Optimal. Leaf size=57 \[ -\frac{2 b}{a^3 (a+b x)}-\frac{b}{2 a^2 (a+b x)^2}-\frac{3 b \log (x)}{a^4}+\frac{3 b \log (a+b x)}{a^4}-\frac{1}{a^3 x} \]
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Rubi [A] time = 0.0292758, antiderivative size = 57, normalized size of antiderivative = 1., 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} \[ -\frac{2 b}{a^3 (a+b x)}-\frac{b}{2 a^2 (a+b x)^2}-\frac{3 b \log (x)}{a^4}+\frac{3 b \log (a+b x)}{a^4}-\frac{1}{a^3 x} \]
Antiderivative was successfully verified.
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Rule 44
Rubi steps
\begin{align*} \int \frac{1}{x^2 (a+b x)^3} \, dx &=\int \left (\frac{1}{a^3 x^2}-\frac{3 b}{a^4 x}+\frac{b^2}{a^2 (a+b x)^3}+\frac{2 b^2}{a^3 (a+b x)^2}+\frac{3 b^2}{a^4 (a+b x)}\right ) \, dx\\ &=-\frac{1}{a^3 x}-\frac{b}{2 a^2 (a+b x)^2}-\frac{2 b}{a^3 (a+b x)}-\frac{3 b \log (x)}{a^4}+\frac{3 b \log (a+b x)}{a^4}\\ \end{align*}
Mathematica [A] time = 0.0767011, size = 53, normalized size = 0.93 \[ -\frac{\frac{a \left (2 a^2+9 a b x+6 b^2 x^2\right )}{x (a+b x)^2}-6 b \log (a+b x)+6 b \log (x)}{2 a^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 56, normalized size = 1. \begin{align*} -{\frac{1}{{a}^{3}x}}-{\frac{b}{2\,{a}^{2} \left ( bx+a \right ) ^{2}}}-2\,{\frac{b}{{a}^{3} \left ( bx+a \right ) }}-3\,{\frac{b\ln \left ( x \right ) }{{a}^{4}}}+3\,{\frac{b\ln \left ( bx+a \right ) }{{a}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08097, size = 93, normalized size = 1.63 \begin{align*} -\frac{6 \, b^{2} x^{2} + 9 \, a b x + 2 \, a^{2}}{2 \,{\left (a^{3} b^{2} x^{3} + 2 \, a^{4} b x^{2} + a^{5} x\right )}} + \frac{3 \, b \log \left (b x + a\right )}{a^{4}} - \frac{3 \, b \log \left (x\right )}{a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.67475, size = 232, normalized size = 4.07 \begin{align*} -\frac{6 \, a b^{2} x^{2} + 9 \, a^{2} b x + 2 \, a^{3} - 6 \,{\left (b^{3} x^{3} + 2 \, a b^{2} x^{2} + a^{2} b x\right )} \log \left (b x + a\right ) + 6 \,{\left (b^{3} x^{3} + 2 \, a b^{2} x^{2} + a^{2} b x\right )} \log \left (x\right )}{2 \,{\left (a^{4} b^{2} x^{3} + 2 \, a^{5} b x^{2} + a^{6} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.643188, size = 65, normalized size = 1.14 \begin{align*} - \frac{2 a^{2} + 9 a b x + 6 b^{2} x^{2}}{2 a^{5} x + 4 a^{4} b x^{2} + 2 a^{3} b^{2} x^{3}} + \frac{3 b \left (- \log{\left (x \right )} + \log{\left (\frac{a}{b} + x \right )}\right )}{a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23607, size = 81, normalized size = 1.42 \begin{align*} \frac{3 \, b \log \left ({\left | b x + a \right |}\right )}{a^{4}} - \frac{3 \, b \log \left ({\left | x \right |}\right )}{a^{4}} - \frac{6 \, a b^{2} x^{2} + 9 \, a^{2} b x + 2 \, a^{3}}{2 \,{\left (b x + a\right )}^{2} a^{4} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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