Optimal. Leaf size=36 \[ -\frac {a A}{2 x^2}-\frac {A b+a B}{x}+B c x+(b B+A c) \log (x) \]
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Rubi [A]
time = 0.02, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {779}
\begin {gather*} -\frac {a B+A b}{x}-\frac {a A}{2 x^2}+\log (x) (A c+b B)+B c x \end {gather*}
Antiderivative was successfully verified.
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Rule 779
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a+b x+c x^2\right )}{x^3} \, dx &=\int \left (B c+\frac {a A}{x^3}+\frac {A b+a B}{x^2}+\frac {b B+A c}{x}\right ) \, dx\\ &=-\frac {a A}{2 x^2}-\frac {A b+a B}{x}+B c x+(b B+A c) \log (x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 37, normalized size = 1.03 \begin {gather*} -\frac {a A}{2 x^2}+\frac {-A b-a B}{x}+B c x+(b B+A c) \log (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 35, normalized size = 0.97
method | result | size |
default | \(B c x -\frac {a A}{2 x^{2}}+\left (A c +B b \right ) \ln \left (x \right )-\frac {A b +B a}{x}\) | \(35\) |
risch | \(B c x +\frac {\left (-A b -B a \right ) x -\frac {A a}{2}}{x^{2}}+A c \ln \left (x \right )+b B \ln \left (x \right )\) | \(36\) |
norman | \(\frac {\left (-A b -B a \right ) x +B c \,x^{3}-\frac {A a}{2}}{x^{2}}+\left (A c +B b \right ) \ln \left (x \right )\) | \(38\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 34, normalized size = 0.94 \begin {gather*} B c x + {\left (B b + A c\right )} \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 = 2.67, size = 41, normalized size = 1.14 \begin {gather*} \frac {2 \, B c x^{3} + 2 \, {\left (B b + A c\right )} 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 = 0.15, size = 36, normalized size = 1.00 \begin {gather*} B c x + \left (A c + B b\right ) \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 = 1.31, size = 35, normalized size = 0.97 \begin {gather*} B c x + {\left (B b + A c\right )} \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 = 1.17, size = 34, normalized size = 0.94 \begin {gather*} \ln \left (x\right )\,\left (A\,c+B\,b\right )-\frac {\frac {A\,a}{2}+x\,\left (A\,b+B\,a\right )}{x^2}+B\,c\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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