3.93 \(\int \frac{(2-3 x+x^2) (d+e x+f x^2)}{(4-5 x^2+x^4)^2} \, dx\)
Optimal. Leaf size=105 \[ -\frac{x (3 d-4 e+6 f)+5 d-6 e+8 f}{12 \left (x^2+3 x+2\right )}-\frac{1}{36} \log (1-x) (d+e+f)+\frac{1}{144} \log (2-x) (d+2 e+4 f)-\frac{1}{36} \log (x+1) (7 d-13 e+19 f)+\frac{1}{144} \log (x+2) (31 d-50 e+76 f) \]
[Out]
-(5*d - 6*e + 8*f + (3*d - 4*e + 6*f)*x)/(12*(2 + 3*x + x^2)) - ((d + e + f)*Log[1 - x])/36 + ((d + 2*e + 4*f)
*Log[2 - x])/144 - ((7*d - 13*e + 19*f)*Log[1 + x])/36 + ((31*d - 50*e + 76*f)*Log[2 + x])/144
________________________________________________________________________________________
Rubi [A] time = 0.319504, antiderivative size = 105, normalized size of antiderivative = 1.,
number of steps used = 9, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161, Rules used =
{1586, 1060, 1072, 632, 31} \[ -\frac{x (3 d-4 e+6 f)+5 d-6 e+8 f}{12 \left (x^2+3 x+2\right )}-\frac{1}{36} \log (1-x) (d+e+f)+\frac{1}{144} \log (2-x) (d+2 e+4 f)-\frac{1}{36} \log (x+1) (7 d-13 e+19 f)+\frac{1}{144} \log (x+2) (31 d-50 e+76 f) \]
Antiderivative was successfully verified.
[In]
Int[((2 - 3*x + x^2)*(d + e*x + f*x^2))/(4 - 5*x^2 + x^4)^2,x]
[Out]
-(5*d - 6*e + 8*f + (3*d - 4*e + 6*f)*x)/(12*(2 + 3*x + x^2)) - ((d + e + f)*Log[1 - x])/36 + ((d + 2*e + 4*f)
*Log[2 - x])/144 - ((7*d - 13*e + 19*f)*Log[1 + x])/36 + ((31*d - 50*e + 76*f)*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 1060
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)*((d_) + (e_.)*(x_) + (f_.)*(x_
)^2)^(q_), x_Symbol] :> Simp[((a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q + 1)*((A*c - a*C)*(2*a*c*e - b*(c
*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) + c*(A*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) - B*(b
*c*d - 2*a*c*e + a*b*f) + C*(b^2*d - a*b*e - 2*a*(c*d - a*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*B - 2*A*c - 2*a*C)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f)
)*(p + 1) + (b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*e + a*B*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - B*c*e - a*C
*f)))*(a*f*(p + 1) - c*d*(p + 2)) - e*((A*c - a*C)*(2*a*c*e - b*(c*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f -
c*(b*e + 2*a*f)))*(p + q + 2) - (2*f*((A*c - a*C)*(2*a*c*e - b*(c*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f - c
*(b*e + 2*a*f)))*(p + q + 2) - (b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*e + a*B*f) + 2*(A*c*(c*d - a*f) - a*(
c*C*d - B*c*e - a*C*f)))*(b*f*(p + 1) - c*e*(2*p + q + 4)))*x - c*f*(b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*
e + a*B*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - B*c*e - a*C*f)))*(2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b,
