Optimal. Leaf size=42 \[ \frac{a^2 \log (x)}{b^3}-\frac{a^2 \log (a x+b)}{b^3}+\frac{a}{b^2 x}-\frac{1}{2 b x^2} \]
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Rubi [A] time = 0.0198733, antiderivative size = 42, normalized size of antiderivative = 1., 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 = {263, 44} \[ \frac{a^2 \log (x)}{b^3}-\frac{a^2 \log (a x+b)}{b^3}+\frac{a}{b^2 x}-\frac{1}{2 b x^2} \]
Antiderivative was successfully verified.
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Rule 263
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{\left (a+\frac{b}{x}\right ) x^4} \, dx &=\int \frac{1}{x^3 (b+a x)} \, dx\\ &=\int \left (\frac{1}{b x^3}-\frac{a}{b^2 x^2}+\frac{a^2}{b^3 x}-\frac{a^3}{b^3 (b+a x)}\right ) \, dx\\ &=-\frac{1}{2 b x^2}+\frac{a}{b^2 x}+\frac{a^2 \log (x)}{b^3}-\frac{a^2 \log (b+a x)}{b^3}\\ \end{align*}
Mathematica [A] time = 0.0044564, size = 42, normalized size = 1. \[ \frac{a^2 \log (x)}{b^3}-\frac{a^2 \log (a x+b)}{b^3}+\frac{a}{b^2 x}-\frac{1}{2 b x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 41, normalized size = 1. \begin{align*} -{\frac{1}{2\,b{x}^{2}}}+{\frac{a}{{b}^{2}x}}+{\frac{{a}^{2}\ln \left ( x \right ) }{{b}^{3}}}-{\frac{{a}^{2}\ln \left ( ax+b \right ) }{{b}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0556, size = 54, normalized size = 1.29 \begin{align*} -\frac{a^{2} \log \left (a x + b\right )}{b^{3}} + \frac{a^{2} \log \left (x\right )}{b^{3}} + \frac{2 \, a x - b}{2 \, b^{2} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.45571, size = 103, normalized size = 2.45 \begin{align*} -\frac{2 \, a^{2} x^{2} \log \left (a x + b\right ) - 2 \, a^{2} x^{2} \log \left (x\right ) - 2 \, a b x + b^{2}}{2 \, b^{3} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.340505, size = 31, normalized size = 0.74 \begin{align*} \frac{a^{2} \left (\log{\left (x \right )} - \log{\left (x + \frac{b}{a} \right )}\right )}{b^{3}} + \frac{2 a x - b}{2 b^{2} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25202, size = 61, normalized size = 1.45 \begin{align*} -\frac{a^{2} \log \left ({\left | a x + b \right |}\right )}{b^{3}} + \frac{a^{2} \log \left ({\left | x \right |}\right )}{b^{3}} + \frac{2 \, a b x - b^{2}}{2 \, b^{3} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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