Optimal. Leaf size=92 \[ -\frac {c^2 (b B-A c) \log \left (b+c x^2\right )}{2 b^4}+\frac {c^2 \log (x) (b B-A c)}{b^4}+\frac {c (b B-A c)}{2 b^3 x^2}-\frac {b B-A c}{4 b^2 x^4}-\frac {A}{6 b x^6} \]
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Rubi [A] time = 0.09, antiderivative size = 92, 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} \[ -\frac {c^2 (b B-A c) \log \left (b+c x^2\right )}{2 b^4}+\frac {c^2 \log (x) (b B-A c)}{b^4}+\frac {c (b B-A c)}{2 b^3 x^2}-\frac {b B-A c}{4 b^2 x^4}-\frac {A}{6 b x^6} \]
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^5 \left (b x^2+c x^4\right )} \, dx &=\int \frac {A+B x^2}{x^7 \left (b+c x^2\right )} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {A+B x}{x^4 (b+c x)} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {A}{b x^4}+\frac {b B-A c}{b^2 x^3}-\frac {c (b B-A c)}{b^3 x^2}+\frac {c^2 (b B-A c)}{b^4 x}-\frac {c^3 (b B-A c)}{b^4 (b+c x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {A}{6 b x^6}-\frac {b B-A c}{4 b^2 x^4}+\frac {c (b B-A c)}{2 b^3 x^2}+\frac {c^2 (b B-A c) \log (x)}{b^4}-\frac {c^2 (b B-A c) \log \left (b+c x^2\right )}{2 b^4}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 96, normalized size = 1.04 \[ \frac {\left (A c^3-b B c^2\right ) \log \left (b+c x^2\right )}{2 b^4}+\frac {\log (x) \left (b B c^2-A c^3\right )}{b^4}+\frac {c (b B-A c)}{2 b^3 x^2}+\frac {A c-b B}{4 b^2 x^4}-\frac {A}{6 b x^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 98, normalized size = 1.07 \[ -\frac {6 \, {\left (B b c^{2} - A c^{3}\right )} x^{6} \log \left (c x^{2} + b\right ) - 12 \, {\left (B b c^{2} - A c^{3}\right )} x^{6} \log \relax (x) - 6 \, {\left (B b^{2} c - A b c^{2}\right )} x^{4} + 2 \, A b^{3} + 3 \, {\left (B b^{3} - A b^{2} c\right )} x^{2}}{12 \, b^{4} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 126, normalized size = 1.37 \[ \frac {{\left (B b c^{2} - A c^{3}\right )} \log \left (x^{2}\right )}{2 \, b^{4}} - \frac {{\left (B b c^{3} - A c^{4}\right )} \log \left ({\left | c x^{2} + b \right |}\right )}{2 \, b^{4} c} - \frac {11 \, B b c^{2} x^{6} - 11 \, A c^{3} x^{6} - 6 \, B b^{2} c x^{4} + 6 \, A b c^{2} x^{4} + 3 \, B b^{3} x^{2} - 3 \, A b^{2} c x^{2} + 2 \, A b^{3}}{12 \, b^{4} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 107, normalized size = 1.16 \[ -\frac {A \,c^{3} \ln \relax (x )}{b^{4}}+\frac {A \,c^{3} \ln \left (c \,x^{2}+b \right )}{2 b^{4}}+\frac {B \,c^{2} \ln \relax (x )}{b^{3}}-\frac {B \,c^{2} \ln \left (c \,x^{2}+b \right )}{2 b^{3}}-\frac {A \,c^{2}}{2 b^{3} x^{2}}+\frac {B c}{2 b^{2} x^{2}}+\frac {A c}{4 b^{2} x^{4}}-\frac {B}{4 b \,x^{4}}-\frac {A}{6 b \,x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.28, size = 96, normalized size = 1.04 \[ -\frac {{\left (B b c^{2} - A c^{3}\right )} \log \left (c x^{2} + b\right )}{2 \, b^{4}} + \frac {{\left (B b c^{2} - A c^{3}\right )} \log \left (x^{2}\right )}{2 \, b^{4}} + \frac {6 \, {\left (B b c - A c^{2}\right )} x^{4} - 2 \, A b^{2} - 3 \, {\left (B b^{2} - A b c\right )} x^{2}}{12 \, b^{3} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.15, size = 92, normalized size = 1.00 \[ \frac {\ln \left (c\,x^2+b\right )\,\left (A\,c^3-B\,b\,c^2\right )}{2\,b^4}-\frac {\frac {A}{6\,b}-\frac {x^2\,\left (A\,c-B\,b\right )}{4\,b^2}+\frac {c\,x^4\,\left (A\,c-B\,b\right )}{2\,b^3}}{x^6}-\frac {\ln \relax (x)\,\left (A\,c^3-B\,b\,c^2\right )}{b^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.41, size = 88, normalized size = 0.96 \[ \frac {- 2 A b^{2} + x^{4} \left (- 6 A c^{2} + 6 B b c\right ) + x^{2} \left (3 A b c - 3 B b^{2}\right )}{12 b^{3} x^{6}} + \frac {c^{2} \left (- A c + B b\right ) \log {\relax (x )}}{b^{4}} - \frac {c^{2} \left (- A c + B b\right ) \log {\left (\frac {b}{c} + x^{2} \right )}}{2 b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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