Optimal. Leaf size=138 \[ -\frac {10 d+e}{7 x^7}-\frac {5 (9 d+2 e)}{6 x^6}-\frac {3 (8 d+3 e)}{x^5}-\frac {15 (7 d+4 e)}{2 x^4}-\frac {14 (6 d+5 e)}{x^3}+\frac {1}{2} x^2 (d+10 e)-\frac {21 (5 d+6 e)}{x^2}+5 x (2 d+9 e)-\frac {30 (4 d+7 e)}{x}+15 (3 d+8 e) \log (x)-\frac {d}{8 x^8}+\frac {e x^3}{3} \]
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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} \begin {gather*} \frac {1}{2} x^2 (d+10 e)-\frac {21 (5 d+6 e)}{x^2}-\frac {14 (6 d+5 e)}{x^3}-\frac {15 (7 d+4 e)}{2 x^4}-\frac {3 (8 d+3 e)}{x^5}-\frac {5 (9 d+2 e)}{6 x^6}-\frac {10 d+e}{7 x^7}+5 x (2 d+9 e)-\frac {30 (4 d+7 e)}{x}+15 (3 d+8 e) \log (x)-\frac {d}{8 x^8}+\frac {e x^3}{3} \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^9} \, dx &=\int \frac {(1+x)^{10} (d+e x)}{x^9} \, dx\\ &=\int \left (5 (2 d+9 e)+\frac {d}{x^9}+\frac {10 d+e}{x^8}+\frac {5 (9 d+2 e)}{x^7}+\frac {15 (8 d+3 e)}{x^6}+\frac {30 (7 d+4 e)}{x^5}+\frac {42 (6 d+5 e)}{x^4}+\frac {42 (5 d+6 e)}{x^3}+\frac {30 (4 d+7 e)}{x^2}+\frac {15 (3 d+8 e)}{x}+(d+10 e) x+e x^2\right ) \, dx\\ &=-\frac {d}{8 x^8}-\frac {10 d+e}{7 x^7}-\frac {5 (9 d+2 e)}{6 x^6}-\frac {3 (8 d+3 e)}{x^5}-\frac {15 (7 d+4 e)}{2 x^4}-\frac {14 (6 d+5 e)}{x^3}-\frac {21 (5 d+6 e)}{x^2}-\frac {30 (4 d+7 e)}{x}+5 (2 d+9 e) x+\frac {1}{2} (d+10 e) x^2+\frac {e x^3}{3}+15 (3 d+8 e) \log (x)\\ \end {align*}
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Mathematica [A] time = 0.04, size = 140, normalized size = 1.01 \begin {gather*} \frac {-10 d-e}{7 x^7}-\frac {5 (9 d+2 e)}{6 x^6}-\frac {3 (8 d+3 e)}{x^5}-\frac {15 (7 d+4 e)}{2 x^4}-\frac {14 (6 d+5 e)}{x^3}+\frac {1}{2} x^2 (d+10 e)-\frac {21 (5 d+6 e)}{x^2}+5 x (2 d+9 e)-\frac {30 (4 d+7 e)}{x}+15 (3 d+8 e) \log (x)-\frac {d}{8 x^8}+\frac {e x^3}{3} \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^9} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.42, size = 131, normalized size = 0.95 \begin {gather*} \frac {56 \, e x^{11} + 84 \, {\left (d + 10 \, e\right )} x^{10} + 840 \, {\left (2 \, d + 9 \, e\right )} x^{9} + 2520 \, {\left (3 \, d + 8 \, e\right )} x^{8} \log \relax (x) - 5040 \, {\left (4 \, d + 7 \, e\right )} x^{7} - 3528 \, {\left (5 \, d + 6 \, e\right )} x^{6} - 2352 \, {\left (6 \, d + 5 \, e\right )} x^{5} - 1260 \, {\left (7 \, d + 4 \, e\right )} x^{4} - 504 \, {\left (8 \, d + 3 \, e\right )} x^{3} - 140 \, {\left (9 \, d + 2 \, e\right )} x^{2} - 24 \, {\left (10 \, d + e\right )} x - 21 \, d}{168 \, x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 139, normalized size = 1.01 \begin {gather*} \frac {1}{3} \, x^{3} e + \frac {1}{2} \, d x^{2} + 5 \, x^{2} e + 10 \, d x + 45 \, x e + 15 \, {\left (3 \, d + 8 \, e\right )} \log \left ({\left | x \right |}\right ) - \frac {5040 \, {\left (4 \, d + 7 \, e\right )} x^{7} + 3528 \, {\left (5 \, d + 6 \, e\right )} x^{6} + 2352 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 1260 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 504 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 140 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 24 \, {\left (10 \, d + e\right )} x + 21 \, d}{168 \, x^{8}} \end {gather*}
