Integrand size = 19, antiderivative size = 140 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^7} \, dx=-\frac {d}{6 x^6}-\frac {10 d+e}{5 x^5}-\frac {5 (9 d+2 e)}{4 x^4}-\frac {5 (8 d+3 e)}{x^3}-\frac {15 (7 d+4 e)}{x^2}-\frac {42 (6 d+5 e)}{x}+30 (4 d+7 e) x+\frac {15}{2} (3 d+8 e) x^2+\frac {5}{3} (2 d+9 e) x^3+\frac {1}{4} (d+10 e) x^4+\frac {e x^5}{5}+42 (5 d+6 e) \log (x) \]
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Time = 0.05 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {27, 77} \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^7} \, dx=-\frac {10 d+e}{5 x^5}+\frac {1}{4} x^4 (d+10 e)-\frac {5 (9 d+2 e)}{4 x^4}+\frac {5}{3} x^3 (2 d+9 e)-\frac {5 (8 d+3 e)}{x^3}+\frac {15}{2} x^2 (3 d+8 e)-\frac {15 (7 d+4 e)}{x^2}+30 x (4 d+7 e)-\frac {42 (6 d+5 e)}{x}+42 (5 d+6 e) \log (x)-\frac {d}{6 x^6}+\frac {e x^5}{5} \]
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Rule 27
Rule 77
Rubi steps \begin{align*} \text {integral}& = \int \frac {(1+x)^{10} (d+e x)}{x^7} \, dx \\ & = \int \left (30 (4 d+7 e)+\frac {d}{x^7}+\frac {10 d+e}{x^6}+\frac {5 (9 d+2 e)}{x^5}+\frac {15 (8 d+3 e)}{x^4}+\frac {30 (7 d+4 e)}{x^3}+\frac {42 (6 d+5 e)}{x^2}+\frac {42 (5 d+6 e)}{x}+15 (3 d+8 e) x+5 (2 d+9 e) x^2+(d+10 e) x^3+e x^4\right ) \, dx \\ & = -\frac {d}{6 x^6}-\frac {10 d+e}{5 x^5}-\frac {5 (9 d+2 e)}{4 x^4}-\frac {5 (8 d+3 e)}{x^3}-\frac {15 (7 d+4 e)}{x^2}-\frac {42 (6 d+5 e)}{x}+30 (4 d+7 e) x+\frac {15}{2} (3 d+8 e) x^2+\frac {5}{3} (2 d+9 e) x^3+\frac {1}{4} (d+10 e) x^4+\frac {e x^5}{5}+42 (5 d+6 e) \log (x) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.01 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^7} \, dx=-\frac {d}{6 x^6}+\frac {-10 d-e}{5 x^5}-\frac {5 (9 d+2 e)}{4 x^4}-\frac {5 (8 d+3 e)}{x^3}-\frac {15 (7 d+4 e)}{x^2}-\frac {42 (6 d+5 e)}{x}+30 (4 d+7 e) x+\frac {15}{2} (3 d+8 e) x^2+\frac {5}{3} (2 d+9 e) x^3+\frac {1}{4} (d+10 e) x^4+\frac {e x^5}{5}+42 (5 d+6 e) \log (x) \]
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Time = 0.10 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.88
method | result | size |
norman | \(\frac {\left (-252 d -210 e \right ) x^{5}+\left (-105 d -60 e \right ) x^{4}+\left (-40 d -15 e \right ) x^{3}+\left (-2 d -\frac {e}{5}\right ) x +\left (120 d +210 e \right ) x^{7}+\left (-\frac {45 d}{4}-\frac {5 e}{2}\right ) x^{2}+\left (\frac {d}{4}+\frac {5 e}{2}\right ) x^{10}+\left (\frac {10 d}{3}+15 e \right ) x^{9}+\left (\frac {45 d}{2}+60 e \right ) x^{8}-\frac {d}{6}+\frac {e \,x^{11}}{5}}{x^{6}}+\left (210 d +252 e \right ) \ln \left (x \right )\) | \(123\) |
risch | \(\frac {e \,x^{5}}{5}+\frac {d \,x^{4}}{4}+\frac {5 e \,x^{4}}{2}+\frac {10 d \,x^{3}}{3}+15 e \,x^{3}+\frac {45 d \,x^{2}}{2}+60 e \,x^{2}+120 d x +210 e x +\frac {\left (-252 d -210 e \right ) x^{5}+\left (-105 d -60 e \right ) x^{4}+\left (-40 d -15 e \right ) x^{3}+\left (-\frac {45 d}{4}-\frac {5 e}{2}\right ) x^{2}+\left (-2 d -\frac {e}{5}\right ) x -\frac {d}{6}}{x^{6}}+210 d \ln \left (x \right )+252 e \ln \left (x \right )\) | \(123\) |
default | \(\frac {e \,x^{5}}{5}+\frac {d \,x^{4}}{4}+\frac {5 e \,x^{4}}{2}+\frac {10 d \,x^{3}}{3}+15 e \,x^{3}+\frac {45 d \,x^{2}}{2}+60 e \,x^{2}+120 d x +210 e x -\frac {d}{6 x^{6}}-\frac {45 d +10 e}{4 x^{4}}-\frac {10 d +e}{5 x^{5}}+\left (210 d +252 e \right ) \ln \left (x \right )-\frac {210 d +120 e}{2 x^{2}}-\frac {252 d +210 e}{x}-\frac {120 d +45 e}{3 x^{3}}\) | \(126\) |
parallelrisch | \(\frac {12 e \,x^{11}+15 d \,x^{10}+150 e \,x^{10}+200 d \,x^{9}+900 e \,x^{9}+1350 d \,x^{8}+3600 e \,x^{8}+12600 \ln \left (x \right ) x^{6} d +15120 \ln \left (x \right ) x^{6} e +7200 d \,x^{7}+12600 e \,x^{7}-15120 d \,x^{5}-12600 e \,x^{5}-6300 d \,x^{4}-3600 e \,x^{4}-2400 d \,x^{3}-900 e \,x^{3}-675 d \,x^{2}-150 e \,x^{2}-120 d x -12 e x -10 d}{60 x^{6}}\) | \(136\) |
