Optimal. Leaf size=14 \[ -\frac {1}{3 b (a+b x)^3} \]
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Rubi [A]
time = 0.00, antiderivative size = 14, normalized size of antiderivative = 1.00, 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}
\begin {gather*} -\frac {1}{3 b (a+b x)^3} \end {gather*}
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*}
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Mathematica [A]
time = 0.00, size = 14, normalized size = 1.00 \begin {gather*} -\frac {1}{3 b (a+b x)^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.65, size = 13, normalized size = 0.93
method | result | size |
default | \(-\frac {1}{3 b \left (b x +a \right )^{3}}\) | \(13\) |
norman | \(-\frac {1}{3 b \left (b x +a \right )^{3}}\) | \(13\) |
risch | \(-\frac {1}{3 b \left (b x +a \right )^{3}}\) | \(13\) |
gosper | \(-\frac {1}{3 \left (b x +a \right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right ) b}\) | \(31\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 35 vs.
\(2 (12) = 24\).
time = 0.31, size = 35, normalized size = 2.50 \begin {gather*} -\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 {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. 35 vs.
\(2 (12) = 24\).
time = 2.36, size = 35, normalized size = 2.50 \begin {gather*} -\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 {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. 37 vs.
\(2 (12) = 24\).
time = 0.10, size = 37, normalized size = 2.64 \begin {gather*} - \frac {1}{3 a^{3} b + 9 a^{2} b^{2} x + 9 a b^{3} x^{2} + 3 b^{4} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.19, size = 12, normalized size = 0.86 \begin {gather*} -\frac {1}{3 \, {\left (b x + a\right )}^{3} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 37, normalized size = 2.64 \begin {gather*} -\frac {1}{3\,a^3\,b+9\,a^2\,b^2\,x+9\,a\,b^3\,x^2+3\,b^4\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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