Optimal. Leaf size=151 \[ -\frac {10 d+e}{19 x^{19}}-\frac {5 (9 d+2 e)}{18 x^{18}}-\frac {15 (8 d+3 e)}{17 x^{17}}-\frac {15 (7 d+4 e)}{8 x^{16}}-\frac {14 (6 d+5 e)}{5 x^{15}}-\frac {3 (5 d+6 e)}{x^{14}}-\frac {30 (4 d+7 e)}{13 x^{13}}-\frac {5 (3 d+8 e)}{4 x^{12}}-\frac {5 (2 d+9 e)}{11 x^{11}}-\frac {d+10 e}{10 x^{10}}-\frac {d}{20 x^{20}}-\frac {e}{9 x^9} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {27, 76} \[ -\frac {d+10 e}{10 x^{10}}-\frac {5 (2 d+9 e)}{11 x^{11}}-\frac {5 (3 d+8 e)}{4 x^{12}}-\frac {30 (4 d+7 e)}{13 x^{13}}-\frac {3 (5 d+6 e)}{x^{14}}-\frac {14 (6 d+5 e)}{5 x^{15}}-\frac {15 (7 d+4 e)}{8 x^{16}}-\frac {15 (8 d+3 e)}{17 x^{17}}-\frac {5 (9 d+2 e)}{18 x^{18}}-\frac {10 d+e}{19 x^{19}}-\frac {d}{20 x^{20}}-\frac {e}{9 x^9} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 27
Rule 76
Rubi steps
\begin {align*} \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{21}} \, dx &=\int \frac {(1+x)^{10} (d+e x)}{x^{21}} \, dx\\ &=\int \left (\frac {d}{x^{21}}+\frac {10 d+e}{x^{20}}+\frac {5 (9 d+2 e)}{x^{19}}+\frac {15 (8 d+3 e)}{x^{18}}+\frac {30 (7 d+4 e)}{x^{17}}+\frac {42 (6 d+5 e)}{x^{16}}+\frac {42 (5 d+6 e)}{x^{15}}+\frac {30 (4 d+7 e)}{x^{14}}+\frac {15 (3 d+8 e)}{x^{13}}+\frac {5 (2 d+9 e)}{x^{12}}+\frac {d+10 e}{x^{11}}+\frac {e}{x^{10}}\right ) \, dx\\ &=-\frac {d}{20 x^{20}}-\frac {10 d+e}{19 x^{19}}-\frac {5 (9 d+2 e)}{18 x^{18}}-\frac {15 (8 d+3 e)}{17 x^{17}}-\frac {15 (7 d+4 e)}{8 x^{16}}-\frac {14 (6 d+5 e)}{5 x^{15}}-\frac {3 (5 d+6 e)}{x^{14}}-\frac {30 (4 d+7 e)}{13 x^{13}}-\frac {5 (3 d+8 e)}{4 x^{12}}-\frac {5 (2 d+9 e)}{11 x^{11}}-\frac {d+10 e}{10 x^{10}}-\frac {e}{9 x^9}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 151, normalized size = 1.00 \[ -\frac {10 d+e}{19 x^{19}}-\frac {5 (9 d+2 e)}{18 x^{18}}-\frac {15 (8 d+3 e)}{17 x^{17}}-\frac {15 (7 d+4 e)}{8 x^{16}}-\frac {14 (6 d+5 e)}{5 x^{15}}-\frac {3 (5 d+6 e)}{x^{14}}-\frac {30 (4 d+7 e)}{13 x^{13}}-\frac {5 (3 d+8 e)}{4 x^{12}}-\frac {5 (2 d+9 e)}{11 x^{11}}-\frac {d+10 e}{10 x^{10}}-\frac {d}{20 x^{20}}-\frac {e}{9 x^9} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.60, size = 129, normalized size = 0.85 \[ -\frac {1847560 \, e x^{11} + 1662804 \, {\left (d + 10 \, e\right )} x^{10} + 7558200 \, {\left (2 \, d + 9 \, e\right )} x^{9} + 20785050 \, {\left (3 \, d + 8 \, e\right )} x^{8} + 38372400 \, {\left (4 \, d + 7 \, e\right )} x^{7} + 49884120 \, {\left (5 \, d + 6 \, e\right )} x^{6} + 46558512 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 31177575 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 14671800 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 4618900 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 875160 \, {\left (10 \, d + e\right )} x + 831402 \, d}{16628040 \, x^{20}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.18, size = 142, normalized size = 0.94 \[ -\frac {1847560 \, x^{11} e + 1662804 \, d x^{10} + 