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Time = 0.28 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.94 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^7} \, dx=\frac {12 \, e x^{11} + 15 \, {\left (d + 10 \, e\right )} x^{10} + 100 \, {\left (2 \, d + 9 \, e\right )} x^{9} + 450 \, {\left (3 \, d + 8 \, e\right )} x^{8} + 1800 \, {\left (4 \, d + 7 \, e\right )} x^{7} + 2520 \, {\left (5 \, d + 6 \, e\right )} x^{6} \log \left (x\right ) - 2520 \, {\left (6 \, d + 5 \, e\right )} x^{5} - 900 \, {\left (7 \, d + 4 \, e\right )} x^{4} - 300 \, {\left (8 \, d + 3 \, e\right )} x^{3} - 75 \, {\left (9 \, d + 2 \, e\right )} x^{2} - 12 \, {\left (10 \, d + e\right )} x - 10 \, d}{60 \, x^{6}} \]
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Time = 1.25 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.91 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^7} \, dx=\frac {e x^{5}}{5} + x^{4} \left (\frac {d}{4} + \frac {5 e}{2}\right ) + x^{3} \cdot \left (\frac {10 d}{3} + 15 e\right ) + x^{2} \cdot \left (\frac {45 d}{2} + 60 e\right ) + x \left (120 d + 210 e\right ) + 42 \cdot \left (5 d + 6 e\right ) \log {\left (x \right )} + \frac {- 10 d + x^{5} \left (- 15120 d - 12600 e\right ) + x^{4} \left (- 6300 d - 3600 e\right ) + x^{3} \left (- 2400 d - 900 e\right ) + x^{2} \left (- 675 d - 150 e\right ) + x \left (- 120 d - 12 e\right )}{60 x^{6}} \]
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Time = 0.19 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.91 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^7} \, dx=\frac {1}{5} \, e x^{5} + \frac {1}{4} \, {\left (d + 10 \, e\right )} x^{4} + \frac {5}{3} \, {\left (2 \, d + 9 \, e\right )} x^{3} + \frac {15}{2} \, {\left (3 \, d + 8 \, e\right )} x^{2} + 30 \, {\left (4 \, d + 7 \, e\right )} x + 42 \, {\left (5 \, d + 6 \, e\right )} \log \left (x\right ) - \frac {2520 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 900 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 300 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 75 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 12 \, {\left (10 \, d + e\right )} x + 10 \, d}{60 \, x^{6}} \]
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Time = 0.28 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.91 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^7} \, dx=\frac {1}{5} \, e x^{5} + \frac {1}{4} \, d x^{4} + \frac {5}{2} \, e x^{4} + \frac {10}{3} \, d x^{3} + 15 \, e x^{3} + \frac {45}{2} \, d x^{2} + 60 \, e x^{2} + 120 \, d x + 210 \, e x + 42 \, {\left (5 \, d + 6 \, e\right )} \log \left ({\left | x \right |}\right ) - \frac {2520 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 900 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 300 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 75 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 12 \, {\left (10 \, d + e\right )} x + 10 \, d}{60 \, x^{6}} \]
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Time = 9.96 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.86 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^7} \, dx=x^4\,\left (\frac {d}{4}+\frac {5\,e}{2}\right )+x^3\,\left (\frac {10\,d}{3}+15\,e\right )+x^2\,\left (\frac {45\,d}{2}+60\,e\right )+\ln \left (x\right )\,\left (210\,d+252\,e\right )-\frac {\left (252\,d+210\,e\right )\,x^5+\left (105\,d+60\,e\right )\,x^4+\left (40\,d+15\,e\right )\,x^3+\left (\frac {45\,d}{4}+\frac {5\,e}{2}\right )\,x^2+\left (2\,d+\frac {e}{5}\right )\,x+\frac {d}{6}}{x^6}+\frac {e\,x^5}{5}+x\,\left (120\,d+210\,e\right ) \]
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