Optimal. Leaf size=68 \[ -\frac{b^2}{2 a^3 x^2}+\frac{b^3}{a^4 x}+\frac{b^4 \log (x)}{a^5}-\frac{b^4 \log (a+b x)}{a^5}+\frac{b}{3 a^2 x^3}-\frac{1}{4 a x^4} \]
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Rubi [A] time = 0.0354385, antiderivative size = 68, 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} \[ -\frac{b^2}{2 a^3 x^2}+\frac{b^3}{a^4 x}+\frac{b^4 \log (x)}{a^5}-\frac{b^4 \log (a+b x)}{a^5}+\frac{b}{3 a^2 x^3}-\frac{1}{4 a x^4} \]
Antiderivative was successfully verified.
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Rule 44
Rubi steps
\begin{align*} \int \frac{1}{x^5 (a+b x)} \, dx &=\int \left (\frac{1}{a x^5}-\frac{b}{a^2 x^4}+\frac{b^2}{a^3 x^3}-\frac{b^3}{a^4 x^2}+\frac{b^4}{a^5 x}-\frac{b^5}{a^5 (a+b x)}\right ) \, dx\\ &=-\frac{1}{4 a x^4}+\frac{b}{3 a^2 x^3}-\frac{b^2}{2 a^3 x^2}+\frac{b^3}{a^4 x}+\frac{b^4 \log (x)}{a^5}-\frac{b^4 \log (a+b x)}{a^5}\\ \end{align*}
Mathematica [A] time = 0.0075066, size = 68, normalized size = 1. \[ -\frac{b^2}{2 a^3 x^2}+\frac{b^3}{a^4 x}+\frac{b^4 \log (x)}{a^5}-\frac{b^4 \log (a+b x)}{a^5}+\frac{b}{3 a^2 x^3}-\frac{1}{4 a x^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 63, normalized size = 0.9 \begin{align*} -{\frac{1}{4\,a{x}^{4}}}+{\frac{b}{3\,{a}^{2}{x}^{3}}}-{\frac{{b}^{2}}{2\,{a}^{3}{x}^{2}}}+{\frac{{b}^{3}}{{a}^{4}x}}+{\frac{{b}^{4}\ln \left ( x \right ) }{{a}^{5}}}-{\frac{{b}^{4}\ln \left ( bx+a \right ) }{{a}^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01518, size = 84, normalized size = 1.24 \begin{align*} -\frac{b^{4} \log \left (b x + a\right )}{a^{5}} + \frac{b^{4} \log \left (x\right )}{a^{5}} + \frac{12 \, b^{3} x^{3} - 6 \, a b^{2} x^{2} + 4 \, a^{2} b x - 3 \, a^{3}}{12 \, a^{4} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.5254, size = 154, normalized size = 2.26 \begin{align*} -\frac{12 \, b^{4} x^{4} \log \left (b x + a\right ) - 12 \, b^{4} x^{4} \log \left (x\right ) - 12 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} - 4 \, a^{3} b x + 3 \, a^{4}}{12 \, a^{5} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.591385, size = 56, normalized size = 0.82 \begin{align*} \frac{- 3 a^{3} + 4 a^{2} b x - 6 a b^{2} x^{2} + 12 b^{3} x^{3}}{12 a^{4} x^{4}} + \frac{b^{4} \left (\log{\left (x \right )} - \log{\left (\frac{a}{b} + x \right )}\right )}{a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18353, size = 90, normalized size = 1.32 \begin{align*} -\frac{b^{4} \log \left ({\left | b x + a \right |}\right )}{a^{5}} + \frac{b^{4} \log \left ({\left | x \right |}\right )}{a^{5}} + \frac{12 \, a b^{3} x^{3} - 6 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x - 3 \, a^{4}}{12 \, a^{5} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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