3.445 \(\int \frac{1}{\left (15+\frac{2}{x^2}+\frac{13}{x}\right ) x^3} \, dx\)

Optimal. Leaf size=27 \[ \frac{\log (x)}{2}+\frac{3}{14} \log (3 x+2)-\frac{5}{7} \log (5 x+1) \]

[Out]

Log[x]/2 + (3*Log[2 + 3*x])/14 - (5*Log[1 + 5*x])/7

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Rubi [A]  time = 0.0437314, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278 \[ \frac{\log (x)}{2}+\frac{3}{14} \log (3 x+2)-\frac{5}{7} \log (5 x+1) \]

Antiderivative was successfully verified.

[In]  Int[1/((15 + 2/x^2 + 13/x)*x^3),x]

[Out]

Log[x]/2 + (3*Log[2 + 3*x])/14 - (5*Log[1 + 5*x])/7

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Rubi in Sympy [A]  time = 12.1255, size = 24, normalized size = 0.89 \[ \frac{\log{\left (x \right )}}{2} + \frac{3 \log{\left (3 x + 2 \right )}}{14} - \frac{5 \log{\left (5 x + 1 \right )}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(15+2/x**2+13/x)/x**3,x)

[Out]

log(x)/2 + 3*log(3*x + 2)/14 - 5*log(5*x + 1)/7

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Mathematica [A]  time = 0.00693499, size = 27, normalized size = 1. \[ \frac{\log (x)}{2}+\frac{3}{14} \log (3 x+2)-\frac{5}{7} \log (5 x+1) \]

Antiderivative was successfully verified.

[In]  Integrate[1/((15 + 2/x^2 + 13/x)*x^3),x]

[Out]

Log[x]/2 + (3*Log[2 + 3*x])/14 - (5*Log[1 + 5*x])/7

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Maple [A]  time = 0.009, size = 22, normalized size = 0.8 \[{\frac{\ln \left ( x \right ) }{2}}+{\frac{3\,\ln \left ( 2+3\,x \right ) }{14}}-{\frac{5\,\ln \left ( 1+5\,x \right ) }{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(15+2/x^2+13/x)/x^3,x)

[Out]

1/2*ln(x)+3/14*ln(2+3*x)-5/7*ln(1+5*x)

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Maxima [A]  time = 0.744873, size = 28, normalized size = 1.04 \[ -\frac{5}{7} \, \log \left (5 \, x + 1\right ) + \frac{3}{14} \, \log \left (3 \, x + 2\right ) + \frac{1}{2} \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^3*(13/x + 2/x^2 + 15)),x, algorithm="maxima")

[Out]

-5/7*log(5*x + 1) + 3/14*log(3*x + 2) + 1/2*log(x)

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Fricas [A]  time = 0.254148, size = 28, normalized size = 1.04 \[ -\frac{5}{7} \, \log \left (5 \, x + 1\right ) + \frac{3}{14} \, \log \left (3 \, x + 2\right ) + \frac{1}{2} \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^3*(13/x + 2/x^2 + 15)),x, algorithm="fricas")

[Out]

-5/7*log(5*x + 1) + 3/14*log(3*x + 2) + 1/2*log(x)

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Sympy [A]  time = 0.313965, size = 24, normalized size = 0.89 \[ \frac{\log{\left (x \right )}}{2} - \frac{5 \log{\left (x + \frac{1}{5} \right )}}{7} + \frac{3 \log{\left (x + \frac{2}{3} \right )}}{14} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(15+2/x**2+13/x)/x**3,x)

[Out]

log(x)/2 - 5*log(x + 1/5)/7 + 3*log(x + 2/3)/14

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GIAC/XCAS [A]  time = 0.265431, size = 32, normalized size = 1.19 \[ -\frac{5}{7} \,{\rm ln}\left ({\left | 5 \, x + 1 \right |}\right ) + \frac{3}{14} \,{\rm ln}\left ({\left | 3 \, x + 2 \right |}\right ) + \frac{1}{2} \,{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^3*(13/x + 2/x^2 + 15)),x, algorithm="giac")

[Out]

-5/7*ln(abs(5*x + 1)) + 3/14*ln(abs(3*x + 2)) + 1/2*ln(abs(x))