Optimal. Leaf size=143 \[ \frac {1}{21} x^{21} (d+10 e)+\frac {5}{19} x^{19} (2 d+9 e)+\frac {15}{17} x^{17} (3 d+8 e)+2 x^{15} (4 d+7 e)+\frac {42}{13} x^{13} (5 d+6 e)+\frac {42}{11} x^{11} (6 d+5 e)+\frac {10}{3} x^9 (7 d+4 e)+\frac {15}{7} x^7 (8 d+3 e)+x^5 (9 d+2 e)+\frac {1}{3} x^3 (10 d+e)+d x+\frac {e x^{23}}{23} \]
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Rubi [A] time = 0.07, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {28, 373} \[ \frac {1}{21} x^{21} (d+10 e)+\frac {5}{19} x^{19} (2 d+9 e)+\frac {15}{17} x^{17} (3 d+8 e)+2 x^{15} (4 d+7 e)+\frac {42}{13} x^{13} (5 d+6 e)+\frac {42}{11} x^{11} (6 d+5 e)+\frac {10}{3} x^9 (7 d+4 e)+\frac {15}{7} x^7 (8 d+3 e)+x^5 (9 d+2 e)+\frac {1}{3} x^3 (10 d+e)+d x+\frac {e x^{23}}{23} \]
Antiderivative was successfully verified.
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Rule 28
Rule 373
Rubi steps
\begin {align*} \int \left (d+e x^2\right ) \left (1+2 x^2+x^4\right )^5 \, dx &=\int \left (1+x^2\right )^{10} \left (d+e x^2\right ) \, dx\\ &=\int \left (d+(10 d+e) x^2+5 (9 d+2 e) x^4+15 (8 d+3 e) x^6+30 (7 d+4 e) x^8+42 (6 d+5 e) x^{10}+42 (5 d+6 e) x^{12}+30 (4 d+7 e) x^{14}+15 (3 d+8 e) x^{16}+5 (2 d+9 e) x^{18}+(d+10 e) x^{20}+e x^{22}\right ) \, dx\\ &=d x+\frac {1}{3} (10 d+e) x^3+(9 d+2 e) x^5+\frac {15}{7} (8 d+3 e) x^7+\frac {10}{3} (7 d+4 e) x^9+\frac {42}{11} (6 d+5 e) x^{11}+\frac {42}{13} (5 d+6 e) x^{13}+2 (4 d+7 e) x^{15}+\frac {15}{17} (3 d+8 e) x^{17}+\frac {5}{19} (2 d+9 e) x^{19}+\frac {1}{21} (d+10 e) x^{21}+\frac {e x^{23}}{23}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 143, normalized size = 1.00 \[ \frac {1}{21} x^{21} (d+10 e)+\frac {5}{19} x^{19} (2 d+9 e)+\frac {15}{17} x^{17} (3 d+8 e)+2 x^{15} (4 d+7 e)+\frac {42}{13} x^{13} (5 d+6 e)+\frac {42}{11} x^{11} (6 d+5 e)+\frac {10}{3} x^9 (7 d+4 e)+\frac {15}{7} x^7 (8 d+3 e)+x^5 (9 d+2 e)+\frac {1}{3} x^3 (10 d+e)+d x+\frac {e x^{23}}{23} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 130, normalized size = 0.91 \[ \frac {1}{23} x^{23} e + \frac {10}{21} x^{21} e + \frac {1}{21} x^{21} d + \frac {45}{19} x^{19} e + \frac {10}{19} x^{19} d + \frac {120}{17} x^{17} e + \frac {45}{17} x^{17} d + 14 x^{15} e + 8 x^{15} d + \frac {252}{13} x^{13} e + \frac {210}{13} x^{13} d + \frac {210}{11} x^{11} e + \frac {252}{11} x^{11} d + \frac {40}{3} x^{9} e + \frac {70}{3} x^{9} d + \frac {45}{7} x^{7} e + \frac {120}{7} x^{7} d + 2 x^{5} e + 9 x^{5} d + \frac {1}{3} x^{3} e + \frac {10}{3} x^{3} d + x d \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 141, normalized size = 0.99 \[ \frac {1}{23} \, x^{23} e + \frac {1}{21} \, d x^{21} + \frac {10}{21} \, x^{21} e + \frac {10}{19} \, d x^{19} + \frac {45}{19} \, x^{19} e + \frac {45}{17} \, d x^{17} + \frac {120}{17} \, x^{17} e + 8 \, d x^{15} + 14 \, x^{15} e + \frac {210}{13} \, d x^{13} + \frac {252}{13} \, x^{13} e + \frac {252}{11} \, d x^{11} + \frac {210}{11} \, x^{11} e + \frac {70}{3} \, d x^{9} + \frac {40}{3} \, x^{9} e + \frac {120}{7} \, d x^{7} + \frac {45}{7} \, x^{7} e + 9 \, d x^{5} + 2 \, x^{5} e + \frac {10}{3} \, d x^{3} + \frac {1}{3} \, x^{3} e + d x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 127, normalized size = 0.89 \[ \frac {e \,x^{23}}{23}+\frac {\left (d +10 e \right ) x^{21}}{21}+\frac {\left (10 d +45 e \right ) x^{19}}{19}+\frac {\left (45 d +120 e \right ) x^{17}}{17}+\frac {\left (120 d +210 e \right ) x^{15}}{15}+\frac {\left (210 d +252 e \right ) x^{13}}{13}+\frac {\left (252 d +210 e \right ) x^{11}}{11}+\frac {\left (210 d +120 e \right ) x^{9}}{9}+\frac {\left (120 d +45 e \right ) x^{7}}{7}+\frac {\left (45 d +10 e \right ) x^{5}}{5}+\frac {\left (10 d +e \right ) x^{3}}{3}+d x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.58, size = 125, normalized size = 0.87 \[ \frac {1}{23} \, e x^{23} + \frac {1}{21} \, {\left (d + 10 \, e\right )} x^{21} + \frac {5}{19} \, {\left (2 \, d + 9 \, e\right )} x^{19} + \frac {15}{17} \, {\left (3 \, d + 8 \, e\right )} x^{17} + 2 \, {\left (4 \, d + 7 \, e\right )} x^{15} + \frac {42}{13} \, {\left (5 \, d + 6 \, e\right )} x^{13} + \frac {42}{11} \, {\left (6 \, d + 5 \, e\right )} x^{11} + \frac {10}{3} \, {\left (7 \, d + 4 \, e\right )} x^{9} + \frac {15}{7} \, {\left (8 \, d + 3 \, e\right )} x^{7} + {\left (9 \, d + 2 \, e\right )} x^{5} + \frac {1}{3} \, {\left (10 \, d + e\right )} x^{3} + d x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 120, normalized size = 0.84 \[ \frac {e\,x^{23}}{23}+\left (\frac {d}{21}+\frac {10\,e}{21}\right )\,x^{21}+\left (\frac {10\,d}{19}+\frac {45\,e}{19}\right )\,x^{19}+\left (\frac {45\,d}{17}+\frac {120\,e}{17}\right )\,x^{17}+\left (8\,d+14\,e\right )\,x^{15}+\left (\frac {210\,d}{13}+\frac {252\,e}{13}\right )\,x^{13}+\left (\frac {252\,d}{11}+\frac {210\,e}{11}\right )\,x^{11}+\left (\frac {70\,d}{3}+\frac {40\,e}{3}\right )\,x^9+\left (\frac {120\,d}{7}+\frac {45\,e}{7}\right )\,x^7+\left (9\,d+2\,e\right )\,x^5+\left (\frac {10\,d}{3}+\frac {e}{3}\right )\,x^3+d\,x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 134, normalized size = 0.94 \[ d x + \frac {e x^{23}}{23} + x^{21} \left (\frac {d}{21} + \frac {10 e}{21}\right ) + x^{19} \left (\frac {10 d}{19} + \frac {45 e}{19}\right ) + x^{17} \left (\frac {45 d}{17} + \frac {120 e}{17}\right ) + x^{15} \left (8 d + 14 e\right ) + x^{13} \left (\frac {210 d}{13} + \frac {252 e}{13}\right ) + x^{11} \left (\frac {252 d}{11} + \frac {210 e}{11}\right ) + x^{9} \left (\frac {70 d}{3} + \frac {40 e}{3}\right ) + x^{7} \left (\frac {120 d}{7} + \frac {45 e}{7}\right ) + x^{5} \left (9 d + 2 e\right ) + x^{3} \left (\frac {10 d}{3} + \frac {e}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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