3.258 \(\int \frac{1+x^5}{-10 x-3 x^2+x^3} \, dx\)

Optimal. Leaf size=42 \[ \frac{x^3}{3}+\frac{3 x^2}{2}+19 x+\frac{3126}{35} \log (5-x)-\frac{\log (x)}{10}-\frac{31}{14} \log (x+2) \]

[Out]

19*x + (3*x^2)/2 + x^3/3 + (3126*Log[5 - x])/35 - Log[x]/10 - (31*Log[2 + x])/14

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Rubi [A]  time = 0.0543834, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{x^3}{3}+\frac{3 x^2}{2}+19 x+\frac{3126}{35} \log (5-x)-\frac{\log (x)}{10}-\frac{31}{14} \log (x+2) \]

Antiderivative was successfully verified.

[In]  Int[(1 + x^5)/(-10*x - 3*x^2 + x^3),x]

[Out]

19*x + (3*x^2)/2 + x^3/3 + (3126*Log[5 - x])/35 - Log[x]/10 - (31*Log[2 + x])/14

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{x^{3}}{3} + 19 x - \frac{\log{\left (x \right )}}{10} + \frac{3126 \log{\left (- x + 5 \right )}}{35} - \frac{31 \log{\left (x + 2 \right )}}{14} + 3 \int x\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**5+1)/(x**3-3*x**2-10*x),x)

[Out]

x**3/3 + 19*x - log(x)/10 + 3126*log(-x + 5)/35 - 31*log(x + 2)/14 + 3*Integral(
x, x)

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Mathematica [A]  time = 0.0106007, size = 42, normalized size = 1. \[ \frac{x^3}{3}+\frac{3 x^2}{2}+19 x+\frac{3126}{35} \log (5-x)-\frac{\log (x)}{10}-\frac{31}{14} \log (x+2) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + x^5)/(-10*x - 3*x^2 + x^3),x]

[Out]

19*x + (3*x^2)/2 + x^3/3 + (3126*Log[5 - x])/35 - Log[x]/10 - (31*Log[2 + x])/14

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Maple [A]  time = 0.012, size = 31, normalized size = 0.7 \[{\frac{{x}^{3}}{3}}+{\frac{3\,{x}^{2}}{2}}+19\,x+{\frac{3126\,\ln \left ( -5+x \right ) }{35}}-{\frac{31\,\ln \left ( 2+x \right ) }{14}}-{\frac{\ln \left ( x \right ) }{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^5+1)/(x^3-3*x^2-10*x),x)

[Out]

1/3*x^3+3/2*x^2+19*x+3126/35*ln(-5+x)-31/14*ln(2+x)-1/10*ln(x)

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Maxima [A]  time = 1.36846, size = 41, normalized size = 0.98 \[ \frac{1}{3} \, x^{3} + \frac{3}{2} \, x^{2} + 19 \, x - \frac{31}{14} \, \log \left (x + 2\right ) + \frac{3126}{35} \, \log \left (x - 5\right ) - \frac{1}{10} \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^5 + 1)/(x^3 - 3*x^2 - 10*x),x, algorithm="maxima")

[Out]

1/3*x^3 + 3/2*x^2 + 19*x - 31/14*log(x + 2) + 3126/35*log(x - 5) - 1/10*log(x)

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Fricas [A]  time = 0.201931, size = 41, normalized size = 0.98 \[ \frac{1}{3} \, x^{3} + \frac{3}{2} \, x^{2} + 19 \, x - \frac{31}{14} \, \log \left (x + 2\right ) + \frac{3126}{35} \, \log \left (x - 5\right ) - \frac{1}{10} \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^5 + 1)/(x^3 - 3*x^2 - 10*x),x, algorithm="fricas")

[Out]

1/3*x^3 + 3/2*x^2 + 19*x - 31/14*log(x + 2) + 3126/35*log(x - 5) - 1/10*log(x)

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Sympy [A]  time = 0.162548, size = 36, normalized size = 0.86 \[ \frac{x^{3}}{3} + \frac{3 x^{2}}{2} + 19 x - \frac{\log{\left (x \right )}}{10} + \frac{3126 \log{\left (x - 5 \right )}}{35} - \frac{31 \log{\left (x + 2 \right )}}{14} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**5+1)/(x**3-3*x**2-10*x),x)

[Out]

x**3/3 + 3*x**2/2 + 19*x - log(x)/10 + 3126*log(x - 5)/35 - 31*log(x + 2)/14

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GIAC/XCAS [A]  time = 0.201086, size = 45, normalized size = 1.07 \[ \frac{1}{3} \, x^{3} + \frac{3}{2} \, x^{2} + 19 \, x - \frac{31}{14} \,{\rm ln}\left ({\left | x + 2 \right |}\right ) + \frac{3126}{35} \,{\rm ln}\left ({\left | x - 5 \right |}\right ) - \frac{1}{10} \,{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^5 + 1)/(x^3 - 3*x^2 - 10*x),x, algorithm="giac")

[Out]

1/3*x^3 + 3/2*x^2 + 19*x - 31/14*ln(abs(x + 2)) + 3126/35*ln(abs(x - 5)) - 1/10*
ln(abs(x))