Optimal. Leaf size=14 \[ -\frac{1}{3 b (a+b x)^3} \]
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Rubi [A] time = 0.0023577, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {27, 32} \[ -\frac{1}{3 b (a+b x)^3} \]
Antiderivative was successfully verified.
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Rule 27
Rule 32
Rubi steps
\begin{align*} \int \frac{1}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac{1}{(a+b x)^4} \, dx\\ &=-\frac{1}{3 b (a+b x)^3}\\ \end{align*}
Mathematica [A] time = 0.0027517, size = 14, normalized size = 1. \[ -\frac{1}{3 b (a+b x)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 13, normalized size = 0.9 \begin{align*} -{\frac{1}{3\,b \left ( bx+a \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.38165, size = 47, normalized size = 3.36 \begin{align*} -\frac{1}{3 \,{\left (b^{4} x^{3} + 3 \, a b^{3} x^{2} + 3 \, a^{2} b^{2} x + a^{3} b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.70062, size = 70, normalized size = 5. \begin{align*} -\frac{1}{3 \,{\left (b^{4} x^{3} + 3 \, a b^{3} x^{2} + 3 \, a^{2} b^{2} x + a^{3} b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.395112, size = 37, normalized size = 2.64 \begin{align*} - \frac{1}{3 a^{3} b + 9 a^{2} b^{2} x + 9 a b^{3} x^{2} + 3 b^{4} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20085, size = 16, normalized size = 1.14 \begin{align*} -\frac{1}{3 \,{\left (b x + a\right )}^{3} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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