Optimal. Leaf size=51 \[ -\frac {A b^2}{4 x^4}-\frac {b (2 A c+b B)}{2 x^2}+c \log (x) (A c+2 b B)+\frac {1}{2} B c^2 x^2 \]
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Rubi [A] time = 0.05, antiderivative size = 51, 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, 76} \[ -\frac {A b^2}{4 x^4}-\frac {b (2 A c+b B)}{2 x^2}+c \log (x) (A c+2 b B)+\frac {1}{2} B c^2 x^2 \]
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 )^2}{x^9} \, dx &=\int \frac {\left (A+B x^2\right ) \left (b+c x^2\right )^2}{x^5} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(A+B x) (b+c x)^2}{x^3} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (B c^2+\frac {A b^2}{x^3}+\frac {b (b B+2 A c)}{x^2}+\frac {c (2 b B+A c)}{x}\right ) \, dx,x,x^2\right )\\ &=-\frac {A b^2}{4 x^4}-\frac {b (b B+2 A c)}{2 x^2}+\frac {1}{2} B c^2 x^2+c (2 b B+A c) \log (x)\\ \end {align*}
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Mathematica [A] time = 0.04, size = 50, normalized size = 0.98 \[ c \log (x) (A c+2 b B)-\frac {A b \left (b+4 c x^2\right )+2 B x^2 \left (b^2-c^2 x^4\right )}{4 x^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 55, normalized size = 1.08 \[ \frac {2 \, B c^{2} x^{6} + 4 \, {\left (2 \, B b c + A c^{2}\right )} x^{4} \log \relax (x) - A b^{2} - 2 \, {\left (B b^{2} + 2 \, A b c\right )} x^{2}}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 72, normalized size = 1.41 \[ \frac {1}{2} \, B c^{2} x^{2} + \frac {1}{2} \, {\left (2 \, B b c + A c^{2}\right )} \log \left (x^{2}\right ) - \frac {6 \, B b c x^{4} + 3 \, A c^{2} x^{4} + 2 \, B b^{2} x^{2} + 4 \, A b c x^{2} + A b^{2}}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 51, normalized size = 1.00 \[ \frac {B \,c^{2} x^{2}}{2}+A \,c^{2} \ln \relax (x )+2 B b c \ln \relax (x )-\frac {A b c}{x^{2}}-\frac {B \,b^{2}}{2 x^{2}}-\frac {A \,b^{2}}{4 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.37, size = 54, normalized size = 1.06 \[ \frac {1}{2} \, B c^{2} x^{2} + \frac {1}{2} \, {\left (2 \, B b c + A c^{2}\right )} \log \left (x^{2}\right ) - \frac {A b^{2} + 2 \, {\left (B b^{2} + 2 \, A b c\right )} x^{2}}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 51, normalized size = 1.00 \[ \ln \relax (x)\,\left (A\,c^2+2\,B\,b\,c\right )-\frac {x^2\,\left (\frac {B\,b^2}{2}+A\,c\,b\right )+\frac {A\,b^2}{4}}{x^4}+\frac {B\,c^2\,x^2}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.55, size = 51, normalized size = 1.00 \[ \frac {B c^{2} x^{2}}{2} + c \left (A c + 2 B b\right ) \log {\relax (x )} + \frac {- A b^{2} + x^{2} \left (- 4 A b c - 2 B b^{2}\right )}{4 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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