c, d, e, f, A, B, C, 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]) && !IGtQ[q, 0]
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{\left (2-3 x+x^2\right ) \left (d+e x+f x^2\right )}{\left (4-5 x^2+x^4\right )^2} \, dx &=\int \frac{d+e x+f x^2}{\left (2-3 x+x^2\right ) \left (2+3 x+x^2\right )^2} \, dx\\ &=-\frac{5 d-6 e+8 f+(3 d-4 e+6 f) x}{12 \left (2+3 x+x^2\right )}-\frac{1}{72} \int \frac{6 (3 d-10 e+12 f)-24 (2 d-3 e+5 f) x+6 (3 d-4 e+6 f) x^2}{\left (2-3 x+x^2\right ) \left (2+3 x+x^2\right )} \, dx\\ &=-\frac{5 d-6 e+8 f+(3 d-4 e+6 f) x}{12 \left (2+3 x+x^2\right )}-\frac{\int \frac{-288 (2 d-3 e+5 f)+108 (3 d-10 e+12 f)+(72 (3 d-4 e+6 f)-36 (3 d-10 e+12 f)) x}{2-3 x+x^2} \, dx}{5184}-\frac{\int \frac{288 (2 d-3 e+5 f)+108 (3 d-10 e+12 f)-(72 (3 d-4 e+6 f)-36 (3 d-10 e+12 f)) x}{2+3 x+x^2} \, dx}{5184}\\ &=-\frac{5 d-6 e+8 f+(3 d-4 e+6 f) x}{12 \left (2+3 x+x^2\right )}-\frac{1}{144} (-31 d+50 e-76 f) \int \frac{1}{2+x} \, dx-\frac{1}{144} (-d-2 e-4 f) \int \frac{1}{-2+x} \, dx-\frac{1}{36} (d+e+f) \int \frac{1}{-1+x} \, dx-\frac{1}{36} (7 d-13 e+19 f) \int \frac{1}{1+x} \, dx\\ &=-\frac{5 d-6 e+8 f+(3 d-4 e+6 f) x}{12 \left (2+3 x+x^2\right )}-\frac{1}{36} (d+e+f) \log (1-x)+\frac{1}{144} (d+2 e+4 f) \log (2-x)-\frac{1}{36} (7 d-13 e+19 f) \log (1+x)+\frac{1}{144} (31 d-50 e+76 f) \log (2+x)\\ \end{align*}
Mathematica [A] time = 0.0743086, size = 97, normalized size = 0.92 \[ \frac{1}{144} \left (-\frac{12 (d (3 x+5)-4 e x-6 e+6 f x+8 f)}{x^2+3 x+2}-4 \log (1-x) (d+e+f)+\log (2-x) (d+2 e+4 f)-4 \log (x+1) (7 d-13 e+19 f)+\log (x+2) (31 d-50 e+76 f)\right ) \]
Antiderivative was successfully verified.
[In]
Integrate[((2 - 3*x + x^2)*(d + e*x + f*x^2))/(4 - 5*x^2 + x^4)^2,x]
[Out]
((-12*(-6*e + 8*f - 4*e*x + 6*f*x + d*(5 + 3*x)))/(2 + 3*x + x^2) - 4*(d + e + f)*Log[1 - x] + (d + 2*e + 4*f)
*Log[2 - x] - 4*(7*d - 13*e + 19*f)*Log[1 + x] + (31*d - 50*e + 76*f)*Log[2 + x])/144
________________________________________________________________________________________
Maple [A] time = 0.013, size = 134, normalized size = 1.3 \begin{align*} -{\frac{d}{24+12\,x}}+{\frac{e}{12+6\,x}}-{\frac{f}{6+3\,x}}+{\frac{31\,\ln \left ( 2+x \right ) d}{144}}-{\frac{25\,\ln \left ( 2+x \right ) e}{72}}+{\frac{19\,\ln \left ( 2+x \right ) f}{36}}-{\frac{7\,\ln \left ( 1+x \right ) d}{36}}+{\frac{13\,\ln \left ( 1+x \right ) e}{36}}-{\frac{19\,\ln \left ( 1+x \right ) f}{36}}-{\frac{d}{6+6\,x}}+{\frac{e}{6+6\,x}}-{\frac{f}{6+6\,x}}+{\frac{\ln \left ( x-2 \right ) d}{144}}+{\frac{\ln \left ( x-2 \right ) e}{72}}+{\frac{\ln \left ( x-2 \right ) f}{36}}-{\frac{\ln \left ( x-1 \right ) d}{36}}-{\frac{\ln \left ( x-1 \right ) e}{36}}-{\frac{\ln \left ( x-1 \right ) f}{36}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((x^2-3*x+2)*(f*x^2+e*x+d)/(x^4-5*x^2+4)^2,x)
