Optimal. Leaf size=55 \[ \frac {1}{13} (x+1)^{13} (d-3 e)-\frac {1}{12} (x+1)^{12} (2 d-3 e)+\frac {1}{11} (x+1)^{11} (d-e)+\frac {1}{14} e (x+1)^{14} \]
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Rubi [A] time = 0.04, antiderivative size = 55, 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}{13} (x+1)^{13} (d-3 e)-\frac {1}{12} (x+1)^{12} (2 d-3 e)+\frac {1}{11} (x+1)^{11} (d-e)+\frac {1}{14} e (x+1)^{14} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 76
Rubi steps
\begin {align*} \int x^2 (d+e x) \left (1+2 x+x^2\right )^5 \, dx &=\int x^2 (1+x)^{10} (d+e x) \, dx\\ &=\int \left ((d-e) (1+x)^{10}+(-2 d+3 e) (1+x)^{11}+(d-3 e) (1+x)^{12}+e (1+x)^{13}\right ) \, dx\\ &=\frac {1}{11} (d-e) (1+x)^{11}-\frac {1}{12} (2 d-3 e) (1+x)^{12}+\frac {1}{13} (d-3 e) (1+x)^{13}+\frac {1}{14} e (1+x)^{14}\\ \end {align*}
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Mathematica [B] time = 0.02, size = 148, normalized size = 2.69 \begin {gather*} \frac {1}{13} x^{13} (d+10 e)+\frac {5}{12} x^{12} (2 d+9 e)+\frac {15}{11} x^{11} (3 d+8 e)+3 x^{10} (4 d+7 e)+\frac {14}{3} x^9 (5 d+6 e)+\frac {21}{4} x^8 (6 d+5 e)+\frac {30}{7} x^7 (7 d+4 e)+\frac {5}{2} x^6 (8 d+3 e)+x^5 (9 d+2 e)+\frac {1}{4} x^4 (10 d+e)+\frac {d x^3}{3}+\frac {e x^{14}}{14} \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^2 (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 = 0.37, size = 133, normalized size = 2.42 \begin {gather*} \frac {1}{14} x^{14} e + \frac {10}{13} x^{13} e + \frac {1}{13} x^{13} d + \frac {15}{4} x^{12} e + \frac {5}{6} x^{12} d + \frac {120}{11} x^{11} e + \frac {45}{11} x^{11} d + 21 x^{10} e + 12 x^{10} d + 28 x^{9} e + \frac {70}{3} x^{9} d + \frac {105}{4} x^{8} e + \frac {63}{2} x^{8} d + \frac {120}{7} x^{7} e + 30 x^{7} d + \frac {15}{2} x^{6} e + 20 x^{6} d + 2 x^{5} e + 9 x^{5} d + \frac {1}{4} x^{4} e + \frac {5}{2} x^{4} d + \frac {1}{3} x^{3} d \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.18, size = 144, normalized size = 2.62 \begin {gather*} \frac {1}{14} \, x^{14} e + \frac {1}{13} \, d x^{13} + \frac {10}{13} \, x^{13} e + \frac {5}{6} \, d x^{12} + \frac {15}{4} \, x^{12} e + \frac {45}{11} \, d x^{11} + \frac {120}{11} \, x^{11} e + 12 \, d x^{10} + 21 \, x^{10} e + \frac {70}{3} \, d x^{9} + 28 \, x^{9} e + \frac {63}{2} \, d x^{8} + \frac {105}{4} \, x^{8} e + 30 \, d x^{7} + \frac {120}{7} \, x^{7} e + 20 \, d x^{6} + \frac {15}{2} \, x^{6} e + 9 \, d x^{5} + 2 \, x^{5} e + \frac {5}{2} \, d x^{4} + \frac {1}{4} \, x^{4} e + \frac {1}{3} \, d x^{3} \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 = 2.36 \begin {gather*} \frac {e \,x^{14}}{14}+\frac {\left (d +10 e \right ) x^{13}}{13}+\frac {\left (10 d +45 e \right ) x^{12}}{12}+\frac {\left (45 d +120 e \right ) x^{11}}{11}+\frac {\left (120 d +210 e \right ) x^{10}}{10}+\frac {\left (210 d +252 e \right ) x^{9}}{9}+\frac {\left (252 d +210 e \right ) x^{8}}{8}+\frac {\left (210 d +120 e \right ) x^{7}}{7}+\frac {\left (120 d +45 e \right ) x^{6}}{6}+\frac {\left (45 d +10 e \right ) x^{5}}{5}+\frac {d \,x^{3}}{3}+\frac {\left (10 d +e \right ) x^{4}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.58, size = 128, normalized size = 2.33 \begin {gather*} \frac {1}{14} \, e x^{14} + \frac {1}{13} \, {\left (d + 10 \, e\right )} x^{13} + \frac {5}{12} \, {\left (2 \, d + 9 \, e\right )} x^{12} + \frac {15}{11} \, {\left (3 \, d + 8 \, e\right )} x^{11} + 3 \, {\left (4 \, d + 7 \, e\right )} x^{10} + \frac {14}{3} \, {\left (5 \, d + 6 \, e\right )} x^{9} + \frac {21}{4} \, {\left (6 \, d + 5 \, e\right )} x^{8} + \frac {30}{7} \, {\left (7 \, d + 4 \, e\right )} x^{7} + \frac {5}{2} \, {\left (8 \, d + 3 \, e\right )} x^{6} + {\left (9 \, d + 2 \, e\right )} x^{5} + \frac {1}{4} \, {\left (10 \, d + e\right )} x^{4} + \frac {1}{3} \, d x^{3} \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 = 2.24 \begin {gather*} \frac {e\,x^{14}}{14}+\left (\frac {d}{13}+\frac {10\,e}{13}\right )\,x^{13}+\left (\frac {5\,d}{6}+\frac {15\,e}{4}\right )\,x^{12}+\left (\frac {45\,d}{11}+\frac {120\,e}{11}\right )\,x^{11}+\left (12\,d+21\,e\right )\,x^{10}+\left (\frac {70\,d}{3}+28\,e\right )\,x^9+\left (\frac {63\,d}{2}+\frac {105\,e}{4}\right )\,x^8+\left (30\,d+\frac {120\,e}{7}\right )\,x^7+\left (20\,d+\frac {15\,e}{2}\right )\,x^6+\left (9\,d+2\,e\right )\,x^5+\left (\frac {5\,d}{2}+\frac {e}{4}\right )\,x^4+\frac {d\,x^3}{3} \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 = 2.42 \begin {gather*} \frac {d x^{3}}{3} + \frac {e x^{14}}{14} + x^{13} \left (\frac {d}{13} + \frac {10 e}{13}\right ) + x^{12} \left (\frac {5 d}{6} + \frac {15 e}{4}\right ) + x^{11} \left (\frac {45 d}{11} + \frac {120 e}{11}\right ) + x^{10} \left (12 d + 21 e\right ) + x^{9} \left (\frac {70 d}{3} + 28 e\right ) + x^{8} \left (\frac {63 d}{2} + \frac {105 e}{4}\right ) + x^{7} \left (30 d + \frac {120 e}{7}\right ) + x^{6} \left (20 d + \frac {15 e}{2}\right ) + x^{5} \left (9 d + 2 e\right ) + x^{4} \left (\frac {5 d}{2} + \frac {e}{4}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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