Optimal. Leaf size=53 \[ -\frac{3 x}{2 a^2 \left (a+\frac{b}{x}\right )}-\frac{3 b \log (a x+b)}{a^4}+\frac{3 x}{a^3}-\frac{x}{2 a \left (a+\frac{b}{x}\right )^2} \]
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Rubi [A] time = 0.0202364, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {192, 193, 43} \[ -\frac{3 x}{2 a^2 \left (a+\frac{b}{x}\right )}-\frac{3 b \log (a x+b)}{a^4}+\frac{3 x}{a^3}-\frac{x}{2 a \left (a+\frac{b}{x}\right )^2} \]
Antiderivative was successfully verified.
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Rule 192
Rule 193
Rule 43
Rubi steps
\begin{align*} \int \frac{1}{\left (a+\frac{b}{x}\right )^3} \, dx &=-\frac{x}{2 a \left (a+\frac{b}{x}\right )^2}+\frac{3 \int \frac{1}{\left (a+\frac{b}{x}\right )^2} \, dx}{2 a}\\ &=-\frac{x}{2 a \left (a+\frac{b}{x}\right )^2}-\frac{3 x}{2 a^2 \left (a+\frac{b}{x}\right )}+\frac{3 \int \frac{1}{a+\frac{b}{x}} \, dx}{a^2}\\ &=-\frac{x}{2 a \left (a+\frac{b}{x}\right )^2}-\frac{3 x}{2 a^2 \left (a+\frac{b}{x}\right )}+\frac{3 \int \frac{x}{b+a x} \, dx}{a^2}\\ &=-\frac{x}{2 a \left (a+\frac{b}{x}\right )^2}-\frac{3 x}{2 a^2 \left (a+\frac{b}{x}\right )}+\frac{3 \int \left (\frac{1}{a}-\frac{b}{a (b+a x)}\right ) \, dx}{a^2}\\ &=\frac{3 x}{a^3}-\frac{x}{2 a \left (a+\frac{b}{x}\right )^2}-\frac{3 x}{2 a^2 \left (a+\frac{b}{x}\right )}-\frac{3 b \log (b+a x)}{a^4}\\ \end{align*}
Mathematica [A] time = 0.037686, size = 40, normalized size = 0.75 \[ -\frac{\frac{b^2 (6 a x+5 b)}{(a x+b)^2}+6 b \log (a x+b)-2 a x}{2 a^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 49, normalized size = 0.9 \begin{align*}{\frac{x}{{a}^{3}}}-3\,{\frac{b\ln \left ( ax+b \right ) }{{a}^{4}}}+{\frac{{b}^{3}}{2\,{a}^{4} \left ( ax+b \right ) ^{2}}}-3\,{\frac{{b}^{2}}{{a}^{4} \left ( ax+b \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10017, size = 77, normalized size = 1.45 \begin{align*} -\frac{6 \, a b^{2} x + 5 \, b^{3}}{2 \,{\left (a^{6} x^{2} + 2 \, a^{5} b x + a^{4} b^{2}\right )}} + \frac{x}{a^{3}} - \frac{3 \, b \log \left (a x + b\right )}{a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.42842, size = 176, normalized size = 3.32 \begin{align*} \frac{2 \, a^{3} x^{3} + 4 \, a^{2} b x^{2} - 4 \, a b^{2} x - 5 \, b^{3} - 6 \,{\left (a^{2} b x^{2} + 2 \, a b^{2} x + b^{3}\right )} \log \left (a x + b\right )}{2 \,{\left (a^{6} x^{2} + 2 \, a^{5} b x + a^{4} b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.398499, size = 56, normalized size = 1.06 \begin{align*} - \frac{6 a b^{2} x + 5 b^{3}}{2 a^{6} x^{2} + 4 a^{5} b x + 2 a^{4} b^{2}} + \frac{x}{a^{3}} - \frac{3 b \log{\left (a x + b \right )}}{a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11505, size = 59, normalized size = 1.11 \begin{align*} \frac{x}{a^{3}} - \frac{3 \, b \log \left ({\left | a x + b \right |}\right )}{a^{4}} - \frac{6 \, a b^{2} x + 5 \, b^{3}}{2 \,{\left (a x + b\right )}^{2} a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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