Optimal. Leaf size=12 \[ -\frac{1}{2 x^2 (a+b)^3} \]
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Rubi [A] time = 0.0030896, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6, 12, 30} \[ -\frac{1}{2 x^2 (a+b)^3} \]
Antiderivative was successfully verified.
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Rule 6
Rule 12
Rule 30
Rubi steps
\begin{align*} \int \frac{1}{(a x+b x)^3} \, dx &=\int \frac{1}{(a+b)^3 x^3} \, dx\\ &=\frac{\int \frac{1}{x^3} \, dx}{(a+b)^3}\\ &=-\frac{1}{2 (a+b)^3 x^2}\\ \end{align*}
Mathematica [A] time = 0.0034175, size = 12, normalized size = 1. \[ -\frac{1}{2 x^2 (a+b)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 11, normalized size = 0.9 \begin{align*} -{\frac{1}{2\, \left ( a+b \right ) ^{3}{x}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.995037, size = 22, normalized size = 1.83 \begin{align*} -\frac{1}{2 \,{\left (a x + b x\right )}^{2}{\left (a + b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.820803, size = 59, normalized size = 4.92 \begin{align*} -\frac{1}{2 \,{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.107827, size = 27, normalized size = 2.25 \begin{align*} - \frac{1}{2 x^{2} \left (a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15199, size = 22, normalized size = 1.83 \begin{align*} -\frac{1}{2 \,{\left (a x + b x\right )}^{2}{\left (a + b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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