Optimal. Leaf size=21 \[ -\frac {a}{2 x^2}+b \log (x)+\frac {c x^2}{2} \]
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Rubi [A] time = 0.01, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {14} \begin {gather*} -\frac {a}{2 x^2}+b \log (x)+\frac {c x^2}{2} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rubi steps
\begin {align*} \int \frac {a+b x^2+c x^4}{x^3} \, dx &=\int \left (\frac {a}{x^3}+\frac {b}{x}+c x\right ) \, dx\\ &=-\frac {a}{2 x^2}+\frac {c x^2}{2}+b \log (x)\\ \end {align*}
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Mathematica [A] time = 0.00, size = 21, normalized size = 1.00 \begin {gather*} -\frac {a}{2 x^2}+b \log (x)+\frac {c x^2}{2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a+b x^2+c x^4}{x^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.02, size = 22, normalized size = 1.05 \begin {gather*} \frac {c x^{4} + 2 \, b x^{2} \log \relax (x) - a}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 26, normalized size = 1.24 \begin {gather*} \frac {1}{2} \, c x^{2} + \frac {1}{2} \, b \log \left (x^{2}\right ) - \frac {b x^{2} + a}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 18, normalized size = 0.86 \begin {gather*} \frac {c \,x^{2}}{2}+b \ln \relax (x )-\frac {a}{2 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 20, normalized size = 0.95 \begin {gather*} \frac {1}{2} \, c x^{2} + \frac {1}{2} \, b \log \left (x^{2}\right ) - \frac {a}{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 = 17, normalized size = 0.81 \begin {gather*} \frac {c\,x^2}{2}-\frac {a}{2\,x^2}+b\,\ln \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 17, normalized size = 0.81 \begin {gather*} - \frac {a}{2 x^{2}} + b \log {\relax (x )} + \frac {c x^{2}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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