3.56 \(\int x^5 \left (d+e x^2\right ) \left (1+2 x^2+x^4\right )^5 \, dx\)

Optimal. Leaf size=63 \[ \frac{1}{26} \left (x^2+1\right )^{13} (d-3 e)-\frac{1}{24} \left (x^2+1\right )^{12} (2 d-3 e)+\frac{1}{22} \left (x^2+1\right )^{11} (d-e)+\frac{1}{28} e \left (x^2+1\right )^{14} \]

[Out]

((d - e)*(1 + x^2)^11)/22 - ((2*d - 3*e)*(1 + x^2)^12)/24 + ((d - 3*e)*(1 + x^2)
^13)/26 + (e*(1 + x^2)^14)/28

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Rubi [A]  time = 0.421653, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13 \[ \frac{1}{26} \left (x^2+1\right )^{13} (d-3 e)-\frac{1}{24} \left (x^2+1\right )^{12} (2 d-3 e)+\frac{1}{22} \left (x^2+1\right )^{11} (d-e)+\frac{1}{28} e \left (x^2+1\right )^{14} \]

Antiderivative was successfully verified.

[In]  Int[x^5*(d + e*x^2)*(1 + 2*x^2 + x^4)^5,x]

[Out]

((d - e)*(1 + x^2)^11)/22 - ((2*d - 3*e)*(1 + x^2)^12)/24 + ((d - 3*e)*(1 + x^2)
^13)/26 + (e*(1 + x^2)^14)/28

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Rubi in Sympy [A]  time = 22.3772, size = 51, normalized size = 0.81 \[ \frac{e \left (x^{2} + 1\right )^{14}}{28} + \left (\frac{d}{26} - \frac{3 e}{26}\right ) \left (x^{2} + 1\right )^{13} + \left (\frac{d}{22} - \frac{e}{22}\right ) \left (x^{2} + 1\right )^{11} - \left (\frac{d}{12} - \frac{e}{8}\right ) \left (x^{2} + 1\right )^{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5*(e*x**2+d)*(x**4+2*x**2+1)**5,x)

[Out]

e*(x**2 + 1)**14/28 + (d/26 - 3*e/26)*(x**2 + 1)**13 + (d/22 - e/22)*(x**2 + 1)*
*11 - (d/12 - e/8)*(x**2 + 1)**12

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Mathematica [B]  time = 0.0415111, size = 153, normalized size = 2.43 \[ \frac{1}{26} x^{26} (d+10 e)+\frac{5}{24} x^{24} (2 d+9 e)+\frac{15}{22} x^{22} (3 d+8 e)+\frac{3}{2} x^{20} (4 d+7 e)+\frac{7}{3} x^{18} (5 d+6 e)+\frac{21}{8} x^{16} (6 d+5 e)+\frac{15}{7} x^{14} (7 d+4 e)+\frac{5}{4} x^{12} (8 d+3 e)+\frac{1}{2} x^{10} (9 d+2 e)+\frac{1}{8} x^8 (10 d+e)+\frac{d x^6}{6}+\frac{e x^{28}}{28} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5*(d + e*x^2)*(1 + 2*x^2 + x^4)^5,x]

[Out]

(d*x^6)/6 + ((10*d + e)*x^8)/8 + ((9*d + 2*e)*x^10)/2 + (5*(8*d + 3*e)*x^12)/4 +
(15*(7*d + 4*e)*x^14)/7 + (21*(6*d + 5*e)*x^16)/8 + (7*(5*d + 6*e)*x^18)/3 + (3
*(4*d + 7*e)*x^20)/2 + (15*(3*d + 8*e)*x^22)/22 + (5*(2*d + 9*e)*x^24)/24 + ((d
+ 10*e)*x^26)/26 + (e*x^28)/28

