Optimal. Leaf size=119 \[ \frac {1}{17} (x+1)^{17} (d-7 e)-\frac {3}{16} (x+1)^{16} (2 d-7 e)+\frac {1}{3} (x+1)^{15} (3 d-7 e)-\frac {5}{14} (x+1)^{14} (4 d-7 e)+\frac {3}{13} (x+1)^{13} (5 d-7 e)-\frac {1}{12} (x+1)^{12} (6 d-7 e)+\frac {1}{11} (x+1)^{11} (d-e)+\frac {1}{18} e (x+1)^{18} \]
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Rubi [A] time = 0.07, antiderivative size = 119, 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 {1}{17} (x+1)^{17} (d-7 e)-\frac {3}{16} (x+1)^{16} (2 d-7 e)+\frac {1}{3} (x+1)^{15} (3 d-7 e)-\frac {5}{14} (x+1)^{14} (4 d-7 e)+\frac {3}{13} (x+1)^{13} (5 d-7 e)-\frac {1}{12} (x+1)^{12} (6 d-7 e)+\frac {1}{11} (x+1)^{11} (d-e)+\frac {1}{18} e (x+1)^{18} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 76
Rubi steps
\begin {align*} \int x^6 (d+e x) \left (1+2 x+x^2\right )^5 \, dx &=\int x^6 (1+x)^{10} (d+e x) \, dx\\ &=\int \left ((d-e) (1+x)^{10}+(-6 d+7 e) (1+x)^{11}+3 (5 d-7 e) (1+x)^{12}-5 (4 d-7 e) (1+x)^{13}+5 (3 d-7 e) (1+x)^{14}-3 (2 d-7 e) (1+x)^{15}+(d-7 e) (1+x)^{16}+e (1+x)^{17}\right ) \, dx\\ &=\frac {1}{11} (d-e) (1+x)^{11}-\frac {1}{12} (6 d-7 e) (1+x)^{12}+\frac {3}{13} (5 d-7 e) (1+x)^{13}-\frac {5}{14} (4 d-7 e) (1+x)^{14}+\frac {1}{3} (3 d-7 e) (1+x)^{15}-\frac {3}{16} (2 d-7 e) (1+x)^{16}+\frac {1}{17} (d-7 e) (1+x)^{17}+\frac {1}{18} e (1+x)^{18}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 150, normalized size = 1.26 \begin {gather*} \frac {1}{17} x^{17} (d+10 e)+\frac {5}{16} x^{16} (2 d+9 e)+x^{15} (3 d+8 e)+\frac {15}{7} x^{14} (4 d+7 e)+\frac {42}{13} x^{13} (5 d+6 e)+\frac {7}{2} x^{12} (6 d+5 e)+\frac {30}{11} x^{11} (7 d+4 e)+\frac {3}{2} x^{10} (8 d+3 e)+\frac {5}{9} x^9 (9 d+2 e)+\frac {1}{8} x^8 (10 d+e)+\frac {d x^7}{7}+\frac {e x^{18}}{18} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^6 (d+e x) \left (1+2 x+x^2\right )^5 \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.35, size = 133, normalized size = 1.12 \begin {gather*} \frac {1}{18} x^{18} e + \frac {10}{17} x^{17} e + \frac {1}{17} x^{17} d + \frac {45}{16} x^{16} e + \frac {5}{8} x^{16} d + 8 x^{15} e + 3 x^{15} d + 15 x^{14} e + \frac {60}{7} x^{14} d + \frac {252}{13} x^{13} e + \frac {210}{13} x^{13} d + \frac {35}{2} x^{12} e + 21 x^{12} d + \frac {120}{11} x^{11} e + \frac {210}{11} x^{11} d + \frac {9}{2} x^{10} e + 12 x^{10} d + \frac {10}{9} x^{9} e + 5 x^{9} d + \frac {1}{8} x^{8} e + \frac {5}{4} x^{8} d + \frac {1}{7} x^{7} d \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 144, normalized size = 1.21 \begin {gather*} \frac {1}{18} \, x^{18} e + \frac {1}{17} \, d x^{17} + \frac {10}{17} \, x^{17} e + \frac {5}{8} \, d x^{16} + \frac {45}{16} \, x^{16} e + 3 \, d x^{15} + 8 \, x^{15} e + \frac {60}{7} \, d x^{14} + 15 \, x^{14} e + \frac {210}{13} \, d x^{13} + \frac {252}{13} \, x^{13} e + 21 \, d x^{12} + \frac {35}{2} \, x^{12} e + \frac {210}{11} \, d