Optimal. Leaf size=49 \[ -\frac {1}{2 b^2 x^2}-\frac {c}{2 b^2 \left (b+c x^2\right )}-\frac {2 c \log (x)}{b^3}+\frac {c \log \left (b+c x^2\right )}{b^3} \]
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Rubi [A]
time = 0.03, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1598, 272, 46}
\begin {gather*} \frac {c \log \left (b+c x^2\right )}{b^3}-\frac {2 c \log (x)}{b^3}-\frac {c}{2 b^2 \left (b+c x^2\right )}-\frac {1}{2 b^2 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 272
Rule 1598
Rubi steps
\begin {align*} \int \frac {x}{\left (b x^2+c x^4\right )^2} \, dx &=\int \frac {1}{x^3 \left (b+c x^2\right )^2} \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{x^2 (b+c x)^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{b^2 x^2}-\frac {2 c}{b^3 x}+\frac {c^2}{b^2 (b+c x)^2}+\frac {2 c^2}{b^3 (b+c x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {1}{2 b^2 x^2}-\frac {c}{2 b^2 \left (b+c x^2\right )}-\frac {2 c \log (x)}{b^3}+\frac {c \log \left (b+c x^2\right )}{b^3}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 41, normalized size = 0.84 \begin {gather*} -\frac {b \left (\frac {1}{x^2}+\frac {c}{b+c x^2}\right )+4 c \log (x)-2 c \log \left (b+c x^2\right )}{2 b^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 55, normalized size = 1.12
method | result | size |
norman | \(\frac {\frac {c^{2} x^{5}}{b^{3}}-\frac {x}{2 b}}{x^{3} \left (c \,x^{2}+b \right )}+\frac {c \ln \left (c \,x^{2}+b \right )}{b^{3}}-\frac {2 c \ln \left (x \right )}{b^{3}}\) | \(53\) |
risch | \(\frac {-\frac {c \,x^{2}}{b^{2}}-\frac {1}{2 b}}{x^{2} \left (c \,x^{2}+b \right )}-\frac {2 c \ln \left (x \right )}{b^{3}}+\frac {c \ln \left (-c \,x^{2}-b \right )}{b^{3}}\) | \(54\) |
default | \(\frac {c^{2} \left (-\frac {b}{c \left (c \,x^{2}+b \right )}+\frac {2 \ln \left (c \,x^{2}+b \right )}{c}\right )}{2 b^{3}}-\frac {1}{2 b^{2} x^{2}}-\frac {2 c \ln \left (x \right )}{b^{3}}\) | \(55\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 52, normalized size = 1.06 \begin {gather*} -\frac {2 \, c x^{2} + b}{2 \, {\left (b^{2} c x^{4} + b^{3} x^{2}\right )}} + \frac {c \log \left (c x^{2} + b\right )}{b^{3}} - \frac {c \log \left (x^{2}\right )}{b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 73, normalized size = 1.49 \begin {gather*} -\frac {2 \, b c x^{2} + b^{2} - 2 \, {\left (c^{2} x^{4} + b c x^{2}\right )} \log \left (c x^{2} + b\right ) + 4 \, {\left (c^{2} x^{4} + b c x^{2}\right )} \log \left (x\right )}{2 \, {\left (b^{3} c x^{4} + b^{4} x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.19, size = 51, normalized size = 1.04 \begin {gather*} \frac {- b - 2 c x^{2}}{2 b^{3} x^{2} + 2 b^{2} c x^{4}} - \frac {2 c \log {\left (x \right )}}{b^{3}} + \frac {c \log {\left (\frac {b}{c} + x^{2} \right )}}{b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.01, size = 50, normalized size = 1.02 \begin {gather*} \frac {c \log \left ({\left | c x^{2} + b \right |}\right )}{b^{3}} - \frac {2 \, c \log \left ({\left | x \right |}\right )}{b^{3}} - \frac {2 \, c x^{2} + b}{2 \, {\left (c x^{4} + b x^{2}\right )} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.21, size = 51, normalized size = 1.04 \begin {gather*} \frac {c\,\ln \left (c\,x^2+b\right )}{b^3}-\frac {\frac {1}{2\,b}+\frac {c\,x^2}{b^2}}{c\,x^4+b\,x^2}-\frac {2\,c\,\ln \left (x\right )}{b^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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