Optimal. Leaf size=52 \[ \frac {(x+1)^{11} (2 d-13 e)}{156 x^{12}}-\frac {(x+1)^{11} (2 d-13 e)}{1716 x^{11}}-\frac {d (x+1)^{11}}{13 x^{13}} \]
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Rubi [A] time = 0.01, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {27, 78, 45, 37} \[ -\frac {(x+1)^{11} (2 d-13 e)}{1716 x^{11}}+\frac {(x+1)^{11} (2 d-13 e)}{156 x^{12}}-\frac {d (x+1)^{11}}{13 x^{13}} \]
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^{14}} \, dx &=\int \frac {(1+x)^{10} (d+e x)}{x^{14}} \, dx\\ &=-\frac {d (1+x)^{11}}{13 x^{13}}-\frac {1}{13} (2 d-13 e) \int \frac {(1+x)^{10}}{x^{13}} \, dx\\ &=-\frac {d (1+x)^{11}}{13 x^{13}}+\frac {(2 d-13 e) (1+x)^{11}}{156 x^{12}}-\frac {1}{156} (-2 d+13 e) \int \frac {(1+x)^{10}}{x^{12}} \, dx\\ &=-\frac {d (1+x)^{11}}{13 x^{13}}+\frac {(2 d-13 e) (1+x)^{11}}{156 x^{12}}-\frac {(2 d-13 e) (1+x)^{11}}{1716 x^{11}}\\ \end {align*}
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Mathematica [B] time = 0.03, size = 115, normalized size = 2.21 \[ -\frac {2 d \left (286 x^{10}+2145 x^9+7722 x^8+17160 x^7+25740 x^6+27027 x^5+20020 x^4+10296 x^3+3510 x^2+715 x+66\right )+13 e x \left (66 x^{10}+440 x^9+1485 x^8+3168 x^7+4620 x^6+4752 x^5+3465 x^4+1760 x^3+594 x^2+120 x+11\right )}{1716 x^{13}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.83, size = 129, normalized size = 2.48 \[ -\frac {858 \, e x^{11} + 572 \, {\left (d + 10 \, e\right )} x^{10} + 2145 \, {\left (2 \, d + 9 \, e\right )} x^{9} + 5148 \, {\left (3 \, d + 8 \, e\right )} x^{8} + 8580 \, {\left (4 \, d + 7 \, e\right )} x^{7} + 10296 \, {\left (5 \, d + 6 \, e\right )} x^{6} + 9009 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 5720 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 2574 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 780 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 143 \, {\left (10 \, d + e\right )} x + 132 \, d}{1716 \, x^{13}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 142, normalized size = 2.73 \[ -\frac {858 \, x^{11} e + 572 \, d x^{10} + 5720 \, x^{10} e + 4290 \, d x^{9} + 19305 \, x^{9} e + 15444 \, d x^{8} + 41184 \, x^{8} e + 34320 \, d x^{7} + 60060 \, x^{7} e + 51480 \, d x^{6} + 61776 \, x^{6} e + 54054 \, d x^{5} + 45045 \, x^{5} e + 40040 \, d x^{4} + 22880 \, x^{4} e + 20592 \, d x^{3} + 7722 \, x^{3} e + 7020 \, d x^{2} + 1560 \, x^{2} e + 1430 \, d x + 143 \, x e + 132 \, d}{1716 \, x^{13}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 130, normalized size = 2.50 \[ -\frac {e}{2 x^{2}}-\frac {d +10 e}{3 x^{3}}-\frac {10 d +45 e}{4 x^{4}}-\frac {45 d +120 e}{5 x^{5}}-\frac {120 d +210 e}{6 x^{6}}-\frac {210 d +252 e}{7 x^{7}}-\frac {252 d +210 e}{8 x^{8}}-\frac {210 d +120 e}{9 x^{9}}-\frac {120 d +45 e}{10 x^{10}}-\frac {45 d +10 e}{11 x^{11}}-\frac {d}{13 x^{13}}-\frac {10 d +e}{12 x^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.55, size = 129, normalized size = 2.48 \[ -\frac {858 \, e x^{11} + 572 \, {\left (d + 10 \, e\right )} x^{10} + 2145 \, {\left (2 \, d + 9 \, e\right )} x^{9} + 5148 \, {\left (3 \, d + 8 \, e\right )} x^{8} + 8580 \, {\left (4 \, d + 7 \, e\right )} x^{7} + 10296 \, {\left (5 \, d + 6 \, e\right )} x^{6} + 9009 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 5720 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 2574 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 780 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 143 \, {\left (10 \, d + e\right )} x + 132 \, d}{1716 \, x^{13}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.14, size = 123, normalized size = 2.37 \[ -\frac {\frac {e\,x^{11}}{2}+\left (\frac {d}{3}+\frac {10\,e}{3}\right )\,x^{10}+\left (\frac {5\,d}{2}+\frac {45\,e}{4}\right )\,x^9+\left (9\,d+24\,e\right )\,x^8+\left (20\,d+35\,e\right )\,x^7+\left (30\,d+36\,e\right )\,x^6+\left (\frac {63\,d}{2}+\frac {105\,e}{4}\right )\,x^5+\left (\frac {70\,d}{3}+\frac {40\,e}{3}\right )\,x^4+\left (12\,d+\frac {9\,e}{2}\right )\,x^3+\left (\frac {45\,d}{11}+\frac {10\,e}{11}\right )\,x^2+\left (\frac {5\,d}{6}+\frac {e}{12}\right )\,x+\frac {d}{13}}{x^{13}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 13.82, size = 131, normalized size = 2.52 \[ \frac {- 132 d - 858 e x^{11} + x^{10} \left (- 572 d - 5720 e\right ) + x^{9} \left (- 4290 d - 19305 e\right ) + x^{8} \left (- 15444 d - 41184 e\right ) + x^{7} \left (- 34320 d - 60060 e\right ) + x^{6} \left (- 51480 d - 61776 e\right ) + x^{5} \left (- 54054 d - 45045 e\right ) + x^{4} \left (- 40040 d - 22880 e\right ) + x^{3} \left (- 20592 d - 7722 e\right ) + x^{2} \left (- 7020 d - 1560 e\right ) + x \left (- 1430 d - 143 e\right )}{1716 x^{13}} \]
Verification of antiderivative is not currently implemented for this CAS.
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