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

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Rubi [A]  time = 0.03, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {1620} \[ \frac {1}{1-x}-\frac {2}{(x+1)^2} \]

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

Rule 1620

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

Rubi steps

\begin {align*} \int \frac {5-5 x+7 x^2+x^3}{(-1+x)^2 (1+x)^3} \, dx &=\int \left (\frac {1}{(-1+x)^2}+\frac {4}{(1+x)^3}\right ) \, dx\\ &=\frac {1}{1-x}-\frac {2}{(1+x)^2}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 15, normalized size = 1.00 \[ -\frac {2}{(x+1)^2}-\frac {1}{x-1} \]

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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fricas [A]  time = 0.88, size = 23, normalized size = 1.53 \[ -\frac {x^{2} + 4 \, x - 1}{x^{3} + x^{2} - x - 1} \]

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, algorithm="fricas")

[Out]

-(x^2 + 4*x - 1)/(x^3 + x^2 - x - 1)

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giac [A]  time = 0.26, size = 30, normalized size = 2.00 \[ -\frac {1}{x - 1} + \frac {\frac {4}{x - 1} + 1}{2 \, {\left (\frac {2}{x - 1} + 1\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 16, normalized size = 1.07 \[ -\frac {2}{\left (x +1\right )^{2}}-\frac {1}{x -1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.27, size = 23, normalized size = 1.53 \[ -\frac {x^{2} + 4 \, x - 1}{x^{3} + x^{2} - x - 1} \]

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, algorithm="maxima")

[Out]

-(x^2 + 4*x - 1)/(x^3 + x^2 - x - 1)

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mupad [B]  time = 2.09, size = 15, normalized size = 1.00 \[ -\frac {1}{x-1}-\frac {2}{{\left (x+1\right )}^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- 1/(x - 1) - 2/(x + 1)^2

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sympy [A]  time = 0.11, size = 17, normalized size = 1.13 \[ \frac {- x^{2} - 4 x + 1}{x^{3} + x^{2} - x - 1} \]

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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