Optimal. Leaf size=27 \[ -\frac {a^2}{2 x^2}+\frac {b^2 x^2}{2}+2 a b \log (x) \]
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Rubi [A]
time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {14}
\begin {gather*} -\frac {a^2}{2 x^2}+2 a b \log (x)+\frac {b^2 x^2}{2} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rubi steps
\begin {align*} \int \frac {a^2+2 a b x^2+b^2 x^4}{x^3} \, dx &=\int \left (\frac {a^2}{x^3}+\frac {2 a b}{x}+b^2 x\right ) \, dx\\ &=-\frac {a^2}{2 x^2}+\frac {b^2 x^2}{2}+2 a b \log (x)\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 27, normalized size = 1.00 \begin {gather*} -\frac {a^2}{2 x^2}+\frac {b^2 x^2}{2}+2 a b \log (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 24, normalized size = 0.89
method | result | size |
default | \(-\frac {a^{2}}{2 x^{2}}+\frac {b^{2} x^{2}}{2}+2 a b \ln \left (x \right )\) | \(24\) |
risch | \(-\frac {a^{2}}{2 x^{2}}+\frac {b^{2} x^{2}}{2}+2 a b \ln \left (x \right )\) | \(24\) |
norman | \(\frac {-\frac {a^{2}}{2}+\frac {b^{2} x^{4}}{2}}{x^{2}}+2 a b \ln \left (x \right )\) | \(26\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 24, normalized size = 0.89 \begin {gather*} \frac {1}{2} \, b^{2} x^{2} + a b \log \left (x^{2}\right ) - \frac {a^{2}}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 27, normalized size = 1.00 \begin {gather*} \frac {b^{2} x^{4} + 4 \, a b x^{2} \log \left (x\right ) - a^{2}}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 24, normalized size = 0.89 \begin {gather*} - \frac {a^{2}}{2 x^{2}} + 2 a b \log {\left (x \right )} + \frac {b^{2} x^{2}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 7.87, size = 32, normalized size = 1.19 \begin {gather*} \frac {1}{2} \, b^{2} x^{2} + a b \log \left (x^{2}\right ) - \frac {2 \, a b x^{2} + a^{2}}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 23, normalized size = 0.85 \begin {gather*} \frac {b^2\,x^2}{2}-\frac {a^2}{2\,x^2}+2\,a\,b\,\ln \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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