Optimal. Leaf size=29 \[ -\frac {A b}{4 x^4}-\frac {b B+A c}{2 x^2}+B c \log (x) \]
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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*} -\frac {A c+b B}{2 x^2}-\frac {A b}{4 x^4}+B c \log (x) \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^7} \, dx &=\int \frac {\left (A+B x^2\right ) \left (b+c x^2\right )}{x^5} \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {(A+B x) (b+c x)}{x^3} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {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 = 0.01, size = 31, normalized size = 1.07 \begin {gather*} -\frac {A b}{4 x^4}+\frac {-b B-A c}{2 x^2}+B 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 | \(-\frac {A b}{4 x^{4}}-\frac {A c +B b}{2 x^{2}}+B c \ln \left (x \right )\) | \(26\) |
risch | \(\frac {\left (-\frac {A c}{2}-\frac {B b}{2}\right ) x^{2}-\frac {A b}{4}}{x^{4}}+B c \ln \left (x \right )\) | \(29\) |
norman | \(\frac {\left (-\frac {A c}{2}-\frac {B b}{2}\right ) x^{4}-\frac {A b \,x^{2}}{4}}{x^{6}}+B c \ln \left (x \right )\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 30, normalized size = 1.03 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.14, size = 31, normalized size = 1.07 \begin {gather*} \frac {4 \, B c x^{4} \log \left (x\right ) - 2 \, {\left (B b + A c\right )} x^{2} - A b}{4 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.18, size = 29, normalized size = 1.00 \begin {gather*} B c \log {\left (x \right )} + \frac {- A b + x^{2} \left (- 2 A c - 2 B b\right )}{4 x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.68, size = 39, normalized size = 1.34 \begin {gather*} \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}} \end {gather*}
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 \begin {gather*} B\,c\,\ln \left (x\right )-\frac {\left (\frac {A\,c}{2}+\frac {B\,b}{2}\right )\,x^2+\frac {A\,b}{4}}{x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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