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 = 0.0270943, 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, 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*}

Mathematica [A]  time = 0.011732, 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 = 0.004, size = 16, normalized size = 1.1 \begin{align*} - \left ( x-1 \right ) ^{-1}-2\, \left ( 1+x \right ) ^{-2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.970745, size = 31, normalized size = 2.07 \begin{align*} -\frac{x^{2} + 4 \, x - 1}{x^{3} + x^{2} - x - 1} \end{align*}

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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Fricas [A]  time = 1.47772, size = 51, normalized size = 3.4 \begin{align*} -\frac{x^{2} + 4 \, x - 1}{x^{3} + x^{2} - x - 1} \end{align*}

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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Sympy [A]  time = 0.110862, size = 19, normalized size = 1.27 \begin{align*} - \frac{x^{2} + 4 x - 1}{x^{3} + x^{2} - x - 1} \end{align*}

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 [B]  time = 1.11392, size = 41, normalized size = 2.73 \begin{align*} -\frac{1}{x - 1} + \frac{\frac{4}{x - 1} + 1}{2 \,{\left (\frac{2}{x - 1} + 1\right )}^{2}} \end{align*}

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