Optimal. Leaf size=10 \[ -\frac {1}{(a+b)^2 x} \]
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Rubi [A]
time = 0.00, antiderivative size = 10, normalized size of antiderivative = 1.00, 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}
\begin {gather*} -\frac {1}{x (a+b)^2} \end {gather*}
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)^2} \, dx &=\int \frac {1}{(a+b)^2 x^2} \, dx\\ &=\frac {\int \frac {1}{x^2} \, dx}{(a+b)^2}\\ &=-\frac {1}{(a+b)^2 x}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 10, normalized size = 1.00 \begin {gather*} -\frac {1}{(a+b)^2 x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 11, normalized size = 1.10
method | result | size |
gosper | \(-\frac {1}{\left (a +b \right )^{2} x}\) | \(11\) |
default | \(-\frac {1}{\left (a +b \right )^{2} x}\) | \(11\) |
norman | \(-\frac {1}{\left (a +b \right )^{2} x}\) | \(11\) |
risch | \(-\frac {1}{\left (a +b \right )^{2} x}\) | \(11\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 16, normalized size = 1.60 \begin {gather*} -\frac {1}{{\left (a x + b x\right )} {\left (a + b\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.23, size = 18, normalized size = 1.80 \begin {gather*} -\frac {1}{{\left (a^{2} + 2 \, a b + b^{2}\right )} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.01, size = 15, normalized size = 1.50 \begin {gather*} - \frac {1}{x \left (a^{2} + 2 a b + b^{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.53, size = 16, normalized size = 1.60 \begin {gather*} -\frac {1}{{\left (a x + b x\right )} {\left (a + b\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 10, normalized size = 1.00 \begin {gather*} -\frac {1}{x\,{\left (a+b\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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