Optimal. Leaf size=86 \[ -\frac{21 a^5 b^2}{2 x^2}-\frac{35 a^4 b^3}{x}+21 a^2 b^5 x+35 a^3 b^4 \log (x)-\frac{7 a^6 b}{3 x^3}-\frac{a^7}{4 x^4}+\frac{7}{2} a b^6 x^2+\frac{b^7 x^3}{3} \]
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Rubi [A] time = 0.0320514, antiderivative size = 86, 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} \[ -\frac{21 a^5 b^2}{2 x^2}-\frac{35 a^4 b^3}{x}+21 a^2 b^5 x+35 a^3 b^4 \log (x)-\frac{7 a^6 b}{3 x^3}-\frac{a^7}{4 x^4}+\frac{7}{2} a b^6 x^2+\frac{b^7 x^3}{3} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b x)^7}{x^5} \, dx &=\int \left (21 a^2 b^5+\frac{a^7}{x^5}+\frac{7 a^6 b}{x^4}+\frac{21 a^5 b^2}{x^3}+\frac{35 a^4 b^3}{x^2}+\frac{35 a^3 b^4}{x}+7 a b^6 x+b^7 x^2\right ) \, dx\\ &=-\frac{a^7}{4 x^4}-\frac{7 a^6 b}{3 x^3}-\frac{21 a^5 b^2}{2 x^2}-\frac{35 a^4 b^3}{x}+21 a^2 b^5 x+\frac{7}{2} a b^6 x^2+\frac{b^7 x^3}{3}+35 a^3 b^4 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0154708, size = 86, normalized size = 1. \[ -\frac{21 a^5 b^2}{2 x^2}-\frac{35 a^4 b^3}{x}+21 a^2 b^5 x+35 a^3 b^4 \log (x)-\frac{7 a^6 b}{3 x^3}-\frac{a^7}{4 x^4}+\frac{7}{2} a b^6 x^2+\frac{b^7 x^3}{3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 77, normalized size = 0.9 \begin{align*} -{\frac{{a}^{7}}{4\,{x}^{4}}}-{\frac{7\,{a}^{6}b}{3\,{x}^{3}}}-{\frac{21\,{a}^{5}{b}^{2}}{2\,{x}^{2}}}-35\,{\frac{{a}^{4}{b}^{3}}{x}}+21\,{a}^{2}{b}^{5}x+{\frac{7\,a{b}^{6}{x}^{2}}{2}}+{\frac{{b}^{7}{x}^{3}}{3}}+35\,{a}^{3}{b}^{4}\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00811, size = 104, normalized size = 1.21 \begin{align*} \frac{1}{3} \, b^{7} x^{3} + \frac{7}{2} \, a b^{6} x^{2} + 21 \, a^{2} b^{5} x + 35 \, a^{3} b^{4} \log \left (x\right ) - \frac{420 \, a^{4} b^{3} x^{3} + 126 \, a^{5} b^{2} x^{2} + 28 \, a^{6} b x + 3 \, a^{7}}{12 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.76785, size = 185, normalized size = 2.15 \begin{align*} \frac{4 \, b^{7} x^{7} + 42 \, a b^{6} x^{6} + 252 \, a^{2} b^{5} x^{5} + 420 \, a^{3} b^{4} x^{4} \log \left (x\right ) - 420 \, a^{4} b^{3} x^{3} - 126 \, a^{5} b^{2} x^{2} - 28 \, a^{6} b x - 3 \, a^{7}}{12 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.592147, size = 83, normalized size = 0.97 \begin{align*} 35 a^{3} b^{4} \log{\left (x \right )} + 21 a^{2} b^{5} x + \frac{7 a b^{6} x^{2}}{2} + \frac{b^{7} x^{3}}{3} - \frac{3 a^{7} + 28 a^{6} b x + 126 a^{5} b^{2} x^{2} + 420 a^{4} b^{3} x^{3}}{12 x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17055, size = 105, normalized size = 1.22 \begin{align*} \frac{1}{3} \, b^{7} x^{3} + \frac{7}{2} \, a b^{6} x^{2} + 21 \, a^{2} b^{5} x + 35 \, a^{3} b^{4} \log \left ({\left | x \right |}\right ) - \frac{420 \, a^{4} b^{3} x^{3} + 126 \, a^{5} b^{2} x^{2} + 28 \, a^{6} b x + 3 \, a^{7}}{12 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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