Optimal. Leaf size=137 \[ \frac {1}{6} x^6 (d+10 e)+x^5 (2 d+9 e)+\frac {15}{4} x^4 (3 d+8 e)+10 x^3 (4 d+7 e)-\frac {10 d+e}{3 x^3}+21 x^2 (5 d+6 e)-\frac {5 (9 d+2 e)}{2 x^2}+42 x (6 d+5 e)-\frac {15 (8 d+3 e)}{x}+30 (7 d+4 e) \log (x)-\frac {d}{4 x^4}+\frac {e x^7}{7} \]
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Rubi [A] time = 0.07, 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 {1}{6} x^6 (d+10 e)+x^5 (2 d+9 e)+\frac {15}{4} x^4 (3 d+8 e)+10 x^3 (4 d+7 e)+21 x^2 (5 d+6 e)-\frac {5 (9 d+2 e)}{2 x^2}-\frac {10 d+e}{3 x^3}+42 x (6 d+5 e)-\frac {15 (8 d+3 e)}{x}+30 (7 d+4 e) \log (x)-\frac {d}{4 x^4}+\frac {e x^7}{7} \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^5} \, dx &=\int \frac {(1+x)^{10} (d+e x)}{x^5} \, dx\\ &=\int \left (42 (6 d+5 e)+\frac {d}{x^5}+\frac {10 d+e}{x^4}+\frac {5 (9 d+2 e)}{x^3}+\frac {15 (8 d+3 e)}{x^2}+\frac {30 (7 d+4 e)}{x}+42 (5 d+6 e) x+30 (4 d+7 e) x^2+15 (3 d+8 e) x^3+5 (2 d+9 e) x^4+(d+10 e) x^5+e x^6\right ) \, dx\\ &=-\frac {d}{4 x^4}-\frac {10 d+e}{3 x^3}-\frac {5 (9 d+2 e)}{2 x^2}-\frac {15 (8 d+3 e)}{x}+42 (6 d+5 e) x+21 (5 d+6 e) x^2+10 (4 d+7 e) x^3+\frac {15}{4} (3 d+8 e) x^4+(2 d+9 e) x^5+\frac {1}{6} (d+10 e) x^6+\frac {e x^7}{7}+30 (7 d+4 e) \log (x)\\ \end {align*}
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Mathematica [A] time = 0.04, size = 139, normalized size = 1.01 \begin {gather*} \frac {1}{6} x^6 (d+10 e)+x^5 (2 d+9 e)+\frac {15}{4} x^4 (3 d+8 e)+10 x^3 (4 d+7 e)+\frac {-10 d-e}{3 x^3}+21 x^2 (5 d+6 e)-\frac {5 (9 d+2 e)}{2 x^2}+42 x (6 d+5 e)-\frac {15 (8 d+3 e)}{x}+30 (7 d+4 e) \log (x)-\frac {d}{4 x^4}+\frac {e x^7}{7} \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^5} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.40, size = 131, normalized size = 0.96 \begin {gather*} \frac {12 \, e x^{11} + 14 \, {\left (d + 10 \, e\right )} x^{10} + 84 \, {\left (2 \, d + 9 \, e\right )} x^{9} + 315 \, {\left (3 \, d + 8 \, e\right )} x^{8} + 840 \, {\left (4 \, d + 7 \, e\right )} x^{7} + 1764 \, {\left (5 \, d + 6 \, e\right )} x^{6} + 3528 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 2520 \, {\left (7 \, d + 4 \, e\right )} x^{4} \log \relax (x) - 1260 \, {\left (8 \, d + 3 \, e\right )} x^{3} - 210 \, {\left (9 \, d + 2 \, e\right )} x^{2} - 28 \, {\left (10 \, d + e\right )} x - 21 \, d}{84 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 139, normalized size = 1.01 \begin {gather*} \frac {1}{7} \, x^{7} e + \frac {1}{6} \, d x^{6} + \frac {5}{3} \, x^{6} e + 2 \, d x^{5} + 9 \, x^{5} e + \frac {45}{4} \, d x^{4} + 30 \, x^{4} e + 40 \, d x^{3} + 70 \, x^{3} e + 105 \, d x^{2} + 126 \, x^{2} e + 252 \, d x + 210 \, x e + 30 \, {\left (7 \, d + 4 \, e\right )} \log \left ({\left | x \right |}\right ) - \frac {180 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 30 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 4 \, {\left (10 \, d + e\right )} x + 3 \, d}{12 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 128, normalized size = 0.93 \begin {gather*} \frac {e \,x^{7}}{7}+\frac {d \,x^{6}}{6}+\frac {5 e \,x^{6}}{3}+2 d \,x^{5}+9 e \,x^{5}+\frac {45 d \,x^{4}}{4}+30 e \,x^{4}+40 d \,x^{3}+70 e \,x^{3}+105 d \,x^{2}+126 e \,x^{2}+252 d x +210 d \ln \relax (x )+210 e x +120 e \ln \relax (x )-\frac {120 d}{x}-\frac {45 e}{x}-\frac {45 d}{2 x^{2}}-\frac {5 e}{x^{2}}-\frac {10 d}{3 x^{3}}-\frac {e}{3 x^{3}}-\frac {d}{4 x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.64, size = 126, normalized size = 0.92 \begin {gather*} \frac {1}{7} \, e x^{7} + \frac {1}{6} \, {\left (d + 10 \, e\right )} x^{6} + {\left (2 \, d + 9 \, e\right )} x^{5} + \frac {15}{4} \, {\left (3 \, d + 8 \, e\right )} x^{4} + 10 \, {\left (4 \, d + 7 \, e\right )} x^{3} + 21 \, {\left (5 \, d + 6 \, e\right )} x^{2} + 42 \, {\left (6 \, d + 5 \, e\right )} x + 30 \, {\left (7 \, d + 4 \, e\right )} \log \relax (x) - \frac {180 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 30 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 4 \, {\left (10 \, d + e\right )} x + 3 \, d}{12 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 121, normalized size = 0.88 \begin {gather*} x^5\,\left (2\,d+9\,e\right )+x^6\,\left (\frac {d}{6}+\frac {5\,e}{3}\right )+x^4\,\left (\frac {45\,d}{4}+30\,e\right )+x^3\,\left (40\,d+70\,e\right )+x^2\,\left (105\,d+126\,e\right )+\ln \relax (x)\,\left (210\,d+120\,e\right )-\frac {\left (120\,d+45\,e\right )\,x^3+\left (\frac {45\,d}{2}+5\,e\right )\,x^2+\left (\frac {10\,d}{3}+\frac {e}{3}\right )\,x+\frac {d}{4}}{x^4}+\frac {e\,x^7}{7}+x\,\left (252\,d+210\,e\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.00, size = 122, normalized size = 0.89 \begin {gather*} \frac {e x^{7}}{7} + x^{6} \left (\frac {d}{6} + \frac {5 e}{3}\right ) + x^{5} \left (2 d + 9 e\right ) + x^{4} \left (\frac {45 d}{4} + 30 e\right ) + x^{3} \left (40 d + 70 e\right ) + x^{2} \left (105 d + 126 e\right ) + x \left (252 d + 210 e\right ) + 30 \left (7 d + 4 e\right ) \log {\relax (x )} + \frac {- 3 d + x^{3} \left (- 1440 d - 540 e\right ) + x^{2} \left (- 270 d - 60 e\right ) + x \left (- 40 d - 4 e\right )}{12 x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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