3.444 \(\int \frac{1}{\left (15+\frac{2}{x^2}+\frac{13}{x}\right ) x^2} \, dx\)

Optimal. Leaf size=23 \[ \frac{1}{7} \log \left (\frac{1}{x}+5\right )-\frac{1}{7} \log \left (\frac{2}{x}+3\right ) \]

[Out]

Log[5 + x^(-1)]/7 - Log[3 + 2/x]/7

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Rubi [A]  time = 0.0302982, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{1}{7} \log \left (\frac{1}{x}+5\right )-\frac{1}{7} \log \left (\frac{2}{x}+3\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/((15 + 2/x^2 + 13/x)*x^2),x]

[Out]

Log[5 + x^(-1)]/7 - Log[3 + 2/x]/7

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Rubi in Sympy [A]  time = 6.82851, size = 15, normalized size = 0.65 \[ - \frac{\log{\left (3 + \frac{2}{x} \right )}}{7} + \frac{\log{\left (5 + \frac{1}{x} \right )}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(15+2/x**2+13/x)/x**2,x)

[Out]

-log(3 + 2/x)/7 + log(5 + 1/x)/7

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Mathematica [A]  time = 0.00448552, size = 21, normalized size = 0.91 \[ \frac{1}{7} \log (5 x+1)-\frac{1}{7} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]  Integrate[1/((15 + 2/x^2 + 13/x)*x^2),x]

[Out]

-Log[2 + 3*x]/7 + Log[1 + 5*x]/7

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Maple [A]  time = 0.007, size = 18, normalized size = 0.8 \[{\frac{\ln \left ( 1+5\,x \right ) }{7}}-{\frac{\ln \left ( 2+3\,x \right ) }{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(15+2/x^2+13/x)/x^2,x)

[Out]

1/7*ln(1+5*x)-1/7*ln(2+3*x)

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Maxima [A]  time = 0.750124, size = 23, normalized size = 1. \[ \frac{1}{7} \, \log \left (5 \, x + 1\right ) - \frac{1}{7} \, \log \left (3 \, x + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^2*(13/x + 2/x^2 + 15)),x, algorithm="maxima")

[Out]

1/7*log(5*x + 1) - 1/7*log(3*x + 2)

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Fricas [A]  time = 0.246921, size = 23, normalized size = 1. \[ \frac{1}{7} \, \log \left (5 \, x + 1\right ) - \frac{1}{7} \, \log \left (3 \, x + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^2*(13/x + 2/x^2 + 15)),x, algorithm="fricas")

[Out]

1/7*log(5*x + 1) - 1/7*log(3*x + 2)

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Sympy [A]  time = 0.216806, size = 15, normalized size = 0.65 \[ \frac{\log{\left (x + \frac{1}{5} \right )}}{7} - \frac{\log{\left (x + \frac{2}{3} \right )}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(15+2/x**2+13/x)/x**2,x)

[Out]

log(x + 1/5)/7 - log(x + 2/3)/7

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GIAC/XCAS [A]  time = 0.301019, size = 26, normalized size = 1.13 \[ \frac{1}{7} \,{\rm ln}\left ({\left | 5 \, x + 1 \right |}\right ) - \frac{1}{7} \,{\rm ln}\left ({\left | 3 \, x + 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^2*(13/x + 2/x^2 + 15)),x, algorithm="giac")

[Out]

1/7*ln(abs(5*x + 1)) - 1/7*ln(abs(3*x + 2))