Optimal. Leaf size=138 \[ -\frac {10 d+e}{9 x^9}-\frac {5 (9 d+2 e)}{8 x^8}-\frac {15 (8 d+3 e)}{7 x^7}-\frac {5 (7 d+4 e)}{x^6}-\frac {42 (6 d+5 e)}{5 x^5}-\frac {21 (5 d+6 e)}{2 x^4}-\frac {10 (4 d+7 e)}{x^3}-\frac {15 (3 d+8 e)}{2 x^2}-\frac {5 (2 d+9 e)}{x}+(d+10 e) \log (x)-\frac {d}{10 x^{10}}+e x \]
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Rubi [A] time = 0.07, antiderivative size = 138, normalized size of antiderivative = 1.00, 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} \[ -\frac {15 (3 d+8 e)}{2 x^2}-\frac {10 (4 d+7 e)}{x^3}-\frac {21 (5 d+6 e)}{2 x^4}-\frac {42 (6 d+5 e)}{5 x^5}-\frac {5 (7 d+4 e)}{x^6}-\frac {15 (8 d+3 e)}{7 x^7}-\frac {5 (9 d+2 e)}{8 x^8}-\frac {10 d+e}{9 x^9}-\frac {5 (2 d+9 e)}{x}+(d+10 e) \log (x)-\frac {d}{10 x^{10}}+e x \]
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^{11}} \, dx &=\int \frac {(1+x)^{10} (d+e x)}{x^{11}} \, dx\\ &=\int \left (e+\frac {d}{x^{11}}+\frac {10 d+e}{x^{10}}+\frac {5 (9 d+2 e)}{x^9}+\frac {15 (8 d+3 e)}{x^8}+\frac {30 (7 d+4 e)}{x^7}+\frac {42 (6 d+5 e)}{x^6}+\frac {42 (5 d+6 e)}{x^5}+\frac {30 (4 d+7 e)}{x^4}+\frac {15 (3 d+8 e)}{x^3}+\frac {5 (2 d+9 e)}{x^2}+\frac {d+10 e}{x}\right ) \, dx\\ &=-\frac {d}{10 x^{10}}-\frac {10 d+e}{9 x^9}-\frac {5 (9 d+2 e)}{8 x^8}-\frac {15 (8 d+3 e)}{7 x^7}-\frac {5 (7 d+4 e)}{x^6}-\frac {42 (6 d+5 e)}{5 x^5}-\frac {21 (5 d+6 e)}{2 x^4}-\frac {10 (4 d+7 e)}{x^3}-\frac {15 (3 d+8 e)}{2 x^2}-\frac {5 (2 d+9 e)}{x}+e x+(d+10 e) \log (x)\\ \end {align*}
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Mathematica [A] time = 0.04, size = 140, normalized size = 1.01 \[ \frac {-10 d-e}{9 x^9}-\frac {5 (9 d+2 e)}{8 x^8}-\frac {15 (8 d+3 e)}{7 x^7}-\frac {5 (7 d+4 e)}{x^6}-\frac {42 (6 d+5 e)}{5 x^5}-\frac {21 (5 d+6 e)}{2 x^4}-\frac {10 (4 d+7 e)}{x^3}-\frac {15 (3 d+8 e)}{2 x^2}-\frac {5 (2 d+9 e)}{x}+(d+10 e) \log (x)-\frac {d}{10 x^{10}}+e x \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 131, normalized size = 0.95 \[ \frac {2520 \, e x^{11} + 2520 \, {\left (d + 10 \, e\right )} x^{10} \log \relax (x) - 12600 \, {\left (2 \, d + 9 \, e\right )} x^{9} - 18900 \, {\left (3 \, d + 8 \, e\right )} x^{8} - 25200 \, {\left (4 \, d + 7 \, e\right )} x^{7} - 26460 \, {\left (5 \, d + 6 \, e\right )} x^{6} - 21168 \, {\left (6 \, d + 5 \, e\right )} x^{5} - 12600 \, {\left (7 \, d + 4 \, e\right )} x^{4} - 5400 \, {\left (8 \, d + 3 \, e\right )} x^{3} - 1575 \, {\left (9 \, d + 2 \, e\right )} x^{2} - 280 \, {\left (10 \, d + e\right )} x - 252 \, d}{2520 \, x^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 137, normalized size = 0.99 \[ x e + {\left (d + 10 \, e\right )} \log \left ({\left | x \right |}\right ) - \frac {12600 \, {\left (2 \, d + 9 \, e\right )} x^{9} + 18900 \, {\left (3 \, d + 8 \, e\right )} x^{8} + 25200 \, {\left (4 \, d + 7 \, e\right )} x^{7} + 26460 \, {\left (5 \, d + 6 \, e\right )} x^{6} + 21168 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 12600 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 5400 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 1575 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 280 \, {\left (10 \, d + e\right )} x + 252 \, d}{2520 \, x^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 128, normalized size = 0.93 \[ d \ln \relax (x )+e x +10 e \ln \relax (x )-\frac {10 d}{x}-\frac {45 e}{x}-\frac {45 d}{2 x^{2}}-\frac {60 e}{x^{2}}-\frac {40 d}{x^{3}}-\frac {70 e}{x^{3}}-\frac {105 d}{2 x^{4}}-\frac {63 e}{x^{4}}-\frac {252 d}{5 x^{5}}-\frac {42 e}{x^{5}}-\frac {35 d}{x^{6}}-\frac {20 e}{x^{6}}-\frac {120 d}{7 x^{7}}-\frac {45 e}{7 x^{7}}-\frac {45 d}{8 x^{8}}-\frac {5 e}{4 x^{8}}-\frac {10 d}{9 x^{9}}-\frac {e}{9 x^{9}}-\frac {d}{10 x^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 125, normalized size = 0.91 \[ e x + {\left (d + 10 \, e\right )} \log \relax (x) - \frac {12600 \, {\left (2 \, d + 9 \, e\right )} x^{9} + 18900 \, {\left (3 \, d + 8 \, e\right )} x^{8} + 25200 \, {\left (4 \, d + 7 \, e\right )} x^{7} + 26460 \, {\left (5 \, d + 6 \, e\right )} x^{6} + 21168 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 12600 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 5400 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 1575 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 280 \, {\left (10 \, d + e\right )} x + 252 \, d}{2520 \, x^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.09, size = 118, normalized size = 0.86 \[ e\,x-\frac {\left (10\,d+45\,e\right )\,x^9+\left (\frac {45\,d}{2}+60\,e\right )\,x^8+\left (40\,d+70\,e\right )\,x^7+\left (\frac {105\,d}{2}+63\,e\right )\,x^6+\left (\frac {252\,d}{5}+42\,e\right )\,x^5+\left (35\,d+20\,e\right )\,x^4+\left (\frac {120\,d}{7}+\frac {45\,e}{7}\right )\,x^3+\left (\frac {45\,d}{8}+\frac {5\,e}{4}\right )\,x^2+\left (\frac {10\,d}{9}+\frac {e}{9}\right )\,x+\frac {d}{10}}{x^{10}}+\ln \relax (x)\,\left (d+10\,e\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.76, size = 124, normalized size = 0.90 \[ e x + \left (d + 10 e\right ) \log {\relax (x )} + \frac {- 252 d + x^{9} \left (- 25200 d - 113400 e\right ) + x^{8} \left (- 56700 d - 151200 e\right ) + x^{7} \left (- 100800 d - 176400 e\right ) + x^{6} \left (- 132300 d - 158760 e\right ) + x^{5} \left (- 127008 d - 105840 e\right ) + x^{4} \left (- 88200 d - 50400 e\right ) + x^{3} \left (- 43200 d - 16200 e\right ) + x^{2} \left (- 14175 d - 3150 e\right ) + x \left (- 2800 d - 280 e\right )}{2520 x^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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