Optimal. Leaf size=20 \[ -\frac {b^2}{x}+a^2 x+2 a b \log (x) \]
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Rubi [A]
time = 0.01, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {199, 45}
\begin {gather*} a^2 x+2 a b \log (x)-\frac {b^2}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 199
Rubi steps
\begin {align*} \int \left (a+\frac {b}{x}\right )^2 \, dx &=\int \frac {(b+a x)^2}{x^2} \, dx\\ &=\int \left (a^2+\frac {b^2}{x^2}+\frac {2 a b}{x}\right ) \, dx\\ &=-\frac {b^2}{x}+a^2 x+2 a b \log (x)\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 20, normalized size = 1.00 \begin {gather*} -\frac {b^2}{x}+a^2 x+2 a b \log (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 21, normalized size = 1.05
method | result | size |
default | \(-\frac {b^{2}}{x}+a^{2} x +2 a b \ln \left (x \right )\) | \(21\) |
risch | \(-\frac {b^{2}}{x}+a^{2} x +2 a b \ln \left (x \right )\) | \(21\) |
norman | \(\frac {a^{2} x^{2}-b^{2}}{x}+2 a b \ln \left (x \right )\) | \(25\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 20, normalized size = 1.00 \begin {gather*} a^{2} x + 2 \, a b \log \left (x\right ) - \frac {b^{2}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 24, normalized size = 1.20 \begin {gather*} \frac {a^{2} x^{2} + 2 \, a b x \log \left (x\right ) - b^{2}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 17, normalized size = 0.85 \begin {gather*} a^{2} x + 2 a b \log {\left (x \right )} - \frac {b^{2}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.96, size = 21, normalized size = 1.05 \begin {gather*} a^{2} x + 2 \, a b \log \left ({\left | x \right |}\right ) - \frac {b^{2}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 20, normalized size = 1.00 \begin {gather*} a^2\,x-\frac {b^2}{x}+2\,a\,b\,\ln \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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