Optimal. Leaf size=34 \[ -\frac {2 b \log (a x+b)}{a^3}+\frac {2 x}{a^2}-\frac {x}{a \left (a+\frac {b}{x}\right )} \]
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Rubi [A] time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {192, 193, 43} \[ -\frac {2 b \log (a x+b)}{a^3}+\frac {2 x}{a^2}-\frac {x}{a \left (a+\frac {b}{x}\right )} \]
Antiderivative was successfully verified.
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Rule 43
Rule 192
Rule 193
Rubi steps
\begin {align*} \int \frac {1}{\left (a+\frac {b}{x}\right )^2} \, dx &=-\frac {x}{a \left (a+\frac {b}{x}\right )}+\frac {2 \int \frac {1}{a+\frac {b}{x}} \, dx}{a}\\ &=-\frac {x}{a \left (a+\frac {b}{x}\right )}+\frac {2 \int \frac {x}{b+a x} \, dx}{a}\\ &=-\frac {x}{a \left (a+\frac {b}{x}\right )}+\frac {2 \int \left (\frac {1}{a}-\frac {b}{a (b+a x)}\right ) \, dx}{a}\\ &=\frac {2 x}{a^2}-\frac {x}{a \left (a+\frac {b}{x}\right )}-\frac {2 b \log (b+a x)}{a^3}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 29, normalized size = 0.85 \[ \frac {-\frac {b^2}{a x+b}-2 b \log (a x+b)+a x}{a^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.07, size = 47, normalized size = 1.38 \[ \frac {a^{2} x^{2} + a b x - b^{2} - 2 \, {\left (a b x + b^{2}\right )} \log \left (a x + b\right )}{a^{4} x + a^{3} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 34, normalized size = 1.00 \[ \frac {x}{a^{2}} - \frac {2 \, b \log \left ({\left | a x + b \right |}\right )}{a^{3}} - \frac {b^{2}}{{\left (a x + b\right )} a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 34, normalized size = 1.00 \[ \frac {x}{a^{2}}-\frac {b^{2}}{\left (a x +b \right ) a^{3}}-\frac {2 b \ln \left (a x +b \right )}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.25, size = 36, normalized size = 1.06 \[ -\frac {b^{2}}{a^{4} x + a^{3} b} + \frac {x}{a^{2}} - \frac {2 \, b \log \left (a x + b\right )}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.06, size = 36, normalized size = 1.06 \[ \frac {x}{a^2}-\frac {b^2}{x\,a^4+b\,a^3}-\frac {2\,b\,\ln \left (b+a\,x\right )}{a^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.17, size = 31, normalized size = 0.91 \[ - \frac {b^{2}}{a^{4} x + a^{3} b} + \frac {x}{a^{2}} - \frac {2 b \log {\left (a x + b \right )}}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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