3.74 \(\int \frac{\left (1+x^2\right ) \left (1+2 x^2+x^4\right )^5}{x^3} \, dx\)

Optimal. Leaf size=80 \[ \frac{x^{20}}{20}+\frac{11 x^{18}}{18}+\frac{55 x^{16}}{16}+\frac{165 x^{14}}{14}+\frac{55 x^{12}}{2}+\frac{231 x^{10}}{5}+\frac{231 x^8}{4}+55 x^6+\frac{165 x^4}{4}+\frac{55 x^2}{2}-\frac{1}{2 x^2}+11 \log (x) \]

[Out]

-1/(2*x^2) + (55*x^2)/2 + (165*x^4)/4 + 55*x^6 + (231*x^8)/4 + (231*x^10)/5 + (5
5*x^12)/2 + (165*x^14)/14 + (55*x^16)/16 + (11*x^18)/18 + x^20/20 + 11*Log[x]

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Rubi [A]  time = 0.0802229, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{x^{20}}{20}+\frac{11 x^{18}}{18}+\frac{55 x^{16}}{16}+\frac{165 x^{14}}{14}+\frac{55 x^{12}}{2}+\frac{231 x^{10}}{5}+\frac{231 x^8}{4}+55 x^6+\frac{165 x^4}{4}+\frac{55 x^2}{2}-\frac{1}{2 x^2}+11 \log (x) \]

Antiderivative was successfully verified.

[In]  Int[((1 + x^2)*(1 + 2*x^2 + x^4)^5)/x^3,x]

[Out]

-1/(2*x^2) + (55*x^2)/2 + (165*x^4)/4 + 55*x^6 + (231*x^8)/4 + (231*x^10)/5 + (5
5*x^12)/2 + (165*x^14)/14 + (55*x^16)/16 + (11*x^18)/18 + x^20/20 + 11*Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{x^{20}}{20} + \frac{11 x^{18}}{18} + \frac{55 x^{16}}{16} + \frac{165 x^{14}}{14} + \frac{55 x^{12}}{2} + \frac{231 x^{10}}{5} + \frac{231 x^{8}}{4} + 55 x^{6} + \frac{55 x^{2}}{2} + \frac{11 \log{\left (x^{2} \right )}}{2} + \frac{165 \int ^{x^{2}} x\, dx}{2} - \frac{1}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**2+1)*(x**4+2*x**2+1)**5/x**3,x)

[Out]

x**20/20 + 11*x**18/18 + 55*x**16/16 + 165*x**14/14 + 55*x**12/2 + 231*x**10/5 +
231*x**8/4 + 55*x**6 + 55*x**2/2 + 11*log(x**2)/2 + 165*Integral(x, (x, x**2))/
2 - 1/(2*x**2)

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Mathematica [A]  time = 0.00445, size = 80, normalized size = 1. \[ \frac{x^{20}}{20}+\frac{11 x^{18}}{18}+\frac{55 x^{16}}{16}+\frac{165 x^{14}}{14}+\frac{55 x^{12}}{2}+\frac{231 x^{10}}{5}+\frac{231 x^8}{4}+55 x^6+\frac{165 x^4}{4}+\frac{55 x^2}{2}-\frac{1}{2 x^2}+11 \log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[((1 + x^2)*(1 + 2*x^2 + x^4)^5)/x^3,x]

[Out]

-1/(2*x^2) + (55*x^2)/2 + (165*x^4)/4 + 55*x^6 + (231*x^8)/4 + (231*x^10)/5 + (5
5*x^12)/2 + (165*x^14)/14 + (55*x^16)/16 + (11*x^18)/18 + x^20/20 + 11*Log[x]

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Maple [A]  time = 0.008, size = 61, normalized size = 0.8 \[ -{\frac{1}{2\,{x}^{2}}}+{\frac{55\,{x}^{2}}{2}}+{\frac{165\,{x}^{4}}{4}}+55\,{x}^{6}+{\frac{231\,{x}^{8}}{4}}+{\frac{231\,{x}^{10}}{5}}+{\frac{55\,{x}^{12}}{2}}+{\frac{165\,{x}^{14}}{14}}+{\frac{55\,{x}^{16}}{16}}+{\frac{11\,{x}^{18}}{18}}+{\frac{{x}^{20}}{20}}+11\,\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^2+1)*(x^4+2*x^2+1)^5/x^3,x)

