Optimal. Leaf size=90 \[ \frac {(x+1)^{11} (4 d-15 e)}{210 x^{14}}-\frac {(x+1)^{11} (4 d-15 e)}{910 x^{13}}+\frac {(x+1)^{11} (4 d-15 e)}{5460 x^{12}}-\frac {(x+1)^{11} (4 d-15 e)}{60060 x^{11}}-\frac {d (x+1)^{11}}{15 x^{15}} \]
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Rubi [A] time = 0.02, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {27, 78, 45, 37} \[ -\frac {(x+1)^{11} (4 d-15 e)}{60060 x^{11}}+\frac {(x+1)^{11} (4 d-15 e)}{5460 x^{12}}-\frac {(x+1)^{11} (4 d-15 e)}{910 x^{13}}+\frac {(x+1)^{11} (4 d-15 e)}{210 x^{14}}-\frac {d (x+1)^{11}}{15 x^{15}} \]
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^{16}} \, dx &=\int \frac {(1+x)^{10} (d+e x)}{x^{16}} \, dx\\ &=-\frac {d (1+x)^{11}}{15 x^{15}}-\frac {1}{15} (4 d-15 e) \int \frac {(1+x)^{10}}{x^{15}} \, dx\\ &=-\frac {d (1+x)^{11}}{15 x^{15}}+\frac {(4 d-15 e) (1+x)^{11}}{210 x^{14}}-\frac {1}{70} (-4 d+15 e) \int \frac {(1+x)^{10}}{x^{14}} \, dx\\ &=-\frac {d (1+x)^{11}}{15 x^{15}}+\frac {(4 d-15 e) (1+x)^{11}}{210 x^{14}}-\frac {(4 d-15 e) (1+x)^{11}}{910 x^{13}}-\frac {1}{455} (4 d-15 e) \int \frac {(1+x)^{10}}{x^{13}} \, dx\\ &=-\frac {d (1+x)^{11}}{15 x^{15}}+\frac {(4 d-15 e) (1+x)^{11}}{210 x^{14}}-\frac {(4 d-15 e) (1+x)^{11}}{910 x^{13}}+\frac {(4 d-15 e) (1+x)^{11}}{5460 x^{12}}-\frac {(-4 d+15 e) \int \frac {(1+x)^{10}}{x^{12}} \, dx}{5460}\\ &=-\frac {d (1+x)^{11}}{15 x^{15}}+\frac {(4 d-15 e) (1+x)^{11}}{210 x^{14}}-\frac {(4 d-15 e) (1+x)^{11}}{910 x^{13}}+\frac {(4 d-15 e) (1+x)^{11}}{5460 x^{12}}-\frac {(4 d-15 e) (1+x)^{11}}{60060 x^{11}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 153, normalized size = 1.70 \[ -\frac {10 d+e}{14 x^{14}}-\frac {5 (9 d+2 e)}{13 x^{13}}-\frac {5 (8 d+3 e)}{4 x^{12}}-\frac {30 (7 d+4 e)}{11 x^{11}}-\frac {21 (6 d+5 e)}{5 x^{10}}-\frac {14 (5 d+6 e)}{3 x^9}-\frac {15 (4 d+7 e)}{4 x^8}-\frac {15 (3 d+8 e)}{7 x^7}-\frac {5 (2 d+9 e)}{6 x^6}-\frac {d+10 e}{5 x^5}-\frac {d}{15 x^{15}}-\frac {e}{4 x^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 129, normalized size = 1.43 \[ -\frac {15015 \, e x^{11} + 12012 \, {\left (d + 10 \, e\right )} x^{10} + 50050 \, {\left (2 \, d + 9 \, e\right )} x^{9} + 128700 \, {\left (3 \, d + 8 \, e\right )} x^{8} + 225225 \, {\left (4 \, d + 7 \, e\right )} x^{7} + 280280 \, {\left (5 \, d + 6 \, e\right )} x^{6} + 252252 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 163800 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 75075 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 23100 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 4290 \, {\left (10 \, d + e\right )} x + 4004 \, d}{60060 \, x^{15}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 142, normalized size = 1.58 \[ -\frac {15015 \, x^{11} e + 12012 \, d x^{10} + 120120 \, x^{10} e + 100100 \, d x^{9} + 