3.2227 \(\int \frac{x^2}{2+13 x+15 x^2} \, dx\)

Optimal. Leaf size=26 \[ \frac{x}{15}-\frac{4}{63} \log (3 x+2)+\frac{1}{175} \log (5 x+1) \]

[Out]

x/15 - (4*Log[2 + 3*x])/63 + Log[1 + 5*x]/175

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Rubi [A]  time = 0.0385346, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ \frac{x}{15}-\frac{4}{63} \log (3 x+2)+\frac{1}{175} \log (5 x+1) \]

Antiderivative was successfully verified.

[In]  Int[x^2/(2 + 13*x + 15*x^2),x]

[Out]

x/15 - (4*Log[2 + 3*x])/63 + Log[1 + 5*x]/175

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Rubi in Sympy [A]  time = 7.28232, size = 20, normalized size = 0.77 \[ \frac{x}{15} - \frac{4 \log{\left (3 x + 2 \right )}}{63} + \frac{\log{\left (5 x + 1 \right )}}{175} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x/15 - 4*log(3*x + 2)/63 + log(5*x + 1)/175

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Mathematica [A]  time = 0.00677276, size = 26, normalized size = 1. \[ \frac{x}{15}-\frac{4}{63} \log (3 x+2)+\frac{1}{175} \log (5 x+1) \]

Antiderivative was successfully verified.

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

[Out]

x/15 - (4*Log[2 + 3*x])/63 + Log[1 + 5*x]/175

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Maple [A]  time = 0.008, size = 21, normalized size = 0.8 \[{\frac{x}{15}}-{\frac{4\,\ln \left ( 2+3\,x \right ) }{63}}+{\frac{\ln \left ( 1+5\,x \right ) }{175}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(15*x^2+13*x+2),x)

[Out]

1/15*x-4/63*ln(2+3*x)+1/175*ln(1+5*x)

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Maxima [A]  time = 0.825219, size = 27, normalized size = 1.04 \[ \frac{1}{15} \, x + \frac{1}{175} \, \log \left (5 \, x + 1\right ) - \frac{4}{63} \, \log \left (3 \, x + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*x + 1/175*log(5*x + 1) - 4/63*log(3*x + 2)

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Fricas [A]  time = 0.213175, size = 27, normalized size = 1.04 \[ \frac{1}{15} \, x + \frac{1}{175} \, \log \left (5 \, x + 1\right ) - \frac{4}{63} \, \log \left (3 \, x + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*x + 1/175*log(5*x + 1) - 4/63*log(3*x + 2)

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Sympy [A]  time = 0.212301, size = 20, normalized size = 0.77 \[ \frac{x}{15} + \frac{\log{\left (x + \frac{1}{5} \right )}}{175} - \frac{4 \log{\left (x + \frac{2}{3} \right )}}{63} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x/15 + log(x + 1/5)/175 - 4*log(x + 2/3)/63

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GIAC/XCAS [A]  time = 0.204557, size = 30, normalized size = 1.15 \[ \frac{1}{15} \, x + \frac{1}{175} \,{\rm ln}\left ({\left | 5 \, x + 1 \right |}\right ) - \frac{4}{63} \,{\rm ln}\left ({\left | 3 \, x + 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*x + 1/175*ln(abs(5*x + 1)) - 4/63*ln(abs(3*x + 2))