Optimal. Leaf size=151 \[ -\frac {10 d+e}{17 x^{17}}-\frac {5 (9 d+2 e)}{16 x^{16}}-\frac {8 d+3 e}{x^{15}}-\frac {15 (7 d+4 e)}{7 x^{14}}-\frac {42 (6 d+5 e)}{13 x^{13}}-\frac {7 (5 d+6 e)}{2 x^{12}}-\frac {30 (4 d+7 e)}{11 x^{11}}-\frac {3 (3 d+8 e)}{2 x^{10}}-\frac {5 (2 d+9 e)}{9 x^9}-\frac {d+10 e}{8 x^8}-\frac {d}{18 x^{18}}-\frac {e}{7 x^7} \]
________________________________________________________________________________________
Rubi [A] time = 0.07, 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} \begin {gather*} -\frac {d+10 e}{8 x^8}-\frac {5 (2 d+9 e)}{9 x^9}-\frac {3 (3 d+8 e)}{2 x^{10}}-\frac {30 (4 d+7 e)}{11 x^{11}}-\frac {7 (5 d+6 e)}{2 x^{12}}-\frac {42 (6 d+5 e)}{13 x^{13}}-\frac {15 (7 d+4 e)}{7 x^{14}}-\frac {8 d+3 e}{x^{15}}-\frac {5 (9 d+2 e)}{16 x^{16}}-\frac {10 d+e}{17 x^{17}}-\frac {d}{18 x^{18}}-\frac {e}{7 x^7} \end {gather*}
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^{19}} \, dx &=\int \frac {(1+x)^{10} (d+e x)}{x^{19}} \, dx\\ &=\int \left (\frac {d}{x^{19}}+\frac {10 d+e}{x^{18}}+\frac {5 (9 d+2 e)}{x^{17}}+\frac {15 (8 d+3 e)}{x^{16}}+\frac {30 (7 d+4 e)}{x^{15}}+\frac {42 (6 d+5 e)}{x^{14}}+\frac {42 (5 d+6 e)}{x^{13}}+\frac {30 (4 d+7 e)}{x^{12}}+\frac {15 (3 d+8 e)}{x^{11}}+\frac {5 (2 d+9 e)}{x^{10}}+\frac {d+10 e}{x^9}+\frac {e}{x^8}\right ) \, dx\\ &=-\frac {d}{18 x^{18}}-\frac {10 d+e}{17 x^{17}}-\frac {5 (9 d+2 e)}{16 x^{16}}-\frac {8 d+3 e}{x^{15}}-\frac {15 (7 d+4 e)}{7 x^{14}}-\frac {42 (6 d+5 e)}{13 x^{13}}-\frac {7 (5 d+6 e)}{2 x^{12}}-\frac {30 (4 d+7 e)}{11 x^{11}}-\frac {3 (3 d+8 e)}{2 x^{10}}-\frac {5 (2 d+9 e)}{9 x^9}-\frac {d+10 e}{8 x^8}-\frac {e}{7 x^7}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 151, normalized size = 1.00 \begin {gather*} -\frac {10 d+e}{17 x^{17}}-\frac {5 (9 d+2 e)}{16 x^{16}}-\frac {8 d+3 e}{x^{15}}-\frac {15 (7 d+4 e)}{7 x^{14}}-\frac {42 (6 d+5 e)}{13 x^{13}}-\frac {7 (5 d+6 e)}{2 x^{12}}-\frac {30 (4 d+7 e)}{11 x^{11}}-\frac {3 (3 d+8 e)}{2 x^{10}}-\frac {5 (2 d+9 e)}{9 x^9}-\frac {d+10 e}{8 x^8}-\frac {d}{18 x^{18}}-\frac {e}{7 x^7} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
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^{19}} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.38, size = 129, normalized size = 0.85 \begin {gather*} -\frac {350064 \, e x^{11} + 306306 \, {\left (d + 10 \, e\right )} x^{10} + 1361360 \, {\left (2 \, d + 9 \, e\right )} x^{9} + 3675672 \, {\left (3 \, d + 8 \, e\right )} x^{8} + 6683040 \, {\left (4 \, d + 7 \, e\right )} x^{7} + 8576568 \, {\left (5 \, d + 6 \, e\right )} x^{6} + 7916832 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 5250960 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 2450448 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 765765 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 144144 \, {\left (10 \, d + e\right )} x + 136136 \, d}{2450448 \, x^{18}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.16, size = 142, normalized size = 