Optimal. Leaf size=73 \[ -\frac {A}{2 b^2 x^2}+\frac {b B-A c}{2 b^2 \left (b+c x^2\right )}+\frac {(b B-2 A c) \log (x)}{b^3}-\frac {(b B-2 A c) \log \left (b+c x^2\right )}{2 b^3} \]
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Rubi [A]
time = 0.06, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1598, 457, 78}
\begin {gather*} -\frac {(b B-2 A c) \log \left (b+c x^2\right )}{2 b^3}+\frac {\log (x) (b B-2 A c)}{b^3}+\frac {b B-A c}{2 b^2 \left (b+c x^2\right )}-\frac {A}{2 b^2 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rule 457
Rule 1598
Rubi steps
\begin {align*} \int \frac {x \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^2} \, dx &=\int \frac {A+B x^2}{x^3 \left (b+c x^2\right )^2} \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {A+B x}{x^2 (b+c x)^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {A}{b^2 x^2}+\frac {b B-2 A c}{b^3 x}-\frac {c (b B-A c)}{b^2 (b+c x)^2}-\frac {c (b B-2 A c)}{b^3 (b+c x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {A}{2 b^2 x^2}+\frac {b B-A c}{2 b^2 \left (b+c x^2\right )}+\frac {(b B-2 A c) \log (x)}{b^3}-\frac {(b B-2 A c) \log \left (b+c x^2\right )}{2 b^3}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 64, normalized size = 0.88 \begin {gather*} \frac {-\frac {A b}{x^2}+\frac {b (b B-A c)}{b+c x^2}+2 (b B-2 A c) \log (x)+(-b B+2 A c) \log \left (b+c x^2\right )}{2 b^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.38, size = 76, normalized size = 1.04
method | result | size |
default | \(\frac {c \left (\frac {\left (2 A c -B b \right ) \ln \left (c \,x^{2}+b \right )}{c}-\frac {b \left (A c -B b \right )}{c \left (c \,x^{2}+b \right )}\right )}{2 b^{3}}-\frac {A}{2 b^{2} x^{2}}+\frac {\left (-2 A c +B b \right ) \ln \left (x \right )}{b^{3}}\) | \(76\) |
norman | \(\frac {-\frac {A x}{2 b}+\frac {c \left (2 A c -B b \right ) x^{5}}{2 b^{3}}}{x^{3} \left (c \,x^{2}+b \right )}-\frac {\left (2 A c -B b \right ) \ln \left (x \right )}{b^{3}}+\frac {\left (2 A c -B b \right ) \ln \left (c \,x^{2}+b \right )}{2 b^{3}}\) | \(79\) |
risch | \(\frac {-\frac {\left (2 A c -B b \right ) x^{2}}{2 b^{2}}-\frac {A}{2 b}}{x^{2} \left (c \,x^{2}+b \right )}-\frac {2 \ln \left (x \right ) A c}{b^{3}}+\frac {\ln \left (x \right ) B}{b^{2}}+\frac {\ln \left (-c \,x^{2}-b \right ) A c}{b^{3}}-\frac {\ln \left (-c \,x^{2}-b \right ) B}{2 b^{2}}\) | \(89\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 76, normalized size = 1.04 \begin {gather*} \frac {{\left (B b - 2 \, A c\right )} x^{2} - A b}{2 \, {\left (b^{2} c x^{4} + b^{3} x^{2}\right )}} - \frac {{\left (B b - 2 \, A c\right )} \log \left (c x^{2} + b\right )}{2 \, b^{3}} + \frac {{\left (B b - 2 \, A c\right )} \log \left (x^{2}\right )}{2 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.02, size = 117, normalized size = 1.60 \begin {gather*} -\frac {A b^{2} - {\left (B b^{2} - 2 \, A b c\right )} x^{2} + {\left ({\left (B b c - 2 \, A c^{2}\right )} x^{4} + {\left (B b^{2} - 2 \, A b c\right )} x^{2}\right )} \log \left (c x^{2} + b\right ) - 2 \, {\left ({\left (B b c - 2 \, A c^{2}\right )} x^{4} + {\left (B b^{2} - 2 \, A b c\right )} 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 = 0.49, size = 70, normalized size = 0.96 \begin {gather*} \frac {- A b + x^{2} \left (- 2 A c + B b\right )}{2 b^{3} x^{2} + 2 b^{2} c x^{4}} + \frac {\left (- 2 A c + B b\right ) \log {\left (x \right )}}{b^{3}} - \frac {\left (- 2 A c + B b\right ) \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 = 1.41, size = 80, normalized size = 1.10 \begin {gather*} \frac {{\left (B b - 2 \, A c\right )} \log \left ({\left | x \right |}\right )}{b^{3}} + \frac {B b x^{2} - 2 \, A c x^{2} - A b}{2 \, {\left (c x^{4} + b x^{2}\right )} b^{2}} - \frac {{\left (B b c - 2 \, A c^{2}\right )} \log \left ({\left | c x^{2} + b \right |}\right )}{2 \, b^{3} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.15, size = 78, normalized size = 1.07 \begin {gather*} \frac {\ln \left (c\,x^2+b\right )\,\left (2\,A\,c-B\,b\right )}{2\,b^3}-\frac {\frac {A}{2\,b}+\frac {x^2\,\left (2\,A\,c-B\,b\right )}{2\,b^2}}{c\,x^4+b\,x^2}-\frac {\ln \left (x\right )\,\left (2\,A\,c-B\,b\right )}{b^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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