[Out]
-1/12/(2+x)*d+1/6/(2+x)*e-1/3/(2+x)*f+31/144*ln(2+x)*d-25/72*ln(2+x)*e+19/36*ln(2+x)*f-7/36*ln(1+x)*d+13/36*ln
(1+x)*e-19/36*ln(1+x)*f-1/6/(1+x)*d+1/6/(1+x)*e-1/6/(1+x)*f+1/144*ln(x-2)*d+1/72*ln(x-2)*e+1/36*ln(x-2)*f-1/36
*ln(x-1)*d-1/36*ln(x-1)*e-1/36*ln(x-1)*f
________________________________________________________________________________________
Maxima [A] time = 0.976987, size = 123, normalized size = 1.17 \begin{align*} \frac{1}{144} \,{\left (31 \, d - 50 \, e + 76 \, f\right )} \log \left (x + 2\right ) - \frac{1}{36} \,{\left (7 \, d - 13 \, e + 19 \, f\right )} \log \left (x + 1\right ) - \frac{1}{36} \,{\left (d + e + f\right )} \log \left (x - 1\right ) + \frac{1}{144} \,{\left (d + 2 \, e + 4 \, f\right )} \log \left (x - 2\right ) - \frac{{\left (3 \, d - 4 \, e + 6 \, f\right )} x + 5 \, d - 6 \, e + 8 \, f}{12 \,{\left (x^{2} + 3 \, x + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x^2-3*x+2)*(f*x^2+e*x+d)/(x^4-5*x^2+4)^2,x, algorithm="maxima")
[Out]
1/144*(31*d - 50*e + 76*f)*log(x + 2) - 1/36*(7*d - 13*e + 19*f)*log(x + 1) - 1/36*(d + e + f)*log(x - 1) + 1/
144*(d + 2*e + 4*f)*log(x - 2) - 1/12*((3*d - 4*e + 6*f)*x + 5*d - 6*e + 8*f)/(x^2 + 3*x + 2)
________________________________________________________________________________________
Fricas [B] time = 2.56752, size = 529, normalized size = 5.04 \begin{align*} -\frac{12 \,{\left (3 \, d - 4 \, e + 6 \, f\right )} x -{\left ({\left (31 \, d - 50 \, e + 76 \, f\right )} x^{2} + 3 \,{\left (31 \, d - 50 \, e + 76 \, f\right )} x + 62 \, d - 100 \, e + 152 \, f\right )} \log \left (x + 2\right ) + 4 \,{\left ({\left (7 \, d - 13 \, e + 19 \, f\right )} x^{2} + 3 \,{\left (7 \, d - 13 \, e + 19 \, f\right )} x + 14 \, d - 26 \, e + 38 \, f\right )} \log \left (x + 1\right ) + 4 \,{\left ({\left (d + e + f\right )} x^{2} + 3 \,{\left (d + e + f\right )} x + 2 \, d + 2 \, e + 2 \, f\right )} \log \left (x - 1\right ) -{\left ({\left (d + 2 \, e + 4 \, f\right )} x^{2} + 3 \,{\left (d + 2 \, e + 4 \, f\right )} x + 2 \, d + 4 \, e + 8 \, f\right )} \log \left (x - 2\right ) + 60 \, d - 72 \, e + 96 \, f}{144 \,{\left (x^{2} + 3 \, x + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x^2-3*x+2)*(f*x^2+e*x+d)/(x^4-5*x^2+4)^2,x, algorithm="fricas")
[Out]
-1/144*(12*(3*d - 4*e + 6*f)*x - ((31*d - 50*e + 76*f)*x^2 + 3*(31*d - 50*e + 76*f)*x + 62*d - 100*e + 152*f)*
log(x + 2) + 4*((7*d - 13*e + 19*f)*x^2 + 3*(7*d - 13*e + 19*f)*x + 14*d - 26*e + 38*f)*log(x + 1) + 4*((d + e
