Optimal. Leaf size=27 \[ -\frac {a B+A b}{x}-\frac {a A}{2 x^2}+b B \log (x) \]
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Rubi [A] time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {76} \begin {gather*} -\frac {a B+A b}{x}-\frac {a A}{2 x^2}+b B \log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 76
Rubi steps
\begin {align*} \int \frac {(a+b x) (A+B x)}{x^3} \, dx &=\int \left (\frac {a A}{x^3}+\frac {A b+a B}{x^2}+\frac {b B}{x}\right ) \, dx\\ &=-\frac {a A}{2 x^2}-\frac {A b+a B}{x}+b B \log (x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 28, normalized size = 1.04 \begin {gather*} \frac {-a B-A b}{x}-\frac {a A}{2 x^2}+b B \log (x) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x) (A+B x)}{x^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.18, size = 29, normalized size = 1.07 \begin {gather*} \frac {2 \, B b x^{2} \log \relax (x) - A a - 2 \, {\left (B a + A b\right )} x}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.22, size = 26, normalized size = 0.96 \begin {gather*} B b \log \left ({\left | x \right |}\right ) - \frac {A a + 2 \, {\left (B a + A b\right )} x}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 28, normalized size = 1.04 \begin {gather*} B b \ln \relax (x )-\frac {A b}{x}-\frac {B a}{x}-\frac {A a}{2 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.12, size = 25, normalized size = 0.93 \begin {gather*} B b \log \relax (x) - \frac {A a + 2 \, {\left (B a + A b\right )} x}{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 = 0.93 \begin {gather*} B\,b\,\ln \relax (x)-\frac {\frac {A\,a}{2}+x\,\left (A\,b+B\,a\right )}{x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.30, size = 27, normalized size = 1.00 \begin {gather*} B b \log {\relax (x )} + \frac {- A a + x \left (- 2 A b - 2 B a\right )}{2 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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