Optimal. Leaf size=38 \[ -\frac{b d-a e}{3 b^2 (a+b x)^3}-\frac{e}{2 b^2 (a+b x)^2} \]
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Rubi [A] time = 0.0219057, antiderivative size = 38, 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} \[ -\frac{b d-a e}{3 b^2 (a+b x)^3}-\frac{e}{2 b^2 (a+b x)^2} \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin{align*} \int \frac{d+e x}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac{d+e x}{(a+b x)^4} \, dx\\ &=\int \left (\frac{b d-a e}{b (a+b x)^4}+\frac{e}{b (a+b x)^3}\right ) \, dx\\ &=-\frac{b d-a e}{3 b^2 (a+b x)^3}-\frac{e}{2 b^2 (a+b x)^2}\\ \end{align*}
Mathematica [A] time = 0.0083979, size = 27, normalized size = 0.71 \[ -\frac{a e+2 b d+3 b e x}{6 b^2 (a+b x)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 35, normalized size = 0.9 \begin{align*} -{\frac{e}{2\,{b}^{2} \left ( bx+a \right ) ^{2}}}-{\frac{-ae+bd}{3\,{b}^{2} \left ( bx+a \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.02562, size = 68, normalized size = 1.79 \begin{align*} -\frac{3 \, b e x + 2 \, b d + a e}{6 \,{\left (b^{5} x^{3} + 3 \, a b^{4} x^{2} + 3 \, a^{2} b^{3} x + a^{3} b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.69501, size = 105, normalized size = 2.76 \begin{align*} -\frac{3 \, b e x + 2 \, b d + a e}{6 \,{\left (b^{5} x^{3} + 3 \, a b^{4} x^{2} + 3 \, a^{2} b^{3} x + a^{3} b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.517849, size = 53, normalized size = 1.39 \begin{align*} - \frac{a e + 2 b d + 3 b e x}{6 a^{3} b^{2} + 18 a^{2} b^{3} x + 18 a b^{4} x^{2} + 6 b^{5} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17718, size = 36, normalized size = 0.95 \begin{align*} -\frac{3 \, b x e + 2 \, b d + a e}{6 \,{\left (b x + a\right )}^{3} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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