Optimal. Leaf size=29 \[ -\frac {A b}{2 x^2}+\frac {1}{2} B c x^2+(b B+A c) \log (x) \]
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Rubi [A]
time = 0.02, 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 = {1598, 457, 77}
\begin {gather*} \log (x) (A c+b B)-\frac {A b}{2 x^2}+\frac {1}{2} B c 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 )}{x^5} \, dx &=\int \frac {\left (A+B x^2\right ) \left (b+c x^2\right )}{x^3} \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {(A+B x) (b+c x)}{x^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (B c+\frac {A b}{x^2}+\frac {b B+A c}{x}\right ) \, dx,x,x^2\right )\\ &=-\frac {A b}{2 x^2}+\frac {1}{2} B c x^2+(b B+A c) \log (x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 29, normalized size = 1.00 \begin {gather*} -\frac {A b}{2 x^2}+\frac {1}{2} B c x^2+(b B+A c) \log (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 26, normalized size = 0.90
method | result | size |
default | \(-\frac {A b}{2 x^{2}}+\frac {B c \,x^{2}}{2}+\left (A c +B b \right ) \ln \left (x \right )\) | \(26\) |
risch | \(-\frac {A b}{2 x^{2}}+\frac {B c \,x^{2}}{2}+A \ln \left (x \right ) c +b B \ln \left (x \right )\) | \(26\) |
norman | \(\frac {-\frac {1}{2} A b \,x^{2}+\frac {1}{2} B c \,x^{6}}{x^{4}}+\left (A c +B b \right ) \ln \left (x \right )\) | \(31\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 28, normalized size = 0.97 \begin {gather*} \frac {1}{2} \, B c x^{2} + \frac {1}{2} \, {\left (B b + A c\right )} \log \left (x^{2}\right ) - \frac {A b}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.33, size = 30, normalized size = 1.03 \begin {gather*} \frac {B c x^{4} + 2 \, {\left (B b + A c\right )} x^{2} \log \left (x\right ) - A b}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.08, size = 26, normalized size = 0.90 \begin {gather*} - \frac {A b}{2 x^{2}} + \frac {B c x^{2}}{2} + \left (A c + B b\right ) \log {\left (x \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.63, size = 42, normalized size = 1.45 \begin {gather*} \frac {1}{2} \, B c x^{2} + \frac {1}{2} \, {\left (B b + A c\right )} \log \left (x^{2}\right ) - \frac {B b x^{2} + A c x^{2} + A b}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 25, normalized size = 0.86 \begin {gather*} \ln \left (x\right )\,\left (A\,c+B\,b\right )-\frac {A\,b}{2\,x^2}+\frac {B\,c\,x^2}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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