450450 \, x^{9} e + 386100 \, d x^{8} + 1029600 \, x^{8} e + 900900 \, d x^{7} + 1576575 \, x^{7} e + 1401400 \, d x^{6} + 1681680 \, x^{6} e + 1513512 \, d x^{5} + 1261260 \, x^{5} e + 1146600 \, d x^{4} + 655200 \, x^{4} e + 600600 \, d x^{3} + 225225 \, x^{3} e + 207900 \, d x^{2} + 46200 \, x^{2} e + 42900 \, d x + 4290 \, x e + 4004 \, d}{60060 \, x^{15}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 130, normalized size = 1.44 \[ -\frac {e}{4 x^{4}}-\frac {d +10 e}{5 x^{5}}-\frac {10 d +45 e}{6 x^{6}}-\frac {45 d +120 e}{7 x^{7}}-\frac {120 d +210 e}{8 x^{8}}-\frac {210 d +252 e}{9 x^{9}}-\frac {252 d +210 e}{10 x^{10}}-\frac {210 d +120 e}{11 x^{11}}-\frac {120 d +45 e}{12 x^{12}}-\frac {45 d +10 e}{13 x^{13}}-\frac {d}{15 x^{15}}-\frac {10 d +e}{14 x^{14}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.71, size = 129, normalized size = 1.43 \[ -\frac {15015 \, e x^{11} + 12012 \, {\left (d + 10 \, e\right )} x^{10} + 50050 \, {\left (2 \, d + 9 \, e\right )} x^{9} + 128700 \, {\left (3 \, d + 8 \, e\right )} x^{8} + 225225 \, {\left (4 \, d + 7 \, e\right )} x^{7} + 280280 \, {\left (5 \, d + 6 \, e\right )} x^{6} + 252252 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 163800 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 75075 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 23100 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 4290 \, {\left (10 \, d + e\right )} x + 4004 \, d}{60060 \, x^{15}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 123, normalized size = 1.37 \[ -\frac {\frac {e\,x^{11}}{4}+\left (\frac {d}{5}+2\,e\right )\,x^{10}+\left (\frac {5\,d}{3}+\frac {15\,e}{2}\right )\,x^9+\left (\frac {45\,d}{7}+\frac {120\,e}{7}\right )\,x^8+\left (15\,d+\frac {105\,e}{4}\right )\,x^7+\left (\frac {70\,d}{3}+28\,e\right )\,x^6+\left (\frac {126\,d}{5}+21\,e\right )\,x^5+\left (\frac {210\,d}{11}+\frac {120\,e}{11}\right )\,x^4+\left (10\,d+\frac {15\,e}{4}\right )\,x^3+\left (\frac {45\,d}{13}+\frac {10\,e}{13}\right )\,x^2+\left (\frac {5\,d}{7}+\frac {e}{14}\right )\,x+\frac {d}{15}}{x^{15}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 19.11, size = 131, normalized size = 1.46 \[ \frac {- 4004 d - 15015 e x^{11} + x^{10} \left (- 12012 d - 120120 e\right ) + x^{9} \left (- 100100 d - 450450 e\right ) + x^{8} \left (- 386100 d - 1029600 e\right ) + x^{7} \left (- 900900 d - 1576575 e\right ) + x^{6} \left (- 1401400 d - 1681680 e\right ) + x^{5} \left (- 1513512 d - 1261260 e\right ) + x^{4} \left (- 1146600 d - 655200 e\right ) + x^{3} \left (- 600600 d - 225225 e\right ) + x^{2} \left (- 207900 d - 46200 e\right ) + x \left (- 42900 d - 4290 e\right )}{60060 x^{15}} \]
Verification of antiderivative is not currently implemented for this CAS.
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