Optimal. Leaf size=27 \[ \frac {x^2}{2 c}-\frac {b \log \left (b+c x^2\right )}{2 c^2} \]
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Rubi [A]
time = 0.02, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {1598, 272, 45}
\begin {gather*} \frac {x^2}{2 c}-\frac {b \log \left (b+c x^2\right )}{2 c^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rule 1598
Rubi steps
\begin {align*} \int \frac {x^5}{b x^2+c x^4} \, dx &=\int \frac {x^3}{b+c x^2} \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {x}{b+c x} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{c}-\frac {b}{c (b+c x)}\right ) \, dx,x,x^2\right )\\ &=\frac {x^2}{2 c}-\frac {b \log \left (b+c x^2\right )}{2 c^2}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 27, normalized size = 1.00 \begin {gather*} \frac {x^2}{2 c}-\frac {b \log \left (b+c x^2\right )}{2 c^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 24, normalized size = 0.89
method | result | size |
default | \(\frac {x^{2}}{2 c}-\frac {b \ln \left (c \,x^{2}+b \right )}{2 c^{2}}\) | \(24\) |
norman | \(\frac {x^{2}}{2 c}-\frac {b \ln \left (c \,x^{2}+b \right )}{2 c^{2}}\) | \(24\) |
risch | \(\frac {x^{2}}{2 c}-\frac {b \ln \left (c \,x^{2}+b \right )}{2 c^{2}}\) | \(24\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 23, normalized size = 0.85 \begin {gather*} \frac {x^{2}}{2 \, c} - \frac {b \log \left (c x^{2} + b\right )}{2 \, c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 22, normalized size = 0.81 \begin {gather*} \frac {c x^{2} - b \log \left (c x^{2} + b\right )}{2 \, c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 20, normalized size = 0.74 \begin {gather*} - \frac {b \log {\left (b + c x^{2} \right )}}{2 c^{2}} + \frac {x^{2}}{2 c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.78, size = 24, normalized size = 0.89 \begin {gather*} \frac {x^{2}}{2 \, c} - \frac {b \log \left ({\left | c x^{2} + b \right |}\right )}{2 \, c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 22, normalized size = 0.81 \begin {gather*} -\frac {b\,\ln \left (c\,x^2+b\right )-c\,x^2}{2\,c^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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