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 \begin {gather*} \frac {e \,x^{3}}{3}+\frac {d \,x^{2}}{2}+5 e \,x^{2}+10 d x +45 d \ln \relax (x )+45 e x +120 e \ln \relax (x )-\frac {120 d}{x}-\frac {210 e}{x}-\frac {105 d}{x^{2}}-\frac {126 e}{x^{2}}-\frac {84 d}{x^{3}}-\frac {70 e}{x^{3}}-\frac {105 d}{2 x^{4}}-\frac {30 e}{x^{4}}-\frac {24 d}{x^{5}}-\frac {9 e}{x^{5}}-\frac {15 d}{2 x^{6}}-\frac {5 e}{3 x^{6}}-\frac {10 d}{7 x^{7}}-\frac {e}{7 x^{7}}-\frac {d}{8 x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 127, normalized size = 0.92 \begin {gather*} \frac {1}{3} \, e x^{3} + \frac {1}{2} \, {\left (d + 10 \, e\right )} x^{2} + 5 \, {\left (2 \, d + 9 \, e\right )} x + 15 \, {\left (3 \, d + 8 \, e\right )} \log \relax (x) - \frac {5040 \, {\left (4 \, d + 7 \, e\right )} x^{7} + 3528 \, {\left (5 \, d + 6 \, e\right )} x^{6} + 2352 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 1260 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 504 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 140 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 24 \, {\left (10 \, d + e\right )} x + 21 \, d}{168 \, x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.08, size = 121, normalized size = 0.88 \begin {gather*} x^2\,\left (\frac {d}{2}+5\,e\right )+\ln \relax (x)\,\left (45\,d+120\,e\right )+\frac {e\,x^3}{3}+x\,\left (10\,d+45\,e\right )-\frac {\left (120\,d+210\,e\right )\,x^7+\left (105\,d+126\,e\right )\,x^6+\left (84\,d+70\,e\right )\,x^5+\left (\frac {105\,d}{2}+30\,e\right )\,x^4+\left (24\,d+9\,e\right )\,x^3+\left (\frac {15\,d}{2}+\frac {5\,e}{3}\right )\,x^2+\left (\frac {10\,d}{7}+\frac {e}{7}\right )\,x+\frac {d}{8}}{x^8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.27, size = 126, normalized size = 0.91 \begin {gather*} \frac {e x^{3}}{3} + x^{2} \left (\frac {d}{2} + 5 e\right ) + x \left (10 d + 45 e\right ) + 15 \left (3 d + 8 e\right ) \log {\relax (x )} + \frac {- 21 d + x^{7} \left (- 20160 d - 35280 e\right ) + x^{6} \left (- 17640 d - 21168 e\right ) + x^{5} \left (- 14112 d - 11760 e\right ) + x^{4} \left (- 8820 d - 5040 e\right ) + x^{3} \left (- 4032 d - 1512 e\right ) + x^{2} \left (- 1260 d - 280 e\right ) + x \left (- 240 d - 24 e\right )}{168 x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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