3.448 \(\int \frac{1}{\left (15+\frac{2}{x^2}+\frac{13}{x}\right ) x^6} \, dx\)

Optimal. Leaf size=48 \[ -\frac{1}{6 x^3}+\frac{13}{8 x^2}-\frac{139}{8 x}-\frac{1417 \log (x)}{16}-\frac{81}{112} \log (3 x+2)+\frac{625}{7} \log (5 x+1) \]

[Out]

-1/(6*x^3) + 13/(8*x^2) - 139/(8*x) - (1417*Log[x])/16 - (81*Log[2 + 3*x])/112 +
(625*Log[1 + 5*x])/7

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Rubi [A]  time = 0.0943076, antiderivative size = 48, 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 \[ -\frac{1}{6 x^3}+\frac{13}{8 x^2}-\frac{139}{8 x}-\frac{1417 \log (x)}{16}-\frac{81}{112} \log (3 x+2)+\frac{625}{7} \log (5 x+1) \]

Antiderivative was successfully verified.

[In]  Int[1/((15 + 2/x^2 + 13/x)*x^6),x]

[Out]

-1/(6*x^3) + 13/(8*x^2) - 139/(8*x) - (1417*Log[x])/16 - (81*Log[2 + 3*x])/112 +
(625*Log[1 + 5*x])/7

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Rubi in Sympy [A]  time = 17.4766, size = 44, normalized size = 0.92 \[ - \frac{1417 \log{\left (x \right )}}{16} - \frac{81 \log{\left (3 x + 2 \right )}}{112} + \frac{625 \log{\left (5 x + 1 \right )}}{7} - \frac{139}{8 x} + \frac{13}{8 x^{2}} - \frac{1}{6 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(15+2/x**2+13/x)/x**6,x)

[Out]

-1417*log(x)/16 - 81*log(3*x + 2)/112 + 625*log(5*x + 1)/7 - 139/(8*x) + 13/(8*x
**2) - 1/(6*x**3)

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Mathematica [A]  time = 0.00803701, size = 48, normalized size = 1. \[ -\frac{1}{6 x^3}+\frac{13}{8 x^2}-\frac{139}{8 x}-\frac{1417 \log (x)}{16}-\frac{81}{112} \log (3 x+2)+\frac{625}{7} \log (5 x+1) \]

Antiderivative was successfully verified.

[In]  Integrate[1/((15 + 2/x^2 + 13/x)*x^6),x]

[Out]

-1/(6*x^3) + 13/(8*x^2) - 139/(8*x) - (1417*Log[x])/16 - (81*Log[2 + 3*x])/112 +
(625*Log[1 + 5*x])/7

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Maple [A]  time = 0.012, size = 37, normalized size = 0.8 \[ -{\frac{1}{6\,{x}^{3}}}+{\frac{13}{8\,{x}^{2}}}-{\frac{139}{8\,x}}-{\frac{1417\,\ln \left ( x \right ) }{16}}-{\frac{81\,\ln \left ( 2+3\,x \right ) }{112}}+{\frac{625\,\ln \left ( 1+5\,x \right ) }{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(15+2/x^2+13/x)/x^6,x)

[Out]

-1/6/x^3+13/8/x^2-139/8/x-1417/16*ln(x)-81/112*ln(2+3*x)+625/7*ln(1+5*x)

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Maxima [A]  time = 0.750557, size = 49, normalized size = 1.02 \[ -\frac{417 \, x^{2} - 39 \, x + 4}{24 \, x^{3}} + \frac{625}{7} \, \log \left (5 \, x + 1\right ) - \frac{81}{112} \, \log \left (3 \, x + 2\right ) - \frac{1417}{16} \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^6*(13/x + 2/x^2 + 15)),x, algorithm="maxima")

[Out]

-1/24*(417*x^2 - 39*x + 4)/x^3 + 625/7*log(5*x + 1) - 81/112*log(3*x + 2) - 1417
/16*log(x)

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Fricas [A]  time = 0.253734, size = 59, normalized size = 1.23 \[ \frac{30000 \, x^{3} \log \left (5 \, x + 1\right ) - 243 \, x^{3} \log \left (3 \, x + 2\right ) - 29757 \, x^{3} \log \left (x\right ) - 5838 \, x^{2} + 546 \, x - 56}{336 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^6*(13/x + 2/x^2 + 15)),x, algorithm="fricas")

[Out]

1/336*(30000*x^3*log(5*x + 1) - 243*x^3*log(3*x + 2) - 29757*x^3*log(x) - 5838*x
^2 + 546*x - 56)/x^3

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Sympy [A]  time = 0.443151, size = 41, normalized size = 0.85 \[ - \frac{1417 \log{\left (x \right )}}{16} + \frac{625 \log{\left (x + \frac{1}{5} \right )}}{7} - \frac{81 \log{\left (x + \frac{2}{3} \right )}}{112} - \frac{417 x^{2} - 39 x + 4}{24 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(15+2/x**2+13/x)/x**6,x)

[Out]

-1417*log(x)/16 + 625*log(x + 1/5)/7 - 81*log(x + 2/3)/112 - (417*x**2 - 39*x +
4)/(24*x**3)

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GIAC/XCAS [A]  time = 0.281253, size = 53, normalized size = 1.1 \[ -\frac{417 \, x^{2} - 39 \, x + 4}{24 \, x^{3}} + \frac{625}{7} \,{\rm ln}\left ({\left | 5 \, x + 1 \right |}\right ) - \frac{81}{112} \,{\rm ln}\left ({\left | 3 \, x + 2 \right |}\right ) - \frac{1417}{16} \,{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^6*(13/x + 2/x^2 + 15)),x, algorithm="giac")

[Out]

-1/24*(417*x^2 - 39*x + 4)/x^3 + 625/7*ln(abs(5*x + 1)) - 81/112*ln(abs(3*x + 2)
) - 1417/16*ln(abs(x))