Optimal. Leaf size=45 \[ \frac {1}{24} \left (x^2+1\right )^{12} (d-2 e)-\frac {1}{22} \left (x^2+1\right )^{11} (d-e)+\frac {1}{26} e \left (x^2+1\right )^{13} \]
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Rubi [A] time = 0.12, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {28, 446, 76} \[ \frac {1}{24} \left (x^2+1\right )^{12} (d-2 e)-\frac {1}{22} \left (x^2+1\right )^{11} (d-e)+\frac {1}{26} e \left (x^2+1\right )^{13} \]
Antiderivative was successfully verified.
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Rule 28
Rule 76
Rule 446
Rubi steps
\begin {align*} \int x^3 \left (d+e x^2\right ) \left (1+2 x^2+x^4\right )^5 \, dx &=\int x^3 \left (1+x^2\right )^{10} \left (d+e x^2\right ) \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int x (1+x)^{10} (d+e x) \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left ((-d+e) (1+x)^{10}+(d-2 e) (1+x)^{11}+e (1+x)^{12}\right ) \, dx,x,x^2\right )\\ &=-\frac {1}{22} (d-e) \left (1+x^2\right )^{11}+\frac {1}{24} (d-2 e) \left (1+x^2\right )^{12}+\frac {1}{26} e \left (1+x^2\right )^{13}\\ \end {align*}
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Mathematica [B] time = 0.02, size = 151, normalized size = 3.36 \[ \frac {1}{24} x^{24} (d+10 e)+\frac {5}{22} x^{22} (2 d+9 e)+\frac {3}{4} x^{20} (3 d+8 e)+\frac {5}{3} x^{18} (4 d+7 e)+\frac {21}{8} x^{16} (5 d+6 e)+3 x^{14} (6 d+5 e)+\frac {5}{2} x^{12} (7 d+4 e)+\frac {3}{2} x^{10} (8 d+3 e)+\frac {5}{8} x^8 (9 d+2 e)+\frac {1}{6} x^6 (10 d+e)+\frac {d x^4}{4}+\frac {e x^{26}}{26} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.65, size = 133, normalized size = 2.96 \[ \frac {1}{26} x^{26} e + \frac {5}{12} x^{24} e + \frac {1}{24} x^{24} d + \frac {45}{22} x^{22} e + \frac {5}{11} x^{22} d + 6 x^{20} e + \frac {9}{4} x^{20} d + \frac {35}{3} x^{18} e + \frac {20}{3} x^{18} d + \frac {63}{4} x^{16} e + \frac {105}{8} x^{16} d + 15 x^{14} e + 18 x^{14} d + 10 x^{12} e + \frac {35}{2} x^{12} d + \frac {9}{2} x^{10} e + 12 x^{10} d + \frac {5}{4} x^{8} e + \frac {45}{8} x^{8} d + \frac {1}{6} x^{6} e + \frac {5}{3} x^{6} d + \frac {1}{4} x^{4} d \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.40, size = 144, normalized size = 3.20 \[ \frac {1}{26} \, x^{26} e + \frac {1}{24} \, d x^{24} + \frac {5}{12} \, x^{24} e + \frac {5}{11} \, d x^{22} + \frac {45}{22} \, x^{22} e + \frac {9}{4} \, d x^{20} + 6 \, x^{20} e + \frac {20}{3} \, d x^{18} + \frac {35}{3} \, x^{18} e + \frac {105}{8} \, d x^{16} + \frac {63}{4} \, x^{16} e + 18 \, d x^{14} + 15 \, x^{14} e + \frac {35}{2} \, d x^{12} + 10 \, x^{12} e + 12 \, d x^{10} + \frac {9}{2} \, x^{10} e + \frac {45}{8} \, d x^{8} + \frac {5}{4} \, x^{8} e + \frac {5}{3} \, d x^{6} + \frac {1}{6} \, x^{6} e + \frac {1}{4} \, d x^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 130, normalized size = 2.89 \[ \frac {e \,x^{26}}{26}+\frac {\left (d +10 e \right ) x^{24}}{24}+\frac {\left (10 d +45 e \right ) x^{22}}{22}+\frac {\left (45 d +120 e \right ) x^{20}}{20}+\frac {\left (120 d +210 e \right ) x^{18}}{18}+\frac {\left (210 d +252 e \right ) x^{16}}{16}+\frac {\left (252 d +210 e \right ) x^{14}}{14}+\frac {\left (210 d +120 e \right ) x^{12}}{12}+\frac {\left (120 d +45 e \right ) x^{10}}{10}+\frac {\left (45 d +10 e \right ) x^{8}}{8}+\frac {\left (10 d +e \right ) x^{6}}{6}+\frac {d \,x^{4}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.65, size = 129, normalized size = 2.87 \[ \frac {1}{26} \, e x^{26} + \frac {1}{24} \, {\left (d + 10 \, e\right )} x^{24} + \frac {5}{22} \, {\left (2 \, d + 9 \, e\right )} x^{22} + \frac {3}{4} \, {\left (3 \, d + 8 \, e\right )} x^{20} + \frac {5}{3} \, {\left (4 \, d + 7 \, e\right )} x^{18} + \frac {21}{8} \, {\left (5 \, d + 6 \, e\right )} x^{16} + 3 \, {\left (6 \, d + 5 \, e\right )} x^{14} + \frac {5}{2} \, {\left (7 \, d + 4 \, e\right )} x^{12} + \frac {3}{2} \, {\left (8 \, d + 3 \, e\right )} x^{10} + \frac {5}{8} \, {\left (9 \, d + 2 \, e\right )} x^{8} + \frac {1}{6} \, {\left (10 \, d + e\right )} x^{6} + \frac {1}{4} \, d x^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 123, normalized size = 2.73 \[ \frac {e\,x^{26}}{26}+\left (\frac {d}{24}+\frac {5\,e}{12}\right )\,x^{24}+\left (\frac {5\,d}{11}+\frac {45\,e}{22}\right )\,x^{22}+\left (\frac {9\,d}{4}+6\,e\right )\,x^{20}+\left (\frac {20\,d}{3}+\frac {35\,e}{3}\right )\,x^{18}+\left (\frac {105\,d}{8}+\frac {63\,e}{4}\right )\,x^{16}+\left (18\,d+15\,e\right )\,x^{14}+\left (\frac {35\,d}{2}+10\,e\right )\,x^{12}+\left (12\,d+\frac {9\,e}{2}\right )\,x^{10}+\left (\frac {45\,d}{8}+\frac {5\,e}{4}\right )\,x^8+\left (\frac {5\,d}{3}+\frac {e}{6}\right )\,x^6+\frac {d\,x^4}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.10, size = 136, normalized size = 3.02 \[ \frac {d x^{4}}{4} + \frac {e x^{26}}{26} + x^{24} \left (\frac {d}{24} + \frac {5 e}{12}\right ) + x^{22} \left (\frac {5 d}{11} + \frac {45 e}{22}\right ) + x^{20} \left (\frac {9 d}{4} + 6 e\right ) + x^{18} \left (\frac {20 d}{3} + \frac {35 e}{3}\right ) + x^{16} \left (\frac {105 d}{8} + \frac {63 e}{4}\right ) + x^{14} \left (18 d + 15 e\right ) + x^{12} \left (\frac {35 d}{2} + 10 e\right ) + x^{10} \left (12 d + \frac {9 e}{2}\right ) + x^{8} \left (\frac {45 d}{8} + \frac {5 e}{4}\right ) + x^{6} \left (\frac {5 d}{3} + \frac {e}{6}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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