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Maple [B]  time = 0.001, size = 130, normalized size = 2.1 \[{\frac{e{x}^{28}}{28}}+{\frac{ \left ( d+10\,e \right ){x}^{26}}{26}}+{\frac{ \left ( 10\,d+45\,e \right ){x}^{24}}{24}}+{\frac{ \left ( 45\,d+120\,e \right ){x}^{22}}{22}}+{\frac{ \left ( 120\,d+210\,e \right ){x}^{20}}{20}}+{\frac{ \left ( 210\,d+252\,e \right ){x}^{18}}{18}}+{\frac{ \left ( 252\,d+210\,e \right ){x}^{16}}{16}}+{\frac{ \left ( 210\,d+120\,e \right ){x}^{14}}{14}}+{\frac{ \left ( 120\,d+45\,e \right ){x}^{12}}{12}}+{\frac{ \left ( 45\,d+10\,e \right ){x}^{10}}{10}}+{\frac{ \left ( 10\,d+e \right ){x}^{8}}{8}}+{\frac{d{x}^{6}}{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5*(e*x^2+d)*(x^4+2*x^2+1)^5,x)

[Out]

1/28*e*x^28+1/26*(d+10*e)*x^26+1/24*(10*d+45*e)*x^24+1/22*(45*d+120*e)*x^22+1/20
*(120*d+210*e)*x^20+1/18*(210*d+252*e)*x^18+1/16*(252*d+210*e)*x^16+1/14*(210*d+
120*e)*x^14+1/12*(120*d+45*e)*x^12+1/10*(45*d+10*e)*x^10+1/8*(10*d+e)*x^8+1/6*d*
x^6

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Maxima [A]  time = 0.694938, size = 174, normalized size = 2.76 \[ \frac{1}{28} \, e x^{28} + \frac{1}{26} \,{\left (d + 10 \, e\right )} x^{26} + \frac{5}{24} \,{\left (2 \, d + 9 \, e\right )} x^{24} + \frac{15}{22} \,{\left (3 \, d + 8 \, e\right )} x^{22} + \frac{3}{2} \,{\left (4 \, d + 7 \, e\right )} x^{20} + \frac{7}{3} \,{\left (5 \, d + 6 \, e\right )} x^{18} + \frac{21}{8} \,{\left (6 \, d + 5 \, e\right )} x^{16} + \frac{15}{7} \,{\left (7 \, d + 4 \, e\right )} x^{14} + \frac{5}{4} \,{\left (8 \, d + 3 \, e\right )} x^{12} + \frac{1}{2} \,{\left (9 \, d + 2 \, e\right )} x^{10} + \frac{1}{8} \,{\left (10 \, d + e\right )} x^{8} + \frac{1}{6} \, d x^{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^4 + 2*x^2 + 1)^5*(e*x^2 + d)*x^5,x, algorithm="maxima")

[Out]

1/28*e*x^28 + 1/26*(d + 10*e)*x^26 + 5/24*(2*d + 9*e)*x^24 + 15/22*(3*d + 8*e)*x
^22 + 3/2*(4*d + 7*e)*x^20 + 7/3*(5*d + 6*e)*x^18 + 21/8*(6*d + 5*e)*x^16 + 15/7
*(7*d + 4*e)*x^14 + 5/4*(8*d + 3*e)*x^12 + 1/2*(9*d + 2*e)*x^10 + 1/8*(10*d + e)
*x^8 + 1/6*d*x^6

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Fricas [A]  time = 0.24395, size = 1, normalized size = 0.02 \[ \frac{1}{28} x^{28} e + \frac{5}{13} x^{26} e + \frac{1}{26} x^{26} d + \frac{15}{8} x^{24} e + \frac{5}{12} x^{24} d + \frac{60}{11} x^{22} e + \frac{45}{22} x^{22} d + \frac{21}{2} x^{20} e + 6 x^{20} d + 14 x^{18} e + \frac{35}{3} x^{18} d + \frac{105}{8} x^{16} e + \frac{63}{4} x^{16} d + \frac{60}{7} x^{14} e + 15 x^{14} d + \frac{15}{4} x^{12} e + 10 x^{12} d + x^{10} e + \frac{9}{2} x^{10} d + \frac{1}{8} x^{8} e + \frac{5}{4} x^{8} d + \frac{1}{6} x^{6} d \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^4 + 2*x^2 + 1)^5*(e*x^2 + d)*x^5,x, algorithm="fricas")

