3.62 \(\int \frac{\left (d+e x^2\right ) \left (1+2 x^2+x^4\right )^5}{x} \, dx\)

Optimal. Leaf size=93 \[ \frac{d x^{20}}{20}+\frac{5 d x^{18}}{9}+\frac{45 d x^{16}}{16}+\frac{60 d x^{14}}{7}+\frac{35 d x^{12}}{2}+\frac{126 d x^{10}}{5}+\frac{105 d x^8}{4}+20 d x^6+\frac{45 d x^4}{4}+5 d x^2+d \log (x)+\frac{1}{22} e \left (x^2+1\right )^{11} \]

[Out]

5*d*x^2 + (45*d*x^4)/4 + 20*d*x^6 + (105*d*x^8)/4 + (126*d*x^10)/5 + (35*d*x^12)
/2 + (60*d*x^14)/7 + (45*d*x^16)/16 + (5*d*x^18)/9 + (d*x^20)/20 + (e*(1 + x^2)^
11)/22 + d*Log[x]

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Rubi [A]  time = 0.108661, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174 \[ \frac{d x^{20}}{20}+\frac{5 d x^{18}}{9}+\frac{45 d x^{16}}{16}+\frac{60 d x^{14}}{7}+\frac{35 d x^{12}}{2}+\frac{126 d x^{10}}{5}+\frac{105 d x^8}{4}+20 d x^6+\frac{45 d x^4}{4}+5 d x^2+d \log (x)+\frac{1}{22} e \left (x^2+1\right )^{11} \]

Antiderivative was successfully verified.

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

[Out]

5*d*x^2 + (45*d*x^4)/4 + 20*d*x^6 + (105*d*x^8)/4 + (126*d*x^10)/5 + (35*d*x^12)
/2 + (60*d*x^14)/7 + (45*d*x^16)/16 + (5*d*x^18)/9 + (d*x^20)/20 + (e*(1 + x^2)^
11)/22 + d*Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{d x^{20}}{20} + \frac{5 d x^{18}}{9} + \frac{45 d x^{16}}{16} + \frac{60 d x^{14}}{7} + \frac{35 d x^{12}}{2} + \frac{126 d x^{10}}{5} + \frac{105 d x^{8}}{4} + 20 d x^{6} + 5 d x^{2} + \frac{d \log{\left (x^{2} \right )}}{2} + \frac{45 d \int ^{x^{2}} x\, dx}{2} + \frac{e \left (x^{2} + 1\right )^{11}}{22} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*x**20/20 + 5*d*x**18/9 + 45*d*x**16/16 + 60*d*x**14/7 + 35*d*x**12/2 + 126*d*x
**10/5 + 105*d*x**8/4 + 20*d*x**6 + 5*d*x**2 + d*log(x**2)/2 + 45*d*Integral(x,
(x, x**2))/2 + e*(x**2 + 1)**11/22

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Mathematica [A]  time = 0.0514577, size = 149, normalized size = 1.6 \[ \frac{1}{20} x^{20} (d+10 e)+\frac{5}{18} x^{18} (2 d+9 e)+\frac{15}{16} x^{16} (3 d+8 e)+\frac{15}{7} x^{14} (4 d+7 e)+\frac{7}{2} x^{12} (5 d+6 e)+\frac{21}{5} x^{10} (6 d+5 e)+\frac{15}{4} x^8 (7 d+4 e)+\frac{5}{2} x^6 (8 d+3 e)+\frac{5}{4} x^4 (9 d+2 e)+\frac{1}{2} x^2 (10 d+e)+d \log (x)+\frac{e x^{22}}{22} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 132, normalized size = 1.4 \[{\frac{e{x}^{22}}{22}}+{\frac{d{x}^{20}}{20}}+{\frac{e{x}^{20}}{2}}+{\frac{5\,d{x}^{18}}{9}}+{\frac{5\,{x}^{18}e}{2}}+{\frac{45\,d{x}^{16}}{16}}+{\frac{15\,{x}^{16}e}{2}}+{\frac{60\,d{x}^{14}}{7}}+15\,{x}^{14}e+{\frac{35\,d{x}^{12}}{2}}+21\,{x}^{12}e+{\frac{126\,d{x}^{10}}{5}}+21\,{x}^{10}e+{\frac{105\,d{x}^{8}}{4}}+15\,{x}^{8}e+20\,d{x}^{6}+{\frac{15\,{x}^{6}e}{2}}+{\frac{45\,d{x}^{4}}{4}}+{\frac{5\,{x}^{4}e}{2}}+5\,d{x}^{2}+{\frac{e{x}^{2}}{2}}+d\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/22*e*x^22+1/20*d*x^20+1/2*e*x^20+5/9*d*x^18+5/2*x^18*e+45/16*d*x^16+15/2*x^16*
e+60/7*d*x^14+15*x^14*e+35/2*d*x^12+21*x^12*e+126/5*d*x^10+21*x^10*e+105/4*d*x^8
+15*x^8*e+20*d*x^6+15/2*x^6*e+45/4*d*x^4+5/2*x^4*e+5*d*x^2+1/2*e*x^2+d*ln(x)