+ f)*x^2 + 3*(d + e + f)*x + 2*d + 2*e + 2*f)*log(x - 1) - ((d + 2*e + 4*f)*x^2 + 3*(d + 2*e + 4*f)*x + 2*d +
4*e + 8*f)*log(x - 2) + 60*d - 72*e + 96*f)/(x^2 + 3*x + 2)
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Sympy [B] time = 117.971, size = 5015, normalized size = 47.76 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x**2-3*x+2)*(f*x**2+e*x+d)/(x**4-5*x**2+4)**2,x)
[Out]
-(d + e + f)*log(x + (-24383100*d**6 + 187408066*d**5*e - 350082438*d**5*f + 10439775*d**5*(d + e + f) - 51159
1980*d**4*e**2 + 2159032460*d**4*e*f - 94132290*d**4*e*(d + e + f) - 2073816840*d**4*f**2 + 137095380*d**4*f*(
d + e + f) + 667200*d**4*(d + e + f)**2 + 469491120*d**3*e**3 - 4339975440*d**3*e**2*f + 333672552*d**3*e**2*(
d + e + f) + 9775163680*d**3*e*f**2 - 980807808*d**3*e*f*(d + e + f) - 2703328*d**3*e*(d + e + f)**2 - 6478462
560*d**3*f**3 + 717713712*d**3*f**2*(d + e + f) + 6408864*d**3*f*(d + e + f)**2 - 198000*d**3*(d + e + f)**3 +
322778400*d**2*e**4 + 1983785760*d**2*e**3*f - 582497712*d**2*e**3*(d + e + f) - 13135991040*d**2*e**2*f**2 +
2589108192*d**2*e**2*f*(d + e + f) + 1752768*d**2*e**2*(d + e + f)**2 + 21641166400*d**2*e*f**3 - 3821636448*
d**2*e*f**2*(d + e + f) - 18174144*d**2*e*f*(d + e + f)**2 + 1107552*d**2*e*(d + e + f)**3 - 11234578560*d**2*
f**4 + 1872896832*d**2*f**3*(d + e + f) + 22905216*d**2*f**2*(d + e + f)**2 - 1601856*d**2*f*(d + e + f)**3 -
863493856*d*e**5 + 3068196000*d*e**4*f + 500776560*d*e**4*(d + e + f) + 901203840*d*e**3*f**2 - 2992140288*d*e
**3*f*(d + e + f) + 4226944*d*e**3*(d + e + f)**2 - 16302840960*d*e**2*f**3 + 6678358848*d*e**2*f**2*(d + e +
f) + 3970944*d*e**2*f*(d + e + f)**2 - 1880640*d*e**2*(d + e + f)**3 + 23260904960*d*e*f**4 - 6599923968*d*e*f
**3*(d + e + f) - 39645696*d*e*f**2*(d + e + f)**2 + 5713152*d*e*f*(d + e + f)**3 - 10226813952*d*f**5 + 24366
22848*d*f**4*(d + e + f) + 36037632*d*f**3*(d + e + f)**2 - 4243968*d*f**2*(d + e + f)**3 + 429000000*e**6 - 2
689492288*e**5*f - 169242912*e**5*(d + e + f) + 5722012800*e**4*f**2 + 1275930432*e**4*f*(d + e + f) - 4538112
*e**4*(d + e + f)**2 - 2792459520*e**3*f**3 - 3830276736*e**3*f**2*(d + e + f) + 13917952*e**3*f*(d + e + f)**
2 + 964224*e**3*(d + e + f)**3 - 6487841280*e**2*f**4 + 5725304064*e**2*f**3*(d + e + f) - 1516032*e**2*f**2*(
d + e + f)**2 - 4637952*e**2*f*(d + e + f)**3 + 9596336128*e*f**5 - 4262274048*e*f**4*(d + e + f) - 27731968*e
*f**3*(d + e + f)**2 + 7246848*e*f**2*(d + e + f)**3 - 3803258880*f**6 + 1264435200*f**5*(d + e + f) + 2101248
0*f**4*(d + e + f)**2 - 3686400*f**3*(d + e + f)**3)/(13474125*d**6 - 102860175*d**5*e + 196999875*d**5*f + 27
4190390*d**4*e**2 - 1214801310*d**4*e*f + 1192261140*d**4*f**2 - 224142072*d**3*e**3 + 2417766776*d**3*e**2*f
