3.92 \(\int \frac{(d+e x) (2-3 x+x^2)}{(4-5 x^2+x^4)^2} \, dx\)
Optimal. Leaf size=89 \[ -\frac{x (3 d-4 e)+5 d-6 e}{12 \left (x^2+3 x+2\right )}-\frac{1}{36} (d+e) \log (1-x)+\frac{1}{144} (d+2 e) \log (2-x)-\frac{1}{36} (7 d-13 e) \log (x+1)+\frac{1}{144} (31 d-50 e) \log (x+2) \]
[Out]
-(5*d - 6*e + (3*d - 4*e)*x)/(12*(2 + 3*x + x^2)) - ((d + e)*Log[1 - x])/36 + ((d + 2*e)*Log[2 - x])/144 - ((7
*d - 13*e)*Log[1 + x])/36 + ((31*d - 50*e)*Log[2 + x])/144
________________________________________________________________________________________
Rubi [A] time = 0.260162, antiderivative size = 89, normalized size of antiderivative = 1.,
number of steps used = 9, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used =
{1586, 1016, 1072, 632, 31} \[ -\frac{x (3 d-4 e)+5 d-6 e}{12 \left (x^2+3 x+2\right )}-\frac{1}{36} (d+e) \log (1-x)+\frac{1}{144} (d+2 e) \log (2-x)-\frac{1}{36} (7 d-13 e) \log (x+1)+\frac{1}{144} (31 d-50 e) \log (x+2) \]
Antiderivative was successfully verified.
[In]
Int[((d + e*x)*(2 - 3*x + x^2))/(4 - 5*x^2 + x^4)^2,x]
[Out]
-(5*d - 6*e + (3*d - 4*e)*x)/(12*(2 + 3*x + x^2)) - ((d + e)*Log[1 - x])/36 + ((d + 2*e)*Log[2 - x])/144 - ((7
*d - 13*e)*Log[1 + x])/36 + ((31*d - 50*e)*Log[2 + x])/144
Rule 1586
Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]
Rule 1016
Int[((g_.) + (h_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (e_.)*(x_) + (f_.)*(x_)^2)^(q_), x_Sy
mbol] :> Simp[((a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q + 1)*(g*c*(2*a*c*e - b*(c*d + a*f)) + (g*b - a*h
)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) + c*(g*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) - h*(b*c*d - 2*a*c*e + a*b*f)
)*x))/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1)), x] + Dist[1/((b^2 - 4*a*c)*((c*d - a*
f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1)), Int[(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^q*Simp[(b*h - 2*g*c)
*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1) + (b^2*(g*f) - b*(h*c*d + g*c*e + a*h*f) + 2*(g*c*(c*d - a*
f) - a*(-(h*c*e))))*(a*f*(p + 1) - c*d*(p + 2)) - e*((g*c)*(2*a*c*e - b*(c*d + a*f)) + (g*b - a*h)*(2*c^2*d +
b^2*f - c*(b*e + 2*a*f)))*(p + q + 2) - (2*f*((g*c)*(2*a*c*e - b*(c*d + a*f)) + (g*b - a*h)*(2*c^2*d + b^2*f -
c*(b*e + 2*a*f)))*(p + q + 2) - (b^2*g*f - b*(h*c*d + g*c*e + a*h*f) + 2*(g*c*(c*d - a*f) - a*(-(h*c*e))))*(b
*f*(p + 1) - c*e*(2*p + q + 4)))*x - c*f*(b^2*(g*f) - b*(h*c*d + g*c*e + a*h*f) + 2*(g*c*(c*d - a*f) + a*h*c*e
))*(2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, q}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2
- 4*d*f, 0] && LtQ[p, -1] && NeQ[(c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f), 0] && !( !IntegerQ[p] && ILtQ[q, -1
])
Rule 1072
Int[((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_) + (e_.)*(x_) + (f_.)*(x_)^2)
), x_Symbol] :> With[{q = c^2*d^2 - b*c*d*e + a*c*e^2 + b^2*d*f - 2*a*c*d*f - a*b*e*f + a^2*f^2}, Dist[1/q, In
t[(A*c^2*d - a*c*C*d - A*b*c*e + a*B*c*e + A*b^2*f - a*b*B*f - a*A*c*f + a^2*C*f + c*(B*c*d - b*C*d - A*c*e +
