Optimal. Leaf size=31 \[ -\frac {d (1+x)^{11}}{12 x^{12}}+\frac {(d-12 e) (1+x)^{11}}{132 x^{11}} \]
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Rubi [A]
time = 0.00, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {27, 79, 37}
\begin {gather*} \frac {(x+1)^{11} (d-12 e)}{132 x^{11}}-\frac {d (x+1)^{11}}{12 x^{12}} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 37
Rule 79
Rubi steps
\begin {align*} \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{13}} \, dx &=\int \frac {(1+x)^{10} (d+e x)}{x^{13}} \, dx\\ &=-\frac {d (1+x)^{11}}{12 x^{12}}-\frac {1}{12} (d-12 e) \int \frac {(1+x)^{10}}{x^{12}} \, dx\\ &=-\frac {d (1+x)^{11}}{12 x^{12}}+\frac {(d-12 e) (1+x)^{11}}{132 x^{11}}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(114\) vs. \(2(31)=62\).
time = 0.02, size = 114, normalized size = 3.68 \begin {gather*} -\frac {12 e x \left (1+11 x+55 x^2+165 x^3+330 x^4+462 x^5+462 x^6+330 x^7+165 x^8+55 x^9+11 x^{10}\right )+d \left (11+120 x+594 x^2+1760 x^3+3465 x^4+4752 x^5+4620 x^6+3168 x^7+1485 x^8+440 x^9+66 x^{10}\right )}{132 x^{12}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(129\) vs.
\(2(27)=54\).
time = 0.68, size = 130, normalized size = 4.19
method | result | size |
norman | \(\frac {-e \,x^{11}+\left (-\frac {d}{2}-5 e \right ) x^{10}+\left (-\frac {10 d}{3}-15 e \right ) x^{9}+\left (-\frac {45 d}{4}-30 e \right ) x^{8}+\left (-24 d -42 e \right ) x^{7}+\left (-35 d -42 e \right ) x^{6}+\left (-36 d -30 e \right ) x^{5}+\left (-\frac {105 d}{4}-15 e \right ) x^{4}+\left (-\frac {40 d}{3}-5 e \right ) x^{3}+\left (-\frac {9 d}{2}-e \right ) x^{2}+\left (-\frac {10 d}{11}-\frac {e}{11}\right ) x -\frac {d}{12}}{x^{12}}\) | \(123\) |
risch | \(\frac {-e \,x^{11}+\left (-\frac {d}{2}-5 e \right ) x^{10}+\left (-\frac {10 d}{3}-15 e \right ) x^{9}+\left (-\frac {45 d}{4}-30 e \right ) x^{8}+\left (-24 d -42 e \right ) x^{7}+\left (-35 d -42 e \right ) x^{6}+\left (-36 d -30 e \right ) x^{5}+\left (-\frac {105 d}{4}-15 e \right ) x^{4}+\left (-\frac {40 d}{3}-5 e \right ) x^{3}+\left (-\frac {9 d}{2}-e \right ) x^{2}+\left (-\frac {10 d}{11}-\frac {e}{11}\right ) x -\frac {d}{12}}{x^{12}}\) | \(123\) |
default | \(-\frac {10 d +e}{11 x^{11}}-\frac {120 d +210 e}{5 x^{5}}-\frac {120 d +45 e}{9 x^{9}}-\frac {45 d +120 e}{4 x^{4}}-\frac {d}{12 x^{12}}-\frac {210 d +252 e}{6 x^{6}}-\frac {d +10 e}{2 x^{2}}-\frac {210 d +120 e}{8 x^{8}}-\frac {252 d +210 e}{7 x^{7}}-\frac {45 d +10 e}{10 x^{10}}-\frac {10 d +45 e}{3 x^{3}}-\frac {e}{x}\) | \(130\) |
gosper | \(-\frac {132 e \,x^{11}+66 d \,x^{10}+660 e \,x^{10}+440 d \,x^{9}+1980 e \,x^{9}+1485 d \,x^{8}+3960 e \,x^{8}+3168 d \,x^{7}+5544 e \,x^{7}+4620 d \,x^{6}+5544 x^{6} e +4752 d \,x^{5}+3960 e \,x^{5}+3465 d \,x^{4}+1980 x^{4} e +1760 d \,x^{3}+660 e \,x^{3}+594 d \,x^{2}+132 e \,x^{2}+120 d x +12 e x +11 d}{132 x^{12}}\) | \(132\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 140 vs.
