Optimal. Leaf size=24 \[ -\frac {a^2}{4 x^4}-\frac {a b}{x^2}+b^2 \log (x) \]
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Rubi [A]
time = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 45}
\begin {gather*} -\frac {a^2}{4 x^4}-\frac {a b}{x^2}+b^2 \log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^2}{x^5} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {(a+b x)^2}{x^3} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {a^2}{x^3}+\frac {2 a b}{x^2}+\frac {b^2}{x}\right ) \, dx,x,x^2\right )\\ &=-\frac {a^2}{4 x^4}-\frac {a b}{x^2}+b^2 \log (x)\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 24, normalized size = 1.00 \begin {gather*} -\frac {a^2}{4 x^4}-\frac {a b}{x^2}+b^2 \log (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 23, normalized size = 0.96
method | result | size |
default | \(-\frac {a^{2}}{4 x^{4}}-\frac {a b}{x^{2}}+b^{2} \ln \left (x \right )\) | \(23\) |
norman | \(\frac {-\frac {1}{4} a^{2}-a b \,x^{2}}{x^{4}}+b^{2} \ln \left (x \right )\) | \(25\) |
risch | \(\frac {-\frac {1}{4} a^{2}-a b \,x^{2}}{x^{4}}+b^{2} \ln \left (x \right )\) | \(25\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 26, normalized size = 1.08 \begin {gather*} \frac {1}{2} \, b^{2} \log \left (x^{2}\right ) - \frac {4 \, a b x^{2} + a^{2}}{4 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.90, size = 28, normalized size = 1.17 \begin {gather*} \frac {4 \, b^{2} x^{4} \log \left (x\right ) - 4 \, a b x^{2} - a^{2}}{4 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 24, normalized size = 1.00 \begin {gather*} b^{2} \log {\left (x \right )} + \frac {- a^{2} - 4 a b x^{2}}{4 x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.35, size = 34, normalized size = 1.42 \begin {gather*} \frac {1}{2} \, b^{2} \log \left (x^{2}\right ) - \frac {3 \, b^{2} x^{4} + 4 \, a b x^{2} + a^{2}}{4 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 24, normalized size = 1.00 \begin {gather*} b^2\,\ln \left (x\right )-\frac {\frac {a^2}{4}+b\,a\,x^2}{x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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