Optimal. Leaf size=140 \[ \frac {1}{5} x^5 (d+10 e)+\frac {5}{4} x^4 (2 d+9 e)-\frac {10 d+e}{4 x^4}+5 x^3 (3 d+8 e)-\frac {5 (9 d+2 e)}{3 x^3}+15 x^2 (4 d+7 e)-\frac {15 (8 d+3 e)}{2 x^2}+42 x (5 d+6 e)-\frac {30 (7 d+4 e)}{x}+42 (6 d+5 e) \log (x)-\frac {d}{5 x^5}+\frac {e x^6}{6} \]
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Rubi [A] time = 0.07, antiderivative size = 140, 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}{5} x^5 (d+10 e)+\frac {5}{4} x^4 (2 d+9 e)+5 x^3 (3 d+8 e)+15 x^2 (4 d+7 e)-\frac {15 (8 d+3 e)}{2 x^2}-\frac {5 (9 d+2 e)}{3 x^3}-\frac {10 d+e}{4 x^4}+42 x (5 d+6 e)-\frac {30 (7 d+4 e)}{x}+42 (6 d+5 e) \log (x)-\frac {d}{5 x^5}+\frac {e x^6}{6} \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^6} \, dx &=\int \frac {(1+x)^{10} (d+e x)}{x^6} \, dx\\ &=\int \left (42 (5 d+6 e)+\frac {d}{x^6}+\frac {10 d+e}{x^5}+\frac {5 (9 d+2 e)}{x^4}+\frac {15 (8 d+3 e)}{x^3}+\frac {30 (7 d+4 e)}{x^2}+\frac {42 (6 d+5 e)}{x}+30 (4 d+7 e) x+15 (3 d+8 e) x^2+5 (2 d+9 e) x^3+(d+10 e) x^4+e x^5\right ) \, dx\\ &=-\frac {d}{5 x^5}-\frac {10 d+e}{4 x^4}-\frac {5 (9 d+2 e)}{3 x^3}-\frac {15 (8 d+3 e)}{2 x^2}-\frac {30 (7 d+4 e)}{x}+42 (5 d+6 e) x+15 (4 d+7 e) x^2+5 (3 d+8 e) x^3+\frac {5}{4} (2 d+9 e) x^4+\frac {1}{5} (d+10 e) x^5+\frac {e x^6}{6}+42 (6 d+5 e) \log (x)\\ \end {align*}
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Mathematica [A] time = 0.04, size = 142, normalized size = 1.01 \begin {gather*} \frac {1}{5} x^5 (d+10 e)+\frac {5}{4} x^4 (2 d+9 e)+\frac {-10 d-e}{4 x^4}+5 x^3 (3 d+8 e)-\frac {5 (9 d+2 e)}{3 x^3}+15 x^2 (4 d+7 e)-\frac {15 (8 d+3 e)}{2 x^2}+42 x (5 d+6 e)-\frac {30 (7 d+4 e)}{x}+42 (6 d+5 e) \log (x)-\frac {d}{5 x^5}+\frac {e x^6}{6} \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^6} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.41, size = 131, normalized size = 0.94 \begin {gather*} \frac {10 \, e x^{11} + 12 \, {\left (d + 10 \, e\right )} x^{10} + 75 \, {\left (2 \, d + 9 \, e\right )} x^{9} + 300 \, {\left (3 \, d + 8 \, e\right )} x^{8} + 900 \, {\left (4 \, d + 7 \, e\right )} x^{7} + 2520 \, {\left (5 \, d + 6 \, e\right )} x^{6} + 2520 \, {\left (6 \, d + 5 \, e\right )} x^{5} \log \relax (x) - 1800 \, {\left (7 \, d + 4 \, e\right )} x^{4} - 450 \, {\left (8 \, d + 3 \, e\right )} x^{3} - 100 \, {\left (9 \, d + 2 \, e\right )} x^{2} - 15 \, {\left (10 \, d + e\right )} x - 12 \, d}{60 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 139, normalized size = 0.99 \begin {gather*} \frac {1}{6} \, x^{6} e + \frac {1}{5} \, d x^{5} + 2 \, x^{5} e + \frac {5}{2} \, d x^{4} + \frac {45}{4} \, x^{4} e + 15 \, d x^{3} + 40 \, x^{3} e + 60 \, d x^{2} + 105 \, x^{2} e + 210 \, d x + 252 \, x e + 42 \, {\left (6 \, d + 5 \, e\right )} \log \left ({\left | x \right |}\right ) - \frac {1800 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 450 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 100 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 15 \, {\left (10 \, d + e\right )} x + 12 \, d}{60 \, x^{5}} \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 = 0.91 \begin {gather*} \frac {e \,x^{6}}{6}+\frac {d \,x^{5}}{5}+2 e \,x^{5}+\frac {5 d \,x^{4}}{2}+\frac {45 e \,x^{4}}{4}+15 d \,x^{3}+40 e \,x^{3}+60 d \,x^{2}+105 e \,x^{2}+210 d x +252 d \ln \relax (x )+252 e x +210 e \ln \relax (x )-\frac {210 d}{x}-\frac {120 e}{x}-\frac {60 d}{x^{2}}-\frac {45 e}{2 x^{2}}-\frac {15 d}{x^{3}}-\frac {10 e}{3 x^{3}}-\frac {5 d}{2 x^{4}}-\frac {e}{4 x^{4}}-\frac {d}{5 x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 127, normalized size = 0.91 \begin {gather*} \frac {1}{6} \, e x^{6} + \frac {1}{5} \, {\left (d + 10 \, e\right )} x^{5} + \frac {5}{4} \, {\left (2 \, d + 9 \, e\right )} x^{4} + 5 \, {\left (3 \, d + 8 \, e\right )} x^{3} + 15 \, {\left (4 \, d + 7 \, e\right )} x^{2} + 42 \, {\left (5 \, d + 6 \, e\right )} x + 42 \, {\left (6 \, d + 5 \, e\right )} \log \relax (x) - \frac {1800 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 450 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 100 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 15 \, {\left (10 \, d + e\right )} x + 12 \, d}{60 \, x^{5}} \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 = 0.86 \begin {gather*} x^5\,\left (\frac {d}{5}+2\,e\right )+x^3\,\left (15\,d+40\,e\right )+x^4\,\left (\frac {5\,d}{2}+\frac {45\,e}{4}\right )+x^2\,\left (60\,d+105\,e\right )+\ln \relax (x)\,\left (252\,d+210\,e\right )-\frac {\left (210\,d+120\,e\right )\,x^4+\left (60\,d+\frac {45\,e}{2}\right )\,x^3+\left (15\,d+\frac {10\,e}{3}\right )\,x^2+\left (\frac {5\,d}{2}+\frac {e}{4}\right )\,x+\frac {d}{5}}{x^5}+\frac {e\,x^6}{6}+x\,\left (210\,d+252\,e\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.42, size = 124, normalized size = 0.89 \begin {gather*} \frac {e x^{6}}{6} + x^{5} \left (\frac {d}{5} + 2 e\right ) + x^{4} \left (\frac {5 d}{2} + \frac {45 e}{4}\right ) + x^{3} \left (15 d + 40 e\right ) + x^{2} \left (60 d + 105 e\right ) + x \left (210 d + 252 e\right ) + 42 \left (6 d + 5 e\right ) \log {\relax (x )} + \frac {- 12 d + x^{4} \left (- 12600 d - 7200 e\right ) + x^{3} \left (- 3600 d - 1350 e\right ) + x^{2} \left (- 900 d - 200 e\right ) + x \left (- 150 d - 15 e\right )}{60 x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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