16628040 \, x^{10} e + 15116400 \, d x^{9} + 68023800 \, x^{9} e + 62355150 \, d x^{8} + 166280400 \, x^{8} e + 153489600 \, d x^{7} + 268606800 \, x^{7} e + 249420600 \, d x^{6} + 299304720 \, x^{6} e + 279351072 \, d x^{5} + 232792560 \, x^{5} e + 218243025 \, d x^{4} + 124710300 \, x^{4} e + 117374400 \, d x^{3} + 44015400 \, x^{3} e + 41570100 \, d x^{2} + 9237800 \, x^{2} e + 8751600 \, d x + 875160 \, x e + 831402 \, d}{16628040 \, x^{20}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 130, normalized size = 0.86 \[ -\frac {e}{9 x^{9}}-\frac {d +10 e}{10 x^{10}}-\frac {10 d +45 e}{11 x^{11}}-\frac {45 d +120 e}{12 x^{12}}-\frac {120 d +210 e}{13 x^{13}}-\frac {210 d +252 e}{14 x^{14}}-\frac {252 d +210 e}{15 x^{15}}-\frac {210 d +120 e}{16 x^{16}}-\frac {120 d +45 e}{17 x^{17}}-\frac {45 d +10 e}{18 x^{18}}-\frac {d}{20 x^{20}}-\frac {10 d +e}{19 x^{19}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.60, size = 129, normalized size = 0.85 \[ -\frac {1847560 \, e x^{11} + 1662804 \, {\left (d + 10 \, e\right )} x^{10} + 7558200 \, {\left (2 \, d + 9 \, e\right )} x^{9} + 20785050 \, {\left (3 \, d + 8 \, e\right )} x^{8} + 38372400 \, {\left (4 \, d + 7 \, e\right )} x^{7} + 49884120 \, {\left (5 \, d + 6 \, e\right )} x^{6} + 46558512 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 31177575 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 14671800 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 4618900 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 875160 \, {\left (10 \, d + e\right )} x + 831402 \, d}{16628040 \, x^{20}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.12, size = 121, normalized size = 0.80 \[ -\frac {\frac {e\,x^{11}}{9}+\left (\frac {d}{10}+e\right )\,x^{10}+\left (\frac {10\,d}{11}+\frac {45\,e}{11}\right )\,x^9+\left (\frac {15\,d}{4}+10\,e\right )\,x^8+\left (\frac {120\,d}{13}+\frac {210\,e}{13}\right )\,x^7+\left (15\,d+18\,e\right )\,x^6+\left (\frac {84\,d}{5}+14\,e\right )\,x^5+\left (\frac {105\,d}{8}+\frac {15\,e}{2}\right )\,x^4+\left (\frac {120\,d}{17}+\frac {45\,e}{17}\right )\,x^3+\left (\frac {5\,d}{2}+\frac {5\,e}{9}\right )\,x^2+\left (\frac {10\,d}{19}+\frac {e}{19}\right )\,x+\frac {d}{20}}{x^{20}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 35.33, size = 131, normalized size = 0.87 \[ \frac {- 831402 d - 1847560 e x^{11} + x^{10} \left (- 1662804 d - 16628040 e\right ) + x^{9} \left (- 15116400 d - 68023800 e\right ) + x^{8} \left (- 62355150 d - 166280400 e\right ) + x^{7} \left (- 153489600 d - 268606800 e\right ) + x^{6} \left (- 249420600 d - 299304720 e\right ) + x^{5} \left (- 279351072 d - 232792560 e\right ) + x^{4} \left (- 218243025 d - 124710300 e\right ) + x^{3} \left (- 117374400 d - 44015400 e\right ) + x^{2} \left (- 41570100 d - 9237800 e\right ) + x \left (- 8751600 d - 875160 e\right )}{16628040 x^{20}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________