Optimal. Leaf size=29 \[ -\frac {A c+b B}{2 x^2}-\frac {A b}{4 x^4}+B c \log (x) \]
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Rubi [A] time = 0.03, antiderivative size = 29, 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 = {1584, 446, 76} \[ -\frac {A c+b B}{2 x^2}-\frac {A b}{4 x^4}+B c \log (x) \]
Antiderivative was successfully verified.
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Rule 76
Rule 446
Rule 1584
Rubi steps
\begin {align*} \int \frac {\left (A+B x^2\right ) \left (b x^2+c x^4\right )}{x^7} \, dx &=\int \frac {\left (A+B x^2\right ) \left (b+c x^2\right )}{x^5} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(A+B x) (b+c x)}{x^3} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {A b}{x^3}+\frac {b B+A c}{x^2}+\frac {B c}{x}\right ) \, dx,x,x^2\right )\\ &=-\frac {A b}{4 x^4}-\frac {b B+A c}{2 x^2}+B c \log (x)\\ \end {align*}
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Mathematica [A] time = 0.03, size = 31, normalized size = 1.07 \[ \frac {-A c-b B}{2 x^2}-\frac {A b}{4 x^4}+B c \log (x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 31, normalized size = 1.07 \[ \frac {4 \, B c x^{4} \log \relax (x) - 2 \, {\left (B b + A c\right )} x^{2} - A b}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 39, normalized size = 1.34 \[ \frac {1}{2} \, B c \log \left (x^{2}\right ) - \frac {3 \, B c x^{4} + 2 \, B b x^{2} + 2 \, A c x^{2} + A b}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 28, normalized size = 0.97 \[ B c \ln \relax (x )-\frac {A c}{2 x^{2}}-\frac {B b}{2 x^{2}}-\frac {A b}{4 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.40, size = 30, normalized size = 1.03 \[ \frac {1}{2} \, B c \log \left (x^{2}\right ) - \frac {2 \, {\left (B b + A c\right )} x^{2} + A b}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 29, normalized size = 1.00 \[ B\,c\,\ln \relax (x)-\frac {\left (\frac {A\,c}{2}+\frac {B\,b}{2}\right )\,x^2+\frac {A\,b}{4}}{x^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.37, size = 29, normalized size = 1.00 \[ B c \log {\relax (x )} + \frac {- A b + x^{2} \left (- 2 A c - 2 B b\right )}{4 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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