Optimal. Leaf size=28 \[ -\frac{(d+e x)^3}{3 (a+b x)^3 (b d-a e)} \]
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Rubi [A] time = 0.0046141, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {27, 37} \[ -\frac{(d+e x)^3}{3 (a+b x)^3 (b d-a e)} \]
Antiderivative was successfully verified.
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Rule 27
Rule 37
Rubi steps
\begin{align*} \int \frac{(d+e x)^2}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac{(d+e x)^2}{(a+b x)^4} \, dx\\ &=-\frac{(d+e x)^3}{3 (b d-a e) (a+b x)^3}\\ \end{align*}
Mathematica [A] time = 0.0238475, size = 53, normalized size = 1.89 \[ -\frac{a^2 e^2+a b e (d+3 e x)+b^2 \left (d^2+3 d e x+3 e^2 x^2\right )}{3 b^3 (a+b x)^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.046, size = 70, normalized size = 2.5 \begin{align*}{\frac{e \left ( ae-bd \right ) }{{b}^{3} \left ( bx+a \right ) ^{2}}}-{\frac{{a}^{2}{e}^{2}-2\,abde+{b}^{2}{d}^{2}}{3\,{b}^{3} \left ( bx+a \right ) ^{3}}}-{\frac{{e}^{2}}{{b}^{3} \left ( bx+a \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.12975, size = 113, normalized size = 4.04 \begin{align*} -\frac{3 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + a b d e + a^{2} e^{2} + 3 \,{\left (b^{2} d e + a b e^{2}\right )} x}{3 \,{\left (b^{6} x^{3} + 3 \, a b^{5} x^{2} + 3 \, a^{2} b^{4} x + a^{3} b^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.74185, size = 170, normalized size = 6.07 \begin{align*} -\frac{3 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + a b d e + a^{2} e^{2} + 3 \,{\left (b^{2} d e + a b e^{2}\right )} x}{3 \,{\left (b^{6} x^{3} + 3 \, a b^{5} x^{2} + 3 \, a^{2} b^{4} x + a^{3} b^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.958547, size = 88, normalized size = 3.14 \begin{align*} - \frac{a^{2} e^{2} + a b d e + b^{2} d^{2} + 3 b^{2} e^{2} x^{2} + x \left (3 a b e^{2} + 3 b^{2} d e\right )}{3 a^{3} b^{3} + 9 a^{2} b^{4} x + 9 a b^{5} x^{2} + 3 b^{6} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.21924, size = 78, normalized size = 2.79 \begin{align*} -\frac{3 \, b^{2} x^{2} e^{2} + 3 \, b^{2} d x e + b^{2} d^{2} + 3 \, a b x e^{2} + a b d e + a^{2} e^{2}}{3 \,{\left (b x + a\right )}^{3} b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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