Optimal. Leaf size=39 \[ \frac {1}{12} (x+1)^{12} (d-2 e)-\frac {1}{11} (x+1)^{11} (d-e)+\frac {1}{13} e (x+1)^{13} \]
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Rubi [A] time = 0.04, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {27, 76} \begin {gather*} \frac {1}{12} (x+1)^{12} (d-2 e)-\frac {1}{11} (x+1)^{11} (d-e)+\frac {1}{13} e (x+1)^{13} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 76
Rubi steps
\begin {align*} \int x (d+e x) \left (1+2 x+x^2\right )^5 \, dx &=\int x (1+x)^{10} (d+e x) \, dx\\ &=\int \left ((-d+e) (1+x)^{10}+(d-2 e) (1+x)^{11}+e (1+x)^{12}\right ) \, dx\\ &=-\frac {1}{11} (d-e) (1+x)^{11}+\frac {1}{12} (d-2 e) (1+x)^{12}+\frac {1}{13} e (1+x)^{13}\\ \end {align*}
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Mathematica [B] time = 0.02, size = 147, normalized size = 3.77 \begin {gather*} \frac {1}{12} x^{12} (d+10 e)+\frac {5}{11} x^{11} (2 d+9 e)+\frac {3}{2} x^{10} (3 d+8 e)+\frac {10}{3} x^9 (4 d+7 e)+\frac {21}{4} x^8 (5 d+6 e)+6 x^7 (6 d+5 e)+5 x^6 (7 d+4 e)+3 x^5 (8 d+3 e)+\frac {5}{4} x^4 (9 d+2 e)+\frac {1}{3} x^3 (10 d+e)+\frac {d x^2}{2}+\frac {e x^{13}}{13} \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 (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 [B] time = 0.36, size = 133, normalized size = 3.41 \begin {gather*} \frac {1}{13} x^{13} e + \frac {5}{6} x^{12} e + \frac {1}{12} x^{12} d + \frac {45}{11} x^{11} e + \frac {10}{11} x^{11} d + 12 x^{10} e + \frac {9}{2} x^{10} d + \frac {70}{3} x^{9} e + \frac {40}{3} x^{9} d + \frac {63}{2} x^{8} e + \frac {105}{4} x^{8} d + 30 x^{7} e + 36 x^{7} d + 20 x^{6} e + 35 x^{6} d + 9 x^{5} e + 24 x^{5} d + \frac {5}{2} x^{4} e + \frac {45}{4} x^{4} d + \frac {1}{3} x^{3} e + \frac {10}{3} x^{3} d + \frac {1}{2} x^{2} d \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 144, normalized size = 3.69 \begin {gather*} \frac {1}{13} \, x^{13} e + \frac {1}{12} \, d x^{12} + \frac {5}{6} \, x^{12} e + \frac {10}{11} \, d x^{11} + \frac {45}{11} \, x^{11} e + \frac {9}{2} \, d x^{10} + 12 \, x^{10} e + \frac {40}{3} \, d x^{9} + \frac {70}{3} \, x^{9} e + \frac {105}{4} \, d x^{8} + \frac {63}{2} \, x^{8} e + 36 \, d x^{7} + 30 \, x^{7} e + 35 \, d x^{6} + 20 \, x^{6} e + 24 \, d x^{5} + 9 \, x^{5} e + \frac {45}{4} \, d x^{4} + \frac {5}{2} \, x^{4} e + \frac {10}{3} \, d x^{3} + \frac {1}{3} \, x^{3} e + \frac {1}{2} \, d x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 130, normalized size = 3.33 \begin {gather*} \frac {e \,x^{13}}{13}+\frac {\left (d +10 e \right ) x^{12}}{12}+\frac {\left (10 d +45 e \right ) x^{11}}{11}+\frac {\left (45 d +120 e \right ) x^{10}}{10}+\frac {\left (120 d +210 e \right ) x^{9}}{9}+\frac {\left (210 d +252 e \right ) x^{8}}{8}+\frac {\left (252 d +210 e \right ) x^{7}}{7}+\frac {\left (210 d +120 e \right ) x^{6}}{6}+\frac {\left (120 d +45 e \right ) x^{5}}{5}+\frac {\left (45 d +10 e \right ) x^{4}}{4}+\frac {d \,x^{2}}{2}+\frac {\left (10 d +e \right ) x^{3}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.56, size = 129, normalized size = 3.31 \begin {gather*} \frac {1}{13} \, e x^{13} + \frac {1}{12} \, {\left (d + 10 \, e\right )} x^{12} + \frac {5}{11} \, {\left (2 \, d + 9 \, e\right )} x^{11} + \frac {3}{2} \, {\left (3 \, d + 8 \, e\right )} x^{10} + \frac {10}{3} \, {\left (4 \, d + 7 \, e\right )} x^{9} + \frac {21}{4} \, {\left (5 \, d + 6 \, e\right )} x^{8} + 6 \, {\left (6 \, d + 5 \, e\right )} x^{7} + 5 \, {\left (7 \, d + 4 \, e\right )} x^{6} + 3 \, {\left (8 \, d + 3 \, e\right )} x^{5} + \frac {5}{4} \, {\left (9 \, d + 2 \, e\right )} x^{4} + \frac {1}{3} \, {\left (10 \, d + e\right )} x^{3} + \frac {1}{2} \, d x^{2} \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 = 3.15 \begin {gather*} \frac {e\,x^{13}}{13}+\left (\frac {d}{12}+\frac {5\,e}{6}\right )\,x^{12}+\left (\frac {10\,d}{11}+\frac {45\,e}{11}\right )\,x^{11}+\left (\frac {9\,d}{2}+12\,e\right )\,x^{10}+\left (\frac {40\,d}{3}+\frac {70\,e}{3}\right )\,x^9+\left (\frac {105\,d}{4}+\frac {63\,e}{2}\right )\,x^8+\left (36\,d+30\,e\right )\,x^7+\left (35\,d+20\,e\right )\,x^6+\left (24\,d+9\,e\right )\,x^5+\left (\frac {45\,d}{4}+\frac {5\,e}{2}\right )\,x^4+\left (\frac {10\,d}{3}+\frac {e}{3}\right )\,x^3+\frac {d\,x^2}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.10, size = 133, normalized size = 3.41 \begin {gather*} \frac {d x^{2}}{2} + \frac {e x^{13}}{13} + x^{12} \left (\frac {d}{12} + \frac {5 e}{6}\right ) + x^{11} \left (\frac {10 d}{11} + \frac {45 e}{11}\right ) + x^{10} \left (\frac {9 d}{2} + 12 e\right ) + x^{9} \left (\frac {40 d}{3} + \frac {70 e}{3}\right ) + x^{8} \left (\frac {105 d}{4} + \frac {63 e}{2}\right ) + x^{7} \left (36 d + 30 e\right ) + x^{6} \left (35 d + 20 e\right ) + x^{5} \left (24 d + 9 e\right ) + x^{4} \left (\frac {45 d}{4} + \frac {5 e}{2}\right ) + x^{3} \left (\frac {10 d}{3} + \frac {e}{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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