Optimal. Leaf size=28 \[ -\frac {b \log (x)}{a^2}+\frac {b \log (a+b x)}{a^2}-\frac {1}{a x} \]
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Rubi [A] time = 0.02, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {1584, 44} \[ -\frac {b \log (x)}{a^2}+\frac {b \log (a+b x)}{a^2}-\frac {1}{a x} \]
Antiderivative was successfully verified.
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Rule 44
Rule 1584
Rubi steps
\begin {align*} \int \frac {x}{a x^3+b x^4} \, dx &=\int \frac {1}{x^2 (a+b x)} \, dx\\ &=\int \left (\frac {1}{a x^2}-\frac {b}{a^2 x}+\frac {b^2}{a^2 (a+b x)}\right ) \, dx\\ &=-\frac {1}{a x}-\frac {b \log (x)}{a^2}+\frac {b \log (a+b x)}{a^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 28, normalized size = 1.00 \[ -\frac {b \log (x)}{a^2}+\frac {b \log (a+b x)}{a^2}-\frac {1}{a x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.38, size = 26, normalized size = 0.93 \[ \frac {b x \log \left (b x + a\right ) - b x \log \relax (x) - a}{a^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 30, normalized size = 1.07 \[ \frac {b \log \left ({\left | b x + a \right |}\right )}{a^{2}} - \frac {b \log \left ({\left | x \right |}\right )}{a^{2}} - \frac {1}{a x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 29, normalized size = 1.04 \[ -\frac {b \ln \relax (x )}{a^{2}}+\frac {b \ln \left (b x +a \right )}{a^{2}}-\frac {1}{a x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 28, normalized size = 1.00 \[ \frac {b \log \left (b x + a\right )}{a^{2}} - \frac {b \log \relax (x)}{a^{2}} - \frac {1}{a x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.21, size = 25, normalized size = 0.89 \[ \frac {2\,b\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )}{a^2}-\frac {1}{a\,x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.21, size = 19, normalized size = 0.68 \[ - \frac {1}{a x} + \frac {b \left (- \log {\relax (x )} + \log {\left (\frac {a}{b} + x \right )}\right )}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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