3.1902 \(\int \left (a+\frac{d}{x^3}+\frac{c}{x^2}+\frac{b}{x}\right ) \, dx\)

Optimal. Leaf size=22 \[ a x+b \log (x)-\frac{c}{x}-\frac{d}{2 x^2} \]

[Out]

-d/(2*x^2) - c/x + a*x + b*Log[x]

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Rubi [A]  time = 0.0125757, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 0, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0. \[ a x+b \log (x)-\frac{c}{x}-\frac{d}{2 x^2} \]

Antiderivative was successfully verified.

[In]  Int[a + d/x^3 + c/x^2 + b/x,x]

[Out]

-d/(2*x^2) - c/x + a*x + b*Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ b \log{\left (x \right )} - \frac{c}{x} - \frac{d}{2 x^{2}} + \int a\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(a+d/x**3+c/x**2+b/x,x)

[Out]

b*log(x) - c/x - d/(2*x**2) + Integral(a, x)

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Mathematica [A]  time = 0.00997067, size = 22, normalized size = 1. \[ a x+b \log (x)-\frac{c}{x}-\frac{d}{2 x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[a + d/x^3 + c/x^2 + b/x,x]

[Out]

-d/(2*x^2) - c/x + a*x + b*Log[x]

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Maple [A]  time = 0.003, size = 21, normalized size = 1. \[ -{\frac{d}{2\,{x}^{2}}}-{\frac{c}{x}}+ax+b\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(a+d/x^3+c/x^2+b/x,x)

[Out]

-1/2*d/x^2-c/x+a*x+b*ln(x)

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Maxima [A]  time = 1.33844, size = 27, normalized size = 1.23 \[ a x + b \log \left (x\right ) - \frac{c}{x} - \frac{d}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(a + b/x + c/x^2 + d/x^3,x, algorithm="maxima")

[Out]

a*x + b*log(x) - c/x - 1/2*d/x^2

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Fricas [A]  time = 0.198006, size = 36, normalized size = 1.64 \[ \frac{2 \, a x^{3} + 2 \, b x^{2} \log \left (x\right ) - 2 \, c x - d}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(a + b/x + c/x^2 + d/x^3,x, algorithm="fricas")

[Out]

1/2*(2*a*x^3 + 2*b*x^2*log(x) - 2*c*x - d)/x^2

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Sympy [A]  time = 0.62331, size = 19, normalized size = 0.86 \[ a x + b \log{\left (x \right )} - \frac{2 c x + d}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(a+d/x**3+c/x**2+b/x,x)

[Out]

a*x + b*log(x) - (2*c*x + d)/(2*x**2)

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GIAC/XCAS [A]  time = 0.26164, size = 28, normalized size = 1.27 \[ a x + b{\rm ln}\left ({\left | x \right |}\right ) - \frac{c}{x} - \frac{d}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(a + b/x + c/x^2 + d/x^3,x, algorithm="giac")

[Out]

a*x + b*ln(abs(x)) - c/x - 1/2*d/x^2