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Maxima [A]  time = 0.687503, size = 176, normalized size = 1.89 \[ \frac{1}{22} \, e x^{22} + \frac{1}{20} \,{\left (d + 10 \, e\right )} x^{20} + \frac{5}{18} \,{\left (2 \, d + 9 \, e\right )} x^{18} + \frac{15}{16} \,{\left (3 \, d + 8 \, e\right )} x^{16} + \frac{15}{7} \,{\left (4 \, d + 7 \, e\right )} x^{14} + \frac{7}{2} \,{\left (5 \, d + 6 \, e\right )} x^{12} + \frac{21}{5} \,{\left (6 \, d + 5 \, e\right )} x^{10} + \frac{15}{4} \,{\left (7 \, d + 4 \, e\right )} x^{8} + \frac{5}{2} \,{\left (8 \, d + 3 \, e\right )} x^{6} + \frac{5}{4} \,{\left (9 \, d + 2 \, e\right )} x^{4} + \frac{1}{2} \,{\left (10 \, d + e\right )} x^{2} + \frac{1}{2} \, d \log \left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.302196, size = 171, normalized size = 1.84 \[ \frac{1}{22} \, e x^{22} + \frac{1}{20} \,{\left (d + 10 \, e\right )} x^{20} + \frac{5}{18} \,{\left (2 \, d + 9 \, e\right )} x^{18} + \frac{15}{16} \,{\left (3 \, d + 8 \, e\right )} x^{16} + \frac{15}{7} \,{\left (4 \, d + 7 \, e\right )} x^{14} + \frac{7}{2} \,{\left (5 \, d + 6 \, e\right )} x^{12} + \frac{21}{5} \,{\left (6 \, d + 5 \, e\right )} x^{10} + \frac{15}{4} \,{\left (7 \, d + 4 \, e\right )} x^{8} + \frac{5}{2} \,{\left (8 \, d + 3 \, e\right )} x^{6} + \frac{5}{4} \,{\left (9 \, d + 2 \, e\right )} x^{4} + \frac{1}{2} \,{\left (10 \, d + e\right )} x^{2} + d \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.927214, size = 131, normalized size = 1.41 \[ d \log{\left (x \right )} + \frac{e x^{22}}{22} + x^{20} \left (\frac{d}{20} + \frac{e}{2}\right ) + x^{18} \left (\frac{5 d}{9} + \frac{5 e}{2}\right ) + x^{16} \left (\frac{45 d}{16} + \frac{15 e}{2}\right ) + x^{14} \left (\frac{60 d}{7} + 15 e\right ) + x^{12} \left (\frac{35 d}{2} + 21 e\right ) + x^{10} \left (\frac{126 d}{5} + 21 e\right ) + x^{8} \left (\frac{105 d}{4} + 15 e\right ) + x^{6} \left (20 d + \frac{15 e}{2}\right ) + x^{4} \left (\frac{45 d}{4} + \frac{5 e}{2}\right ) + x^{2} \left (5 d + \frac{e}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*log(x) + e*x**22/22 + x**20*(d/20 + e/2) + x**18*(5*d/9 + 5*e/2) + x**16*(45*d
/16 + 15*e/2) + x**14*(60*d/7 + 15*e) + x**12*(35*d/2 + 21*e) + x**10*(126*d/5 +
21*e) + x**8*(105*d/4 + 15*e) + x**6*(20*d + 15*e/2) + x**4*(45*d/4 + 5*e/2) +
x**2*(5*d + e/2)

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GIAC/XCAS [A]  time = 0.272075, size = 196, normalized size = 2.11 \[ \frac{1}{22} \, x^{22} e + \frac{1}{20} \, d x^{20} + \frac{1}{2} \, x^{20} e + \frac{5}{9} \, d x^{18} + \frac{5}{2} \, x^{18} e + \frac{45}{16} \, d x^{16} + \frac{15}{2} \, x^{16} e + \frac{60}{7} \, d x^{14} + 15 \, x^{14} e + \frac{35}{2} \, d x^{12} + 21 \, x^{12} e + \frac{126}{5} \, d x^{10} + 21 \, x^{10} e + \frac{105}{4} \, d x^{8} + 15 \, x^{8} e + 20 \, d x^{6} + \frac{15}{2} \, x^{6} e + \frac{45}{4} \, d x^{4} + \frac{5}{2} \, x^{4} e + 5 \, d x^{2} + \frac{1}{2} \, x^{2} e + \frac{1}{2} \, d{\rm ln}\left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/22*x^22*e + 1/20*d*x^20 + 1/2*x^20*e + 5/9*d*x^18 + 5/2*x^18*e + 45/16*d*x^16
+ 15/2*x^16*e + 60/7*d*x^14 + 15*x^14*e + 35/2*d*x^12 + 21*x^12*e + 126/5*d*x^10
+ 21*x^10*e + 105/4*d*x^8 + 15*x^8*e + 20*d*x^6 + 15/2*x^6*e + 45/4*d*x^4 + 5/2
*x^4*e + 5*d*x^2 + 1/2*x^2*e + 1/2*d*ln(x^2)