Optimal. Leaf size=42 \[ \frac {b^2 \log (x)}{a^3}-\frac {b^2 \log (a+b x)}{a^3}+\frac {b}{a^2 x}-\frac {1}{2 a x^2} \]
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Rubi [A] time = 0.02, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1593, 44} \[ \frac {b^2 \log (x)}{a^3}-\frac {b^2 \log (a+b x)}{a^3}+\frac {b}{a^2 x}-\frac {1}{2 a x^2} \]
Antiderivative was successfully verified.
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Rule 44
Rule 1593
Rubi steps
\begin {align*} \int \frac {1}{a x^3+b x^4} \, dx &=\int \frac {1}{x^3 (a+b x)} \, dx\\ &=\int \left (\frac {1}{a x^3}-\frac {b}{a^2 x^2}+\frac {b^2}{a^3 x}-\frac {b^3}{a^3 (a+b x)}\right ) \, dx\\ &=-\frac {1}{2 a x^2}+\frac {b}{a^2 x}+\frac {b^2 \log (x)}{a^3}-\frac {b^2 \log (a+b x)}{a^3}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 42, normalized size = 1.00 \[ \frac {b^2 \log (x)}{a^3}-\frac {b^2 \log (a+b x)}{a^3}+\frac {b}{a^2 x}-\frac {1}{2 a x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 41, normalized size = 0.98 \[ -\frac {2 \, b^{2} x^{2} \log \left (b x + a\right ) - 2 \, b^{2} x^{2} \log \relax (x) - 2 \, a b x + a^{2}}{2 \, a^{3} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 45, normalized size = 1.07 \[ -\frac {b^{2} \log \left ({\left | b x + a \right |}\right )}{a^{3}} + \frac {b^{2} \log \left ({\left | x \right |}\right )}{a^{3}} + \frac {2 \, a b x - a^{2}}{2 \, a^{3} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 41, normalized size = 0.98 \[ \frac {b^{2} \ln \relax (x )}{a^{3}}-\frac {b^{2} \ln \left (b x +a \right )}{a^{3}}+\frac {b}{a^{2} x}-\frac {1}{2 a \,x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 40, normalized size = 0.95 \[ -\frac {b^{2} \log \left (b x + a\right )}{a^{3}} + \frac {b^{2} \log \relax (x)}{a^{3}} + \frac {2 \, b x - a}{2 \, a^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.72, size = 38, normalized size = 0.90 \[ -\frac {\frac {a^2}{2}-a\,b\,x}{a^3\,x^2}-\frac {2\,b^2\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )}{a^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.22, size = 31, normalized size = 0.74 \[ \frac {- a + 2 b x}{2 a^{2} x^{2}} + \frac {b^{2} \left (\log {\relax (x )} - \log {\left (\frac {a}{b} + x \right )}\right )}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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