Optimal. Leaf size=131 \[ \frac {1}{18} (x+1)^{18} (d-8 e)-\frac {7}{17} (x+1)^{17} (d-4 e)+\frac {7}{16} (x+1)^{16} (3 d-8 e)-\frac {7}{3} (x+1)^{15} (d-2 e)+\frac {1}{2} (x+1)^{14} (5 d-8 e)-\frac {7}{13} (x+1)^{13} (3 d-4 e)+\frac {1}{12} (x+1)^{12} (7 d-8 e)-\frac {1}{11} (x+1)^{11} (d-e)+\frac {1}{19} e (x+1)^{19} \]
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Rubi [A] time = 0.13, antiderivative size = 131, 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}{18} (x+1)^{18} (d-8 e)-\frac {7}{17} (x+1)^{17} (d-4 e)+\frac {7}{16} (x+1)^{16} (3 d-8 e)-\frac {7}{3} (x+1)^{15} (d-2 e)+\frac {1}{2} (x+1)^{14} (5 d-8 e)-\frac {7}{13} (x+1)^{13} (3 d-4 e)+\frac {1}{12} (x+1)^{12} (7 d-8 e)-\frac {1}{11} (x+1)^{11} (d-e)+\frac {1}{19} e (x+1)^{19} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 76
Rubi steps
\begin {align*} \int x^7 (d+e x) \left (1+2 x+x^2\right )^5 \, dx &=\int x^7 (1+x)^{10} (d+e x) \, dx\\ &=\int \left ((-d+e) (1+x)^{10}+(7 d-8 e) (1+x)^{11}-7 (3 d-4 e) (1+x)^{12}+7 (5 d-8 e) (1+x)^{13}-35 (d-2 e) (1+x)^{14}+7 (3 d-8 e) (1+x)^{15}-7 (d-4 e) (1+x)^{16}+(d-8 e) (1+x)^{17}+e (1+x)^{18}\right ) \, dx\\ &=-\frac {1}{11} (d-e) (1+x)^{11}+\frac {1}{12} (7 d-8 e) (1+x)^{12}-\frac {7}{13} (3 d-4 e) (1+x)^{13}+\frac {1}{2} (5 d-8 e) (1+x)^{14}-\frac {7}{3} (d-2 e) (1+x)^{15}+\frac {7}{16} (3 d-8 e) (1+x)^{16}-\frac {7}{17} (d-4 e) (1+x)^{17}+\frac {1}{18} (d-8 e) (1+x)^{18}+\frac {1}{19} e (1+x)^{19}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 149, normalized size = 1.14 \begin {gather*} \frac {1}{18} x^{18} (d+10 e)+\frac {5}{17} x^{17} (2 d+9 e)+\frac {15}{16} x^{16} (3 d+8 e)+2 x^{15} (4 d+7 e)+3 x^{14} (5 d+6 e)+\frac {42}{13} x^{13} (6 d+5 e)+\frac {5}{2} x^{12} (7 d+4 e)+\frac {15}{11} x^{11} (8 d+3 e)+\frac {1}{2} x^{10} (9 d+2 e)+\frac {1}{9} x^9 (10 d+e)+\frac {d x^8}{8}+\frac {e x^{19}}{19} \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^7 (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 = 0.37, size = 132, normalized size = 1.01 \begin {gather*} \frac {1}{19} x^{19} e + \frac {5}{9} x^{18} e + \frac {1}{18} x^{18} d + \frac {45}{17} x^{17} e + \frac {10}{17} x^{17} d + \frac {15}{2} x^{16} e + \frac {45}{16} x^{16} d + 14 x^{15} e + 8 x^{15} d + 18 x^{14} e + 15 x^{14} d + \frac {210}{13} x^{13} e + \frac {252}{13} x^{13} d + 10 x^{12} e + \frac {35}{2} x^{12} d + \frac {45}{11} x^{11} e + \frac {120}{11} x^{11} d + x^{10} e + \frac {9}{2} x^{10} d + \frac {1}{9} x^{9} e + \frac {10}{9} x^{9} d + \frac {1}{8} x^{8} d \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 143, normalized size = 1.09 \begin {gather*} \frac {1}{19} \, x^{19} e + \frac {1}{18} \, d x^{18} + \frac {5}{9} \, x^{18} e + \frac {10}{17} \, d x^{17} + \frac {45}{17} \, x^{17} e + \frac {45}{16} \, d x^{16} + \frac {15}{2} \, x^{16} e + 8 \, d x^{15} + 14 \, x^{15} e + 15 \, d x^{14} + 18 \, x^{14} e + \frac {252}{13} \, d x^{13} + \frac {210}{13} \, x^{13} e + \frac {35}{2} \, d x^{12} + 10 \, x^{12} e + \frac {120}{11} \, d x^{11} + \frac {45}{11} \, x^{11} e + \frac {9}{2} \, d x^{10} + x^{10} e + \frac {10}{9} \, d x^{9} + \frac {1}{9} \, x^{9} e + \frac {1}{8} \, d x^{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 130, normalized size = 0.99 \begin {gather*} \frac {e \,x^{19}}{19}+\frac {\left (d +10 e \right ) x^{18}}{18}+\frac {\left (10 d +45 e \right ) x^{17}}{17}+\frac {\left (45 d +120 e \right ) x^{16}}{16}+\frac {\left (120 d +210 e \right ) x^{15}}{15}+\frac {\left (210 d +252 e \right ) x^{14}}{14}+\frac {\left (252 d +210 e \right ) x^{13}}{13}+\frac {\left (210 d +120 e \right ) x^{12}}{12}+\frac {\left (120 d +45 e \right ) x^{11}}{11}+\frac {\left (45 d +10 e \right ) x^{10}}{10}+\frac {d \,x^{8}}{8}+\frac {\left (10 d +e \right ) x^{9}}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 129, normalized size = 0.98 \begin {gather*} \frac {1}{19} \, e x^{19} + \frac {1}{18} \, {\left (d + 10 \, e\right )} x^{18} + \frac {5}{17} \, {\left (2 \, d + 9 \, e\right )} x^{17} + \frac {15}{16} \, {\left (3 \, d + 8 \, e\right )} x^{16} + 2 \, {\left (4 \, d + 7 \, e\right )} x^{15} + 3 \, {\left (5 \, d + 6 \, e\right )} x^{14} + \frac {42}{13} \, {\left (6 \, d + 5 \, e\right )} x^{13} + \frac {5}{2} \, {\left (7 \, d + 4 \, e\right )} x^{12} + \frac {15}{11} \, {\left (8 \, d + 3 \, e\right )} x^{11} + \frac {1}{2} \, {\left (9 \, d + 2 \, e\right )} x^{10} + \frac {1}{9} \, {\left (10 \, d + e\right )} x^{9} + \frac {1}{8} \, d x^{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.12, size = 121, normalized size = 0.92 \begin {gather*} \frac {e\,x^{19}}{19}+\left (\frac {d}{18}+\frac {5\,e}{9}\right )\,x^{18}+\left (\frac {10\,d}{17}+\frac {45\,e}{17}\right )\,x^{17}+\left (\frac {45\,d}{16}+\frac {15\,e}{2}\right )\,x^{16}+\left (8\,d+14\,e\right )\,x^{15}+\left (15\,d+18\,e\right )\,x^{14}+\left (\frac {252\,d}{13}+\frac {210\,e}{13}\right )\,x^{13}+\left (\frac {35\,d}{2}+10\,e\right )\,x^{12}+\left (\frac {120\,d}{11}+\frac {45\,e}{11}\right )\,x^{11}+\left (\frac {9\,d}{2}+e\right )\,x^{10}+\left (\frac {10\,d}{9}+\frac {e}{9}\right )\,x^9+\frac {d\,x^8}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 133, normalized size = 1.02 \begin {gather*} \frac {d x^{8}}{8} + \frac {e x^{19}}{19} + x^{18} \left (\frac {d}{18} + \frac {5 e}{9}\right ) + x^{17} \left (\frac {10 d}{17} + \frac {45 e}{17}\right ) + x^{16} \left (\frac {45 d}{16} + \frac {15 e}{2}\right ) + x^{15} \left (8 d + 14 e\right ) + x^{14} \left (15 d + 18 e\right ) + x^{13} \left (\frac {252 d}{13} + \frac {210 e}{13}\right ) + x^{12} \left (\frac {35 d}{2} + 10 e\right ) + x^{11} \left (\frac {120 d}{11} + \frac {45 e}{11}\right ) + x^{10} \left (\frac {9 d}{2} + e\right ) + x^{9} \left (\frac {10 d}{9} + \frac {e}{9}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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