Optimal. Leaf size=69 \[ -\frac{a^3}{b^4 (a x+b)}-\frac{3 a^2}{b^4 x}-\frac{4 a^3 \log (x)}{b^5}+\frac{4 a^3 \log (a x+b)}{b^5}+\frac{a}{b^3 x^2}-\frac{1}{3 b^2 x^3} \]
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Rubi [A] time = 0.0379721, antiderivative size = 69, normalized size of antiderivative = 1., 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, 44} \[ -\frac{a^3}{b^4 (a x+b)}-\frac{3 a^2}{b^4 x}-\frac{4 a^3 \log (x)}{b^5}+\frac{4 a^3 \log (a x+b)}{b^5}+\frac{a}{b^3 x^2}-\frac{1}{3 b^2 x^3} \]
Antiderivative was successfully verified.
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Rule 263
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{\left (a+\frac{b}{x}\right )^2 x^6} \, dx &=\int \frac{1}{x^4 (b+a x)^2} \, dx\\ &=\int \left (\frac{1}{b^2 x^4}-\frac{2 a}{b^3 x^3}+\frac{3 a^2}{b^4 x^2}-\frac{4 a^3}{b^5 x}+\frac{a^4}{b^4 (b+a x)^2}+\frac{4 a^4}{b^5 (b+a x)}\right ) \, dx\\ &=-\frac{1}{3 b^2 x^3}+\frac{a}{b^3 x^2}-\frac{3 a^2}{b^4 x}-\frac{a^3}{b^4 (b+a x)}-\frac{4 a^3 \log (x)}{b^5}+\frac{4 a^3 \log (b+a x)}{b^5}\\ \end{align*}
Mathematica [A] time = 0.0527656, size = 66, normalized size = 0.96 \[ -\frac{\frac{b \left (6 a^2 b x^2+12 a^3 x^3-2 a b^2 x+b^3\right )}{x^3 (a x+b)}-12 a^3 \log (a x+b)+12 a^3 \log (x)}{3 b^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 68, normalized size = 1. \begin{align*} -{\frac{1}{3\,{b}^{2}{x}^{3}}}+{\frac{a}{{b}^{3}{x}^{2}}}-3\,{\frac{{a}^{2}}{{b}^{4}x}}-{\frac{{a}^{3}}{{b}^{4} \left ( ax+b \right ) }}-4\,{\frac{{a}^{3}\ln \left ( x \right ) }{{b}^{5}}}+4\,{\frac{{a}^{3}\ln \left ( ax+b \right ) }{{b}^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01542, size = 99, normalized size = 1.43 \begin{align*} -\frac{12 \, a^{3} x^{3} + 6 \, a^{2} b x^{2} - 2 \, a b^{2} x + b^{3}}{3 \,{\left (a b^{4} x^{4} + b^{5} x^{3}\right )}} + \frac{4 \, a^{3} \log \left (a x + b\right )}{b^{5}} - \frac{4 \, a^{3} \log \left (x\right )}{b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.47069, size = 204, normalized size = 2.96 \begin{align*} -\frac{12 \, a^{3} b x^{3} + 6 \, a^{2} b^{2} x^{2} - 2 \, a b^{3} x + b^{4} - 12 \,{\left (a^{4} x^{4} + a^{3} b x^{3}\right )} \log \left (a x + b\right ) + 12 \,{\left (a^{4} x^{4} + a^{3} b x^{3}\right )} \log \left (x\right )}{3 \,{\left (a b^{5} x^{4} + b^{6} x^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.461942, size = 66, normalized size = 0.96 \begin{align*} \frac{4 a^{3} \left (- \log{\left (x \right )} + \log{\left (x + \frac{b}{a} \right )}\right )}{b^{5}} - \frac{12 a^{3} x^{3} + 6 a^{2} b x^{2} - 2 a b^{2} x + b^{3}}{3 a b^{4} x^{4} + 3 b^{5} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12123, size = 99, normalized size = 1.43 \begin{align*} \frac{4 \, a^{3} \log \left ({\left | a x + b \right |}\right )}{b^{5}} - \frac{4 \, a^{3} \log \left ({\left | x \right |}\right )}{b^{5}} - \frac{12 \, a^{3} b x^{3} + 6 \, a^{2} b^{2} x^{2} - 2 \, a b^{3} x + b^{4}}{3 \,{\left (a x + b\right )} b^{5} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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