Optimal. Leaf size=87 \[ \frac {d x^{10}}{10}+\frac {10 d x^9}{9}+\frac {45 d x^8}{8}+\frac {120 d x^7}{7}+35 d x^6+\frac {252 d x^5}{5}+\frac {105 d x^4}{2}+40 d x^3+\frac {45 d x^2}{2}+10 d x+d \log (x)+\frac {1}{11} e (x+1)^{11} \]
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Rubi [A] time = 0.02, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {27, 80, 43} \[ \frac {d x^{10}}{10}+\frac {10 d x^9}{9}+\frac {45 d x^8}{8}+\frac {120 d x^7}{7}+35 d x^6+\frac {252 d x^5}{5}+\frac {105 d x^4}{2}+40 d x^3+\frac {45 d x^2}{2}+10 d x+d \log (x)+\frac {1}{11} e (x+1)^{11} \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rule 80
Rubi steps
\begin {align*} \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x} \, dx &=\int \frac {(1+x)^{10} (d+e x)}{x} \, dx\\ &=\frac {1}{11} e (1+x)^{11}+d \int \frac {(1+x)^{10}}{x} \, dx\\ &=\frac {1}{11} e (1+x)^{11}+d \int \left (10+\frac {1}{x}+45 x+120 x^2+210 x^3+252 x^4+210 x^5+120 x^6+45 x^7+10 x^8+x^9\right ) \, dx\\ &=10 d x+\frac {45 d x^2}{2}+40 d x^3+\frac {105 d x^4}{2}+\frac {252 d x^5}{5}+35 d x^6+\frac {120 d x^7}{7}+\frac {45 d x^8}{8}+\frac {10 d x^9}{9}+\frac {d x^{10}}{10}+\frac {1}{11} e (1+x)^{11}+d \log (x)\\ \end {align*}
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Mathematica [A] time = 0.03, size = 85, normalized size = 0.98 \[ d \left (\frac {x^{10}}{10}+\frac {10 x^9}{9}+\frac {45 x^8}{8}+\frac {120 x^7}{7}+35 x^6+\frac {252 x^5}{5}+\frac {105 x^4}{2}+40 x^3+\frac {45 x^2}{2}+10 x+\frac {7381}{2520}\right )+d \log (-x)+\frac {1}{11} e (x+1)^{11} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.10, size = 124, normalized size = 1.43 \[ \frac {1}{11} \, e x^{11} + \frac {1}{10} \, {\left (d + 10 \, e\right )} x^{10} + \frac {5}{9} \, {\left (2 \, d + 9 \, e\right )} x^{9} + \frac {15}{8} \, {\left (3 \, d + 8 \, e\right )} x^{8} + \frac {30}{7} \, {\left (4 \, d + 7 \, e\right )} x^{7} + 7 \, {\left (5 \, d + 6 \, e\right )} x^{6} + \frac {42}{5} \, {\left (6 \, d + 5 \, e\right )} x^{5} + \frac {15}{2} \, {\left (7 \, d + 4 \, e\right )} x^{4} + 5 \, {\left (8 \, d + 3 \, e\right )} x^{3} + \frac {5}{2} \, {\left (9 \, d + 2 \, e\right )} x^{2} + {\left (10 \, d + e\right )} x + d \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 137, normalized size = 1.57 \[ \frac {1}{11} \, x^{11} e + \frac {1}{10} \, d x^{10} + x^{10} e + \frac {10}{9} \, d x^{9} + 5 \, x^{9} e + \frac {45}{8} \, d x^{8} + 15 \, x^{8} e + \frac {120}{7} \, d x^{7} + 30 \, x^{7} e + 35 \, d x^{6} + 42 \, x^{6} e + \frac {252}{5} \, d x^{5} + 42 \, x^{5} e + \frac {105}{2} \, d x^{4} + 30 \, x^{4} e + 40 \, d x^{3} + 15 \, x^{3} e + \frac {45}{2} \, d x^{2} + 5 \, x^{2} e + 10 \, d x + x e + d \log \left ({\left | x \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 126, normalized size = 1.45 \[ \frac {e \,x^{11}}{11}+\frac {d \,x^{10}}{10}+e \,x^{10}+\frac {10 d \,x^{9}}{9}+5 e \,x^{9}+\frac {45 d \,x^{8}}{8}+15 e \,x^{8}+\frac {120 d \,x^{7}}{7}+30 e \,x^{7}+35 d \,x^{6}+42 e \,x^{6}+\frac {252 d \,x^{5}}{5}+42 e \,x^{5}+\frac {105 d \,x^{4}}{2}+30 e \,x^{4}+40 d \,x^{3}+15 e \,x^{3}+\frac {45 d \,x^{2}}{2}+5 e \,x^{2}+10 d x +d \ln \relax (x )+e x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.63, size = 124, normalized size = 1.43 \[ \frac {1}{11} \, e x^{11} + \frac {1}{10} \, {\left (d + 10 \, e\right )} x^{10} + \frac {5}{9} \, {\left (2 \, d + 9 \, e\right )} x^{9} + \frac {15}{8} \, {\left (3 \, d + 8 \, e\right )} x^{8} + \frac {30}{7} \, {\left (4 \, d + 7 \, e\right )} x^{7} + 7 \, {\left (5 \, d + 6 \, e\right )} x^{6} + \frac {42}{5} \, {\left (6 \, d + 5 \, e\right )} x^{5} + \frac {15}{2} \, {\left (7 \, d + 4 \, e\right )} x^{4} + 5 \, {\left (8 \, d + 3 \, e\right )} x^{3} + \frac {5}{2} \, {\left (9 \, d + 2 \, e\right )} x^{2} + {\left (10 \, d + e\right )} x + d \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 115, normalized size = 1.32 \[ x^9\,\left (\frac {10\,d}{9}+5\,e\right )+x^2\,\left (\frac {45\,d}{2}+5\,e\right )+x^3\,\left (40\,d+15\,e\right )+x^8\,\left (\frac {45\,d}{8}+15\,e\right )+x^6\,\left (35\,d+42\,e\right )+x^4\,\left (\frac {105\,d}{2}+30\,e\right )+x^7\,\left (\frac {120\,d}{7}+30\,e\right )+x^5\,\left (\frac {252\,d}{5}+42\,e\right )+x\,\left (10\,d+e\right )+\frac {e\,x^{11}}{11}+d\,\ln \relax (x)+x^{10}\,\left (\frac {d}{10}+e\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.31, size = 117, normalized size = 1.34 \[ d \log {\relax (x )} + \frac {e x^{11}}{11} + x^{10} \left (\frac {d}{10} + e\right ) + x^{9} \left (\frac {10 d}{9} + 5 e\right ) + x^{8} \left (\frac {45 d}{8} + 15 e\right ) + x^{7} \left (\frac {120 d}{7} + 30 e\right ) + x^{6} \left (35 d + 42 e\right ) + x^{5} \left (\frac {252 d}{5} + 42 e\right ) + x^{4} \left (\frac {105 d}{2} + 30 e\right ) + x^{3} \left (40 d + 15 e\right ) + x^{2} \left (\frac {45 d}{2} + 5 e\right ) + x \left (10 d + e\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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