3.446 \(\int \frac{1}{\left (15+\frac{2}{x^2}+\frac{13}{x}\right ) x^4} \, dx\)

Optimal. Leaf size=34 \[ -\frac{1}{2 x}-\frac{13 \log (x)}{4}-\frac{9}{28} \log (3 x+2)+\frac{25}{7} \log (5 x+1) \]

[Out]

-1/(2*x) - (13*Log[x])/4 - (9*Log[2 + 3*x])/28 + (25*Log[1 + 5*x])/7

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Rubi [A]  time = 0.0751486, antiderivative size = 34, 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 \[ -\frac{1}{2 x}-\frac{13 \log (x)}{4}-\frac{9}{28} \log (3 x+2)+\frac{25}{7} \log (5 x+1) \]

Antiderivative was successfully verified.

[In]  Int[1/((15 + 2/x^2 + 13/x)*x^4),x]

[Out]

-1/(2*x) - (13*Log[x])/4 - (9*Log[2 + 3*x])/28 + (25*Log[1 + 5*x])/7

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Rubi in Sympy [A]  time = 17.3745, size = 31, normalized size = 0.91 \[ - \frac{13 \log{\left (x \right )}}{4} - \frac{9 \log{\left (3 x + 2 \right )}}{28} + \frac{25 \log{\left (5 x + 1 \right )}}{7} - \frac{1}{2 x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(15+2/x**2+13/x)/x**4,x)

[Out]

-13*log(x)/4 - 9*log(3*x + 2)/28 + 25*log(5*x + 1)/7 - 1/(2*x)

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Mathematica [A]  time = 0.0064387, size = 34, normalized size = 1. \[ -\frac{1}{2 x}-\frac{13 \log (x)}{4}-\frac{9}{28} \log (3 x+2)+\frac{25}{7} \log (5 x+1) \]

Antiderivative was successfully verified.

[In]  Integrate[1/((15 + 2/x^2 + 13/x)*x^4),x]

[Out]

-1/(2*x) - (13*Log[x])/4 - (9*Log[2 + 3*x])/28 + (25*Log[1 + 5*x])/7

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Maple [A]  time = 0.012, size = 27, normalized size = 0.8 \[ -{\frac{1}{2\,x}}-{\frac{13\,\ln \left ( x \right ) }{4}}-{\frac{9\,\ln \left ( 2+3\,x \right ) }{28}}+{\frac{25\,\ln \left ( 1+5\,x \right ) }{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(15+2/x^2+13/x)/x^4,x)

[Out]

-1/2/x-13/4*ln(x)-9/28*ln(2+3*x)+25/7*ln(1+5*x)

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Maxima [A]  time = 0.742661, size = 35, normalized size = 1.03 \[ -\frac{1}{2 \, x} + \frac{25}{7} \, \log \left (5 \, x + 1\right ) - \frac{9}{28} \, \log \left (3 \, x + 2\right ) - \frac{13}{4} \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^4*(13/x + 2/x^2 + 15)),x, algorithm="maxima")

[Out]

-1/2/x + 25/7*log(5*x + 1) - 9/28*log(3*x + 2) - 13/4*log(x)

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Fricas [A]  time = 0.257558, size = 41, normalized size = 1.21 \[ \frac{100 \, x \log \left (5 \, x + 1\right ) - 9 \, x \log \left (3 \, x + 2\right ) - 91 \, x \log \left (x\right ) - 14}{28 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^4*(13/x + 2/x^2 + 15)),x, algorithm="fricas")

[Out]

1/28*(100*x*log(5*x + 1) - 9*x*log(3*x + 2) - 91*x*log(x) - 14)/x

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Sympy [A]  time = 0.364334, size = 31, normalized size = 0.91 \[ - \frac{13 \log{\left (x \right )}}{4} + \frac{25 \log{\left (x + \frac{1}{5} \right )}}{7} - \frac{9 \log{\left (x + \frac{2}{3} \right )}}{28} - \frac{1}{2 x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(15+2/x**2+13/x)/x**4,x)

[Out]

-13*log(x)/4 + 25*log(x + 1/5)/7 - 9*log(x + 2/3)/28 - 1/(2*x)

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GIAC/XCAS [A]  time = 0.284736, size = 39, normalized size = 1.15 \[ -\frac{1}{2 \, x} + \frac{25}{7} \,{\rm ln}\left ({\left | 5 \, x + 1 \right |}\right ) - \frac{9}{28} \,{\rm ln}\left ({\left | 3 \, x + 2 \right |}\right ) - \frac{13}{4} \,{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^4*(13/x + 2/x^2 + 15)),x, algorithm="giac")

[Out]

-1/2/x + 25/7*ln(abs(5*x + 1)) - 9/28*ln(abs(3*x + 2)) - 13/4*ln(abs(x))