- 5675135904*d**3*e*f**2 + 3821016960*d**3*f**3 - 245084096*d**2*e**4 - 1010456304*d**2*e**3*f + 7752467424*d*
*2*e**2*f**2 - 13083849984*d**2*e*f**3 + 6834458880*d**2*f**4 + 535797456*d*e**5 - 1875889936*d*e**4*f - 75853
1712*d*e**3*f**2 + 10575845888*d*e**2*f**3 - 14846923776*d*e*f**4 + 6462996480*d*f**5 - 256183200*e**6 + 15797
08320*e**5*f - 3185673920*e**4*f**2 + 909173760*e**3*f**3 + 5054735360*e**2*f**4 - 6608977920*e*f**5 + 2521497
600*f**6))/36 + (d + 2*e + 4*f)*log(x + (-24383100*d**6 + 187408066*d**5*e - 350082438*d**5*f - 10439775*d**5*
(d + 2*e + 4*f)/4 - 511591980*d**4*e**2 + 2159032460*d**4*e*f + 47066145*d**4*e*(d + 2*e + 4*f)/2 - 2073816840
*d**4*f**2 - 34273845*d**4*f*(d + 2*e + 4*f) + 41700*d**4*(d + 2*e + 4*f)**2 + 469491120*d**3*e**3 - 433997544
0*d**3*e**2*f - 83418138*d**3*e**2*(d + 2*e + 4*f) + 9775163680*d**3*e*f**2 + 245201952*d**3*e*f*(d + 2*e + 4*
f) - 168958*d**3*e*(d + 2*e + 4*f)**2 - 6478462560*d**3*f**3 - 179428428*d**3*f**2*(d + 2*e + 4*f) + 400554*d*
*3*f*(d + 2*e + 4*f)**2 + 12375*d**3*(d + 2*e + 4*f)**3/4 + 322778400*d**2*e**4 + 1983785760*d**2*e**3*f + 145
624428*d**2*e**3*(d + 2*e + 4*f) - 13135991040*d**2*e**2*f**2 - 647277048*d**2*e**2*f*(d + 2*e + 4*f) + 109548
*d**2*e**2*(d + 2*e + 4*f)**2 + 21641166400*d**2*e*f**3 + 955409112*d**2*e*f**2*(d + 2*e + 4*f) - 1135884*d**2
*e*f*(d + 2*e + 4*f)**2 - 34611*d**2*e*(d + 2*e + 4*f)**3/2 - 11234578560*d**2*f**4 - 468224208*d**2*f**3*(d +
2*e + 4*f) + 1431576*d**2*f**2*(d + 2*e + 4*f)**2 + 25029*d**2*f*(d + 2*e + 4*f)**3 - 863493856*d*e**5 + 3068
196000*d*e**4*f - 125194140*d*e**4*(d + 2*e + 4*f) + 901203840*d*e**3*f**2 + 748035072*d*e**3*f*(d + 2*e + 4*f
) + 264184*d*e**3*(d + 2*e + 4*f)**2 - 16302840960*d*e**2*f**3 - 1669589712*d*e**2*f**2*(d + 2*e + 4*f) + 2481
84*d*e**2*f*(d + 2*e + 4*f)**2 + 29385*d*e**2*(d + 2*e + 4*f)**3 + 23260904960*d*e*f**4 + 1649980992*d*e*f**3*
(d + 2*e + 4*f) - 2477856*d*e*f**2*(d + 2*e + 4*f)**2 - 89268*d*e*f*(d + 2*e + 4*f)**3 - 10226813952*d*f**5 -
609155712*d*f**4*(d + 2*e + 4*f) + 2252352*d*f**3*(d + 2*e + 4*f)**2 + 66312*d*f**2*(d + 2*e + 4*f)**3 + 42900
0000*e**6 - 2689492288*e**5*f + 42310728*e**5*(d + 2*e + 4*f) + 5722012800*e**4*f**2 - 318982608*e**4*f*(d + 2
*e + 4*f) - 283632*e**4*(d + 2*e + 4*f)**2 - 2792459520*e**3*f**3 + 957569184*e**3*f**2*(d + 2*e + 4*f) + 8698
72*e**3*f*(d + 2*e + 4*f)**2 - 15066*e**3*(d + 2*e + 4*f)**3 - 6487841280*e**2*f**4 - 1431326016*e**2*f**3*(d
+ 2*e + 4*f) - 94752*e**2*f**2*(d + 2*e + 4*f)**2 + 72468*e**2*f*(d + 2*e + 4*f)**3 + 9596336128*e*f**5 + 1065
568512*e*f**4*(d + 2*e + 4*f) - 1733248*e*f**3*(d + 2*e + 4*f)**2 - 113232*e*f**2*(d + 2*e + 4*f)**3 - 3803258
