Optimal. Leaf size=138 \[ \frac {1}{8} x^8 (d+10 e)+\frac {5}{7} x^7 (2 d+9 e)+\frac {5}{2} x^6 (3 d+8 e)+6 x^5 (4 d+7 e)+\frac {21}{2} x^4 (5 d+6 e)+14 x^3 (6 d+5 e)+15 x^2 (7 d+4 e)+15 x (8 d+3 e)-\frac {10 d+e}{x}+5 (9 d+2 e) \log (x)-\frac {d}{2 x^2}+\frac {e x^9}{9} \]
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Rubi [A] time = 0.07, antiderivative size = 138, 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} \[ \frac {1}{8} x^8 (d+10 e)+\frac {5}{7} x^7 (2 d+9 e)+\frac {5}{2} x^6 (3 d+8 e)+6 x^5 (4 d+7 e)+\frac {21}{2} x^4 (5 d+6 e)+14 x^3 (6 d+5 e)+15 x^2 (7 d+4 e)+15 x (8 d+3 e)-\frac {10 d+e}{x}+5 (9 d+2 e) \log (x)-\frac {d}{2 x^2}+\frac {e x^9}{9} \]
Antiderivative was successfully verified.
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Rule 27
Rule 76
Rubi steps
\begin {align*} \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^3} \, dx &=\int \frac {(1+x)^{10} (d+e x)}{x^3} \, dx\\ &=\int \left (15 (8 d+3 e)+\frac {d}{x^3}+\frac {10 d+e}{x^2}+\frac {5 (9 d+2 e)}{x}+30 (7 d+4 e) x+42 (6 d+5 e) x^2+42 (5 d+6 e) x^3+30 (4 d+7 e) x^4+15 (3 d+8 e) x^5+5 (2 d+9 e) x^6+(d+10 e) x^7+e x^8\right ) \, dx\\ &=-\frac {d}{2 x^2}-\frac {10 d+e}{x}+15 (8 d+3 e) x+15 (7 d+4 e) x^2+14 (6 d+5 e) x^3+\frac {21}{2} (5 d+6 e) x^4+6 (4 d+7 e) x^5+\frac {5}{2} (3 d+8 e) x^6+\frac {5}{7} (2 d+9 e) x^7+\frac {1}{8} (d+10 e) x^8+\frac {e x^9}{9}+5 (9 d+2 e) \log (x)\\ \end {align*}
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Mathematica [A] time = 0.04, size = 139, normalized size = 1.01 \[ \frac {1}{8} x^8 (d+10 e)+\frac {5}{7} x^7 (2 d+9 e)+\frac {5}{2} x^6 (3 d+8 e)+6 x^5 (4 d+7 e)+\frac {21}{2} x^4 (5 d+6 e)+14 x^3 (6 d+5 e)+15 x^2 (7 d+4 e)+15 x (8 d+3 e)+\frac {-10 d-e}{x}+5 (9 d+2 e) \log (x)-\frac {d}{2 x^2}+\frac {e x^9}{9} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 131, normalized size = 0.95 \[ \frac {56 \, e x^{11} + 63 \, {\left (d + 10 \, e\right )} x^{10} + 360 \, {\left (2 \, d + 9 \, e\right )} x^{9} + 1260 \, {\left (3 \, d + 8 \, e\right )} x^{8} + 3024 \, {\left (4 \, d + 7 \, e\right )} x^{7} + 5292 \, {\left (5 \, d + 6 \, e\right )} x^{6} + 7056 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 7560 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 7560 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 2520 \, {\left (9 \, d + 2 \, e\right )} x^{2} \log \relax (x) - 504 \, {\left (10 \, d + e\right )} x - 252 \, d}{504 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 137, normalized size = 0.99 \[ \frac {1}{9} \, x^{9} e + \frac {1}{8} \, d x^{8} + \frac {5}{4} \, x^{8} e + \frac {10}{7} \, d x^{7} + \frac {45}{7} \, x^{7} e + \frac {15}{2} \, d x^{6} + 20 \, x^{6} e + 24 \, d x^{5} + 42 \, x^{5} e + \frac {105}{2} \, d x^{4} + 63 \, x^{4} e + 84 \, d x^{3} + 70 \, x^{3} e + 105 \, d x^{2} + 60 \, x^{2} e + 120 \, d x + 45 \, x e + 5 \, {\left (9 \, d + 2 \, e\right )} \log \left ({\left | x \right |}\right ) - \frac {2 \, {\left (10 \, d + e\right )} x + d}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 128, normalized size = 0.93 \[ \frac {e \,x^{9}}{9}+\frac {d \,x^{8}}{8}+\frac {5 e \,x^{8}}{4}+\frac {10 d \,x^{7}}{7}+\frac {45 e \,x^{7}}{7}+\frac {15 d \,x^{6}}{2}+20 e \,x^{6}+24 d \,x^{5}+42 e \,x^{5}+\frac {105 d \,x^{4}}{2}+63 e \,x^{4}+84 d \,x^{3}+70 e \,x^{3}+105 d \,x^{2}+60 e \,x^{2}+120 d x +45 d \ln \relax (x )+45 e x +10 e \ln \relax (x )-\frac {10 d}{x}-\frac {e}{x}-\frac {d}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 125, normalized size = 0.91 \[ \frac {1}{9} \, e x^{9} + \frac {1}{8} \, {\left (d + 10 \, e\right )} x^{8} + \frac {5}{7} \, {\left (2 \, d + 9 \, e\right )} x^{7} + \frac {5}{2} \, {\left (3 \, d + 8 \, e\right )} x^{6} + 6 \, {\left (4 \, d + 7 \, e\right )} x^{5} + \frac {21}{2} \, {\left (5 \, d + 6 \, e\right )} x^{4} + 14 \, {\left (6 \, d + 5 \, e\right )} x^{3} + 15 \, {\left (7 \, d + 4 \, e\right )} x^{2} + 15 \, {\left (8 \, d + 3 \, e\right )} x + 5 \, {\left (9 \, d + 2 \, e\right )} \log \relax (x) - \frac {2 \, {\left (10 \, d + e\right )} x + d}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 119, normalized size = 0.86 \[ x^8\,\left (\frac {d}{8}+\frac {5\,e}{4}\right )+x^6\,\left (\frac {15\,d}{2}+20\,e\right )+x^5\,\left (24\,d+42\,e\right )+x^7\,\left (\frac {10\,d}{7}+\frac {45\,e}{7}\right )+x^3\,\left (84\,d+70\,e\right )+x^2\,\left (105\,d+60\,e\right )+x^4\,\left (\frac {105\,d}{2}+63\,e\right )+\ln \relax (x)\,\left (45\,d+10\,e\right )+\frac {e\,x^9}{9}-\frac {\frac {d}{2}+x\,\left (10\,d+e\right )}{x^2}+x\,\left (120\,d+45\,e\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.45, size = 122, normalized size = 0.88 \[ \frac {e x^{9}}{9} + x^{8} \left (\frac {d}{8} + \frac {5 e}{4}\right ) + x^{7} \left (\frac {10 d}{7} + \frac {45 e}{7}\right ) + x^{6} \left (\frac {15 d}{2} + 20 e\right ) + x^{5} \left (24 d + 42 e\right ) + x^{4} \left (\frac {105 d}{2} + 63 e\right ) + x^{3} \left (84 d + 70 e\right ) + x^{2} \left (105 d + 60 e\right ) + x \left (120 d + 45 e\right ) + 5 \left (9 d + 2 e\right ) \log {\relax (x )} + \frac {- d + x \left (- 20 d - 2 e\right )}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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