Optimal. Leaf size=19 \[ \frac {1}{2} \log \left (a-b x^2-c x^4\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1247, 628} \begin {gather*} \frac {1}{2} \log \left (a-b x^2-c x^4\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 628
Rule 1247
Rubi steps
\begin {align*} \int \frac {x \left (b+2 c x^2\right )}{-a+b x^2+c x^4} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {b+2 c x}{-a+b x+c x^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \log \left (a-b x^2-c x^4\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 19, normalized size = 1.00 \begin {gather*} \frac {1}{2} \log \left (-a+b x^2+c x^4\right ) \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 {x \left (b+2 c x^2\right )}{-a+b x^2+c x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.63, size = 17, normalized size = 0.89 \begin {gather*} \frac {1}{2} \, \log \left (c x^{4} + b x^{2} - a\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.62, size = 18, normalized size = 0.95 \begin {gather*} \frac {1}{2} \, \log \left ({\left | c x^{4} + b x^{2} - a \right |}\right ) \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 = 0.95 \begin {gather*} \frac {\ln \left (c \,x^{4}+b \,x^{2}-a \right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 17, normalized size = 0.89 \begin {gather*} \frac {1}{2} \, \log \left (c x^{4} + b x^{2} - a\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 17, normalized size = 0.89 \begin {gather*} \frac {\ln \left (c\,x^4+b\,x^2-a\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.28, size = 14, normalized size = 0.74 \begin {gather*} \frac {\log {\left (- a + b x^{2} + c x^{4} \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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