880*f**6 - 316108800*f**5*(d + 2*e + 4*f) + 1313280*f**4*(d + 2*e + 4*f)**2 + 57600*f**3*(d + 2*e + 4*f)**3)/(
13474125*d**6 - 102860175*d**5*e + 196999875*d**5*f + 274190390*d**4*e**2 - 1214801310*d**4*e*f + 1192261140*d
**4*f**2 - 224142072*d**3*e**3 + 2417766776*d**3*e**2*f - 5675135904*d**3*e*f**2 + 3821016960*d**3*f**3 - 2450
84096*d**2*e**4 - 1010456304*d**2*e**3*f + 7752467424*d**2*e**2*f**2 - 13083849984*d**2*e*f**3 + 6834458880*d*
*2*f**4 + 535797456*d*e**5 - 1875889936*d*e**4*f - 758531712*d*e**3*f**2 + 10575845888*d*e**2*f**3 - 148469237
76*d*e*f**4 + 6462996480*d*f**5 - 256183200*e**6 + 1579708320*e**5*f - 3185673920*e**4*f**2 + 909173760*e**3*f
**3 + 5054735360*e**2*f**4 - 6608977920*e*f**5 + 2521497600*f**6))/144 - (7*d - 13*e + 19*f)*log(x + (-2438310
0*d**6 + 187408066*d**5*e - 350082438*d**5*f + 10439775*d**5*(7*d - 13*e + 19*f) - 511591980*d**4*e**2 + 21590
32460*d**4*e*f - 94132290*d**4*e*(7*d - 13*e + 19*f) - 2073816840*d**4*f**2 + 137095380*d**4*f*(7*d - 13*e + 1
9*f) + 667200*d**4*(7*d - 13*e + 19*f)**2 + 469491120*d**3*e**3 - 4339975440*d**3*e**2*f + 333672552*d**3*e**2
*(7*d - 13*e + 19*f) + 9775163680*d**3*e*f**2 - 980807808*d**3*e*f*(7*d - 13*e + 19*f) - 2703328*d**3*e*(7*d -
13*e + 19*f)**2 - 6478462560*d**3*f**3 + 717713712*d**3*f**2*(7*d - 13*e + 19*f) + 6408864*d**3*f*(7*d - 13*e
+ 19*f)**2 - 198000*d**3*(7*d - 13*e + 19*f)**3 + 322778400*d**2*e**4 + 1983785760*d**2*e**3*f - 582497712*d*
*2*e**3*(7*d - 13*e + 19*f) - 13135991040*d**2*e**2*f**2 + 2589108192*d**2*e**2*f*(7*d - 13*e + 19*f) + 175276
8*d**2*e**2*(7*d - 13*e + 19*f)**2 + 21641166400*d**2*e*f**3 - 3821636448*d**2*e*f**2*(7*d - 13*e + 19*f) - 18
174144*d**2*e*f*(7*d - 13*e + 19*f)**2 + 1107552*d**2*e*(7*d - 13*e + 19*f)**3 - 11234578560*d**2*f**4 + 18728
96832*d**2*f**3*(7*d - 13*e + 19*f) + 22905216*d**2*f**2*(7*d - 13*e + 19*f)**2 - 1601856*d**2*f*(7*d - 13*e +
19*f)**3 - 863493856*d*e**5 + 3068196000*d*e**4*f + 500776560*d*e**4*(7*d - 13*e + 19*f) + 901203840*d*e**3*f
**2 - 2992140288*d*e**3*f*(7*d - 13*e + 19*f) + 4226944*d*e**3*(7*d - 13*e + 19*f)**2 - 16302840960*d*e**2*f**
3 + 6678358848*d*e**2*f**2*(7*d - 13*e + 19*f) + 3970944*d*e**2*f*(7*d - 13*e + 19*f)**2 - 1880640*d*e**2*(7*d
- 13*e + 19*f)**3 + 23260904960*d*e*f**4 - 6599923968*d*e*f**3*(7*d - 13*e + 19*f) - 39645696*d*e*f**2*(7*d -
13*e + 19*f)**2 + 5713152*d*e*f*(7*d - 13*e + 19*f)**3 - 10226813952*d*f**5 + 2436622848*d*f**4*(7*d - 13*e +
19*f) + 36037632*d*f**3*(7*d - 13*e + 19*f)**2 - 4243968*d*f**2*(7*d - 13*e + 19*f)**3 + 429000000*e**6 - 268
9492288*e**5*f - 169242912*e**5*(7*d - 13*e + 19*f) + 5722012800*e**4*f**2 + 1275930432*e**4*f*(7*d - 13*e + 1
9*f) - 4538112*e**4*(7*d - 13*e + 19*f)**2 - 2792459520*e**3*f**3 - 3830276736*e**3*f**2*(7*d - 13*e + 19*f) +
