Optimal. Leaf size=29 \[ -\frac {a A}{4 x^4}-\frac {A b+a B}{2 x^2}+b B \log (x) \]
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Rubi [A]
time = 0.02, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {457, 77}
\begin {gather*} -\frac {a B+A b}{2 x^2}-\frac {a A}{4 x^4}+b B \log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rule 457
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right ) \left (A+B x^2\right )}{x^5} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {(a+b x) (A+B x)}{x^3} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {a A}{x^3}+\frac {A b+a B}{x^2}+\frac {b B}{x}\right ) \, dx,x,x^2\right )\\ &=-\frac {a A}{4 x^4}-\frac {A b+a B}{2 x^2}+b B \log (x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 31, normalized size = 1.07 \begin {gather*} -\frac {a A}{4 x^4}+\frac {-A b-a B}{2 x^2}+b B \log (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 26, normalized size = 0.90
method | result | size |
default | \(-\frac {a A}{4 x^{4}}-\frac {A b +B a}{2 x^{2}}+b B \ln \left (x \right )\) | \(26\) |
norman | \(\frac {\left (-\frac {A b}{2}-\frac {B a}{2}\right ) x^{2}-\frac {A a}{4}}{x^{4}}+b B \ln \left (x \right )\) | \(29\) |
risch | \(\frac {\left (-\frac {A b}{2}-\frac {B a}{2}\right ) x^{2}-\frac {A a}{4}}{x^{4}}+b B \ln \left (x \right )\) | \(29\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 30, normalized size = 1.03 \begin {gather*} \frac {1}{2} \, B b \log \left (x^{2}\right ) - \frac {2 \, {\left (B a + A b\right )} x^{2} + A a}{4 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.18, size = 31, normalized size = 1.07 \begin {gather*} \frac {4 \, B b x^{4} \log \left (x\right ) - 2 \, {\left (B a + A b\right )} x^{2} - A a}{4 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.17, size = 29, normalized size = 1.00 \begin {gather*} B b \log {\left (x \right )} + \frac {- A a + x^{2} \left (- 2 A b - 2 B a\right )}{4 x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.80, size = 39, normalized size = 1.34 \begin {gather*} \frac {1}{2} \, B b \log \left (x^{2}\right ) - \frac {3 \, B b x^{4} + 2 \, B a x^{2} + 2 \, A b x^{2} + A a}{4 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 29, normalized size = 1.00 \begin {gather*} B\,b\,\ln \left (x\right )-\frac {\left (\frac {A\,b}{2}+\frac {B\,a}{2}\right )\,x^2+\frac {A\,a}{4}}{x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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