Optimal. Leaf size=97 \[ \frac {c (2 b B-3 A c) \log \left (b+c x^2\right )}{2 b^4}-\frac {c \log (x) (2 b B-3 A c)}{b^4}-\frac {c (b B-A c)}{2 b^3 \left (b+c x^2\right )}-\frac {b B-2 A c}{2 b^3 x^2}-\frac {A}{4 b^2 x^4} \]
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Rubi [A] time = 0.11, antiderivative size = 97, 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 = {1584, 446, 77} \[ -\frac {c (b B-A c)}{2 b^3 \left (b+c x^2\right )}-\frac {b B-2 A c}{2 b^3 x^2}+\frac {c (2 b B-3 A c) \log \left (b+c x^2\right )}{2 b^4}-\frac {c \log (x) (2 b B-3 A c)}{b^4}-\frac {A}{4 b^2 x^4} \]
Antiderivative was successfully verified.
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Rule 77
Rule 446
Rule 1584
Rubi steps
\begin {align*} \int \frac {A+B x^2}{x \left (b x^2+c x^4\right )^2} \, dx &=\int \frac {A+B x^2}{x^5 \left (b+c x^2\right )^2} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {A+B x}{x^3 (b+c x)^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {A}{b^2 x^3}+\frac {b B-2 A c}{b^3 x^2}-\frac {c (2 b B-3 A c)}{b^4 x}+\frac {c^2 (b B-A c)}{b^3 (b+c x)^2}+\frac {c^2 (2 b B-3 A c)}{b^4 (b+c x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {A}{4 b^2 x^4}-\frac {b B-2 A c}{2 b^3 x^2}-\frac {c (b B-A c)}{2 b^3 \left (b+c x^2\right )}-\frac {c (2 b B-3 A c) \log (x)}{b^4}+\frac {c (2 b B-3 A c) \log \left (b+c x^2\right )}{2 b^4}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 85, normalized size = 0.88 \[ -\frac {\frac {A b^2}{x^4}+\frac {2 b c (b B-A c)}{b+c x^2}+\frac {2 b (b B-2 A c)}{x^2}+2 c (3 A c-2 b B) \log \left (b+c x^2\right )-4 c \log (x) (3 A c-2 b B)}{4 b^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 154, normalized size = 1.59 \[ -\frac {2 \, {\left (2 \, B b^{2} c - 3 \, A b c^{2}\right )} x^{4} + A b^{3} + {\left (2 \, B b^{3} - 3 \, A b^{2} c\right )} x^{2} - 2 \, {\left ({\left (2 \, B b c^{2} - 3 \, A c^{3}\right )} x^{6} + {\left (2 \, B b^{2} c - 3 \, A b c^{2}\right )} x^{4}\right )} \log \left (c x^{2} + b\right ) + 4 \, {\left ({\left (2 \, B b c^{2} - 3 \, A c^{3}\right )} x^{6} + {\left (2 \, B b^{2} c - 3 \, A b c^{2}\right )} x^{4}\right )} \log \relax (x)}{4 \, {\left (b^{4} c x^{6} + b^{5} x^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 150, normalized size = 1.55 \[ -\frac {{\left (2 \, B b c - 3 \, A c^{2}\right )} \log \left (x^{2}\right )}{2 \, b^{4}} + \frac {{\left (2 \, B b c^{2} - 3 \, A c^{3}\right )} \log \left ({\left | c x^{2} + b \right |}\right )}{2 \, b^{4} c} - \frac {2 \, B b c^{2} x^{2} - 3 \, A c^{3} x^{2} + 3 \, B b^{2} c - 4 \, A b c^{2}}{2 \, {\left (c x^{2} + b\right )} b^{4}} + \frac {6 \, B b c x^{4} - 9 \, A c^{2} x^{4} - 2 \, B b^{2} x^{2} + 4 \, A b c x^{2} - A b^{2}}{4 \, b^{4} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 114, normalized size = 1.18 \[ \frac {A \,c^{2}}{2 \left (c \,x^{2}+b \right ) b^{3}}+\frac {3 A \,c^{2} \ln \relax (x )}{b^{4}}-\frac {3 A \,c^{2} \ln \left (c \,x^{2}+b \right )}{2 b^{4}}-\frac {B c}{2 \left (c \,x^{2}+b \right ) b^{2}}-\frac {2 B c \ln \relax (x )}{b^{3}}+\frac {B c \ln \left (c \,x^{2}+b \right )}{b^{3}}+\frac {A c}{b^{3} x^{2}}-\frac {B}{2 b^{2} x^{2}}-\frac {A}{4 b^{2} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 106, normalized size = 1.09 \[ -\frac {2 \, {\left (2 \, B b c - 3 \, A c^{2}\right )} x^{4} + A b^{2} + {\left (2 \, B b^{2} - 3 \, A b c\right )} x^{2}}{4 \, {\left (b^{3} c x^{6} + b^{4} x^{4}\right )}} + \frac {{\left (2 \, B b c - 3 \, A c^{2}\right )} \log \left (c x^{2} + b\right )}{2 \, b^{4}} - \frac {{\left (2 \, B b c - 3 \, A c^{2}\right )} \log \left (x^{2}\right )}{2 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 100, normalized size = 1.03 \[ \frac {\frac {x^2\,\left (3\,A\,c-2\,B\,b\right )}{4\,b^2}-\frac {A}{4\,b}+\frac {c\,x^4\,\left (3\,A\,c-2\,B\,b\right )}{2\,b^3}}{c\,x^6+b\,x^4}-\frac {\ln \left (c\,x^2+b\right )\,\left (3\,A\,c^2-2\,B\,b\,c\right )}{2\,b^4}+\frac {\ln \relax (x)\,\left (3\,A\,c^2-2\,B\,b\,c\right )}{b^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.04, size = 100, normalized size = 1.03 \[ \frac {- A b^{2} + x^{4} \left (6 A c^{2} - 4 B b c\right ) + x^{2} \left (3 A b c - 2 B b^{2}\right )}{4 b^{4} x^{4} + 4 b^{3} c x^{6}} - \frac {c \left (- 3 A c + 2 B b\right ) \log {\relax (x )}}{b^{4}} + \frac {c \left (- 3 A c + 2 B b\right ) \log {\left (\frac {b}{c} + x^{2} \right )}}{2 b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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