Optimal. Leaf size=71 \[ \frac {(x+1)^{11} (3 d-14 e)}{182 x^{13}}-\frac {(x+1)^{11} (3 d-14 e)}{1092 x^{12}}+\frac {(x+1)^{11} (3 d-14 e)}{12012 x^{11}}-\frac {d (x+1)^{11}}{14 x^{14}} \]
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Rubi [A] time = 0.02, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 5, 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} (3 d-14 e)}{12012 x^{11}}-\frac {(x+1)^{11} (3 d-14 e)}{1092 x^{12}}+\frac {(x+1)^{11} (3 d-14 e)}{182 x^{13}}-\frac {d (x+1)^{11}}{14 x^{14}} \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^{15}} \, dx &=\int \frac {(1+x)^{10} (d+e x)}{x^{15}} \, dx\\ &=-\frac {d (1+x)^{11}}{14 x^{14}}-\frac {1}{14} (3 d-14 e) \int \frac {(1+x)^{10}}{x^{14}} \, dx\\ &=-\frac {d (1+x)^{11}}{14 x^{14}}+\frac {(3 d-14 e) (1+x)^{11}}{182 x^{13}}-\frac {1}{91} (-3 d+14 e) \int \frac {(1+x)^{10}}{x^{13}} \, dx\\ &=-\frac {d (1+x)^{11}}{14 x^{14}}+\frac {(3 d-14 e) (1+x)^{11}}{182 x^{13}}-\frac {(3 d-14 e) (1+x)^{11}}{1092 x^{12}}-\frac {(3 d-14 e) \int \frac {(1+x)^{10}}{x^{12}} \, dx}{1092}\\ &=-\frac {d (1+x)^{11}}{14 x^{14}}+\frac {(3 d-14 e) (1+x)^{11}}{182 x^{13}}-\frac {(3 d-14 e) (1+x)^{11}}{1092 x^{12}}+\frac {(3 d-14 e) (1+x)^{11}}{12012 x^{11}}\\ \end {align*}
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Mathematica [B] time = 0.04, size = 149, normalized size = 2.10 \begin {gather*} -\frac {10 d+e}{13 x^{13}}-\frac {5 (9 d+2 e)}{12 x^{12}}-\frac {15 (8 d+3 e)}{11 x^{11}}-\frac {3 (7 d+4 e)}{x^{10}}-\frac {14 (6 d+5 e)}{3 x^9}-\frac {21 (5 d+6 e)}{4 x^8}-\frac {30 (4 d+7 e)}{7 x^7}-\frac {5 (3 d+8 e)}{2 x^6}-\frac {2 d+9 e}{x^5}-\frac {d+10 e}{4 x^4}-\frac {d}{14 x^{14}}-\frac {e}{3 x^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^{15}} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.40, size = 129, normalized size = 1.82 \begin {gather*} -\frac {4004 \, e x^{11} + 3003 \, {\left (d + 10 \, e\right )} x^{10} + 12012 \, {\left (2 \, d + 9 \, e\right )} x^{9} + 30030 \, {\left (3 \, d + 8 \, e\right )} x^{8} + 51480 \, {\left (4 \, d + 7 \, e\right )} x^{7} + 63063 \, {\left (5 \, d + 6 \, e\right )} x^{6} + 56056 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 36036 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 16380 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 5005 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 924 \, {\left (10 \, d + e\right )} x + 858 \, d}{12012 \, x^{14}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 142, normalized size = 2.00 \begin {gather*} -\frac {4004 \, x^{11} e + 3003 \, d x^{10} + 30030 \, x^{10} e + 24024 \, d x^{9} + 108108 \, x^{9} e + 90090 \, d x^{8} + 240240 \, x^{8} e + 205920 \, d x^{7} + 360360 \, x^{7} e + 315315 \, d x^{6} + 378378 \, x^{6} e + 336336 \, d x^{5} + 280280 \, x^{5} e + 252252 \, d x^{4} + 144144 \, x^{4} e + 131040 \, d x^{3} + 49140 \, x^{3} e + 45045 \, d x^{2} + 10010 \, x^{2} e + 9240 \, d x + 924 \, x e + 858 \, d}{12012 \, x^{14}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 130, normalized size = 1.83 \begin {gather*} -\frac {e}{3 x^{3}}-\frac {d +10 e}{4 x^{4}}-\frac {10 d +45 e}{5 x^{5}}-\frac {45 d +120 e}{6 x^{6}}-\frac {120 d +210 e}{7 x^{7}}-\frac {210 d +252 e}{8 x^{8}}-\frac {252 d +210 e}{9 x^{9}}-\frac {210 d +120 e}{10 x^{10}}-\frac {120 d +45 e}{11 x^{11}}-\frac {45 d +10 e}{12 x^{12}}-\frac {d}{14 x^{14}}-\frac {10 d +e}{13 x^{13}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.72, size = 129, normalized size = 1.82 \begin {gather*} -\frac {4004 \, e x^{11} + 3003 \, {\left (d + 10 \, e\right )} x^{10} + 12012 \, {\left (2 \, d + 9 \, e\right )} x^{9} + 30030 \, {\left (3 \, d + 8 \, e\right )} x^{8} + 51480 \, {\left (4 \, d + 7 \, e\right )} x^{7} + 63063 \, {\left (5 \, d + 6 \, e\right )} x^{6} + 56056 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 36036 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 16380 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 5005 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 924 \, {\left (10 \, d + e\right )} x + 858 \, d}{12012 \, x^{14}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 123, normalized size = 1.73 \begin {gather*} -\frac {\frac {e\,x^{11}}{3}+\left (\frac {d}{4}+\frac {5\,e}{2}\right )\,x^{10}+\left (2\,d+9\,e\right )\,x^9+\left (\frac {15\,d}{2}+20\,e\right )\,x^8+\left (\frac {120\,d}{7}+30\,e\right )\,x^7+\left (\frac {105\,d}{4}+\frac {63\,e}{2}\right )\,x^6+\left (28\,d+\frac {70\,e}{3}\right )\,x^5+\left (21\,d+12\,e\right )\,x^4+\left (\frac {120\,d}{11}+\frac {45\,e}{11}\right )\,x^3+\left (\frac {15\,d}{4}+\frac {5\,e}{6}\right )\,x^2+\left (\frac {10\,d}{13}+\frac {e}{13}\right )\,x+\frac {d}{14}}{x^{14}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 16.09, size = 131, normalized size = 1.85 \begin {gather*} \frac {- 858 d - 4004 e x^{11} + x^{10} \left (- 3003 d - 30030 e\right ) + x^{9} \left (- 24024 d - 108108 e\right ) + x^{8} \left (- 90090 d - 240240 e\right ) + x^{7} \left (- 205920 d - 360360 e\right ) + x^{6} \left (- 315315 d - 378378 e\right ) + x^{5} \left (- 336336 d - 280280 e\right ) + x^{4} \left (- 252252 d - 144144 e\right ) + x^{3} \left (- 131040 d - 49140 e\right ) + x^{2} \left (- 45045 d - 10010 e\right ) + x \left (- 9240 d - 924 e\right )}{12012 x^{14}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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