13917952*e**3*f*(7*d - 13*e + 19*f)**2 + 964224*e**3*(7*d - 13*e + 19*f)**3 - 6487841280*e**2*f**4 + 57253040
64*e**2*f**3*(7*d - 13*e + 19*f) - 1516032*e**2*f**2*(7*d - 13*e + 19*f)**2 - 4637952*e**2*f*(7*d - 13*e + 19*
f)**3 + 9596336128*e*f**5 - 4262274048*e*f**4*(7*d - 13*e + 19*f) - 27731968*e*f**3*(7*d - 13*e + 19*f)**2 + 7
246848*e*f**2*(7*d - 13*e + 19*f)**3 - 3803258880*f**6 + 1264435200*f**5*(7*d - 13*e + 19*f) + 21012480*f**4*(
7*d - 13*e + 19*f)**2 - 3686400*f**3*(7*d - 13*e + 19*f)**3)/(13474125*d**6 - 102860175*d**5*e + 196999875*d**
5*f + 274190390*d**4*e**2 - 1214801310*d**4*e*f + 1192261140*d**4*f**2 - 224142072*d**3*e**3 + 2417766776*d**3
*e**2*f - 5675135904*d**3*e*f**2 + 3821016960*d**3*f**3 - 245084096*d**2*e**4 - 1010456304*d**2*e**3*f + 77524
67424*d**2*e**2*f**2 - 13083849984*d**2*e*f**3 + 6834458880*d**2*f**4 + 535797456*d*e**5 - 1875889936*d*e**4*f
- 758531712*d*e**3*f**2 + 10575845888*d*e**2*f**3 - 14846923776*d*e*f**4 + 6462996480*d*f**5 - 256183200*e**6
+ 1579708320*e**5*f - 3185673920*e**4*f**2 + 909173760*e**3*f**3 + 5054735360*e**2*f**4 - 6608977920*e*f**5 +
2521497600*f**6))/36 + (31*d - 50*e + 76*f)*log(x + (-24383100*d**6 + 187408066*d**5*e - 350082438*d**5*f - 1
0439775*d**5*(31*d - 50*e + 76*f)/4 - 511591980*d**4*e**2 + 2159032460*d**4*e*f + 47066145*d**4*e*(31*d - 50*e
+ 76*f)/2 - 2073816840*d**4*f**2 - 34273845*d**4*f*(31*d - 50*e + 76*f) + 41700*d**4*(31*d - 50*e + 76*f)**2
+ 469491120*d**3*e**3 - 4339975440*d**3*e**2*f - 83418138*d**3*e**2*(31*d - 50*e + 76*f) + 9775163680*d**3*e*f
**2 + 245201952*d**3*e*f*(31*d - 50*e + 76*f) - 168958*d**3*e*(31*d - 50*e + 76*f)**2 - 6478462560*d**3*f**3 -
179428428*d**3*f**2*(31*d - 50*e + 76*f) + 400554*d**3*f*(31*d - 50*e + 76*f)**2 + 12375*d**3*(31*d - 50*e +
76*f)**3/4 + 322778400*d**2*e**4 + 1983785760*d**2*e**3*f + 145624428*d**2*e**3*(31*d - 50*e + 76*f) - 1313599
1040*d**2*e**2*f**2 - 647277048*d**2*e**2*f*(31*d - 50*e + 76*f) + 109548*d**2*e**2*(31*d - 50*e + 76*f)**2 +
21641166400*d**2*e*f**3 + 955409112*d**2*e*f**2*(31*d - 50*e + 76*f) - 1135884*d**2*e*f*(31*d - 50*e + 76*f)**
2 - 34611*d**2*e*(31*d - 50*e + 76*f)**3/2 - 11234578560*d**2*f**4 - 468224208*d**2*f**3*(31*d - 50*e + 76*f)
+ 1431576*d**2*f**2*(31*d - 50*e + 76*f)**2 + 25029*d**2*f*(31*d - 50*e + 76*f)**3 - 863493856*d*e**5 + 306819
6000*d*e**4*f - 125194140*d*e**4*(31*d - 50*e + 76*f) + 901203840*d*e**3*f**2 + 748035072*d*e**3*f*(31*d - 50*
e + 76*f) + 264184*d*e**3*(31*d - 50*e + 76*f)**2 - 16302840960*d*e**2*f**3 - 1669589712*d*e**2*f**2*(31*d - 5
0*e + 76*f) + 248184*d*e**2*f*(31*d - 50*e + 76*f)**2 + 29385*d*e**2*(31*d - 50*e + 76*f)**3 + 23260904960*d*e
