Optimal. Leaf size=61 \[ -\frac{10 a^3 b^2}{3 x^3}-\frac{5 a^2 b^3}{x^2}-\frac{5 a^4 b}{4 x^4}-\frac{a^5}{5 x^5}-\frac{5 a b^4}{x}+b^5 \log (x) \]
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Rubi [A] time = 0.0209423, antiderivative size = 61, 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 = {43} \[ -\frac{10 a^3 b^2}{3 x^3}-\frac{5 a^2 b^3}{x^2}-\frac{5 a^4 b}{4 x^4}-\frac{a^5}{5 x^5}-\frac{5 a b^4}{x}+b^5 \log (x) \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b x)^5}{x^6} \, dx &=\int \left (\frac{a^5}{x^6}+\frac{5 a^4 b}{x^5}+\frac{10 a^3 b^2}{x^4}+\frac{10 a^2 b^3}{x^3}+\frac{5 a b^4}{x^2}+\frac{b^5}{x}\right ) \, dx\\ &=-\frac{a^5}{5 x^5}-\frac{5 a^4 b}{4 x^4}-\frac{10 a^3 b^2}{3 x^3}-\frac{5 a^2 b^3}{x^2}-\frac{5 a b^4}{x}+b^5 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0053105, size = 61, normalized size = 1. \[ -\frac{10 a^3 b^2}{3 x^3}-\frac{5 a^2 b^3}{x^2}-\frac{5 a^4 b}{4 x^4}-\frac{a^5}{5 x^5}-\frac{5 a b^4}{x}+b^5 \log (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 56, normalized size = 0.9 \begin{align*} -{\frac{{a}^{5}}{5\,{x}^{5}}}-{\frac{5\,{a}^{4}b}{4\,{x}^{4}}}-{\frac{10\,{a}^{3}{b}^{2}}{3\,{x}^{3}}}-5\,{\frac{{a}^{2}{b}^{3}}{{x}^{2}}}-5\,{\frac{a{b}^{4}}{x}}+{b}^{5}\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02984, size = 76, normalized size = 1.25 \begin{align*} b^{5} \log \left (x\right ) - \frac{300 \, a b^{4} x^{4} + 300 \, a^{2} b^{3} x^{3} + 200 \, a^{3} b^{2} x^{2} + 75 \, a^{4} b x + 12 \, a^{5}}{60 \, x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.62891, size = 140, normalized size = 2.3 \begin{align*} \frac{60 \, b^{5} x^{5} \log \left (x\right ) - 300 \, a b^{4} x^{4} - 300 \, a^{2} b^{3} x^{3} - 200 \, a^{3} b^{2} x^{2} - 75 \, a^{4} b x - 12 \, a^{5}}{60 \, x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.620298, size = 58, normalized size = 0.95 \begin{align*} b^{5} \log{\left (x \right )} - \frac{12 a^{5} + 75 a^{4} b x + 200 a^{3} b^{2} x^{2} + 300 a^{2} b^{3} x^{3} + 300 a b^{4} x^{4}}{60 x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20627, size = 77, normalized size = 1.26 \begin{align*} b^{5} \log \left ({\left | x \right |}\right ) - \frac{300 \, a b^{4} x^{4} + 300 \, a^{2} b^{3} x^{3} + 200 \, a^{3} b^{2} x^{2} + 75 \, a^{4} b x + 12 \, a^{5}}{60 \, x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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