Optimal. Leaf size=22 \[ -\frac {A b}{x}+B c x+(b B+A c) \log (x) \]
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Rubi [A]
time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {779}
\begin {gather*} \log (x) (A c+b B)-\frac {A b}{x}+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 (b x+c x^2\right )}{x^3} \, dx &=\int \left (B c+\frac {A b}{x^2}+\frac {b B+A c}{x}\right ) \, dx\\ &=-\frac {A b}{x}+B c x+(b B+A c) \log (x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 22, normalized size = 1.00 \begin {gather*} -\frac {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 = 0.04, size = 23, normalized size = 1.05
method | result | size |
default | \(-\frac {A b}{x}+B c x +\left (A c +B b \right ) \ln \left (x \right )\) | \(23\) |
risch | \(-\frac {A b}{x}+B c x +A c \ln \left (x \right )+b B \ln \left (x \right )\) | \(23\) |
norman | \(\frac {B c \,x^{3}-A b x}{x^{2}}+\left (A c +B b \right ) \ln \left (x \right )\) | \(28\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 22, normalized size = 1.00 \begin {gather*} B c x + {\left (B b + A c\right )} \log \left (x\right ) - \frac {A b}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.73, size = 26, normalized size = 1.18 \begin {gather*} \frac {B c x^{2} + {\left (B b + A c\right )} x \log \left (x\right ) - A b}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 19, normalized size = 0.86 \begin {gather*} - \frac {A b}{x} + B c x + \left (A c + B b\right ) \log {\left (x \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.26, size = 23, normalized size = 1.05 \begin {gather*} B c x + {\left (B b + A c\right )} \log \left ({\left | x \right |}\right ) - \frac {A b}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 22, normalized size = 1.00 \begin {gather*} \ln \left (x\right )\,\left (A\,c+B\,b\right )+B\,c\,x-\frac {A\,b}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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