2} e^{\left (-3\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{{\left (4 \,{\left (b^{2} d - a b e\right )} x +{\left (3 \, b^{2} d^{2} - 2 \, a b d e - a^{2} e^{2}\right )} e^{\left (-1\right )}\right )} e^{\left (-2\right )}}{2 \,{\left (x e + d\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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