Optimal. Leaf size=29 \[ \frac {1}{22} \left (x^2+1\right )^{11} (d-e)+\frac {1}{24} e \left (x^2+1\right )^{12} \]
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Rubi [A] time = 0.05, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {28, 444, 43} \[ \frac {1}{22} \left (x^2+1\right )^{11} (d-e)+\frac {1}{24} e \left (x^2+1\right )^{12} \]
Antiderivative was successfully verified.
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Rule 28
Rule 43
Rule 444
Rubi steps
\begin {align*} \int x \left (d+e x^2\right ) \left (1+2 x^2+x^4\right )^5 \, dx &=\int x \left (1+x^2\right )^{10} \left (d+e x^2\right ) \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int (1+x)^{10} (d+e x) \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left ((d-e) (1+x)^{10}+e (1+x)^{11}\right ) \, dx,x,x^2\right )\\ &=\frac {1}{22} (d-e) \left (1+x^2\right )^{11}+\frac {1}{24} e \left (1+x^2\right )^{12}\\ \end {align*}
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Mathematica [B] time = 0.01, size = 149, normalized size = 5.14 \[ \frac {1}{22} x^{22} (d+10 e)+\frac {1}{4} x^{20} (2 d+9 e)+\frac {5}{6} x^{18} (3 d+8 e)+\frac {15}{8} x^{16} (4 d+7 e)+3 x^{14} (5 d+6 e)+\frac {7}{2} x^{12} (6 d+5 e)+3 x^{10} (7 d+4 e)+\frac {15}{8} x^8 (8 d+3 e)+\frac {5}{6} x^6 (9 d+2 e)+\frac {1}{4} x^4 (10 d+e)+\frac {d x^2}{2}+\frac {e x^{24}}{24} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.64, size = 133, normalized size = 4.59 \[ \frac {1}{24} x^{24} e + \frac {5}{11} x^{22} e + \frac {1}{22} x^{22} d + \frac {9}{4} x^{20} e + \frac {1}{2} x^{20} d + \frac {20}{3} x^{18} e + \frac {5}{2} x^{18} d + \frac {105}{8} x^{16} e + \frac {15}{2} x^{16} d + 18 x^{14} e + 15 x^{14} d + \frac {35}{2} x^{12} e + 21 x^{12} d + 12 x^{10} e + 21 x^{10} d + \frac {45}{8} x^{8} e + 15 x^{8} d + \frac {5}{3} x^{6} e + \frac {15}{2} x^{6} d + \frac {1}{4} x^{4} e + \frac {5}{2} x^{4} d + \frac {1}{2} x^{2} d \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.31, size = 144, normalized size = 4.97 \[ \frac {1}{24} \, x^{24} e + \frac {1}{22} \, d x^{22} + \frac {5}{11} \, x^{22} e + \frac {1}{2} \, d x^{20} + \frac {9}{4} \, x^{20} e + \frac {5}{2} \, d x^{18} + \frac {20}{3} \, x^{18} e + \frac {15}{2} \, d x^{16} + \frac {105}{8} \, x^{16} e + 15 \, d x^{14} + 18 \, x^{14} e + 21 \, d x^{12} + \frac {35}{2} \, x^{12} e + 21 \, d x^{10} + 12 \, x^{10} e + 15 \, d x^{8} + \frac {45}{8} \, x^{8} e + \frac {15}{2} \, d x^{6} + \frac {5}{3} \, x^{6} e + \frac {5}{2} \, d x^{4} + \frac {1}{4} \, x^{4} e + \frac {1}{2} \, d x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 130, normalized size = 4.48 \[ \frac {e \,x^{24}}{24}+\frac {\left (d +10 e \right ) x^{22}}{22}+\frac {\left (10 d +45 e \right ) x^{20}}{20}+\frac {\left (45 d +120 e \right ) x^{18}}{18}+\frac {\left (120 d +210 e \right ) x^{16}}{16}+\frac {\left (210 d +252 e \right ) x^{14}}{14}+\frac {\left (252 d +210 e \right ) x^{12}}{12}+\frac {\left (210 d +120 e \right ) x^{10}}{10}+\frac {\left (120 d +45 e \right ) x^{8}}{8}+\frac {\left (45 d +10 e \right ) x^{6}}{6}+\frac {\left (10 d +e \right ) x^{4}}{4}+\frac {d \,x^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.65, size = 129, normalized size = 4.45 \[ \frac {1}{24} \, e x^{24} + \frac {1}{22} \, {\left (d + 10 \, e\right )} x^{22} + \frac {1}{4} \, {\left (2 \, d + 9 \, e\right )} x^{20} + \frac {5}{6} \, {\left (3 \, d + 8 \, e\right )} x^{18} + \frac {15}{8} \, {\left (4 \, d + 7 \, e\right )} x^{16} + 3 \, {\left (5 \, d + 6 \, e\right )} x^{14} + \frac {7}{2} \, {\left (6 \, d + 5 \, e\right )} x^{12} + 3 \, {\left (7 \, d + 4 \, e\right )} x^{10} + \frac {15}{8} \, {\left (8 \, d + 3 \, e\right )} x^{8} + \frac {5}{6} \, {\left (9 \, d + 2 \, e\right )} x^{6} + \frac {1}{4} \, {\left (10 \, d + e\right )} x^{4} + \frac {1}{2} \, d x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 123, normalized size = 4.24 \[ \frac {e\,x^{24}}{24}+\left (\frac {d}{22}+\frac {5\,e}{11}\right )\,x^{22}+\left (\frac {d}{2}+\frac {9\,e}{4}\right )\,x^{20}+\left (\frac {5\,d}{2}+\frac {20\,e}{3}\right )\,x^{18}+\left (\frac {15\,d}{2}+\frac {105\,e}{8}\right )\,x^{16}+\left (15\,d+18\,e\right )\,x^{14}+\left (21\,d+\frac {35\,e}{2}\right )\,x^{12}+\left (21\,d+12\,e\right )\,x^{10}+\left (15\,d+\frac {45\,e}{8}\right )\,x^8+\left (\frac {15\,d}{2}+\frac {5\,e}{3}\right )\,x^6+\left (\frac {5\,d}{2}+\frac {e}{4}\right )\,x^4+\frac {d\,x^2}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.10, size = 133, normalized size = 4.59 \[ \frac {d x^{2}}{2} + \frac {e x^{24}}{24} + x^{22} \left (\frac {d}{22} + \frac {5 e}{11}\right ) + x^{20} \left (\frac {d}{2} + \frac {9 e}{4}\right ) + x^{18} \left (\frac {5 d}{2} + \frac {20 e}{3}\right ) + x^{16} \left (\frac {15 d}{2} + \frac {105 e}{8}\right ) + x^{14} \left (15 d + 18 e\right ) + x^{12} \left (21 d + \frac {35 e}{2}\right ) + x^{10} \left (21 d + 12 e\right ) + x^{8} \left (15 d + \frac {45 e}{8}\right ) + x^{6} \left (\frac {15 d}{2} + \frac {5 e}{3}\right ) + x^{4} \left (\frac {5 d}{2} + \frac {e}{4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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