3.303 \(\int \frac{5-5 x+7 x^2+x^3}{(-1+x)^2 (1+x)^3} \, dx\)

Optimal. Leaf size=15 \[ \frac{1}{1-x}-\frac{2}{(x+1)^2} \]

[Out]

(1 - x)^(-1) - 2/(1 + x)^2

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Rubi [A]  time = 0.0370854, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ \frac{1}{1-x}-\frac{2}{(x+1)^2} \]

Antiderivative was successfully verified.

[In]  Int[(5 - 5*x + 7*x^2 + x^3)/((-1 + x)^2*(1 + x)^3),x]

[Out]

(1 - x)^(-1) - 2/(1 + x)^2

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**3+7*x**2-5*x+5)/(-1+x)**2/(1+x)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 0.0191993, size = 15, normalized size = 1. \[ -\frac{2}{(x+1)^2}-\frac{1}{x-1} \]

Antiderivative was successfully verified.

[In]  Integrate[(5 - 5*x + 7*x^2 + x^3)/((-1 + x)^2*(1 + x)^3),x]

[Out]

-(-1 + x)^(-1) - 2/(1 + x)^2

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Maple [A]  time = 0.008, size = 16, normalized size = 1.1 \[ - \left ( -1+x \right ) ^{-1}-2\, \left ( 1+x \right ) ^{-2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^3+7*x^2-5*x+5)/(-1+x)^2/(1+x)^3,x)

[Out]

-1/(-1+x)-2/(1+x)^2

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Maxima [A]  time = 0.797168, size = 31, normalized size = 2.07 \[ -\frac{x^{2} + 4 \, x - 1}{x^{3} + x^{2} - x - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^3 + 7*x^2 - 5*x + 5)/((x + 1)^3*(x - 1)^2),x, algorithm="maxima")

[Out]

-(x^2 + 4*x - 1)/(x^3 + x^2 - x - 1)

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Fricas [A]  time = 0.242329, size = 31, normalized size = 2.07 \[ -\frac{x^{2} + 4 \, x - 1}{x^{3} + x^{2} - x - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^3 + 7*x^2 - 5*x + 5)/((x + 1)^3*(x - 1)^2),x, algorithm="fricas")

[Out]

-(x^2 + 4*x - 1)/(x^3 + x^2 - x - 1)

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Sympy [A]  time = 0.274454, size = 19, normalized size = 1.27 \[ - \frac{x^{2} + 4 x - 1}{x^{3} + x^{2} - x - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**3+7*x**2-5*x+5)/(-1+x)**2/(1+x)**3,x)

[Out]

-(x**2 + 4*x - 1)/(x**3 + x**2 - x - 1)

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GIAC/XCAS [A]  time = 0.262833, size = 41, normalized size = 2.73 \[ -\frac{1}{x - 1} + \frac{\frac{4}{x - 1} + 1}{2 \,{\left (\frac{2}{x - 1} + 1\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^3 + 7*x^2 - 5*x + 5)/((x + 1)^3*(x - 1)^2),x, algorithm="giac")

[Out]

-1/(x - 1) + 1/2*(4/(x - 1) + 1)/(2/(x - 1) + 1)^2