0.94 \begin {gather*} -\frac {350064 \, x^{11} e + 306306 \, d x^{10} + 3063060 \, x^{10} e + 2722720 \, d x^{9} + 12252240 \, x^{9} e + 11027016 \, d x^{8} + 29405376 \, x^{8} e + 26732160 \, d x^{7} + 46781280 \, x^{7} e + 42882840 \, d x^{6} + 51459408 \, x^{6} e + 47500992 \, d x^{5} + 39584160 \, x^{5} e + 36756720 \, d x^{4} + 21003840 \, x^{4} e + 19603584 \, d x^{3} + 7351344 \, x^{3} e + 6891885 \, d x^{2} + 1531530 \, x^{2} e + 1441440 \, d x + 144144 \, x e + 136136 \, d}{2450448 \, x^{18}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 130, normalized size = 0.86 \begin {gather*} -\frac {e}{7 x^{7}}-\frac {d +10 e}{8 x^{8}}-\frac {10 d +45 e}{9 x^{9}}-\frac {45 d +120 e}{10 x^{10}}-\frac {120 d +210 e}{11 x^{11}}-\frac {210 d +252 e}{12 x^{12}}-\frac {252 d +210 e}{13 x^{13}}-\frac {210 d +120 e}{14 x^{14}}-\frac {120 d +45 e}{15 x^{15}}-\frac {45 d +10 e}{16 x^{16}}-\frac {d}{18 x^{18}}-\frac {10 d +e}{17 x^{17}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.57, size = 129, normalized size = 0.85 \begin {gather*} -\frac {350064 \, e x^{11} + 306306 \, {\left (d + 10 \, e\right )} x^{10} + 1361360 \, {\left (2 \, d + 9 \, e\right )} x^{9} + 3675672 \, {\left (3 \, d + 8 \, e\right )} x^{8} + 6683040 \, {\left (4 \, d + 7 \, e\right )} x^{7} + 8576568 \, {\left (5 \, d + 6 \, e\right )} x^{6} + 7916832 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 5250960 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 2450448 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 765765 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 144144 \, {\left (10 \, d + e\right )} x + 136136 \, d}{2450448 \, x^{18}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.12, size = 123, normalized size = 0.81 \begin {gather*} -\frac {\frac {e\,x^{11}}{7}+\left (\frac {d}{8}+\frac {5\,e}{4}\right )\,x^{10}+\left (\frac {10\,d}{9}+5\,e\right )\,x^9+\left (\frac {9\,d}{2}+12\,e\right )\,x^8+\left (\frac {120\,d}{11}+\frac {210\,e}{11}\right )\,x^7+\left (\frac {35\,d}{2}+21\,e\right )\,x^6+\left (\frac {252\,d}{13}+\frac {210\,e}{13}\right )\,x^5+\left (15\,d+\frac {60\,e}{7}\right )\,x^4+\left (8\,d+3\,e\right )\,x^3+\left (\frac {45\,d}{16}+\frac {5\,e}{8}\right )\,x^2+\left (\frac {10\,d}{17}+\frac {e}{17}\right )\,x+\frac {d}{18}}{x^{18}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 27.79, size = 131, normalized size = 0.87 \begin {gather*} \frac {- 136136 d - 350064 e x^{11} + x^{10} \left (- 306306 d - 3063060 e\right ) + x^{9} \left (- 2722720 d - 12252240 e\right ) + x^{8} \left (- 11027016 d - 29405376 e\right ) + x^{7} \left (- 26732160 d - 46781280 e\right ) + x^{6} \left (- 42882840 d - 51459408 e\right ) + x^{5} \left (- 47500992 d - 39584160 e\right ) + x^{4} \left (- 36756720 d - 21003840 e\right ) + x^{3} \left (- 19603584 d - 7351344 e\right ) + x^{2} \left (- 6891885 d - 1531530 e\right ) + x \left (- 1441440 d - 144144 e\right )}{2450448 x^{18}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________