Optimal. Leaf size=12 \[ -\frac {1}{2 (a+b)^3 x^2} \]
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Rubi [A]
time = 0.00, antiderivative size = 12, 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}{2 x^2 (a+b)^3} \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)^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*}
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Mathematica [A]
time = 0.00, size = 12, normalized size = 1.00 \begin {gather*} -\frac {1}{2 (a+b)^3 x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 11, normalized size = 0.92
method | result | size |
gosper | \(-\frac {1}{2 \left (a +b \right )^{3} x^{2}}\) | \(11\) |
default | \(-\frac {1}{2 \left (a +b \right )^{3} x^{2}}\) | \(11\) |
norman | \(-\frac {1}{2 \left (a +b \right )^{3} x^{2}}\) | \(11\) |
risch | \(-\frac {1}{2 \left (a +b \right )^{3} x^{2}}\) | \(11\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 16, normalized size = 1.33 \begin {gather*} -\frac {1}{2 \, {\left (a x + b x\right )}^{2} {\left (a + b\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 26 vs.
\(2 (10) = 20\).
time = 2.33, size = 26, normalized size = 2.17 \begin {gather*} -\frac {1}{2 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 27 vs.
\(2 (12) = 24\).
time = 0.02, size = 27, normalized size = 2.25 \begin {gather*} - \frac {1}{2 x^{2} \left (a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.54, size = 16, normalized size = 1.33 \begin {gather*} -\frac {1}{2 \, {\left (a x + b x\right )}^{2} {\left (a + b\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 26, normalized size = 2.17 \begin {gather*} -\frac {1}{2\,x^2\,\left (a^3+3\,a^2\,b+3\,a\,b^2+b^3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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