Optimal. Leaf size=84 \[ -\frac{3 b^2}{2 a^4 x^2}+\frac{b^4}{a^5 (a+b x)}+\frac{4 b^3}{a^5 x}+\frac{5 b^4 \log (x)}{a^6}-\frac{5 b^4 \log (a+b x)}{a^6}+\frac{2 b}{3 a^3 x^3}-\frac{1}{4 a^2 x^4} \]
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Rubi [A] time = 0.0428869, antiderivative size = 84, 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} \[ -\frac{3 b^2}{2 a^4 x^2}+\frac{b^4}{a^5 (a+b x)}+\frac{4 b^3}{a^5 x}+\frac{5 b^4 \log (x)}{a^6}-\frac{5 b^4 \log (a+b x)}{a^6}+\frac{2 b}{3 a^3 x^3}-\frac{1}{4 a^2 x^4} \]
Antiderivative was successfully verified.
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Rule 44
Rubi steps
\begin{align*} \int \frac{1}{x^5 (a+b x)^2} \, dx &=\int \left (\frac{1}{a^2 x^5}-\frac{2 b}{a^3 x^4}+\frac{3 b^2}{a^4 x^3}-\frac{4 b^3}{a^5 x^2}+\frac{5 b^4}{a^6 x}-\frac{b^5}{a^5 (a+b x)^2}-\frac{5 b^5}{a^6 (a+b x)}\right ) \, dx\\ &=-\frac{1}{4 a^2 x^4}+\frac{2 b}{3 a^3 x^3}-\frac{3 b^2}{2 a^4 x^2}+\frac{4 b^3}{a^5 x}+\frac{b^4}{a^5 (a+b x)}+\frac{5 b^4 \log (x)}{a^6}-\frac{5 b^4 \log (a+b x)}{a^6}\\ \end{align*}
Mathematica [A] time = 0.0792331, size = 79, normalized size = 0.94 \[ \frac{\frac{a \left (-10 a^2 b^2 x^2+5 a^3 b x-3 a^4+30 a b^3 x^3+60 b^4 x^4\right )}{x^4 (a+b x)}-60 b^4 \log (a+b x)+60 b^4 \log (x)}{12 a^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 79, normalized size = 0.9 \begin{align*} -{\frac{1}{4\,{a}^{2}{x}^{4}}}+{\frac{2\,b}{3\,{a}^{3}{x}^{3}}}-{\frac{3\,{b}^{2}}{2\,{a}^{4}{x}^{2}}}+4\,{\frac{{b}^{3}}{{a}^{5}x}}+{\frac{{b}^{4}}{{a}^{5} \left ( bx+a \right ) }}+5\,{\frac{{b}^{4}\ln \left ( x \right ) }{{a}^{6}}}-5\,{\frac{{b}^{4}\ln \left ( bx+a \right ) }{{a}^{6}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05882, size = 116, normalized size = 1.38 \begin{align*} \frac{60 \, b^{4} x^{4} + 30 \, a b^{3} x^{3} - 10 \, a^{2} b^{2} x^{2} + 5 \, a^{3} b x - 3 \, a^{4}}{12 \,{\left (a^{5} b x^{5} + a^{6} x^{4}\right )}} - \frac{5 \, b^{4} \log \left (b x + a\right )}{a^{6}} + \frac{5 \, b^{4} \log \left (x\right )}{a^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.4977, size = 231, normalized size = 2.75 \begin{align*} \frac{60 \, a b^{4} x^{4} + 30 \, a^{2} b^{3} x^{3} - 10 \, a^{3} b^{2} x^{2} + 5 \, a^{4} b x - 3 \, a^{5} - 60 \,{\left (b^{5} x^{5} + a b^{4} x^{4}\right )} \log \left (b x + a\right ) + 60 \,{\left (b^{5} x^{5} + a b^{4} x^{4}\right )} \log \left (x\right )}{12 \,{\left (a^{6} b x^{5} + a^{7} x^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.704667, size = 80, normalized size = 0.95 \begin{align*} \frac{- 3 a^{4} + 5 a^{3} b x - 10 a^{2} b^{2} x^{2} + 30 a b^{3} x^{3} + 60 b^{4} x^{4}}{12 a^{6} x^{4} + 12 a^{5} b x^{5}} + \frac{5 b^{4} \left (\log{\left (x \right )} - \log{\left (\frac{a}{b} + x \right )}\right )}{a^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19309, size = 140, normalized size = 1.67 \begin{align*} \frac{5 \, b^{4} \log \left ({\left | -\frac{a}{b x + a} + 1 \right |}\right )}{a^{6}} + \frac{b^{4}}{{\left (b x + a\right )} a^{5}} - \frac{\frac{260 \, a b^{4}}{b x + a} - \frac{300 \, a^{2} b^{4}}{{\left (b x + a\right )}^{2}} + \frac{120 \, a^{3} b^{4}}{{\left (b x + a\right )}^{3}} - 77 \, b^{4}}{12 \, a^{6}{\left (\frac{a}{b x + a} - 1\right )}^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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