Optimal. Leaf size=42 \[ -\frac {2 a \log (x)}{b^3}+\frac {2 a \log (a x+b)}{b^3}-\frac {a}{b^2 (a x+b)}-\frac {1}{b^2 x} \]
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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 = {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 44
Rule 263
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*}
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Mathematica [A] time = 0.04, 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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fricas [A] time = 0.90, size = 63, normalized size = 1.50 \[ -\frac {2 \, a b x + b^{2} - 2 \, {\left (a^{2} x^{2} + a b x\right )} \log \left (a x + b\right ) + 2 \, {\left (a^{2} x^{2} + a b x\right )} \log \relax (x)}{a b^{3} x^{2} + b^{4} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 45, normalized size = 1.07 \[ \frac {2 \, a \log \left ({\left | a x + b \right |}\right )}{b^{3}} - \frac {2 \, a \log \left ({\left | x \right |}\right )}{b^{3}} - \frac {2 \, a x + b}{{\left (a x^{2} + b x\right )} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 43, normalized size = 1.02 \[ -\frac {a}{\left (a x +b \right ) b^{2}}-\frac {2 a \ln \relax (x )}{b^{3}}+\frac {2 a \ln \left (a x +b \right )}{b^{3}}-\frac {1}{b^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.14, size = 45, normalized size = 1.07 \[ -\frac {2 \, a x + b}{a b^{2} x^{2} + b^{3} x} + \frac {2 \, a \log \left (a x + b\right )}{b^{3}} - \frac {2 \, a \log \relax (x)}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.07, size = 41, normalized size = 0.98 \[ \frac {4\,a\,\mathrm {atanh}\left (\frac {2\,a\,x}{b}+1\right )}{b^3}-\frac {\frac {1}{b}+\frac {2\,a\,x}{b^2}}{a\,x^2+b\,x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.28, size = 37, normalized size = 0.88 \[ \frac {2 a \left (- \log {\relax (x )} + \log {\left (x + \frac {b}{a} \right )}\right )}{b^{3}} + \frac {- 2 a x - b}{a b^{2} x^{2} + b^{3} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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