Optimal. Leaf size=128 \[ \frac {(x+1)^{11} (6 d-17 e)}{272 x^{16}}-\frac {(x+1)^{11} (6 d-17 e)}{816 x^{15}}+\frac {(x+1)^{11} (6 d-17 e)}{2856 x^{14}}-\frac {(x+1)^{11} (6 d-17 e)}{12376 x^{13}}+\frac {(x+1)^{11} (6 d-17 e)}{74256 x^{12}}-\frac {(x+1)^{11} (6 d-17 e)}{816816 x^{11}}-\frac {d (x+1)^{11}}{17 x^{17}} \]
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Rubi [A] time = 0.03, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {27, 78, 45, 37} \begin {gather*} -\frac {(x+1)^{11} (6 d-17 e)}{816816 x^{11}}+\frac {(x+1)^{11} (6 d-17 e)}{74256 x^{12}}-\frac {(x+1)^{11} (6 d-17 e)}{12376 x^{13}}+\frac {(x+1)^{11} (6 d-17 e)}{2856 x^{14}}-\frac {(x+1)^{11} (6 d-17 e)}{816 x^{15}}+\frac {(x+1)^{11} (6 d-17 e)}{272 x^{16}}-\frac {d (x+1)^{11}}{17 x^{17}} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 37
Rule 45
Rule 78
Rubi steps
\begin {align*} \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{18}} \, dx &=\int \frac {(1+x)^{10} (d+e x)}{x^{18}} \, dx\\ &=-\frac {d (1+x)^{11}}{17 x^{17}}-\frac {1}{17} (6 d-17 e) \int \frac {(1+x)^{10}}{x^{17}} \, dx\\ &=-\frac {d (1+x)^{11}}{17 x^{17}}+\frac {(6 d-17 e) (1+x)^{11}}{272 x^{16}}+\frac {1}{272} (5 (6 d-17 e)) \int \frac {(1+x)^{10}}{x^{16}} \, dx\\ &=-\frac {d (1+x)^{11}}{17 x^{17}}+\frac {(6 d-17 e) (1+x)^{11}}{272 x^{16}}-\frac {(6 d-17 e) (1+x)^{11}}{816 x^{15}}+\frac {1}{204} (-6 d+17 e) \int \frac {(1+x)^{10}}{x^{15}} \, dx\\ &=-\frac {d (1+x)^{11}}{17 x^{17}}+\frac {(6 d-17 e) (1+x)^{11}}{272 x^{16}}-\frac {(6 d-17 e) (1+x)^{11}}{816 x^{15}}+\frac {(6 d-17 e) (1+x)^{11}}{2856 x^{14}}+\frac {1}{952} (6 d-17 e) \int \frac {(1+x)^{10}}{x^{14}} \, dx\\ &=-\frac {d (1+x)^{11}}{17 x^{17}}+\frac {(6 d-17 e) (1+x)^{11}}{272 x^{16}}-\frac {(6 d-17 e) (1+x)^{11}}{816 x^{15}}+\frac {(6 d-17 e) (1+x)^{11}}{2856 x^{14}}-\frac {(6 d-17 e) (1+x)^{11}}{12376 x^{13}}+\frac {(-6 d+17 e) \int \frac {(1+x)^{10}}{x^{13}} \, dx}{6188}\\ &=-\frac {d (1+x)^{11}}{17 x^{17}}+\frac {(6 d-17 e) (1+x)^{11}}{272 x^{16}}-\frac {(6 d-17 e) (1+x)^{11}}{816 x^{15}}+\frac {(6 d-17 e) (1+x)^{11}}{2856 x^{14}}-\frac {(6 d-17 e) (1+x)^{11}}{12376 x^{13}}+\frac {(6 d-17 e) (1+x)^{11}}{74256 x^{12}}+\frac {(6 d-17 e) \int \frac {(1+x)^{10}}{x^{12}} \, dx}{74256}\\ &=-\frac {d (1+x)^{11}}{17 x^{17}}+\frac {(6 d-17 e) (1+x)^{11}}{272 x^{16}}-\frac {(6 d-17 e) (1+x)^{11}}{816 x^{15}}+\frac {(6 d-17 e) (1+x)^{11}}{2856 x^{14}}-\frac {(6 d-17 e) (1+x)^{11}}{12376 x^{13}}+\frac {(6 d-17 e) (1+x)^{11}}{74256 x^{12}}-\frac {(6 d-17 e) (1+x)^{11}}{816816 x^{11}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 151, normalized size = 1.18 \begin {gather*} -\frac {10 d+e}{16 x^{16}}-\frac {9 d+2 e}{3 x^{15}}-\frac {15 (8 d+3 e)}{14 x^{14}}-\frac {30 (7 d+4 e)}{13 x^{13}}-\frac {7 (6 d+5 e)}{2 x^{12}}-\frac {42 (5 d+6 e)}{11 x^{11}}-\frac {3 (4 d+7 e)}{x^{10}}-\frac {5 (3 d+8 e)}{3 x^9}-\frac {5 (2 d+9 e)}{8 x^8}-\frac {d+10 e}{7 x^7}-\frac {d}{17 x^{17}}-\frac {e}{6 x^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^{18}} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.40, size = 129, normalized size = 1.01 \begin {gather*} -\frac {136136 \, e x^{11} + 116688 \, {\left (d + 10 \, e\right )} x^{10} + 510510 \, {\left (2 \, d + 9 \, e\right )} x^{9} + 1361360 \, {\left (3 \, d + 8 \, e\right )} x^{8} + 2450448 \, {\left (4 \, d + 7 \, e\right )} x^{7} + 3118752 \, {\left (5 \, d + 6 \, e\right )} x^{6} + 