Optimal. Leaf size=138 \[ \frac{1}{7} x^7 (d+10 e)+\frac{5}{6} x^6 (2 d+9 e)+3 x^5 (3 d+8 e)+\frac{15}{2} x^4 (4 d+7 e)+14 x^3 (5 d+6 e)+21 x^2 (6 d+5 e)-\frac{10 d+e}{2 x^2}+30 x (7 d+4 e)-\frac{5 (9 d+2 e)}{x}+15 (8 d+3 e) \log (x)-\frac{d}{3 x^3}+\frac{e x^8}{8} \]
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Rubi [A] time = 0.0710799, antiderivative size = 138, 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} \[ \frac{1}{7} x^7 (d+10 e)+\frac{5}{6} x^6 (2 d+9 e)+3 x^5 (3 d+8 e)+\frac{15}{2} x^4 (4 d+7 e)+14 x^3 (5 d+6 e)+21 x^2 (6 d+5 e)-\frac{10 d+e}{2 x^2}+30 x (7 d+4 e)-\frac{5 (9 d+2 e)}{x}+15 (8 d+3 e) \log (x)-\frac{d}{3 x^3}+\frac{e x^8}{8} \]
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^4} \, dx &=\int \frac{(1+x)^{10} (d+e x)}{x^4} \, dx\\ &=\int \left (30 (7 d+4 e)+\frac{d}{x^4}+\frac{10 d+e}{x^3}+\frac{5 (9 d+2 e)}{x^2}+\frac{15 (8 d+3 e)}{x}+42 (6 d+5 e) x+42 (5 d+6 e) x^2+30 (4 d+7 e) x^3+15 (3 d+8 e) x^4+5 (2 d+9 e) x^5+(d+10 e) x^6+e x^7\right ) \, dx\\ &=-\frac{d}{3 x^3}-\frac{10 d+e}{2 x^2}-\frac{5 (9 d+2 e)}{x}+30 (7 d+4 e) x+21 (6 d+5 e) x^2+14 (5 d+6 e) x^3+\frac{15}{2} (4 d+7 e) x^4+3 (3 d+8 e) x^5+\frac{5}{6} (2 d+9 e) x^6+\frac{1}{7} (d+10 e) x^7+\frac{e x^8}{8}+15 (8 d+3 e) \log (x)\\ \end{align*}
Mathematica [A] time = 0.0384766, size = 140, normalized size = 1.01 \[ \frac{1}{7} x^7 (d+10 e)+\frac{5}{6} x^6 (2 d+9 e)+3 x^5 (3 d+8 e)+\frac{15}{2} x^4 (4 d+7 e)+14 x^3 (5 d+6 e)+21 x^2 (6 d+5 e)+\frac{-10 d-e}{2 x^2}+30 x (7 d+4 e)-\frac{5 (9 d+2 e)}{x}+15 (8 d+3 e) \log (x)-\frac{d}{3 x^3}+\frac{e x^8}{8} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 128, normalized size = 0.9 \begin{align*}{\frac{e{x}^{8}}{8}}+{\frac{d{x}^{7}}{7}}+{\frac{10\,e{x}^{7}}{7}}+{\frac{5\,d{x}^{6}}{3}}+{\frac{15\,e{x}^{6}}{2}}+9\,d{x}^{5}+24\,e{x}^{5}+30\,d{x}^{4}+{\frac{105\,e{x}^{4}}{2}}+70\,d{x}^{3}+84\,e{x}^{3}+126\,d{x}^{2}+105\,e{x}^{2}+210\,dx+120\,ex+120\,d\ln \left ( x \right ) +45\,e\ln \left ( x \right ) -{\frac{d}{3\,{x}^{3}}}-5\,{\frac{d}{{x}^{2}}}-{\frac{e}{2\,{x}^{2}}}-45\,{\frac{d}{x}}-10\,{\frac{e}{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00056, size = 171, normalized size = 1.24 \begin{align*} \frac{1}{8} \, e x^{8} + \frac{1}{7} \,{\left (d + 10 \, e\right )} x^{7} + \frac{5}{6} \,{\left (2 \, d + 9 \, e\right )} x^{6} + 3 \,{\left (3 \, d + 8 \, e\right )} x^{5} + \frac{15}{2} \,{\left (4 \, d + 7 \, e\right )} x^{4} + 14 \,{\left (5 \, d + 6 \, e\right )} x^{3} + 21 \,{\left (6 \, d + 5 \, e\right )} x^{2} + 30 \,{\left (7 \, d + 4 \, e\right )} x + 15 \,{\left (8 \, d + 3 \, e\right )} \log \left (x\right ) - \frac{30 \,{\left (9 \, d + 2 \, e\right )} x^{2} + 3 \,{\left (10 \, d + e\right )} x + 2 \, d}{6 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.3835, size = 347, normalized size = 2.51 \begin{align*} \frac{21 \, e x^{11} + 24 \,{\left (d + 10 \, e\right )} x^{10} + 140 \,{\left (2 \, d + 9 \, e\right )} x^{9} + 504 \,{\left (3 \, d + 8 \, e\right )} x^{8} + 1260 \,{\left (4 \, d + 7 \, e\right )} x^{7} + 2352 \,{\left (5 \, d + 6 \, e\right )} x^{6} + 3528 \,{\left (6 \, d + 5 \, e\right )} x^{5} + 5040 \,{\left (7 \, d + 4 \, e\right )} x^{4} + 2520 \,{\left (8 \, d + 3 \, e\right )} x^{3} \log \left (x\right ) - 840 \,{\left (9 \, d + 2 \, e\right )} x^{2} - 84 \,{\left (10 \, d + e\right )} x - 56 \, d}{168 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.52523, size = 121, normalized size = 0.88 \begin{align*} \frac{e x^{8}}{8} + x^{7} \left (\frac{d}{7} + \frac{10 e}{7}\right ) + x^{6} \left (\frac{5 d}{3} + \frac{15 e}{2}\right ) + x^{5} \left (9 d + 24 e\right ) + x^{4} \left (30 d + \frac{105 e}{2}\right ) + x^{3} \left (70 d + 84 e\right ) + x^{2} \left (126 d + 105 e\right ) + x \left (210 d + 120 e\right ) + 15 \left (8 d + 3 e\right ) \log{\left (x \right )} - \frac{2 d + x^{2} \left (270 d + 60 e\right ) + x \left (30 d + 3 e\right )}{6 x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18666, size = 188, normalized size = 1.36 \begin{align*} \frac{1}{8} \, x^{8} e + \frac{1}{7} \, d x^{7} + \frac{10}{7} \, x^{7} e + \frac{5}{3} \, d x^{6} + \frac{15}{2} \, x^{6} e + 9 \, d x^{5} + 24 \, x^{5} e + 30 \, d x^{4} + \frac{105}{2} \, x^{4} e + 70 \, d x^{3} + 84 \, x^{3} e + 126 \, d x^{2} + 105 \, x^{2} e + 210 \, d x + 120 \, x e + 15 \,{\left (8 \, d + 3 \, e\right )} \log \left ({\left | x \right |}\right ) - \frac{30 \,{\left (9 \, d + 2 \, e\right )} x^{2} + 3 \,{\left (10 \, d + e\right )} x + 2 \, d}{6 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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