Optimal. Leaf size=49 \[ -\frac {A}{2 b x^2}+\frac {(b B-A c) \log (x)}{b^2}-\frac {(b B-A c) \log \left (b+c x^2\right )}{2 b^2} \]
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Rubi [A]
time = 0.04, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1598, 457, 78}
\begin {gather*} -\frac {(b B-A c) \log \left (b+c x^2\right )}{2 b^2}+\frac {\log (x) (b B-A c)}{b^2}-\frac {A}{2 b x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rule 457
Rule 1598
Rubi steps
\begin {align*} \int \frac {A+B x^2}{x \left (b x^2+c x^4\right )} \, dx &=\int \frac {A+B x^2}{x^3 \left (b+c x^2\right )} \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {A+B x}{x^2 (b+c x)} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {A}{b x^2}+\frac {b B-A c}{b^2 x}-\frac {c (b B-A c)}{b^2 (b+c x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {A}{2 b x^2}+\frac {(b B-A c) \log (x)}{b^2}-\frac {(b B-A c) \log \left (b+c x^2\right )}{2 b^2}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 49, normalized size = 1.00 \begin {gather*} -\frac {A}{2 b x^2}+\frac {(b B-A c) \log (x)}{b^2}+\frac {(-b B+A c) \log \left (b+c x^2\right )}{2 b^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.44, size = 46, normalized size = 0.94
method | result | size |
default | \(\frac {\left (A c -B b \right ) \ln \left (c \,x^{2}+b \right )}{2 b^{2}}-\frac {A}{2 b \,x^{2}}+\frac {\left (-A c +B b \right ) \ln \left (x \right )}{b^{2}}\) | \(46\) |
norman | \(-\frac {A}{2 b \,x^{2}}-\frac {\left (A c -B b \right ) \ln \left (x \right )}{b^{2}}+\frac {\left (A c -B b \right ) \ln \left (c \,x^{2}+b \right )}{2 b^{2}}\) | \(47\) |
risch | \(-\frac {A}{2 b \,x^{2}}-\frac {\ln \left (x \right ) A c}{b^{2}}+\frac {\ln \left (x \right ) B}{b}+\frac {\ln \left (-c \,x^{2}-b \right ) A c}{2 b^{2}}-\frac {\ln \left (-c \,x^{2}-b \right ) B}{2 b}\) | \(62\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 48, normalized size = 0.98 \begin {gather*} -\frac {{\left (B b - A c\right )} \log \left (c x^{2} + b\right )}{2 \, b^{2}} + \frac {{\left (B b - A c\right )} \log \left (x^{2}\right )}{2 \, b^{2}} - \frac {A}{2 \, b x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.81, size = 47, normalized size = 0.96 \begin {gather*} -\frac {{\left (B b - A c\right )} x^{2} \log \left (c x^{2} + b\right ) - 2 \, {\left (B b - A c\right )} x^{2} \log \left (x\right ) + A b}{2 \, b^{2} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.39, size = 41, normalized size = 0.84 \begin {gather*} - \frac {A}{2 b x^{2}} + \frac {\left (- A c + B b\right ) \log {\left (x \right )}}{b^{2}} - \frac {\left (- A c + B b\right ) \log {\left (\frac {b}{c} + x^{2} \right )}}{2 b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.85, size = 71, normalized size = 1.45 \begin {gather*} \frac {{\left (B b - A c\right )} \log \left (x^{2}\right )}{2 \, b^{2}} - \frac {{\left (B b c - A c^{2}\right )} \log \left ({\left | c x^{2} + b \right |}\right )}{2 \, b^{2} c} - \frac {B b x^{2} - A c x^{2} + A b}{2 \, b^{2} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.14, size = 46, normalized size = 0.94 \begin {gather*} \frac {\ln \left (c\,x^2+b\right )\,\left (A\,c-B\,b\right )}{2\,b^2}-\frac {A}{2\,b\,x^2}-\frac {\ln \left (x\right )\,\left (A\,c-B\,b\right )}{b^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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