a*C*e + A*b*f - a*B*f)*x)/(a + b*x + c*x^2), x], x] + Dist[1/q, Int[(c*C*d^2 - B*c*d*e + A*c*e^2 + b*B*d*f - A
*c*d*f - a*C*d*f - A*b*e*f + a*A*f^2 - f*(B*c*d - b*C*d - A*c*e + a*C*e + A*b*f - a*B*f)*x)/(d + e*x + f*x^2),
x], x] /; NeQ[q, 0]] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0]
Rule 632
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[
(c*d - e*(b/2 - q/2))/q, Int[1/(b/2 - q/2 + c*x), x], x] - Dist[(c*d - e*(b/2 + q/2))/q, Int[1/(b/2 + q/2 + c*
x), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] && NiceSqrtQ[b^2 - 4*a*
c]
Rule 31
Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]
Rubi steps
\begin{align*} \int \frac{(d+e x) \left (2-3 x+x^2\right )}{\left (4-5 x^2+x^4\right )^2} \, dx &=\int \frac{d+e x}{\left (2-3 x+x^2\right ) \left (2+3 x+x^2\right )^2} \, dx\\ &=-\frac{5 d-6 e+(3 d-4 e) x}{12 \left (2+3 x+x^2\right )}-\frac{1}{72} \int \frac{6 (3 d-10 e)-24 (2 d-3 e) x+6 (3 d-4 e) x^2}{\left (2-3 x+x^2\right ) \left (2+3 x+x^2\right )} \, dx\\ &=-\frac{5 d-6 e+(3 d-4 e) x}{12 \left (2+3 x+x^2\right )}-\frac{\int \frac{108 (3 d-10 e)-288 (2 d-3 e)+(-36 (3 d-10 e)+72 (3 d-4 e)) x}{2-3 x+x^2} \, dx}{5184}-\frac{\int \frac{108 (3 d-10 e)+288 (2 d-3 e)-(-36 (3 d-10 e)+72 (3 d-4 e)) x}{2+3 x+x^2} \, dx}{5184}\\ &=-\frac{5 d-6 e+(3 d-4 e) x}{12 \left (2+3 x+x^2\right )}-\frac{1}{36} (7 d-13 e) \int \frac{1}{1+x} \, dx-\frac{1}{144} (-d-2 e) \int \frac{1}{-2+x} \, dx-\frac{1}{36} (d+e) \int \frac{1}{-1+x} \, dx-\frac{1}{144} (-31 d+50 e) \int \frac{1}{2+x} \, dx\\ &=-\frac{5 d-6 e+(3 d-4 e) x}{12 \left (2+3 x+x^2\right )}-\frac{1}{36} (d+e) \log (1-x)+\frac{1}{144} (d+2 e) \log (2-x)-\frac{1}{36} (7 d-13 e) \log (1+x)+\frac{1}{144} (31 d-50 e) \log (2+x)\\ \end{align*}
Mathematica [A] time = 0.0507321, size = 80, normalized size = 0.9 \[ \frac{1}{144} \left (\frac{12 (-3 d x-5 d+4 e x+6 e)}{x^2+3 x+2}-4 (d+e) \log (1-x)+(d+2 e) \log (2-x)+4 (13 e-7 d) \log (x+1)+(31 d-50 e) \log (x+2)\right ) \]
Antiderivative was successfully verified.
[In]
Integrate[((d + e*x)*(2 - 3*x + x^2))/(4 - 5*x^2 + x^4)^2,x]
[Out]
((12*(-5*d + 6*e - 3*d*x + 4*e*x))/(2 + 3*x + x^2) - 4*(d + e)*Log[1 - x] + (d + 2*e)*Log[2 - x] + 4*(-7*d + 1
3*e)*Log[1 + x] + (31*d - 50*e)*Log[2 + x])/144
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Maple [A] time = 0.011, size = 90, normalized size = 1. \begin{align*} -{\frac{d}{24+12\,x}}+{\frac{e}{12+6\,x}}+{\frac{31\,\ln \left ( 2+x \right ) d}{144}}-{\frac{25\,\ln \left ( 2+x \right ) e}{72}}-{\frac{7\,\ln \left ( 1+x \right ) d}{36}}+{\frac{13\,\ln \left ( 1+x \right ) e}{36}}-{\frac{d}{6+6\,x}}+{\frac{e}{6+6\,x}}+{\frac{\ln \left ( x-2 \right ) d}{144}}+{\frac{\ln \left ( x-2 \right ) e}{72}}-{\frac{\ln \left ( x-1 \right ) d}{36}}-{\frac{\ln \left ( x-1 \right ) e}{36}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((e*x+d)*(x^2-3*x+2)/(x^4-5*x^2+4)^2,x)
[Out]
-1/12/(2+x)*d+1/6/(2+x)*e+31/144*ln(2+x)*d-25/72*ln(2+x)*e-7/36*ln(1+x)*d+13/36*ln(1+x)*e-1/6/(1+x)*d+1/6/(1+x
)*e+1/144*ln(x-2)*d+1/72*ln(x-2)*e-1/36*ln(x-1)*d-1/36*ln(x-1)*e
