Optimal. Leaf size=41 \[ -\frac {b^2}{2 a^3 (a x+b)^2}+\frac {2 b}{a^3 (a x+b)}+\frac {\log (a x+b)}{a^3} \]
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Rubi [A] time = 0.02, antiderivative size = 41, 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, 43} \[ -\frac {b^2}{2 a^3 (a x+b)^2}+\frac {2 b}{a^3 (a x+b)}+\frac {\log (a x+b)}{a^3} \]
Antiderivative was successfully verified.
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Rule 43
Rule 263
Rubi steps
\begin {align*} \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x} \, dx &=\int \frac {x^2}{(b+a x)^3} \, dx\\ &=\int \left (\frac {b^2}{a^2 (b+a x)^3}-\frac {2 b}{a^2 (b+a x)^2}+\frac {1}{a^2 (b+a x)}\right ) \, dx\\ &=-\frac {b^2}{2 a^3 (b+a x)^2}+\frac {2 b}{a^3 (b+a x)}+\frac {\log (b+a x)}{a^3}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 33, normalized size = 0.80 \[ \frac {\frac {b (4 a x+3 b)}{(a x+b)^2}+2 \log (a x+b)}{2 a^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 61, normalized size = 1.49 \[ \frac {4 \, a b x + 3 \, b^{2} + 2 \, {\left (a^{2} x^{2} + 2 \, a b x + b^{2}\right )} \log \left (a x + b\right )}{2 \, {\left (a^{5} x^{2} + 2 \, a^{4} b x + a^{3} b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 37, normalized size = 0.90 \[ \frac {\log \left ({\left | a x + b \right |}\right )}{a^{3}} + \frac {4 \, b x + \frac {3 \, b^{2}}{a}}{2 \, {\left (a x + b\right )}^{2} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 40, normalized size = 0.98 \[ -\frac {b^{2}}{2 \left (a x +b \right )^{2} a^{3}}+\frac {2 b}{\left (a x +b \right ) a^{3}}+\frac {\ln \left (a x +b \right )}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.98, size = 48, normalized size = 1.17 \[ \frac {4 \, a b x + 3 \, b^{2}}{2 \, {\left (a^{5} x^{2} + 2 \, a^{4} b x + a^{3} b^{2}\right )}} + \frac {\log \left (a x + b\right )}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.07, size = 46, normalized size = 1.12 \[ \frac {\ln \left (b+a\,x\right )}{a^3}+\frac {\frac {3\,b^2}{2\,a^3}+\frac {2\,b\,x}{a^2}}{a^2\,x^2+2\,a\,b\,x+b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.22, size = 46, normalized size = 1.12 \[ \frac {4 a b x + 3 b^{2}}{2 a^{5} x^{2} + 4 a^{4} b x + 2 a^{3} b^{2}} + \frac {\log {\left (a x + b \right )}}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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