Optimal. Leaf size=63 \[ -\frac {1}{6 b x^6}+\frac {c}{4 b^2 x^4}-\frac {c^2}{2 b^3 x^2}-\frac {c^3 \log (x)}{b^4}+\frac {c^3 \log \left (b+c x^2\right )}{2 b^4} \]
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Rubi [A]
time = 0.03, antiderivative size = 63, 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^3 \log \left (b+c x^2\right )}{2 b^4}-\frac {c^3 \log (x)}{b^4}-\frac {c^2}{2 b^3 x^2}+\frac {c}{4 b^2 x^4}-\frac {1}{6 b x^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 272
Rule 1598
Rubi steps
\begin {align*} \int \frac {1}{x^5 \left (b x^2+c x^4\right )} \, dx &=\int \frac {1}{x^7 \left (b+c x^2\right )} \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{x^4 (b+c x)} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{b x^4}-\frac {c}{b^2 x^3}+\frac {c^2}{b^3 x^2}-\frac {c^3}{b^4 x}+\frac {c^4}{b^4 (b+c x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {1}{6 b x^6}+\frac {c}{4 b^2 x^4}-\frac {c^2}{2 b^3 x^2}-\frac {c^3 \log (x)}{b^4}+\frac {c^3 \log \left (b+c x^2\right )}{2 b^4}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 63, normalized size = 1.00 \begin {gather*} -\frac {1}{6 b x^6}+\frac {c}{4 b^2 x^4}-\frac {c^2}{2 b^3 x^2}-\frac {c^3 \log (x)}{b^4}+\frac {c^3 \log \left (b+c x^2\right )}{2 b^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 56, normalized size = 0.89
method | result | size |
default | \(-\frac {1}{6 b \,x^{6}}+\frac {c}{4 b^{2} x^{4}}-\frac {c^{2}}{2 b^{3} x^{2}}-\frac {c^{3} \ln \left (x \right )}{b^{4}}+\frac {c^{3} \ln \left (c \,x^{2}+b \right )}{2 b^{4}}\) | \(56\) |
norman | \(\frac {-\frac {1}{6 b}+\frac {c \,x^{2}}{4 b^{2}}-\frac {c^{2} x^{4}}{2 b^{3}}}{x^{6}}-\frac {c^{3} \ln \left (x \right )}{b^{4}}+\frac {c^{3} \ln \left (c \,x^{2}+b \right )}{2 b^{4}}\) | \(58\) |
risch | \(\frac {-\frac {1}{6 b}+\frac {c \,x^{2}}{4 b^{2}}-\frac {c^{2} x^{4}}{2 b^{3}}}{x^{6}}-\frac {c^{3} \ln \left (x \right )}{b^{4}}+\frac {c^{3} \ln \left (-c \,x^{2}-b \right )}{2 b^{4}}\) | \(61\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 58, normalized size = 0.92 \begin {gather*} \frac {c^{3} \log \left (c x^{2} + b\right )}{2 \, b^{4}} - \frac {c^{3} \log \left (x^{2}\right )}{2 \, b^{4}} - \frac {6 \, c^{2} x^{4} - 3 \, b c x^{2} + 2 \, b^{2}}{12 \, b^{3} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 58, normalized size = 0.92 \begin {gather*} \frac {6 \, c^{3} x^{6} \log \left (c x^{2} + b\right ) - 12 \, c^{3} x^{6} \log \left (x\right ) - 6 \, b c^{2} x^{4} + 3 \, b^{2} c x^{2} - 2 \, b^{3}}{12 \, b^{4} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.17, size = 56, normalized size = 0.89 \begin {gather*} \frac {- 2 b^{2} + 3 b c x^{2} - 6 c^{2} x^{4}}{12 b^{3} x^{6}} - \frac {c^{3} \log {\left (x \right )}}{b^{4}} + \frac {c^{3} \log {\left (\frac {b}{c} + x^{2} \right )}}{2 b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.27, size = 70, normalized size = 1.11 \begin {gather*} -\frac {c^{3} \log \left (x^{2}\right )}{2 \, b^{4}} + \frac {c^{3} \log \left ({\left | c x^{2} + b \right |}\right )}{2 \, b^{4}} + \frac {11 \, c^{3} x^{6} - 6 \, b c^{2} x^{4} + 3 \, b^{2} c x^{2} - 2 \, b^{3}}{12 \, b^{4} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 58, normalized size = 0.92 \begin {gather*} \frac {c^3\,\ln \left (c\,x^2+b\right )}{2\,b^4}-\frac {\frac {1}{6\,b}-\frac {c\,x^2}{4\,b^2}+\frac {c^2\,x^4}{2\,b^3}}{x^6}-\frac {c^3\,\ln \left (x\right )}{b^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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