x^{11} + \frac {120}{11} \, x^{11} e + 12 \, d x^{10} + \frac {9}{2} \, x^{10} e + 5 \, d x^{9} + \frac {10}{9} \, x^{9} e + \frac {5}{4} \, d x^{8} + \frac {1}{8} \, x^{8} e + \frac {1}{7} \, d x^{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 130, normalized size = 1.09 \begin {gather*} \frac {e \,x^{18}}{18}+\frac {\left (d +10 e \right ) x^{17}}{17}+\frac {\left (10 d +45 e \right ) x^{16}}{16}+\frac {\left (45 d +120 e \right ) x^{15}}{15}+\frac {\left (120 d +210 e \right ) x^{14}}{14}+\frac {\left (210 d +252 e \right ) x^{13}}{13}+\frac {\left (252 d +210 e \right ) x^{12}}{12}+\frac {\left (210 d +120 e \right ) x^{11}}{11}+\frac {\left (120 d +45 e \right ) x^{10}}{10}+\frac {\left (45 d +10 e \right ) x^{9}}{9}+\frac {d \,x^{7}}{7}+\frac {\left (10 d +e \right ) x^{8}}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.63, size = 128, normalized size = 1.08 \begin {gather*} \frac {1}{18} \, e x^{18} + \frac {1}{17} \, {\left (d + 10 \, e\right )} x^{17} + \frac {5}{16} \, {\left (2 \, d + 9 \, e\right )} x^{16} + {\left (3 \, d + 8 \, e\right )} x^{15} + \frac {15}{7} \, {\left (4 \, d + 7 \, e\right )} x^{14} + \frac {42}{13} \, {\left (5 \, d + 6 \, e\right )} x^{13} + \frac {7}{2} \, {\left (6 \, d + 5 \, e\right )} x^{12} + \frac {30}{11} \, {\left (7 \, d + 4 \, e\right )} x^{11} + \frac {3}{2} \, {\left (8 \, d + 3 \, e\right )} x^{10} + \frac {5}{9} \, {\left (9 \, d + 2 \, e\right )} x^{9} + \frac {1}{8} \, {\left (10 \, d + e\right )} x^{8} + \frac {1}{7} \, d x^{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 123, normalized size = 1.03 \begin {gather*} \frac {e\,x^{18}}{18}+\left (\frac {d}{17}+\frac {10\,e}{17}\right )\,x^{17}+\left (\frac {5\,d}{8}+\frac {45\,e}{16}\right )\,x^{16}+\left (3\,d+8\,e\right )\,x^{15}+\left (\frac {60\,d}{7}+15\,e\right )\,x^{14}+\left (\frac {210\,d}{13}+\frac {252\,e}{13}\right )\,x^{13}+\left (21\,d+\frac {35\,e}{2}\right )\,x^{12}+\left (\frac {210\,d}{11}+\frac {120\,e}{11}\right )\,x^{11}+\left (12\,d+\frac {9\,e}{2}\right )\,x^{10}+\left (5\,d+\frac {10\,e}{9}\right )\,x^9+\left (\frac {5\,d}{4}+\frac {e}{8}\right )\,x^8+\frac {d\,x^7}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 134, normalized size = 1.13 \begin {gather*} \frac {d x^{7}}{7} + \frac {e x^{18}}{18} + x^{17} \left (\frac {d}{17} + \frac {10 e}{17}\right ) + x^{16} \left (\frac {5 d}{8} + \frac {45 e}{16}\right ) + x^{15} \left (3 d + 8 e\right ) + x^{14} \left (\frac {60 d}{7} + 15 e\right ) + x^{13} \left (\frac {210 d}{13} + \frac {252 e}{13}\right ) + x^{12} \left (21 d + \frac {35 e}{2}\right ) + x^{11} \left (\frac {210 d}{11} + \frac {120 e}{11}\right ) + x^{10} \left (12 d + \frac {9 e}{2}\right ) + x^{9} \left (5 d + \frac {10 e}{9}\right ) + x^{8} \left (\frac {5 d}{4} + \frac {e}{8}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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