Optimal. Leaf size=137 \[ -\frac {10 d+e}{8 x^8}-\frac {5 (9 d+2 e)}{7 x^7}-\frac {5 (8 d+3 e)}{2 x^6}-\frac {6 (7 d+4 e)}{x^5}-\frac {21 (6 d+5 e)}{2 x^4}-\frac {14 (5 d+6 e)}{x^3}-\frac {15 (4 d+7 e)}{x^2}+x (d+10 e)-\frac {15 (3 d+8 e)}{x}+5 (2 d+9 e) \log (x)-\frac {d}{9 x^9}+\frac {e x^2}{2} \]
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Rubi [A] time = 0.08, antiderivative size = 137, 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} \begin {gather*} -\frac {15 (4 d+7 e)}{x^2}-\frac {14 (5 d+6 e)}{x^3}-\frac {21 (6 d+5 e)}{2 x^4}-\frac {6 (7 d+4 e)}{x^5}-\frac {5 (8 d+3 e)}{2 x^6}-\frac {5 (9 d+2 e)}{7 x^7}-\frac {10 d+e}{8 x^8}+x (d+10 e)-\frac {15 (3 d+8 e)}{x}+5 (2 d+9 e) \log (x)-\frac {d}{9 x^9}+\frac {e x^2}{2} \end {gather*}
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^{10}} \, dx &=\int \frac {(1+x)^{10} (d+e x)}{x^{10}} \, dx\\ &=\int \left (d \left (1+\frac {10 e}{d}\right )+\frac {d}{x^{10}}+\frac {10 d+e}{x^9}+\frac {5 (9 d+2 e)}{x^8}+\frac {15 (8 d+3 e)}{x^7}+\frac {30 (7 d+4 e)}{x^6}+\frac {42 (6 d+5 e)}{x^5}+\frac {42 (5 d+6 e)}{x^4}+\frac {30 (4 d+7 e)}{x^3}+\frac {15 (3 d+8 e)}{x^2}+\frac {5 (2 d+9 e)}{x}+e x\right ) \, dx\\ &=-\frac {d}{9 x^9}-\frac {10 d+e}{8 x^8}-\frac {5 (9 d+2 e)}{7 x^7}-\frac {5 (8 d+3 e)}{2 x^6}-\frac {6 (7 d+4 e)}{x^5}-\frac {21 (6 d+5 e)}{2 x^4}-\frac {14 (5 d+6 e)}{x^3}-\frac {15 (4 d+7 e)}{x^2}-\frac {15 (3 d+8 e)}{x}+(d+10 e) x+\frac {e x^2}{2}+5 (2 d+9 e) \log (x)\\ \end {align*}
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Mathematica [A] time = 0.04, size = 139, normalized size = 1.01 \begin {gather*} \frac {-10 d-e}{8 x^8}-\frac {5 (9 d+2 e)}{7 x^7}-\frac {5 (8 d+3 e)}{2 x^6}-\frac {6 (7 d+4 e)}{x^5}-\frac {21 (6 d+5 e)}{2 x^4}-\frac {14 (5 d+6 e)}{x^3}-\frac {15 (4 d+7 e)}{x^2}+x (d+10 e)-\frac {15 (3 d+8 e)}{x}+5 (2 d+9 e) \log (x)-\frac {d}{9 x^9}+\frac {e x^2}{2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{10}} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.42, size = 131, normalized size = 0.96 \begin {gather*} \frac {252 \, e x^{11} + 504 \, {\left (d + 10 \, e\right )} x^{10} + 2520 \, {\left (2 \, d + 9 \, e\right )} x^{9} \log \relax (x) - 7560 \, {\left (3 \, d + 8 \, e\right )} x^{8} - 7560 \, {\left (4 \, d + 7 \, e\right )} x^{7} - 7056 \, {\left (5 \, d + 6 \, e\right )} x^{6} - 5292 \, {\left (6 \, d + 5 \, e\right )} x^{5} - 3024 \, {\left (7 \, d + 4 \, e\right )} x^{4} - 1260 \, {\left (8 \, d + 3 \, e\right )} x^{3} - 360 \, {\left (9 \, d + 2 \, e\right )} x^{2} - 63 \, {\left (10 \, d + e\right )} x - 56 \, d}{504 \, x^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 138, normalized size = 1.01 \begin {gather*} \frac {1}{2} \, x^{2} e + d x + 10 \, x e + 5 \, {\left (2 \, d + 9 \, e\right )} \log \left ({\left | x \right |}\right ) - \frac {7560 \, {\left (3 \, d + 8 \, e\right )} x^{8} + 7560 \, {\left (4 \, d + 7 \, e\right )} x^{7} + 7056 \, {\left (5 \, d + 6 \, e\right )} x^{6} + 5292 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 3024 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 1260 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 360 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 63 \, {\left (10 \, d + e\right )} x + 56 \, d}{504 \, x^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 127, normalized size = 0.93 \begin {gather*} \frac {e \,x^{2}}{2}+d x +10 d \ln \relax (x )+10 e x +45 e \ln \relax (x )-\frac {45 d}{x}-\frac {120 e}{x}-\frac {60 d}{x^{2}}-\frac {105 e}{x^{2}}-\frac {70 d}{x^{3}}-\frac {84 e}{x^{3}}-\frac {63 d}{x^{4}}-\frac {105 e}{2 x^{4}}-\frac {42 d}{x^{5}}-\frac {24 e}{x^{5}}-\frac {20 d}{x^{6}}-\frac {15 e}{2 x^{6}}-\frac {45 d}{7 x^{7}}-\frac {10 e}{7 x^{7}}-\frac {5 d}{4 x^{8}}-\frac {e}{8 x^{8}}-\frac {d}{9 x^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 126, normalized size = 0.92 \begin {gather*} \frac {1}{2} \, e x^{2} + {\left (d + 10 \, e\right )} x + 5 \, {\left (2 \, d + 9 \, e\right )} \log \relax (x) - \frac {7560 \, {\left (3 \, d + 8 \, e\right )} x^{8} + 7560 \, {\left (4 \, d + 7 \, e\right )} x^{7} + 7056 \, {\left (5 \, d + 6 \, e\right )} x^{6} + 5292 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 3024 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 1260 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 360 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 63 \, {\left (10 \, d + e\right )} x + 56 \, d}{504 \, x^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 119, normalized size = 0.87 \begin {gather*} \ln \relax (x)\,\left (10\,d+45\,e\right )+x\,\left (d+10\,e\right )+\frac {e\,x^2}{2}-\frac {\left (45\,d+120\,e\right )\,x^8+\left (60\,d+105\,e\right )\,x^7+\left (70\,d+84\,e\right )\,x^6+\left (63\,d+\frac {105\,e}{2}\right )\,x^5+\left (42\,d+24\,e\right )\,x^4+\left (20\,d+\frac {15\,e}{2}\right )\,x^3+\left (\frac {45\,d}{7}+\frac {10\,e}{7}\right )\,x^2+\left (\frac {5\,d}{4}+\frac {e}{8}\right )\,x+\frac {d}{9}}{x^9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.59, size = 126, normalized size = 0.92 \begin {gather*} \frac {e x^{2}}{2} + x \left (d + 10 e\right ) + 5 \left (2 d + 9 e\right ) \log {\relax (x )} + \frac {- 56 d + x^{8} \left (- 22680 d - 60480 e\right ) + x^{7} \left (- 30240 d - 52920 e\right ) + x^{6} \left (- 35280 d - 42336 e\right ) + x^{5} \left (- 31752 d - 26460 e\right ) + x^{4} \left (- 21168 d - 12096 e\right ) + x^{3} \left (- 10080 d - 3780 e\right ) + x^{2} \left (- 3240 d - 720 e\right ) + x \left (- 630 d - 63 e\right )}{504 x^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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