Optimal. Leaf size=27 \[ -\frac {a A}{2 x^2}-\frac {A b+a B}{x}+b B \log (x) \]
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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 = {77}
\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 77
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 = 0.01, size = 28, normalized size = 1.04 \begin {gather*} -\frac {a A}{2 x^2}+\frac {-A b-a B}{x}+b B \log (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 26, normalized size = 0.96
method | result | size |
default | \(-\frac {A b +B a}{x}-\frac {a A}{2 x^{2}}+b B \ln \left (x \right )\) | \(26\) |
norman | \(\frac {\left (-A b -B a \right ) x -\frac {A a}{2}}{x^{2}}+b B \ln \left (x \right )\) | \(27\) |
risch | \(\frac {\left (-A b -B a \right ) x -\frac {A a}{2}}{x^{2}}+b B \ln \left (x \right )\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 25, normalized size = 0.93 \begin {gather*} B b \log \left (x\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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Fricas [A]
time = 0.96, size = 29, normalized size = 1.07 \begin {gather*} \frac {2 \, B b x^{2} \log \left (x\right ) - 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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Sympy [A]
time = 0.11, size = 27, normalized size = 1.00 \begin {gather*} B b \log {\left (x \right )} + \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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Giac [A]
time = 1.73, 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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Mupad [B]
time = 0.04, size = 25, normalized size = 0.93 \begin {gather*} B\,b\,\ln \left (x\right )-\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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