Optimal. Leaf size=140 \[ \frac{1}{4} x^4 (d+10 e)+\frac{5}{3} x^3 (2 d+9 e)+\frac{15}{2} x^2 (3 d+8 e)-\frac{15 (7 d+4 e)}{x^2}-\frac{5 (8 d+3 e)}{x^3}-\frac{5 (9 d+2 e)}{4 x^4}-\frac{10 d+e}{5 x^5}+30 x (4 d+7 e)-\frac{42 (6 d+5 e)}{x}+42 (5 d+6 e) \log (x)-\frac{d}{6 x^6}+\frac{e x^5}{5} \]
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Rubi [A] time = 0.0724965, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {27, 76} \[ \frac{1}{4} x^4 (d+10 e)+\frac{5}{3} x^3 (2 d+9 e)+\frac{15}{2} x^2 (3 d+8 e)-\frac{15 (7 d+4 e)}{x^2}-\frac{5 (8 d+3 e)}{x^3}-\frac{5 (9 d+2 e)}{4 x^4}-\frac{10 d+e}{5 x^5}+30 x (4 d+7 e)-\frac{42 (6 d+5 e)}{x}+42 (5 d+6 e) \log (x)-\frac{d}{6 x^6}+\frac{e x^5}{5} \]
Antiderivative was successfully verified.
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Rule 27
Rule 76
Rubi steps
\begin{align*} \int \frac{(d+e x) \left (1+2 x+x^2\right )^5}{x^7} \, dx &=\int \frac{(1+x)^{10} (d+e x)}{x^7} \, dx\\ &=\int \left (30 (4 d+7 e)+\frac{d}{x^7}+\frac{10 d+e}{x^6}+\frac{5 (9 d+2 e)}{x^5}+\frac{15 (8 d+3 e)}{x^4}+\frac{30 (7 d+4 e)}{x^3}+\frac{42 (6 d+5 e)}{x^2}+\frac{42 (5 d+6 e)}{x}+15 (3 d+8 e) x+5 (2 d+9 e) x^2+(d+10 e) x^3+e x^4\right ) \, dx\\ &=-\frac{d}{6 x^6}-\frac{10 d+e}{5 x^5}-\frac{5 (9 d+2 e)}{4 x^4}-\frac{5 (8 d+3 e)}{x^3}-\frac{15 (7 d+4 e)}{x^2}-\frac{42 (6 d+5 e)}{x}+30 (4 d+7 e) x+\frac{15}{2} (3 d+8 e) x^2+\frac{5}{3} (2 d+9 e) x^3+\frac{1}{4} (d+10 e) x^4+\frac{e x^5}{5}+42 (5 d+6 e) \log (x)\\ \end{align*}
Mathematica [A] time = 0.0359995, size = 142, normalized size = 1.01 \[ \frac{1}{4} x^4 (d+10 e)+\frac{5}{3} x^3 (2 d+9 e)+\frac{15}{2} x^2 (3 d+8 e)-\frac{15 (7 d+4 e)}{x^2}-\frac{5 (8 d+3 e)}{x^3}-\frac{5 (9 d+2 e)}{4 x^4}+\frac{-10 d-e}{5 x^5}+30 x (4 d+7 e)-\frac{42 (6 d+5 e)}{x}+42 (5 d+6 e) \log (x)-\frac{d}{6 x^6}+\frac{e x^5}{5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 128, normalized size = 0.9 \begin{align*}{\frac{e{x}^{5}}{5}}+{\frac{d{x}^{4}}{4}}+{\frac{5\,e{x}^{4}}{2}}+{\frac{10\,d{x}^{3}}{3}}+15\,e{x}^{3}+{\frac{45\,d{x}^{2}}{2}}+60\,e{x}^{2}+120\,dx+210\,ex+210\,d\ln \left ( x \right ) +252\,e\ln \left ( x \right ) -40\,{\frac{d}{{x}^{3}}}-15\,{\frac{e}{{x}^{3}}}-105\,{\frac{d}{{x}^{2}}}-60\,{\frac{e}{{x}^{2}}}-252\,{\frac{d}{x}}-210\,{\frac{e}{x}}-2\,{\frac{d}{{x}^{5}}}-{\frac{e}{5\,{x}^{5}}}-{\frac{45\,d}{4\,{x}^{4}}}-{\frac{5\,e}{2\,{x}^{4}}}-{\frac{d}{6\,{x}^{6}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02685, size = 171, normalized size = 1.22 \begin{align*} \frac{1}{5} \, e x^{5} + \frac{1}{4} \,{\left (d + 10 \, e\right )} x^{4} + \frac{5}{3} \,{\left (2 \, d + 9 \, e\right )} x^{3} + \frac{15}{2} \,{\left (3 \, d + 8 \, e\right )} x^{2} + 30 \,{\left (4 \, d + 7 \, e\right )} x + 42 \,{\left (5 \, d + 6 \, e\right )} \log \left (x\right ) - \frac{2520 \,{\left (6 \, d + 5 \, e\right )} x^{5} + 900 \,{\left (7 \, d + 4 \, e\right )} x^{4} + 300 \,{\left (8 \, d + 3 \, e\right )} x^{3} + 75 \,{\left (9 \, d + 2 \, e\right )} x^{2} + 12 \,{\left (10 \, d + e\right )} x + 10 \, d}{60 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.29145, size = 342, normalized size = 2.44 \begin{align*} \frac{12 \, e x^{11} + 15 \,{\left (d + 10 \, e\right )} x^{10} + 100 \,{\left (2 \, d + 9 \, e\right )} x^{9} + 450 \,{\left (3 \, d + 8 \, e\right )} x^{8} + 1800 \,{\left (4 \, d + 7 \, e\right )} x^{7} + 2520 \,{\left (5 \, d + 6 \, e\right )} x^{6} \log \left (x\right ) - 2520 \,{\left (6 \, d + 5 \, e\right )} x^{5} - 900 \,{\left (7 \, d + 4 \, e\right )} x^{4} - 300 \,{\left (8 \, d + 3 \, e\right )} x^{3} - 75 \,{\left (9 \, d + 2 \, e\right )} x^{2} - 12 \,{\left (10 \, d + e\right )} x - 10 \, d}{60 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.98129, size = 119, normalized size = 0.85 \begin{align*} \frac{e x^{5}}{5} + x^{4} \left (\frac{d}{4} + \frac{5 e}{2}\right ) + x^{3} \left (\frac{10 d}{3} + 15 e\right ) + x^{2} \left (\frac{45 d}{2} + 60 e\right ) + x \left (120 d + 210 e\right ) + 42 \left (5 d + 6 e\right ) \log{\left (x \right )} - \frac{10 d + x^{5} \left (15120 d + 12600 e\right ) + x^{4} \left (6300 d + 3600 e\right ) + x^{3} \left (2400 d + 900 e\right ) + x^{2} \left (675 d + 150 e\right ) + x \left (120 d + 12 e\right )}{60 x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15464, size = 188, normalized size = 1.34 \begin{align*} \frac{1}{5} \, x^{5} e + \frac{1}{4} \, d x^{4} + \frac{5}{2} \, x^{4} e + \frac{10}{3} \, d x^{3} + 15 \, x^{3} e + \frac{45}{2} \, d x^{2} + 60 \, x^{2} e + 120 \, d x + 210 \, x e + 42 \,{\left (5 \, d + 6 \, e\right )} \log \left ({\left | x \right |}\right ) - \frac{2520 \,{\left (6 \, d + 5 \, e\right )} x^{5} + 900 \,{\left (7 \, d + 4 \, e\right )} x^{4} + 300 \,{\left (8 \, d + 3 \, e\right )} x^{3} + 75 \,{\left (9 \, d + 2 \, e\right )} x^{2} + 12 \,{\left (10 \, d + e\right )} x + 10 \, d}{60 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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