3.440 \(\int \frac{x^2}{15+\frac{2}{x^2}+\frac{13}{x}} \, dx\)

Optimal. Leaf size=40 \[ \frac{x^3}{45}-\frac{13 x^2}{450}+\frac{139 x}{3375}-\frac{16}{567} \log (3 x+2)+\frac{\log (5 x+1)}{4375} \]

[Out]

(139*x)/3375 - (13*x^2)/450 + x^3/45 - (16*Log[2 + 3*x])/567 + Log[1 + 5*x]/4375

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Rubi [A]  time = 0.0570015, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{x^3}{45}-\frac{13 x^2}{450}+\frac{139 x}{3375}-\frac{16}{567} \log (3 x+2)+\frac{\log (5 x+1)}{4375} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(15 + 2/x^2 + 13/x),x]

[Out]

(139*x)/3375 - (13*x^2)/450 + x^3/45 - (16*Log[2 + 3*x])/567 + Log[1 + 5*x]/4375

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{x^{3}}{45} - \frac{16 \log{\left (3 x + 2 \right )}}{567} + \frac{\log{\left (5 x + 1 \right )}}{4375} + \int \frac{139}{3375}\, dx - \frac{13 \int x\, dx}{225} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(15+2/x**2+13/x),x)

[Out]

x**3/45 - 16*log(3*x + 2)/567 + log(5*x + 1)/4375 + Integral(139/3375, x) - 13*I
ntegral(x, x)/225

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Mathematica [A]  time = 0.00880849, size = 40, normalized size = 1. \[ \frac{x^3}{45}-\frac{13 x^2}{450}+\frac{139 x}{3375}-\frac{16}{567} \log (3 x+2)+\frac{\log (5 x+1)}{4375} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(15 + 2/x^2 + 13/x),x]

[Out]

(139*x)/3375 - (13*x^2)/450 + x^3/45 - (16*Log[2 + 3*x])/567 + Log[1 + 5*x]/4375

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Maple [A]  time = 0.009, size = 31, normalized size = 0.8 \[{\frac{139\,x}{3375}}-{\frac{13\,{x}^{2}}{450}}+{\frac{{x}^{3}}{45}}-{\frac{16\,\ln \left ( 2+3\,x \right ) }{567}}+{\frac{\ln \left ( 1+5\,x \right ) }{4375}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(15+2/x^2+13/x),x)

[Out]

139/3375*x-13/450*x^2+1/45*x^3-16/567*ln(2+3*x)+1/4375*ln(1+5*x)

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Maxima [A]  time = 0.738535, size = 41, normalized size = 1.02 \[ \frac{1}{45} \, x^{3} - \frac{13}{450} \, x^{2} + \frac{139}{3375} \, x + \frac{1}{4375} \, \log \left (5 \, x + 1\right ) - \frac{16}{567} \, \log \left (3 \, x + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(13/x + 2/x^2 + 15),x, algorithm="maxima")

[Out]

1/45*x^3 - 13/450*x^2 + 139/3375*x + 1/4375*log(5*x + 1) - 16/567*log(3*x + 2)

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Fricas [A]  time = 0.263365, size = 41, normalized size = 1.02 \[ \frac{1}{45} \, x^{3} - \frac{13}{450} \, x^{2} + \frac{139}{3375} \, x + \frac{1}{4375} \, \log \left (5 \, x + 1\right ) - \frac{16}{567} \, \log \left (3 \, x + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(13/x + 2/x^2 + 15),x, algorithm="fricas")

[Out]

1/45*x^3 - 13/450*x^2 + 139/3375*x + 1/4375*log(5*x + 1) - 16/567*log(3*x + 2)

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Sympy [A]  time = 0.246941, size = 34, normalized size = 0.85 \[ \frac{x^{3}}{45} - \frac{13 x^{2}}{450} + \frac{139 x}{3375} + \frac{\log{\left (x + \frac{1}{5} \right )}}{4375} - \frac{16 \log{\left (x + \frac{2}{3} \right )}}{567} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(15+2/x**2+13/x),x)

[Out]

x**3/45 - 13*x**2/450 + 139*x/3375 + log(x + 1/5)/4375 - 16*log(x + 2/3)/567

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GIAC/XCAS [A]  time = 0.269196, size = 43, normalized size = 1.08 \[ \frac{1}{45} \, x^{3} - \frac{13}{450} \, x^{2} + \frac{139}{3375} \, x + \frac{1}{4375} \,{\rm ln}\left ({\left | 5 \, x + 1 \right |}\right ) - \frac{16}{567} \,{\rm ln}\left ({\left | 3 \, x + 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(13/x + 2/x^2 + 15),x, algorithm="giac")

[Out]

1/45*x^3 - 13/450*x^2 + 139/3375*x + 1/4375*ln(abs(5*x + 1)) - 16/567*ln(abs(3*x
+ 2))