Optimal. Leaf size=14 \[ -\frac{1}{4 b (a+b x)^4} \]
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Rubi [A] time = 0.002937, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {27, 32} \[ -\frac{1}{4 b (a+b x)^4} \]
Antiderivative was successfully verified.
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Rule 27
Rule 32
Rubi steps
\begin{align*} \int \frac{a+b x}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac{1}{(a+b x)^5} \, dx\\ &=-\frac{1}{4 b (a+b x)^4}\\ \end{align*}
Mathematica [A] time = 0.0031205, size = 14, normalized size = 1. \[ -\frac{1}{4 b (a+b x)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 13, normalized size = 0.9 \begin{align*} -{\frac{1}{4\,b \left ( bx+a \right ) ^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.963375, size = 31, normalized size = 2.21 \begin{align*} -\frac{1}{4 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{2} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.42221, size = 92, normalized size = 6.57 \begin{align*} -\frac{1}{4 \,{\left (b^{5} x^{4} + 4 \, a b^{4} x^{3} + 6 \, a^{2} b^{3} x^{2} + 4 \, a^{3} b^{2} x + a^{4} b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.43722, size = 49, normalized size = 3.5 \begin{align*} - \frac{1}{4 a^{4} b + 16 a^{3} b^{2} x + 24 a^{2} b^{3} x^{2} + 16 a b^{4} x^{3} + 4 b^{5} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13992, size = 16, normalized size = 1.14 \begin{align*} -\frac{1}{4 \,{\left (b x + a\right )}^{4} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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