Optimal. Leaf size=51 \[ -\frac {A b^2}{4 x^4}-\frac {b (b B+2 A c)}{2 x^2}+\frac {1}{2} B c^2 x^2+c (2 b B+A c) \log (x) \]
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Rubi [A]
time = 0.03, 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 = {1598, 457, 77}
\begin {gather*} -\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 \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rule 457
Rule 1598
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} \text {Subst}\left (\int \frac {(A+B x) (b+c x)^2}{x^3} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {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 = 0.03, size = 50, normalized size = 0.98 \begin {gather*} -\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}+c (2 b B+A c) \log (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.36, size = 46, normalized size = 0.90
method | result | size |
default | \(-\frac {A \,b^{2}}{4 x^{4}}-\frac {b \left (2 A c +B b \right )}{2 x^{2}}+\frac {B \,c^{2} x^{2}}{2}+c \left (A c +2 B b \right ) \ln \left (x \right )\) | \(46\) |
risch | \(\frac {B \,c^{2} x^{2}}{2}+\frac {\left (-A b c -\frac {1}{2} b^{2} B \right ) x^{2}-\frac {b^{2} A}{4}}{x^{4}}+A \ln \left (x \right ) c^{2}+2 B \ln \left (x \right ) b c\) | \(52\) |
norman | \(\frac {\left (-A b c -\frac {1}{2} b^{2} B \right ) x^{6}-\frac {A \,b^{2} x^{4}}{4}+\frac {B \,c^{2} x^{10}}{2}}{x^{8}}+\left (A \,c^{2}+2 b B c \right ) \ln \left (x \right )\) | \(55\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 54, normalized size = 1.06 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.57, size = 55, normalized size = 1.08 \begin {gather*} \frac {2 \, B c^{2} x^{6} + 4 \, {\left (2 \, B b c + A c^{2}\right )} x^{4} \log \left (x\right ) - A b^{2} - 2 \, {\left (B b^{2} + 2 \, A b c\right )} x^{2}}{4 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.27, size = 51, normalized size = 1.00 \begin {gather*} \frac {B c^{2} x^{2}}{2} + c \left (A c + 2 B b\right ) \log {\left (x \right )} + \frac {- A b^{2} + x^{2} \left (- 4 A b c - 2 B b^{2}\right )}{4 x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.61, size = 72, normalized size = 1.41 \begin {gather*} \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}} \end {gather*}
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 \begin {gather*} \ln \left (x\right )\,\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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