Optimal. Leaf size=42 \[ -\frac{a}{b^2 (a x+b)}-\frac{2 a \log (x)}{b^3}+\frac{2 a \log (a x+b)}{b^3}-\frac{1}{b^2 x} \]
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Rubi [A] time = 0.0240158, 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} \[ -\frac{a}{b^2 (a x+b)}-\frac{2 a \log (x)}{b^3}+\frac{2 a \log (a x+b)}{b^3}-\frac{1}{b^2 x} \]
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 )^2 x^4} \, dx &=\int \frac{1}{x^2 (b+a x)^2} \, dx\\ &=\int \left (\frac{1}{b^2 x^2}-\frac{2 a}{b^3 x}+\frac{a^2}{b^2 (b+a x)^2}+\frac{2 a^2}{b^3 (b+a x)}\right ) \, dx\\ &=-\frac{1}{b^2 x}-\frac{a}{b^2 (b+a x)}-\frac{2 a \log (x)}{b^3}+\frac{2 a \log (b+a x)}{b^3}\\ \end{align*}
Mathematica [A] time = 0.0386083, size = 35, normalized size = 0.83 \[ -\frac{b \left (\frac{a}{a x+b}+\frac{1}{x}\right )-2 a \log (a x+b)+2 a \log (x)}{b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 43, normalized size = 1. \begin{align*} -{\frac{1}{{b}^{2}x}}-{\frac{a}{{b}^{2} \left ( ax+b \right ) }}-2\,{\frac{a\ln \left ( x \right ) }{{b}^{3}}}+2\,{\frac{a\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 = 0.988525, size = 61, normalized size = 1.45 \begin{align*} -\frac{2 \, a x + b}{a b^{2} x^{2} + b^{3} x} + \frac{2 \, a \log \left (a x + b\right )}{b^{3}} - \frac{2 \, a \log \left (x\right )}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.4455, size = 138, normalized size = 3.29 \begin{align*} -\frac{2 \, a b x + b^{2} - 2 \,{\left (a^{2} x^{2} + a b x\right )} \log \left (a x + b\right ) + 2 \,{\left (a^{2} x^{2} + a b x\right )} \log \left (x\right )}{a b^{3} x^{2} + b^{4} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.388862, size = 36, normalized size = 0.86 \begin{align*} \frac{2 a \left (- \log{\left (x \right )} + \log{\left (x + \frac{b}{a} \right )}\right )}{b^{3}} - \frac{2 a x + b}{a b^{2} x^{2} + b^{3} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11049, size = 61, normalized size = 1.45 \begin{align*} \frac{2 \, a \log \left ({\left | a x + b \right |}\right )}{b^{3}} - \frac{2 \, a \log \left ({\left | x \right |}\right )}{b^{3}} - \frac{2 \, a x + b}{{\left (a x^{2} + b x\right )} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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