\(2 (28) = 56\).
time = 0.27, size = 140, normalized size = 4.52 \begin {gather*} -\frac {132 \, x^{11} e + 66 \, {\left (d + 10 \, e\right )} x^{10} + 220 \, {\left (2 \, d + 9 \, e\right )} x^{9} + 495 \, {\left (3 \, d + 8 \, e\right )} x^{8} + 792 \, {\left (4 \, d + 7 \, e\right )} x^{7} + 924 \, {\left (5 \, d + 6 \, e\right )} x^{6} + 792 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 495 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 220 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 66 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 12 \, {\left (10 \, d + e\right )} x + 11 \, d}{132 \, x^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 123 vs.
\(2 (28) = 56\).
time = 1.92, size = 123, normalized size = 3.97 \begin {gather*} -\frac {66 \, d x^{10} + 440 \, d x^{9} + 1485 \, d x^{8} + 3168 \, d x^{7} + 4620 \, d x^{6} + 4752 \, d x^{5} + 3465 \, d x^{4} + 1760 \, d x^{3} + 594 \, d x^{2} + 120 \, d x + 12 \, {\left (11 \, x^{11} + 55 \, x^{10} + 165 \, x^{9} + 330 \, x^{8} + 462 \, x^{7} + 462 \, x^{6} + 330 \, x^{5} + 165 \, x^{4} + 55 \, x^{3} + 11 \, x^{2} + x\right )} e + 11 \, d}{132 \, x^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 131 vs.
\(2 (26) = 52\).
time = 8.44, size = 131, normalized size = 4.23 \begin {gather*} \frac {- 11 d - 132 e x^{11} + x^{10} \left (- 66 d - 660 e\right ) + x^{9} \left (- 440 d - 1980 e\right ) + x^{8} \left (- 1485 d - 3960 e\right ) + x^{7} \left (- 3168 d - 5544 e\right ) + x^{6} \left (- 4620 d - 5544 e\right ) + x^{5} \left (- 4752 d - 3960 e\right ) + x^{4} \left (- 3465 d - 1980 e\right ) + x^{3} \left (- 1760 d - 660 e\right ) + x^{2} \left (- 594 d - 132 e\right ) + x \left (- 120 d - 12 e\right )}{132 x^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 142 vs.
\(2 (28) = 56\).
time = 1.58, size = 142, normalized size = 4.58 \begin {gather*} -\frac {132 \, x^{11} e + 66 \, d x^{10} + 660 \, x^{10} e + 440 \, d x^{9} + 1980 \, x^{9} e + 1485 \, d x^{8} + 3960 \, x^{8} e + 3168 \, d x^{7} + 5544 \, x^{7} e + 4620 \, d x^{6} + 5544 \, x^{6} e + 4752 \, d x^{5} + 3960 \, x^{5} e + 3465 \, d x^{4} + 1980 \, x^{4} e + 1760 \, d x^{3} + 660 \, x^{3} e + 594 \, d x^{2} + 132 \, x^{2} e + 120 \, d x + 12 \, x e + 11 \, d}{132 \, x^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.12, size = 120, normalized size = 3.87 \begin {gather*} -\frac {e\,x^{11}+\left (\frac {d}{2}+5\,e\right )\,x^{10}+\left (\frac {10\,d}{3}+15\,e\right )\,x^9+\left (\frac {45\,d}{4}+30\,e\right )\,x^8+\left (24\,d+42\,e\right )\,x^7+\left (35\,d+42\,e\right )\,x^6+\left (36\,d+30\,e\right )\,x^5+\left (\frac {105\,d}{4}+15\,e\right )\,x^4+\left (\frac {40\,d}{3}+5\,e\right )\,x^3+\left (\frac {9\,d}{2}+e\right )\,x^2+\left (\frac {10\,d}{11}+\frac {e}{11}\right )\,x+\frac {d}{12}}{x^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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