________________________________________________________________________________________
Maxima [A] time = 0.958643, size = 101, normalized size = 1.13 \begin{align*} \frac{1}{144} \,{\left (31 \, d - 50 \, e\right )} \log \left (x + 2\right ) - \frac{1}{36} \,{\left (7 \, d - 13 \, e\right )} \log \left (x + 1\right ) - \frac{1}{36} \,{\left (d + e\right )} \log \left (x - 1\right ) + \frac{1}{144} \,{\left (d + 2 \, e\right )} \log \left (x - 2\right ) - \frac{{\left (3 \, d - 4 \, e\right )} x + 5 \, d - 6 \, e}{12 \,{\left (x^{2} + 3 \, x + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)*(x^2-3*x+2)/(x^4-5*x^2+4)^2,x, algorithm="maxima")
[Out]
1/144*(31*d - 50*e)*log(x + 2) - 1/36*(7*d - 13*e)*log(x + 1) - 1/36*(d + e)*log(x - 1) + 1/144*(d + 2*e)*log(
x - 2) - 1/12*((3*d - 4*e)*x + 5*d - 6*e)/(x^2 + 3*x + 2)
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Fricas [A] time = 1.94788, size = 410, normalized size = 4.61 \begin{align*} -\frac{12 \,{\left (3 \, d - 4 \, e\right )} x -{\left ({\left (31 \, d - 50 \, e\right )} x^{2} + 3 \,{\left (31 \, d - 50 \, e\right )} x + 62 \, d - 100 \, e\right )} \log \left (x + 2\right ) + 4 \,{\left ({\left (7 \, d - 13 \, e\right )} x^{2} + 3 \,{\left (7 \, d - 13 \, e\right )} x + 14 \, d - 26 \, e\right )} \log \left (x + 1\right ) + 4 \,{\left ({\left (d + e\right )} x^{2} + 3 \,{\left (d + e\right )} x + 2 \, d + 2 \, e\right )} \log \left (x - 1\right ) -{\left ({\left (d + 2 \, e\right )} x^{2} + 3 \,{\left (d + 2 \, e\right )} x + 2 \, d + 4 \, e\right )} \log \left (x - 2\right ) + 60 \, d - 72 \, e}{144 \,{\left (x^{2} + 3 \, x + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)*(x^2-3*x+2)/(x^4-5*x^2+4)^2,x, algorithm="fricas")
[Out]
-1/144*(12*(3*d - 4*e)*x - ((31*d - 50*e)*x^2 + 3*(31*d - 50*e)*x + 62*d - 100*e)*log(x + 2) + 4*((7*d - 13*e)
*x^2 + 3*(7*d - 13*e)*x + 14*d - 26*e)*log(x + 1) + 4*((d + e)*x^2 + 3*(d + e)*x + 2*d + 2*e)*log(x - 1) - ((d
+ 2*e)*x^2 + 3*(d + 2*e)*x + 2*d + 4*e)*log(x - 2) + 60*d - 72*e)/(x^2 + 3*x + 2)
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Sympy [B] time = 5.75008, size = 1255, normalized size = 14.1 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)*(x**2-3*x+2)/(x**4-5*x**2+4)**2,x)
[Out]
-(d + e)*log(x + (-24383100*d**6 + 187408066*d**5*e + 10439775*d**5*(d + e) - 511591980*d**4*e**2 - 94132290*d
**4*e*(d + e) + 667200*d**4*(d + e)**2 + 469491120*d**3*e**3 + 333672552*d**3*e**2*(d + e) - 2703328*d**3*e*(d
+ e)**2 - 198000*d**3*(d + e)**3 + 322778400*d**2*e**4 - 582497712*d**2*e**3*(d + e) + 1752768*d**2*e**2*(d +
e)**2 + 1107552*d**2*e*(d + e)**3 - 863493856*d*e**5 + 500776560*d*e**4*(d + e) + 4226944*d*e**3*(d + e)**2 -
1880640*d*e**2*(d + e)**3 + 429000000*e**6 - 169242912*e**5*(d + e) - 4538112*e**4*(d + e)**2 + 964224*e**3*(
d + e)**3)/(13474125*d**6 - 102860175*d**5*e + 274190390*d**4*e**2 - 224142072*d**3*e**3 - 245084096*d**2*e**4
+ 535797456*d*e**5 - 256183200*e**6))/36 + (d + 2*e)*log(x + (-24383100*d**6 + 187408066*d**5*e - 10439775*d*
*5*(d + 2*e)/4 - 511591980*d**4*e**2 + 47066145*d**4*e*(d + 2*e)/2 + 41700*d**4*(d + 2*e)**2 + 469491120*d**3*
