Optimal. Leaf size=49 \[ -\frac {1}{4 b x^4}+\frac {c}{2 b^2 x^2}+\frac {c^2 \log (x)}{b^3}-\frac {c^2 \log \left (b+c x^2\right )}{2 b^3} \]
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Rubi [A]
time = 0.02, antiderivative size = 49, 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, 46}
\begin {gather*} -\frac {c^2 \log \left (b+c x^2\right )}{2 b^3}+\frac {c^2 \log (x)}{b^3}+\frac {c}{2 b^2 x^2}-\frac {1}{4 b x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 272
Rule 1598
Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (b x^2+c x^4\right )} \, dx &=\int \frac {1}{x^5 \left (b+c x^2\right )} \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{x^3 (b+c x)} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{b x^3}-\frac {c}{b^2 x^2}+\frac {c^2}{b^3 x}-\frac {c^3}{b^3 (b+c x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {1}{4 b x^4}+\frac {c}{2 b^2 x^2}+\frac {c^2 \log (x)}{b^3}-\frac {c^2 \log \left (b+c x^2\right )}{2 b^3}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 49, normalized size = 1.00 \begin {gather*} -\frac {1}{4 b x^4}+\frac {c}{2 b^2 x^2}+\frac {c^2 \log (x)}{b^3}-\frac {c^2 \log \left (b+c x^2\right )}{2 b^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 44, normalized size = 0.90
method | result | size |
default | \(-\frac {1}{4 b \,x^{4}}+\frac {c}{2 b^{2} x^{2}}+\frac {c^{2} \ln \left (x \right )}{b^{3}}-\frac {c^{2} \ln \left (c \,x^{2}+b \right )}{2 b^{3}}\) | \(44\) |
norman | \(\frac {-\frac {1}{4 b}+\frac {c \,x^{2}}{2 b^{2}}}{x^{4}}+\frac {c^{2} \ln \left (x \right )}{b^{3}}-\frac {c^{2} \ln \left (c \,x^{2}+b \right )}{2 b^{3}}\) | \(46\) |
risch | \(\frac {-\frac {1}{4 b}+\frac {c \,x^{2}}{2 b^{2}}}{x^{4}}+\frac {c^{2} \ln \left (x \right )}{b^{3}}-\frac {c^{2} \ln \left (c \,x^{2}+b \right )}{2 b^{3}}\) | \(46\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 47, normalized size = 0.96 \begin {gather*} -\frac {c^{2} \log \left (c x^{2} + b\right )}{2 \, b^{3}} + \frac {c^{2} \log \left (x^{2}\right )}{2 \, b^{3}} + \frac {2 \, c x^{2} - b}{4 \, b^{2} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 45, normalized size = 0.92 \begin {gather*} -\frac {2 \, c^{2} x^{4} \log \left (c x^{2} + b\right ) - 4 \, c^{2} x^{4} \log \left (x\right ) - 2 \, b c x^{2} + b^{2}}{4 \, b^{3} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.14, size = 42, normalized size = 0.86 \begin {gather*} \frac {- b + 2 c x^{2}}{4 b^{2} x^{4}} + \frac {c^{2} \log {\left (x \right )}}{b^{3}} - \frac {c^{2} \log {\left (\frac {b}{c} + x^{2} \right )}}{2 b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.21, size = 57, normalized size = 1.16 \begin {gather*} \frac {c^{2} \log \left (x^{2}\right )}{2 \, b^{3}} - \frac {c^{2} \log \left ({\left | c x^{2} + b \right |}\right )}{2 \, b^{3}} - \frac {3 \, c^{2} x^{4} - 2 \, b c x^{2} + b^{2}}{4 \, b^{3} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 46, normalized size = 0.94 \begin {gather*} \frac {c^2\,\ln \left (x\right )}{b^3}-\frac {c^2\,\ln \left (c\,x^2+b\right )}{2\,b^3}-\frac {\frac {1}{4\,b}-\frac {c\,x^2}{2\,b^2}}{x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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