Optimal. Leaf size=109 \[ \frac{(x+1)^{11} (5 d-16 e)}{240240 x^{11}}-\frac{(x+1)^{11} (5 d-16 e)}{21840 x^{12}}+\frac{(x+1)^{11} (5 d-16 e)}{3640 x^{13}}-\frac{(x+1)^{11} (5 d-16 e)}{840 x^{14}}+\frac{(x+1)^{11} (5 d-16 e)}{240 x^{15}}-\frac{d (x+1)^{11}}{16 x^{16}} \]
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Rubi [A] time = 0.0274371, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {27, 78, 45, 37} \[ \frac{(x+1)^{11} (5 d-16 e)}{240240 x^{11}}-\frac{(x+1)^{11} (5 d-16 e)}{21840 x^{12}}+\frac{(x+1)^{11} (5 d-16 e)}{3640 x^{13}}-\frac{(x+1)^{11} (5 d-16 e)}{840 x^{14}}+\frac{(x+1)^{11} (5 d-16 e)}{240 x^{15}}-\frac{d (x+1)^{11}}{16 x^{16}} \]
Antiderivative was successfully verified.
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Rule 27
Rule 78
Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{(d+e x) \left (1+2 x+x^2\right )^5}{x^{17}} \, dx &=\int \frac{(1+x)^{10} (d+e x)}{x^{17}} \, dx\\ &=-\frac{d (1+x)^{11}}{16 x^{16}}-\frac{1}{16} (5 d-16 e) \int \frac{(1+x)^{10}}{x^{16}} \, dx\\ &=-\frac{d (1+x)^{11}}{16 x^{16}}+\frac{(5 d-16 e) (1+x)^{11}}{240 x^{15}}-\frac{1}{60} (-5 d+16 e) \int \frac{(1+x)^{10}}{x^{15}} \, dx\\ &=-\frac{d (1+x)^{11}}{16 x^{16}}+\frac{(5 d-16 e) (1+x)^{11}}{240 x^{15}}-\frac{(5 d-16 e) (1+x)^{11}}{840 x^{14}}-\frac{1}{280} (5 d-16 e) \int \frac{(1+x)^{10}}{x^{14}} \, dx\\ &=-\frac{d (1+x)^{11}}{16 x^{16}}+\frac{(5 d-16 e) (1+x)^{11}}{240 x^{15}}-\frac{(5 d-16 e) (1+x)^{11}}{840 x^{14}}+\frac{(5 d-16 e) (1+x)^{11}}{3640 x^{13}}-\frac{(-5 d+16 e) \int \frac{(1+x)^{10}}{x^{13}} \, dx}{1820}\\ &=-\frac{d (1+x)^{11}}{16 x^{16}}+\frac{(5 d-16 e) (1+x)^{11}}{240 x^{15}}-\frac{(5 d-16 e) (1+x)^{11}}{840 x^{14}}+\frac{(5 d-16 e) (1+x)^{11}}{3640 x^{13}}-\frac{(5 d-16 e) (1+x)^{11}}{21840 x^{12}}-\frac{(5 d-16 e) \int \frac{(1+x)^{10}}{x^{12}} \, dx}{21840}\\ &=-\frac{d (1+x)^{11}}{16 x^{16}}+\frac{(5 d-16 e) (1+x)^{11}}{240 x^{15}}-\frac{(5 d-16 e) (1+x)^{11}}{840 x^{14}}+\frac{(5 d-16 e) (1+x)^{11}}{3640 x^{13}}-\frac{(5 d-16 e) (1+x)^{11}}{21840 x^{12}}+\frac{(5 d-16 e) (1+x)^{11}}{240240 x^{11}}\\ \end{align*}
Mathematica [A] time = 0.0352043, size = 153, normalized size = 1.4 \[ -\frac{d+10 e}{6 x^6}-\frac{5 (2 d+9 e)}{7 x^7}-\frac{15 (3 d+8 e)}{8 x^8}-\frac{10 (4 d+7 e)}{3 x^9}-\frac{21 (5 d+6 e)}{5 x^{10}}-\frac{42 (6 d+5 e)}{11 x^{11}}-\frac{5 (7 d+4 e)}{2 x^{12}}-\frac{15 (8 d+3 e)}{13 x^{13}}-\frac{5 (9 d+2 e)}{14 x^{14}}-\frac{10 d+e}{15 x^{15}}-\frac{d}{16 x^{16}}-\frac{e}{5 x^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 130, normalized size = 1.2 \begin{align*} -{\frac{210\,d+120\,e}{12\,{x}^{12}}}-{\frac{120\,d+210\,e}{9\,{x}^{9}}}-{\frac{d}{16\,{x}^{16}}}-{\frac{e}{5\,{x}^{5}}}-{\frac{45\,d+10\,e}{14\,{x}^{14}}}-{\frac{d+10\,e}{6\,{x}^{6}}}-{\frac{120\,d+45\,e}{13\,{x}^{13}}}-{\frac{10\,d+45\,e}{7\,{x}^{7}}}-{\frac{10\,d+e}{15\,{x}^{15}}}-{\frac{45\,d+120\,e}{8\,{x}^{8}}}-{\frac{252\,d+210\,e}{11\,{x}^{11}}}-{\frac{210\,d+252\,e}{10\,{x}^{10}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.977521, size = 174, normalized size = 1.6 \begin{align*} -\frac{48048 \, e x^{11} + 40040 \,{\left (d + 10 \, e\right )} x^{10} + 171600 \,{\left (2 \, d + 9 \, e\right )} x^{9} + 450450 \,{\left (3 \, d + 8 \, e\right )} x^{8} + 800800 \,{\left (4 \, d + 7 \, e\right )} x^{7} + 1009008 \,{\left (5 \, d + 6 \, e\right )} x^{6} + 917280 \,{\left (6 \, d + 5 \, e\right )} x^{5} + 600600 \,{\left (7 \, d + 4 \, e\right )} x^{4} + 277200 \,{\left (8 \, d + 3 \, e\right )} x^{3} + 85800 \,{\left (9 \, d + 2 \, e\right )} x^{2} + 16016 \,{\left (10 \, d + e\right )} x + 15015 \, d}{240240 \, x^{16}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.26997, size = 386, normalized size = 3.54 \begin{align*} -\frac{48048 \, e x^{11} + 40040 \,{\left (d + 10 \, e\right )} x^{10} + 171600 \,{\left (2 \, d + 9 \, e\right )} x^{9} + 450450 \,{\left (3 \, d + 8 \, e\right )} x^{8} + 800800 \,{\left (4 \, d + 7 \, e\right )} x^{7} + 1009008 \,{\left (5 \, d + 6 \, e\right )} x^{6} + 917280 \,{\left (6 \, d + 5 \, e\right )} x^{5} + 600600 \,{\left (7 \, d + 4 \, e\right )} x^{4} + 277200 \,{\left (8 \, d + 3 \, e\right )} x^{3} + 85800 \,{\left (9 \, d + 2 \, e\right )} x^{2} + 16016 \,{\left (10 \, d + e\right )} x + 15015 \, d}{240240 \, x^{16}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 22.071, size = 116, normalized size = 1.06 \begin{align*} - \frac{15015 d + 48048 e x^{11} + x^{10} \left (40040 d + 400400 e\right ) + x^{9} \left (343200 d + 1544400 e\right ) + x^{8} \left (1351350 d + 3603600 e\right ) + x^{7} \left (3203200 d + 5605600 e\right ) + x^{6} \left (5045040 d + 6054048 e\right ) + x^{5} \left (5503680 d + 4586400 e\right ) + x^{4} \left (4204200 d + 2402400 e\right ) + x^{3} \left (2217600 d + 831600 e\right ) + x^{2} \left (772200 d + 171600 e\right ) + x \left (160160 d + 16016 e\right )}{240240 x^{16}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12987, size = 192, normalized size = 1.76 \begin{align*} -\frac{48048 \, x^{11} e + 40040 \, d x^{10} + 400400 \, x^{10} e + 343200 \, d x^{9} + 1544400 \, x^{9} e + 1351350 \, d x^{8} + 3603600 \, x^{8} e + 3203200 \, d x^{7} + 5605600 \, x^{7} e + 5045040 \, d x^{6} + 6054048 \, x^{6} e + 5503680 \, d x^{5} + 4586400 \, x^{5} e + 4204200 \, d x^{4} + 2402400 \, x^{4} e + 2217600 \, d x^{3} + 831600 \, x^{3} e + 772200 \, d x^{2} + 171600 \, x^{2} e + 160160 \, d x + 16016 \, x e + 15015 \, d}{240240 \, x^{16}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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