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

Optimal. Leaf size=21 \[ \frac{2}{21} \log (3 x+2)-\frac{1}{35} \log (5 x+1) \]

[Out]

(2*Log[2 + 3*x])/21 - Log[1 + 5*x]/35

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Rubi [A]  time = 0.030687, antiderivative size = 21, 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 \[ \frac{2}{21} \log (3 x+2)-\frac{1}{35} \log (5 x+1) \]

Antiderivative was successfully verified.

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

[Out]

(2*Log[2 + 3*x])/21 - Log[1 + 5*x]/35

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Rubi in Sympy [A]  time = 9.10233, size = 17, normalized size = 0.81 \[ \frac{2 \log{\left (3 x + 2 \right )}}{21} - \frac{\log{\left (5 x + 1 \right )}}{35} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*log(3*x + 2)/21 - log(5*x + 1)/35

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Mathematica [A]  time = 0.00506181, size = 21, normalized size = 1. \[ \frac{2}{21} \log (3 x+2)-\frac{1}{35} \log (5 x+1) \]

Antiderivative was successfully verified.

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

[Out]

(2*Log[2 + 3*x])/21 - Log[1 + 5*x]/35

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Maple [A]  time = 0.008, size = 18, normalized size = 0.9 \[{\frac{2\,\ln \left ( 2+3\,x \right ) }{21}}-{\frac{\ln \left ( 1+5\,x \right ) }{35}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(15+2/x^2+13/x)/x,x)

[Out]

2/21*ln(2+3*x)-1/35*ln(1+5*x)

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Maxima [A]  time = 0.743437, size = 23, normalized size = 1.1 \[ -\frac{1}{35} \, \log \left (5 \, x + 1\right ) + \frac{2}{21} \, \log \left (3 \, x + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/35*log(5*x + 1) + 2/21*log(3*x + 2)

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Fricas [A]  time = 0.286427, size = 23, normalized size = 1.1 \[ -\frac{1}{35} \, \log \left (5 \, x + 1\right ) + \frac{2}{21} \, \log \left (3 \, x + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/35*log(5*x + 1) + 2/21*log(3*x + 2)

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Sympy [A]  time = 0.217867, size = 17, normalized size = 0.81 \[ - \frac{\log{\left (x + \frac{1}{5} \right )}}{35} + \frac{2 \log{\left (x + \frac{2}{3} \right )}}{21} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-log(x + 1/5)/35 + 2*log(x + 2/3)/21

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GIAC/XCAS [A]  time = 0.297453, size = 26, normalized size = 1.24 \[ -\frac{1}{35} \,{\rm ln}\left ({\left | 5 \, x + 1 \right |}\right ) + \frac{2}{21} \,{\rm ln}\left ({\left | 3 \, x + 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/35*ln(abs(5*x + 1)) + 2/21*ln(abs(3*x + 2))