*f**4 + 1649980992*d*e*f**3*(31*d - 50*e + 76*f) - 2477856*d*e*f**2*(31*d - 50*e + 76*f)**2 - 89268*d*e*f*(31*
d - 50*e + 76*f)**3 - 10226813952*d*f**5 - 609155712*d*f**4*(31*d - 50*e + 76*f) + 2252352*d*f**3*(31*d - 50*e
+ 76*f)**2 + 66312*d*f**2*(31*d - 50*e + 76*f)**3 + 429000000*e**6 - 2689492288*e**5*f + 42310728*e**5*(31*d
- 50*e + 76*f) + 5722012800*e**4*f**2 - 318982608*e**4*f*(31*d - 50*e + 76*f) - 283632*e**4*(31*d - 50*e + 76*
f)**2 - 2792459520*e**3*f**3 + 957569184*e**3*f**2*(31*d - 50*e + 76*f) + 869872*e**3*f*(31*d - 50*e + 76*f)**
2 - 15066*e**3*(31*d - 50*e + 76*f)**3 - 6487841280*e**2*f**4 - 1431326016*e**2*f**3*(31*d - 50*e + 76*f) - 94
752*e**2*f**2*(31*d - 50*e + 76*f)**2 + 72468*e**2*f*(31*d - 50*e + 76*f)**3 + 9596336128*e*f**5 + 1065568512*
e*f**4*(31*d - 50*e + 76*f) - 1733248*e*f**3*(31*d - 50*e + 76*f)**2 - 113232*e*f**2*(31*d - 50*e + 76*f)**3 -
3803258880*f**6 - 316108800*f**5*(31*d - 50*e + 76*f) + 1313280*f**4*(31*d - 50*e + 76*f)**2 + 57600*f**3*(31
*d - 50*e + 76*f)**3)/(13474125*d**6 - 102860175*d**5*e + 196999875*d**5*f + 274190390*d**4*e**2 - 1214801310*
d**4*e*f + 1192261140*d**4*f**2 - 224142072*d**3*e**3 + 2417766776*d**3*e**2*f - 5675135904*d**3*e*f**2 + 3821
016960*d**3*f**3 - 245084096*d**2*e**4 - 1010456304*d**2*e**3*f + 7752467424*d**2*e**2*f**2 - 13083849984*d**2
*e*f**3 + 6834458880*d**2*f**4 + 535797456*d*e**5 - 1875889936*d*e**4*f - 758531712*d*e**3*f**2 + 10575845888*
d*e**2*f**3 - 14846923776*d*e*f**4 + 6462996480*d*f**5 - 256183200*e**6 + 1579708320*e**5*f - 3185673920*e**4*
f**2 + 909173760*e**3*f**3 + 5054735360*e**2*f**4 - 6608977920*e*f**5 + 2521497600*f**6))/144 - (5*d - 6*e + 8
*f + x*(3*d - 4*e + 6*f))/(12*x**2 + 36*x + 24)
________________________________________________________________________________________
Giac [A] time = 1.09447, size = 136, normalized size = 1.3 \begin{align*} \frac{1}{144} \,{\left (31 \, d + 76 \, f - 50 \, e\right )} \log \left ({\left | x + 2 \right |}\right ) - \frac{1}{36} \,{\left (7 \, d + 19 \, f - 13 \, e\right )} \log \left ({\left | x + 1 \right |}\right ) - \frac{1}{36} \,{\left (d + f + e\right )} \log \left ({\left | x - 1 \right |}\right ) + \frac{1}{144} \,{\left (d + 4 \, f + 2 \, e\right )} \log \left ({\left | x - 2 \right |}\right ) - \frac{{\left (3 \, d + 6 \, f - 4 \, e\right )} x + 5 \, d + 8 \, f - 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((x^2-3*x+2)*(f*x^2+e*x+d)/(x^4-5*x^2+4)^2,x, algorithm="giac")
[Out]
1/144*(31*d + 76*f - 50*e)*log(abs(x + 2)) - 1/36*(7*d + 19*f - 13*e)*log(abs(x + 1)) - 1/36*(d + f + e)*log(a
bs(x - 1)) + 1/144*(d + 4*f + 2*e)*log(abs(x - 2)) - 1/12*((3*d + 6*f - 4*e)*x + 5*d + 8*f - 6*e)/((x + 2)*(x
+ 1))