3.447 \(\int \frac{1}{\left (15+\frac{2}{x^2}+\frac{13}{x}\right ) x^5} \, dx\)

Optimal. Leaf size=41 \[ -\frac{1}{4 x^2}+\frac{13}{4 x}+\frac{139 \log (x)}{8}+\frac{27}{56} \log (3 x+2)-\frac{125}{7} \log (5 x+1) \]

[Out]

-1/(4*x^2) + 13/(4*x) + (139*Log[x])/8 + (27*Log[2 + 3*x])/56 - (125*Log[1 + 5*x
])/7

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Rubi [A]  time = 0.0817365, antiderivative size = 41, 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 \[ -\frac{1}{4 x^2}+\frac{13}{4 x}+\frac{139 \log (x)}{8}+\frac{27}{56} \log (3 x+2)-\frac{125}{7} \log (5 x+1) \]

Antiderivative was successfully verified.

[In]  Int[1/((15 + 2/x^2 + 13/x)*x^5),x]

[Out]

-1/(4*x^2) + 13/(4*x) + (139*Log[x])/8 + (27*Log[2 + 3*x])/56 - (125*Log[1 + 5*x
])/7

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Rubi in Sympy [A]  time = 17.6277, size = 37, normalized size = 0.9 \[ \frac{139 \log{\left (x \right )}}{8} + \frac{27 \log{\left (3 x + 2 \right )}}{56} - \frac{125 \log{\left (5 x + 1 \right )}}{7} + \frac{13}{4 x} - \frac{1}{4 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(15+2/x**2+13/x)/x**5,x)

[Out]

139*log(x)/8 + 27*log(3*x + 2)/56 - 125*log(5*x + 1)/7 + 13/(4*x) - 1/(4*x**2)

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Mathematica [A]  time = 0.00765143, size = 41, normalized size = 1. \[ -\frac{1}{4 x^2}+\frac{13}{4 x}+\frac{139 \log (x)}{8}+\frac{27}{56} \log (3 x+2)-\frac{125}{7} \log (5 x+1) \]

Antiderivative was successfully verified.

[In]  Integrate[1/((15 + 2/x^2 + 13/x)*x^5),x]

[Out]

-1/(4*x^2) + 13/(4*x) + (139*Log[x])/8 + (27*Log[2 + 3*x])/56 - (125*Log[1 + 5*x
])/7

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Maple [A]  time = 0.012, size = 32, normalized size = 0.8 \[ -{\frac{1}{4\,{x}^{2}}}+{\frac{13}{4\,x}}+{\frac{139\,\ln \left ( x \right ) }{8}}+{\frac{27\,\ln \left ( 2+3\,x \right ) }{56}}-{\frac{125\,\ln \left ( 1+5\,x \right ) }{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(15+2/x^2+13/x)/x^5,x)

[Out]

-1/4/x^2+13/4/x+139/8*ln(x)+27/56*ln(2+3*x)-125/7*ln(1+5*x)

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Maxima [A]  time = 0.740028, size = 42, normalized size = 1.02 \[ \frac{13 \, x - 1}{4 \, x^{2}} - \frac{125}{7} \, \log \left (5 \, x + 1\right ) + \frac{27}{56} \, \log \left (3 \, x + 2\right ) + \frac{139}{8} \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^5*(13/x + 2/x^2 + 15)),x, algorithm="maxima")

[Out]

1/4*(13*x - 1)/x^2 - 125/7*log(5*x + 1) + 27/56*log(3*x + 2) + 139/8*log(x)

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Fricas [A]  time = 0.253013, size = 53, normalized size = 1.29 \[ -\frac{1000 \, x^{2} \log \left (5 \, x + 1\right ) - 27 \, x^{2} \log \left (3 \, x + 2\right ) - 973 \, x^{2} \log \left (x\right ) - 182 \, x + 14}{56 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^5*(13/x + 2/x^2 + 15)),x, algorithm="fricas")

[Out]

-1/56*(1000*x^2*log(5*x + 1) - 27*x^2*log(3*x + 2) - 973*x^2*log(x) - 182*x + 14
)/x^2

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Sympy [A]  time = 0.396951, size = 36, normalized size = 0.88 \[ \frac{139 \log{\left (x \right )}}{8} - \frac{125 \log{\left (x + \frac{1}{5} \right )}}{7} + \frac{27 \log{\left (x + \frac{2}{3} \right )}}{56} + \frac{13 x - 1}{4 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(15+2/x**2+13/x)/x**5,x)

[Out]

139*log(x)/8 - 125*log(x + 1/5)/7 + 27*log(x + 2/3)/56 + (13*x - 1)/(4*x**2)

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GIAC/XCAS [A]  time = 0.280195, size = 46, normalized size = 1.12 \[ \frac{13 \, x - 1}{4 \, x^{2}} - \frac{125}{7} \,{\rm ln}\left ({\left | 5 \, x + 1 \right |}\right ) + \frac{27}{56} \,{\rm ln}\left ({\left | 3 \, x + 2 \right |}\right ) + \frac{139}{8} \,{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^5*(13/x + 2/x^2 + 15)),x, algorithm="giac")

[Out]

1/4*(13*x - 1)/x^2 - 125/7*ln(abs(5*x + 1)) + 27/56*ln(abs(3*x + 2)) + 139/8*ln(
abs(x))