[Out]

-1/2/x^2+55/2*x^2+165/4*x^4+55*x^6+231/4*x^8+231/5*x^10+55/2*x^12+165/14*x^14+55
/16*x^16+11/18*x^18+1/20*x^20+11*ln(x)

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Maxima [A]  time = 0.69737, size = 84, normalized size = 1.05 \[ \frac{1}{20} \, x^{20} + \frac{11}{18} \, x^{18} + \frac{55}{16} \, x^{16} + \frac{165}{14} \, x^{14} + \frac{55}{2} \, x^{12} + \frac{231}{5} \, x^{10} + \frac{231}{4} \, x^{8} + 55 \, x^{6} + \frac{165}{4} \, x^{4} + \frac{55}{2} \, x^{2} - \frac{1}{2 \, x^{2}} + \frac{11}{2} \, \log \left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^4 + 2*x^2 + 1)^5*(x^2 + 1)/x^3,x, algorithm="maxima")

[Out]

1/20*x^20 + 11/18*x^18 + 55/16*x^16 + 165/14*x^14 + 55/2*x^12 + 231/5*x^10 + 231
/4*x^8 + 55*x^6 + 165/4*x^4 + 55/2*x^2 - 1/2/x^2 + 11/2*log(x^2)

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Fricas [A]  time = 0.249939, size = 86, normalized size = 1.08 \[ \frac{252 \, x^{22} + 3080 \, x^{20} + 17325 \, x^{18} + 59400 \, x^{16} + 138600 \, x^{14} + 232848 \, x^{12} + 291060 \, x^{10} + 277200 \, x^{8} + 207900 \, x^{6} + 138600 \, x^{4} + 55440 \, x^{2} \log \left (x\right ) - 2520}{5040 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^4 + 2*x^2 + 1)^5*(x^2 + 1)/x^3,x, algorithm="fricas")

[Out]

1/5040*(252*x^22 + 3080*x^20 + 17325*x^18 + 59400*x^16 + 138600*x^14 + 232848*x^
12 + 291060*x^10 + 277200*x^8 + 207900*x^6 + 138600*x^4 + 55440*x^2*log(x) - 252
0)/x^2

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Sympy [A]  time = 0.228366, size = 75, normalized size = 0.94 \[ \frac{x^{20}}{20} + \frac{11 x^{18}}{18} + \frac{55 x^{16}}{16} + \frac{165 x^{14}}{14} + \frac{55 x^{12}}{2} + \frac{231 x^{10}}{5} + \frac{231 x^{8}}{4} + 55 x^{6} + \frac{165 x^{4}}{4} + \frac{55 x^{2}}{2} + 11 \log{\left (x \right )} - \frac{1}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**2+1)*(x**4+2*x**2+1)**5/x**3,x)

[Out]

x**20/20 + 11*x**18/18 + 55*x**16/16 + 165*x**14/14 + 55*x**12/2 + 231*x**10/5 +
231*x**8/4 + 55*x**6 + 165*x**4/4 + 55*x**2/2 + 11*log(x) - 1/(2*x**2)

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GIAC/XCAS [A]  time = 0.263279, size = 93, normalized size = 1.16 \[ \frac{1}{20} \, x^{20} + \frac{11}{18} \, x^{18} + \frac{55}{16} \, x^{16} + \frac{165}{14} \, x^{14} + \frac{55}{2} \, x^{12} + \frac{231}{5} \, x^{10} + \frac{231}{4} \, x^{8} + 55 \, x^{6} + \frac{165}{4} \, x^{4} + \frac{55}{2} \, x^{2} - \frac{11 \, x^{2} + 1}{2 \, x^{2}} + \frac{11}{2} \,{\rm ln}\left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^4 + 2*x^2 + 1)^5*(x^2 + 1)/x^3,x, algorithm="giac")

[Out]

1/20*x^20 + 11/18*x^18 + 55/16*x^16 + 165/14*x^14 + 55/2*x^12 + 231/5*x^10 + 231
/4*x^8 + 55*x^6 + 165/4*x^4 + 55/2*x^2 - 1/2*(11*x^2 + 1)/x^2 + 11/2*ln(x^2)