e**3 - 83418138*d**3*e**2*(d + 2*e) - 168958*d**3*e*(d + 2*e)**2 + 12375*d**3*(d + 2*e)**3/4 + 322778400*d**2*
e**4 + 145624428*d**2*e**3*(d + 2*e) + 109548*d**2*e**2*(d + 2*e)**2 - 34611*d**2*e*(d + 2*e)**3/2 - 863493856
*d*e**5 - 125194140*d*e**4*(d + 2*e) + 264184*d*e**3*(d + 2*e)**2 + 29385*d*e**2*(d + 2*e)**3 + 429000000*e**6
+ 42310728*e**5*(d + 2*e) - 283632*e**4*(d + 2*e)**2 - 15066*e**3*(d + 2*e)**3)/(13474125*d**6 - 102860175*d*
*5*e + 274190390*d**4*e**2 - 224142072*d**3*e**3 - 245084096*d**2*e**4 + 535797456*d*e**5 - 256183200*e**6))/1
44 - (7*d - 13*e)*log(x + (-24383100*d**6 + 187408066*d**5*e + 10439775*d**5*(7*d - 13*e) - 511591980*d**4*e**
2 - 94132290*d**4*e*(7*d - 13*e) + 667200*d**4*(7*d - 13*e)**2 + 469491120*d**3*e**3 + 333672552*d**3*e**2*(7*
d - 13*e) - 2703328*d**3*e*(7*d - 13*e)**2 - 198000*d**3*(7*d - 13*e)**3 + 322778400*d**2*e**4 - 582497712*d**
2*e**3*(7*d - 13*e) + 1752768*d**2*e**2*(7*d - 13*e)**2 + 1107552*d**2*e*(7*d - 13*e)**3 - 863493856*d*e**5 +
500776560*d*e**4*(7*d - 13*e) + 4226944*d*e**3*(7*d - 13*e)**2 - 1880640*d*e**2*(7*d - 13*e)**3 + 429000000*e*
*6 - 169242912*e**5*(7*d - 13*e) - 4538112*e**4*(7*d - 13*e)**2 + 964224*e**3*(7*d - 13*e)**3)/(13474125*d**6
- 102860175*d**5*e + 274190390*d**4*e**2 - 224142072*d**3*e**3 - 245084096*d**2*e**4 + 535797456*d*e**5 - 2561
83200*e**6))/36 + (31*d - 50*e)*log(x + (-24383100*d**6 + 187408066*d**5*e - 10439775*d**5*(31*d - 50*e)/4 - 5
11591980*d**4*e**2 + 47066145*d**4*e*(31*d - 50*e)/2 + 41700*d**4*(31*d - 50*e)**2 + 469491120*d**3*e**3 - 834
18138*d**3*e**2*(31*d - 50*e) - 168958*d**3*e*(31*d - 50*e)**2 + 12375*d**3*(31*d - 50*e)**3/4 + 322778400*d**
2*e**4 + 145624428*d**2*e**3*(31*d - 50*e) + 109548*d**2*e**2*(31*d - 50*e)**2 - 34611*d**2*e*(31*d - 50*e)**3
/2 - 863493856*d*e**5 - 125194140*d*e**4*(31*d - 50*e) + 264184*d*e**3*(31*d - 50*e)**2 + 29385*d*e**2*(31*d -
50*e)**3 + 429000000*e**6 + 42310728*e**5*(31*d - 50*e) - 283632*e**4*(31*d - 50*e)**2 - 15066*e**3*(31*d - 5
0*e)**3)/(13474125*d**6 - 102860175*d**5*e + 274190390*d**4*e**2 - 224142072*d**3*e**3 - 245084096*d**2*e**4 +
535797456*d*e**5 - 256183200*e**6))/144 - (5*d - 6*e + x*(3*d - 4*e))/(12*x**2 + 36*x + 24)
________________________________________________________________________________________
Giac [A] time = 1.07827, size = 115, normalized size = 1.29 \begin{align*} \frac{1}{144} \,{\left (31 \, d - 50 \, e\right )} \log \left ({\left | x + 2 \right |}\right ) - \frac{1}{36} \,{\left (7 \, d - 13 \, e\right )} \log \left ({\left | x + 1 \right |}\right ) - \frac{1}{36} \,{\left (d + e\right )} \log \left ({\left | x - 1 \right |}\right ) + \frac{1}{144} \,{\left (d + 2 \, e\right )} \log \left ({\left | x - 2 \right |}\right ) - \frac{{\left (3 \, d - 4 \, e\right )} x + 5 \, d - 6 \, e}{12 \,{\left (x + 2\right )}{\left (x + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)*(x^2-3*x+2)/(x^4-5*x^2+4)^2,x, algorithm="giac")
[Out]
1/144*(31*d - 50*e)*log(abs(x + 2)) - 1/36*(7*d - 13*e)*log(abs(x + 1)) - 1/36*(d + e)*log(abs(x - 1)) + 1/144
*(d + 2*e)*log(abs(x - 2)) - 1/12*((3*d - 4*e)*x + 5*d - 6*e)/((x + 2)*(x + 1))