3.465 \(\int \frac{-4 x^3+3 x^5}{(-1+x^2)^5} \, dx\)

Optimal. Leaf size=40 \[ -\frac{3}{4 \left (1-x^2\right )^2}+\frac{1}{3 \left (1-x^2\right )^3}+\frac{1}{8 \left (1-x^2\right )^4} \]

[Out]

1/(8*(1 - x^2)^4) + 1/(3*(1 - x^2)^3) - 3/(4*(1 - x^2)^2)

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Rubi [A]  time = 0.0353437, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {1593, 446, 77} \[ -\frac{3}{4 \left (1-x^2\right )^2}+\frac{1}{3 \left (1-x^2\right )^3}+\frac{1}{8 \left (1-x^2\right )^4} \]

Antiderivative was successfully verified.

[In]

Int[(-4*x^3 + 3*x^5)/(-1 + x^2)^5,x]

[Out]

1/(8*(1 - x^2)^4) + 1/(3*(1 - x^2)^3) - 3/(4*(1 - x^2)^2)

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int \frac{-4 x^3+3 x^5}{\left (-1+x^2\right )^5} \, dx &=\int \frac{x^3 \left (-4+3 x^2\right )}{\left (-1+x^2\right )^5} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x (-4+3 x)}{(-1+x)^5} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{1}{(-1+x)^5}+\frac{2}{(-1+x)^4}+\frac{3}{(-1+x)^3}\right ) \, dx,x,x^2\right )\\ &=\frac{1}{8 \left (1-x^2\right )^4}+\frac{1}{3 \left (1-x^2\right )^3}-\frac{3}{4 \left (1-x^2\right )^2}\\ \end{align*}

Mathematica [A]  time = 0.0081096, size = 23, normalized size = 0.57 \[ \frac{-18 x^4+28 x^2-7}{24 \left (x^2-1\right )^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(-4*x^3 + 3*x^5)/(-1 + x^2)^5,x]

[Out]

(-7 + 28*x^2 - 18*x^4)/(24*(-1 + x^2)^4)

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Maple [A]  time = 0.01, size = 58, normalized size = 1.5 \begin{align*}{\frac{1}{128\, \left ( 1+x \right ) ^{4}}}+{\frac{11}{192\, \left ( 1+x \right ) ^{3}}}-{\frac{27}{256\, \left ( 1+x \right ) ^{2}}}-{\frac{27}{256+256\,x}}+{\frac{1}{128\, \left ( -1+x \right ) ^{4}}}-{\frac{11}{192\, \left ( -1+x \right ) ^{3}}}-{\frac{27}{256\, \left ( -1+x \right ) ^{2}}}+{\frac{27}{-256+256\,x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^5-4*x^3)/(x^2-1)^5,x)

[Out]

1/128/(1+x)^4+11/192/(1+x)^3-27/256/(1+x)^2-27/256/(1+x)+1/128/(-1+x)^4-11/192/(-1+x)^3-27/256/(-1+x)^2+27/256
/(-1+x)

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Maxima [A]  time = 0.933901, size = 49, normalized size = 1.22 \begin{align*} -\frac{18 \, x^{4} - 28 \, x^{2} + 7}{24 \,{\left (x^{8} - 4 \, x^{6} + 6 \, x^{4} - 4 \, x^{2} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^5-4*x^3)/(x^2-1)^5,x, algorithm="maxima")

[Out]

-1/24*(18*x^4 - 28*x^2 + 7)/(x^8 - 4*x^6 + 6*x^4 - 4*x^2 + 1)

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Fricas [A]  time = 1.94571, size = 85, normalized size = 2.12 \begin{align*} -\frac{18 \, x^{4} - 28 \, x^{2} + 7}{24 \,{\left (x^{8} - 4 \, x^{6} + 6 \, x^{4} - 4 \, x^{2} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^5-4*x^3)/(x^2-1)^5,x, algorithm="fricas")

[Out]

-1/24*(18*x^4 - 28*x^2 + 7)/(x^8 - 4*x^6 + 6*x^4 - 4*x^2 + 1)

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Sympy [A]  time = 0.129208, size = 34, normalized size = 0.85 \begin{align*} - \frac{18 x^{4} - 28 x^{2} + 7}{24 x^{8} - 96 x^{6} + 144 x^{4} - 96 x^{2} + 24} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x**5-4*x**3)/(x**2-1)**5,x)

[Out]

-(18*x**4 - 28*x**2 + 7)/(24*x**8 - 96*x**6 + 144*x**4 - 96*x**2 + 24)

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Giac [A]  time = 1.05319, size = 28, normalized size = 0.7 \begin{align*} -\frac{18 \, x^{4} - 28 \, x^{2} + 7}{24 \,{\left (x^{2} - 1\right )}^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^5-4*x^3)/(x^2-1)^5,x, algorithm="giac")

[Out]

-1/24*(18*x^4 - 28*x^2 + 7)/(x^2 - 1)^4