Optimal. Leaf size=99 \[ \frac {1}{16} (x+1)^{16} (d-6 e)-\frac {1}{3} (x+1)^{15} (d-3 e)+\frac {5}{7} (x+1)^{14} (d-2 e)-\frac {5}{13} (x+1)^{13} (2 d-3 e)+\frac {1}{12} (x+1)^{12} (5 d-6 e)-\frac {1}{11} (x+1)^{11} (d-e)+\frac {1}{17} e (x+1)^{17} \]
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Rubi [A] time = 0.06, antiderivative size = 99, 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}{16} (x+1)^{16} (d-6 e)-\frac {1}{3} (x+1)^{15} (d-3 e)+\frac {5}{7} (x+1)^{14} (d-2 e)-\frac {5}{13} (x+1)^{13} (2 d-3 e)+\frac {1}{12} (x+1)^{12} (5 d-6 e)-\frac {1}{11} (x+1)^{11} (d-e)+\frac {1}{17} e (x+1)^{17} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 76
Rubi steps
\begin {align*} \int x^5 (d+e x) \left (1+2 x+x^2\right )^5 \, dx &=\int x^5 (1+x)^{10} (d+e x) \, dx\\ &=\int \left ((-d+e) (1+x)^{10}+(5 d-6 e) (1+x)^{11}-5 (2 d-3 e) (1+x)^{12}+10 (d-2 e) (1+x)^{13}-5 (d-3 e) (1+x)^{14}+(d-6 e) (1+x)^{15}+e (1+x)^{16}\right ) \, dx\\ &=-\frac {1}{11} (d-e) (1+x)^{11}+\frac {1}{12} (5 d-6 e) (1+x)^{12}-\frac {5}{13} (2 d-3 e) (1+x)^{13}+\frac {5}{7} (d-2 e) (1+x)^{14}-\frac {1}{3} (d-3 e) (1+x)^{15}+\frac {1}{16} (d-6 e) (1+x)^{16}+\frac {1}{17} e (1+x)^{17}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 151, normalized size = 1.53 \begin {gather*} \frac {1}{16} x^{16} (d+10 e)+\frac {1}{3} x^{15} (2 d+9 e)+\frac {15}{14} x^{14} (3 d+8 e)+\frac {30}{13} x^{13} (4 d+7 e)+\frac {7}{2} x^{12} (5 d+6 e)+\frac {42}{11} x^{11} (6 d+5 e)+3 x^{10} (7 d+4 e)+\frac {5}{3} x^9 (8 d+3 e)+\frac {5}{8} x^8 (9 d+2 e)+\frac {1}{7} x^7 (10 d+e)+\frac {d x^6}{6}+\frac {e x^{17}}{17} \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^5 (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 = 0.42, size = 133, normalized size = 1.34 \begin {gather*} \frac {1}{17} x^{17} e + \frac {5}{8} x^{16} e + \frac {1}{16} x^{16} d + 3 x^{15} e + \frac {2}{3} x^{15} d + \frac {60}{7} x^{14} e + \frac {45}{14} x^{14} d + \frac {210}{13} x^{13} e + \frac {120}{13} x^{13} d + 21 x^{12} e + \frac {35}{2} x^{12} d + \frac {210}{11} x^{11} e + \frac {252}{11} x^{11} d + 12 x^{10} e + 21 x^{10} d + 5 x^{9} e + \frac {40}{3} x^{9} d + \frac {5}{4} x^{8} e + \frac {45}{8} x^{8} d + \frac {1}{7} x^{7} e + \frac {10}{7} x^{7} d + \frac {1}{6} x^{6} d \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 144, normalized size = 1.45 \begin {gather*} \frac {1}{17} \, x^{17} e + \frac {1}{16} \, d x^{16} + \frac {5}{8} \, x^{16} e + \frac {2}{3} \, d x^{15} + 3 \, x^{15} e + \frac {45}{14} \, d x^{14} + \frac {60}{7} \, x^{14} e + \frac {120}{13} \, d x^{13} + \frac {210}{13} \, x^{13} e + \frac {35}{2} \, d x^{12} + 21 \, x^{12} e + \frac {252}{11} \, d x^{11} + \frac {210}{11} \, x^{11} e + 21 \, d x^{10} + 12 \, x^{10} e + \frac {40}{3} \, d x^{9} + 5 \, x^{9} e + \frac {45}{8} \, d x^{8} + \frac {5}{4} \, x^{8} e + \frac {10}{7} \, d x^{7} + \frac {1}{7} \, x^{7} e + \frac {1}{6} \, d x^{6} \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 = 1.31 \begin {gather*} \frac {e \,x^{17}}{17}+\frac {\left (d +10 e \right ) x^{16}}{16}+\frac {\left (10 d +45 e \right ) x^{15}}{15}+\frac {\left (45 d +120 e \right ) x^{14}}{14}+\frac {\left (120 d +210 e \right ) x^{13}}{13}+\frac {\left (210 d +252 e \right ) x^{12}}{12}+\frac {\left (252 d +210 e \right ) x^{11}}{11}+\frac {\left (210 d +120 e \right ) x^{10}}{10}+\frac {\left (120 d +45 e \right ) x^{9}}{9}+\frac {\left (45 d +10 e \right ) x^{8}}{8}+\frac {d \,x^{6}}{6}+\frac {\left (10 d +e \right ) x^{7}}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.68, size = 129, normalized size = 1.30 \begin {gather*} \frac {1}{17} \, e x^{17} + \frac {1}{16} \, {\left (d + 10 \, e\right )} x^{16} + \frac {1}{3} \, {\left (2 \, d + 9 \, e\right )} x^{15} + \frac {15}{14} \, {\left (3 \, d + 8 \, e\right )} x^{14} + \frac {30}{13} \, {\left (4 \, d + 7 \, e\right )} x^{13} + \frac {7}{2} \, {\left (5 \, d + 6 \, e\right )} x^{12} + \frac {42}{11} \, {\left (6 \, d + 5 \, e\right )} x^{11} + 3 \, {\left (7 \, d + 4 \, e\right )} x^{10} + \frac {5}{3} \, {\left (8 \, d + 3 \, e\right )} x^{9} + \frac {5}{8} \, {\left (9 \, d + 2 \, e\right )} x^{8} + \frac {1}{7} \, {\left (10 \, d + e\right )} x^{7} + \frac {1}{6} \, d x^{6} \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 = 1.24 \begin {gather*} \frac {e\,x^{17}}{17}+\left (\frac {d}{16}+\frac {5\,e}{8}\right )\,x^{16}+\left (\frac {2\,d}{3}+3\,e\right )\,x^{15}+\left (\frac {45\,d}{14}+\frac {60\,e}{7}\right )\,x^{14}+\left (\frac {120\,d}{13}+\frac {210\,e}{13}\right )\,x^{13}+\left (\frac {35\,d}{2}+21\,e\right )\,x^{12}+\left (\frac {252\,d}{11}+\frac {210\,e}{11}\right )\,x^{11}+\left (21\,d+12\,e\right )\,x^{10}+\left (\frac {40\,d}{3}+5\,e\right )\,x^9+\left (\frac {45\,d}{8}+\frac {5\,e}{4}\right )\,x^8+\left (\frac {10\,d}{7}+\frac {e}{7}\right )\,x^7+\frac {d\,x^6}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 136, normalized size = 1.37 \begin {gather*} \frac {d x^{6}}{6} + \frac {e x^{17}}{17} + x^{16} \left (\frac {d}{16} + \frac {5 e}{8}\right ) + x^{15} \left (\frac {2 d}{3} + 3 e\right ) + x^{14} \left (\frac {45 d}{14} + \frac {60 e}{7}\right ) + x^{13} \left (\frac {120 d}{13} + \frac {210 e}{13}\right ) + x^{12} \left (\frac {35 d}{2} + 21 e\right ) + x^{11} \left (\frac {252 d}{11} + \frac {210 e}{11}\right ) + x^{10} \left (21 d + 12 e\right ) + x^{9} \left (\frac {40 d}{3} + 5 e\right ) + x^{8} \left (\frac {45 d}{8} + \frac {5 e}{4}\right ) + x^{7} \left (\frac {10 d}{7} + \frac {e}{7}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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