Optimal. Leaf size=16 \[ \frac {1}{2} \log \left (b x^2+c x^4\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 15, normalized size of antiderivative = 0.94, number of steps
used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {1598, 457, 78}
\begin {gather*} \frac {1}{2} \log \left (b+c x^2\right )+\log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rule 457
Rule 1598
Rubi steps
\begin {align*} \int \frac {x \left (b+2 c x^2\right )}{b x^2+c x^4} \, dx &=\int \frac {b+2 c x^2}{x \left (b+c x^2\right )} \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {b+2 c x}{x (b+c x)} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{x}+\frac {c}{b+c x}\right ) \, dx,x,x^2\right )\\ &=\log (x)+\frac {1}{2} \log \left (b+c x^2\right )\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 15, normalized size = 0.94 \begin {gather*} \log (x)+\frac {1}{2} \log \left (b+c x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 14, normalized size = 0.88
method | result | size |
default | \(\ln \left (x \right )+\frac {\ln \left (c \,x^{2}+b \right )}{2}\) | \(14\) |
norman | \(\ln \left (x \right )+\frac {\ln \left (c \,x^{2}+b \right )}{2}\) | \(14\) |
risch | \(\ln \left (x \right )+\frac {\ln \left (c \,x^{2}+b \right )}{2}\) | \(14\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 17, normalized size = 1.06 \begin {gather*} \frac {1}{2} \, \log \left (c x^{2} + b\right ) + \frac {1}{2} \, \log \left (x^{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 13, normalized size = 0.81 \begin {gather*} \frac {1}{2} \, \log \left (c x^{2} + b\right ) + \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.10, size = 12, normalized size = 0.75 \begin {gather*} \log {\left (x \right )} + \frac {\log {\left (\frac {b}{c} + x^{2} \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.41, size = 15, normalized size = 0.94 \begin {gather*} \frac {1}{2} \, \log \left ({\left | c x^{4} + b x^{2} \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 13, normalized size = 0.81 \begin {gather*} \frac {\ln \left (c\,x^2+b\right )}{2}+\ln \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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