Optimal. Leaf size=93 \[ \frac {d x^{20}}{20}+\frac {5 d x^{18}}{9}+\frac {45 d x^{16}}{16}+\frac {60 d x^{14}}{7}+\frac {35 d x^{12}}{2}+\frac {126 d x^{10}}{5}+\frac {105 d x^8}{4}+20 d x^6+\frac {45 d x^4}{4}+5 d x^2+d \log (x)+\frac {1}{22} e \left (x^2+1\right )^{11} \]
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Rubi [A] time = 0.05, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {28, 446, 80, 43} \[ \frac {d x^{20}}{20}+\frac {5 d x^{18}}{9}+\frac {45 d x^{16}}{16}+\frac {60 d x^{14}}{7}+\frac {35 d x^{12}}{2}+\frac {126 d x^{10}}{5}+\frac {105 d x^8}{4}+20 d x^6+\frac {45 d x^4}{4}+5 d x^2+d \log (x)+\frac {1}{22} e \left (x^2+1\right )^{11} \]
Antiderivative was successfully verified.
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Rule 28
Rule 43
Rule 80
Rule 446
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right ) \left (1+2 x^2+x^4\right )^5}{x} \, dx &=\int \frac {\left (1+x^2\right )^{10} \left (d+e x^2\right )}{x} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(1+x)^{10} (d+e x)}{x} \, dx,x,x^2\right )\\ &=\frac {1}{22} e \left (1+x^2\right )^{11}+\frac {1}{2} d \operatorname {Subst}\left (\int \frac {(1+x)^{10}}{x} \, dx,x,x^2\right )\\ &=\frac {1}{22} e \left (1+x^2\right )^{11}+\frac {1}{2} d \operatorname {Subst}\left (\int \left (10+\frac {1}{x}+45 x+120 x^2+210 x^3+252 x^4+210 x^5+120 x^6+45 x^7+10 x^8+x^9\right ) \, dx,x,x^2\right )\\ &=5 d x^2+\frac {45 d x^4}{4}+20 d x^6+\frac {105 d x^8}{4}+\frac {126 d x^{10}}{5}+\frac {35 d x^{12}}{2}+\frac {60 d x^{14}}{7}+\frac {45 d x^{16}}{16}+\frac {5 d x^{18}}{9}+\frac {d x^{20}}{20}+\frac {1}{22} e \left (1+x^2\right )^{11}+d \log (x)\\ \end {align*}
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Mathematica [A] time = 0.03, size = 149, normalized size = 1.60 \[ \frac {1}{20} x^{20} (d+10 e)+\frac {5}{18} x^{18} (2 d+9 e)+\frac {15}{16} x^{16} (3 d+8 e)+\frac {15}{7} x^{14} (4 d+7 e)+\frac {7}{2} x^{12} (5 d+6 e)+\frac {21}{5} x^{10} (6 d+5 e)+\frac {15}{4} x^8 (7 d+4 e)+\frac {5}{2} x^6 (8 d+3 e)+\frac {5}{4} x^4 (9 d+2 e)+\frac {1}{2} x^2 (10 d+e)+d \log (x)+\frac {e x^{22}}{22} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 127, normalized size = 1.37 \[ \frac {1}{22} \, e x^{22} + \frac {1}{20} \, {\left (d + 10 \, e\right )} x^{20} + \frac {5}{18} \, {\left (2 \, d + 9 \, e\right )} x^{18} + \frac {15}{16} \, {\left (3 \, d + 8 \, e\right )} x^{16} + \frac {15}{7} \, {\left (4 \, d + 7 \, e\right )} x^{14} + \frac {7}{2} \, {\left (5 \, d + 6 \, e\right )} x^{12} + \frac {21}{5} \, {\left (6 \, d + 5 \, e\right )} x^{10} + \frac {15}{4} \, {\left (7 \, d + 4 \, e\right )} x^{8} + \frac {5}{2} \, {\left (8 \, d + 3 \, e\right )} x^{6} + \frac {5}{4} \, {\left (9 \, d + 2 \, e\right )} x^{4} + \frac {1}{2} \, {\left (10 \, d + e\right )} x^{2} + d \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 145, normalized size = 1.56 \[ \frac {1}{22} \, x^{22} e + \frac {1}{20} \, d x^{20} + \frac {1}{2} \, x^{20} e + \frac {5}{9} \, d x^{18} + \frac {5}{2} \, x^{18} e + \frac {45}{16} \, d x^{16} + \frac {15}{2} \, x^{16} e + \frac {60}{7} \, d x^{14} + 15 \, x^{14} e + \frac {35}{2} \, d x^{12} + 21 \, x^{12} e + \frac {126}{5} \, d x^{10} + 21 \, x^{10} e + \frac {105}{4} \, d x^{8} + 15 \, x^{8} e + 20 \, d x^{6} + \frac {15}{2} \, x^{6} e + \frac {45}{4} \, d x^{4} + \frac {5}{2} \, x^{4} e + 5 \, d x^{2} + \frac {1}{2} \, x^{2} e + \frac {1}{2} \, d \log \left (x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 132, normalized size = 1.42 \[ \frac {e \,x^{22}}{22}+\frac {d \,x^{20}}{20}+\frac {e \,x^{20}}{2}+\frac {5 d \,x^{18}}{9}+\frac {5 e \,x^{18}}{2}+\frac {45 d \,x^{16}}{16}+\frac {15 e \,x^{16}}{2}+\frac {60 d \,x^{14}}{7}+15 e \,x^{14}+\frac {35 d \,x^{12}}{2}+21 e \,x^{12}+\frac {126 d \,x^{10}}{5}+21 e \,x^{10}+\frac {105 d \,x^{8}}{4}+15 e \,x^{8}+20 d \,x^{6}+\frac {15 e \,x^{6}}{2}+\frac {45 d \,x^{4}}{4}+\frac {5 e \,x^{4}}{2}+5 d \,x^{2}+\frac {e \,x^{2}}{2}+d \ln \relax (x ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 130, normalized size = 1.40 \[ \frac {1}{22} \, e x^{22} + \frac {1}{20} \, {\left (d + 10 \, e\right )} x^{20} + \frac {5}{18} \, {\left (2 \, d + 9 \, e\right )} x^{18} + \frac {15}{16} \, {\left (3 \, d + 8 \, e\right )} x^{16} + \frac {15}{7} \, {\left (4 \, d + 7 \, e\right )} x^{14} + \frac {7}{2} \, {\left (5 \, d + 6 \, e\right )} x^{12} + \frac {21}{5} \, {\left (6 \, d + 5 \, e\right )} x^{10} + \frac {15}{4} \, {\left (7 \, d + 4 \, e\right )} x^{8} + \frac {5}{2} \, {\left (8 \, d + 3 \, e\right )} x^{6} + \frac {5}{4} \, {\left (9 \, d + 2 \, e\right )} x^{4} + \frac {1}{2} \, {\left (10 \, d + e\right )} x^{2} + \frac {1}{2} \, d \log \left (x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 121, normalized size = 1.30 \[ x^2\,\left (5\,d+\frac {e}{2}\right )+x^{18}\,\left (\frac {5\,d}{9}+\frac {5\,e}{2}\right )+x^6\,\left (20\,d+\frac {15\,e}{2}\right )+x^{20}\,\left (\frac {d}{20}+\frac {e}{2}\right )+x^4\,\left (\frac {45\,d}{4}+\frac {5\,e}{2}\right )+x^{12}\,\left (\frac {35\,d}{2}+21\,e\right )+x^{16}\,\left (\frac {45\,d}{16}+\frac {15\,e}{2}\right )+x^{14}\,\left (\frac {60\,d}{7}+15\,e\right )+x^8\,\left (\frac {105\,d}{4}+15\,e\right )+x^{10}\,\left (\frac {126\,d}{5}+21\,e\right )+\frac {e\,x^{22}}{22}+d\,\ln \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.33, size = 131, normalized size = 1.41 \[ d \log {\relax (x )} + \frac {e x^{22}}{22} + x^{20} \left (\frac {d}{20} + \frac {e}{2}\right ) + x^{18} \left (\frac {5 d}{9} + \frac {5 e}{2}\right ) + x^{16} \left (\frac {45 d}{16} + \frac {15 e}{2}\right ) + x^{14} \left (\frac {60 d}{7} + 15 e\right ) + x^{12} \left (\frac {35 d}{2} + 21 e\right ) + x^{10} \left (\frac {126 d}{5} + 21 e\right ) + x^{8} \left (\frac {105 d}{4} + 15 e\right ) + x^{6} \left (20 d + \frac {15 e}{2}\right ) + x^{4} \left (\frac {45 d}{4} + \frac {5 e}{2}\right ) + x^{2} \left (5 d + \frac {e}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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