[Out]

1/28*x^28*e + 5/13*x^26*e + 1/26*x^26*d + 15/8*x^24*e + 5/12*x^24*d + 60/11*x^22
*e + 45/22*x^22*d + 21/2*x^20*e + 6*x^20*d + 14*x^18*e + 35/3*x^18*d + 105/8*x^1
6*e + 63/4*x^16*d + 60/7*x^14*e + 15*x^14*d + 15/4*x^12*e + 10*x^12*d + x^10*e +
9/2*x^10*d + 1/8*x^8*e + 5/4*x^8*d + 1/6*x^6*d

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Sympy [A]  time = 0.178714, size = 134, normalized size = 2.13 \[ \frac{d x^{6}}{6} + \frac{e x^{28}}{28} + x^{26} \left (\frac{d}{26} + \frac{5 e}{13}\right ) + x^{24} \left (\frac{5 d}{12} + \frac{15 e}{8}\right ) + x^{22} \left (\frac{45 d}{22} + \frac{60 e}{11}\right ) + x^{20} \left (6 d + \frac{21 e}{2}\right ) + x^{18} \left (\frac{35 d}{3} + 14 e\right ) + x^{16} \left (\frac{63 d}{4} + \frac{105 e}{8}\right ) + x^{14} \left (15 d + \frac{60 e}{7}\right ) + x^{12} \left (10 d + \frac{15 e}{4}\right ) + x^{10} \left (\frac{9 d}{2} + e\right ) + x^{8} \left (\frac{5 d}{4} + \frac{e}{8}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5*(e*x**2+d)*(x**4+2*x**2+1)**5,x)

[Out]

d*x**6/6 + e*x**28/28 + x**26*(d/26 + 5*e/13) + x**24*(5*d/12 + 15*e/8) + x**22*
(45*d/22 + 60*e/11) + x**20*(6*d + 21*e/2) + x**18*(35*d/3 + 14*e) + x**16*(63*d
/4 + 105*e/8) + x**14*(15*d + 60*e/7) + x**12*(10*d + 15*e/4) + x**10*(9*d/2 + e
) + x**8*(5*d/4 + e/8)

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GIAC/XCAS [A]  time = 0.260514, size = 193, normalized size = 3.06 \[ \frac{1}{28} \, x^{28} e + \frac{1}{26} \, d x^{26} + \frac{5}{13} \, x^{26} e + \frac{5}{12} \, d x^{24} + \frac{15}{8} \, x^{24} e + \frac{45}{22} \, d x^{22} + \frac{60}{11} \, x^{22} e + 6 \, d x^{20} + \frac{21}{2} \, x^{20} e + \frac{35}{3} \, d x^{18} + 14 \, x^{18} e + \frac{63}{4} \, d x^{16} + \frac{105}{8} \, x^{16} e + 15 \, d x^{14} + \frac{60}{7} \, x^{14} e + 10 \, d x^{12} + \frac{15}{4} \, x^{12} e + \frac{9}{2} \, d x^{10} + x^{10} e + \frac{5}{4} \, d x^{8} + \frac{1}{8} \, x^{8} e + \frac{1}{6} \, d x^{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^4 + 2*x^2 + 1)^5*(e*x^2 + d)*x^5,x, algorithm="giac")

[Out]

1/28*x^28*e + 1/26*d*x^26 + 5/13*x^26*e + 5/12*d*x^24 + 15/8*x^24*e + 45/22*d*x^
22 + 60/11*x^22*e + 6*d*x^20 + 21/2*x^20*e + 35/3*d*x^18 + 14*x^18*e + 63/4*d*x^
16 + 105/8*x^16*e + 15*d*x^14 + 60/7*x^14*e + 10*d*x^12 + 15/4*x^12*e + 9/2*d*x^
10 + x^10*e + 5/4*d*x^8 + 1/8*x^8*e + 1/6*d*x^6