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

Optimal. Leaf size=33 \[ \frac{x^2}{30}-\frac{13 x}{225}+\frac{8}{189} \log (3 x+2)-\frac{1}{875} \log (5 x+1) \]

[Out]

(-13*x)/225 + x^2/30 + (8*Log[2 + 3*x])/189 - Log[1 + 5*x]/875

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Rubi [A]  time = 0.057556, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{x^2}{30}-\frac{13 x}{225}+\frac{8}{189} \log (3 x+2)-\frac{1}{875} \log (5 x+1) \]

Antiderivative was successfully verified.

[In]  Int[x^2/(13 + 2/x + 15*x),x]

[Out]

(-13*x)/225 + x^2/30 + (8*Log[2 + 3*x])/189 - Log[1 + 5*x]/875

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{8 \log{\left (3 x + 2 \right )}}{189} - \frac{\log{\left (5 x + 1 \right )}}{875} + \int \left (- \frac{13}{225}\right )\, dx + \frac{\int x\, dx}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

8*log(3*x + 2)/189 - log(5*x + 1)/875 + Integral(-13/225, x) + Integral(x, x)/15

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Mathematica [A]  time = 0.00672796, size = 33, normalized size = 1. \[ \frac{x^2}{30}-\frac{13 x}{225}+\frac{8}{189} \log (3 x+2)-\frac{1}{875} \log (5 x+1) \]

Antiderivative was successfully verified.

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

[Out]

(-13*x)/225 + x^2/30 + (8*Log[2 + 3*x])/189 - Log[1 + 5*x]/875

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Maple [A]  time = 0.008, size = 26, normalized size = 0.8 \[ -{\frac{13\,x}{225}}+{\frac{{x}^{2}}{30}}+{\frac{8\,\ln \left ( 2+3\,x \right ) }{189}}-{\frac{\ln \left ( 1+5\,x \right ) }{875}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(13+2/x+15*x),x)

[Out]

-13/225*x+1/30*x^2+8/189*ln(2+3*x)-1/875*ln(1+5*x)

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Maxima [A]  time = 0.829853, size = 34, normalized size = 1.03 \[ \frac{1}{30} \, x^{2} - \frac{13}{225} \, x - \frac{1}{875} \, \log \left (5 \, x + 1\right ) + \frac{8}{189} \, \log \left (3 \, x + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/30*x^2 - 13/225*x - 1/875*log(5*x + 1) + 8/189*log(3*x + 2)

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Fricas [A]  time = 0.267203, size = 34, normalized size = 1.03 \[ \frac{1}{30} \, x^{2} - \frac{13}{225} \, x - \frac{1}{875} \, \log \left (5 \, x + 1\right ) + \frac{8}{189} \, \log \left (3 \, x + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/30*x^2 - 13/225*x - 1/875*log(5*x + 1) + 8/189*log(3*x + 2)

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Sympy [A]  time = 0.241234, size = 27, normalized size = 0.82 \[ \frac{x^{2}}{30} - \frac{13 x}{225} - \frac{\log{\left (x + \frac{1}{5} \right )}}{875} + \frac{8 \log{\left (x + \frac{2}{3} \right )}}{189} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(13+2/x+15*x),x)

[Out]

x**2/30 - 13*x/225 - log(x + 1/5)/875 + 8*log(x + 2/3)/189

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GIAC/XCAS [A]  time = 0.261983, size = 36, normalized size = 1.09 \[ \frac{1}{30} \, x^{2} - \frac{13}{225} \, x - \frac{1}{875} \,{\rm ln}\left ({\left | 5 \, x + 1 \right |}\right ) + \frac{8}{189} \,{\rm ln}\left ({\left | 3 \, x + 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/30*x^2 - 13/225*x - 1/875*ln(abs(5*x + 1)) + 8/189*ln(abs(3*x + 2))