Optimal. Leaf size=43 \[ -\frac {\log (a+b x)}{a^3}+\frac {\log (x)}{a^3}+\frac {1}{a^2 (a+b x)}+\frac {1}{2 a (a+b x)^2} \]
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Rubi [A] time = 0.02, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {44} \[ \frac {1}{a^2 (a+b x)}-\frac {\log (a+b x)}{a^3}+\frac {\log (x)}{a^3}+\frac {1}{2 a (a+b x)^2} \]
Antiderivative was successfully verified.
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Rule 44
Rubi steps
\begin {align*} \int \frac {1}{x (a+b x)^3} \, dx &=\int \left (\frac {1}{a^3 x}-\frac {b}{a (a+b x)^3}-\frac {b}{a^2 (a+b x)^2}-\frac {b}{a^3 (a+b x)}\right ) \, dx\\ &=\frac {1}{2 a (a+b x)^2}+\frac {1}{a^2 (a+b x)}+\frac {\log (x)}{a^3}-\frac {\log (a+b x)}{a^3}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 37, normalized size = 0.86 \[ \frac {\frac {a (3 a+2 b x)}{(a+b x)^2}-2 \log (a+b x)+2 \log (x)}{2 a^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 80, normalized size = 1.86 \[ \frac {2 \, a b x + 3 \, a^{2} - 2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \log \left (b x + a\right ) + 2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \log \relax (x)}{2 \, {\left (a^{3} b^{2} x^{2} + 2 \, a^{4} b x + a^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.03, size = 43, normalized size = 1.00 \[ -\frac {\log \left ({\left | b x + a \right |}\right )}{a^{3}} + \frac {\log \left ({\left | x \right |}\right )}{a^{3}} + \frac {2 \, a b x + 3 \, a^{2}}{2 \, {\left (b x + a\right )}^{2} a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 42, normalized size = 0.98 \[ \frac {1}{2 \left (b x +a \right )^{2} a}+\frac {1}{\left (b x +a \right ) a^{2}}+\frac {\ln \relax (x )}{a^{3}}-\frac {\ln \left (b x +a \right )}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 51, normalized size = 1.19 \[ \frac {2 \, b x + 3 \, a}{2 \, {\left (a^{2} b^{2} x^{2} + 2 \, a^{3} b x + a^{4}\right )}} - \frac {\log \left (b x + a\right )}{a^{3}} + \frac {\log \relax (x)}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 43, normalized size = 1.00 \[ \frac {\frac {1}{a^2+b\,x\,a}-\frac {\ln \left (\frac {a+b\,x}{x}\right )}{a^2}}{a}+\frac {1}{2\,a\,{\left (a+b\,x\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.35, size = 46, normalized size = 1.07 \[ \frac {3 a + 2 b x}{2 a^{4} + 4 a^{3} b x + 2 a^{2} b^{2} x^{2}} + \frac {\log {\relax (x )} - \log {\left (\frac {a}{b} + x \right )}}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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