2858856 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 1884960 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 875160 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 272272 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 51051 \, {\left (10 \, d + e\right )} x + 48048 \, d}{816816 \, x^{17}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 142, normalized size = 1.11 \begin {gather*} -\frac {136136 \, x^{11} e + 116688 \, d x^{10} + 1166880 \, x^{10} e + 1021020 \, d x^{9} + 4594590 \, x^{9} e + 4084080 \, d x^{8} + 10890880 \, x^{8} e + 9801792 \, d x^{7} + 17153136 \, x^{7} e + 15593760 \, d x^{6} + 18712512 \, x^{6} e + 17153136 \, d x^{5} + 14294280 \, x^{5} e + 13194720 \, d x^{4} + 7539840 \, x^{4} e + 7001280 \, d x^{3} + 2625480 \, x^{3} e + 2450448 \, d x^{2} + 544544 \, x^{2} e + 510510 \, d x + 51051 \, x e + 48048 \, d}{816816 \, x^{17}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 130, normalized size = 1.02 \begin {gather*} -\frac {e}{6 x^{6}}-\frac {d +10 e}{7 x^{7}}-\frac {10 d +45 e}{8 x^{8}}-\frac {45 d +120 e}{9 x^{9}}-\frac {120 d +210 e}{10 x^{10}}-\frac {210 d +252 e}{11 x^{11}}-\frac {252 d +210 e}{12 x^{12}}-\frac {210 d +120 e}{13 x^{13}}-\frac {120 d +45 e}{14 x^{14}}-\frac {45 d +10 e}{15 x^{15}}-\frac {d}{17 x^{17}}-\frac {10 d +e}{16 x^{16}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 129, normalized size = 1.01 \begin {gather*} -\frac {136136 \, e x^{11} + 116688 \, {\left (d + 10 \, e\right )} x^{10} + 510510 \, {\left (2 \, d + 9 \, e\right )} x^{9} + 1361360 \, {\left (3 \, d + 8 \, e\right )} x^{8} + 2450448 \, {\left (4 \, d + 7 \, e\right )} x^{7} + 3118752 \, {\left (5 \, d + 6 \, e\right )} x^{6} + 2858856 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 1884960 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 875160 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 272272 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 51051 \, {\left (10 \, d + e\right )} x + 48048 \, d}{816816 \, x^{17}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.13, size = 123, normalized size = 0.96 \begin {gather*} -\frac {\frac {e\,x^{11}}{6}+\left (\frac {d}{7}+\frac {10\,e}{7}\right )\,x^{10}+\left (\frac {5\,d}{4}+\frac {45\,e}{8}\right )\,x^9+\left (5\,d+\frac {40\,e}{3}\right )\,x^8+\left (12\,d+21\,e\right )\,x^7+\left (\frac {210\,d}{11}+\frac {252\,e}{11}\right )\,x^6+\left (21\,d+\frac {35\,e}{2}\right )\,x^5+\left (\frac {210\,d}{13}+\frac {120\,e}{13}\right )\,x^4+\left (\frac {60\,d}{7}+\frac {45\,e}{14}\right )\,x^3+\left (3\,d+\frac {2\,e}{3}\right )\,x^2+\left (\frac {5\,d}{8}+\frac {e}{16}\right )\,x+\frac {d}{17}}{x^{17}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 24.87, size = 131, normalized size = 1.02 \begin {gather*} \frac {- 48048 d - 136136 e x^{11} + x^{10} \left (- 116688 d - 1166880 e\right ) + x^{9} \left (- 1021020 d - 4594590 e\right ) + x^{8} \left (- 4084080 d - 10890880 e\right ) + x^{7} \left (- 9801792 d - 17153136 e\right ) + x^{6} \left (- 15593760 d - 18712512 e\right ) + x^{5} \left (- 17153136 d - 14294280 e\right ) + x^{4} \left (- 13194720 d - 7539840 e\right ) + x^{3} \left (- 7001280 d - 2625480 e\right ) + x^{2} \left (- 2450448 d - 544544 e\right ) + x \left (- 510510 d - 51051 e\right )}{816816 x^{17}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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