\(\int \frac {d+e x+f x^2}{(4-5 x^2+x^4)^3} \, dx\) [43]

Optimal result

Integrand size = 23, antiderivative size = 175 \[ \int \frac {d+e x+f x^2}{\left (4-5 x^2+x^4\right )^3} \, dx=\frac {e \left (5-2 x^2\right )}{36 \left (4-5 x^2+x^4\right )^2}+\frac {x \left (17 d+20 f-(5 d+8 f) x^2\right )}{144 \left (4-5 x^2+x^4\right )^2}-\frac {e \left (5-2 x^2\right )}{54 \left (4-5 x^2+x^4\right )}-\frac {x \left (59 d+380 f-35 (d+4 f) x^2\right )}{3456 \left (4-5 x^2+x^4\right )}-\frac {(313 d+820 f) \text {arctanh}\left (\frac {x}{2}\right )}{20736}+\frac {1}{648} (13 d+25 f) \text {arctanh}(x)-\frac {1}{81} e \log \left (1-x^2\right )+\frac {1}{81} e \log \left (4-x^2\right ) \]

[Out]

1/36*e*(-2*x^2+5)/(x^4-5*x^2+4)^2+1/144*x*(17*d+20*f-(5*d+8*f)*x^2)/(x^4-5*x^2+4)^2-1/54*e*(-2*x^2+5)/(x^4-5*x
^2+4)-1/3456*x*(59*d+380*f-35*(d+4*f)*x^2)/(x^4-5*x^2+4)-1/20736*(313*d+820*f)*arctanh(1/2*x)+1/648*(13*d+25*f
)*arctanh(x)-1/81*e*ln(-x^2+1)+1/81*e*ln(-x^2+4)

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {1687, 1192, 1180, 213, 12, 1121, 628, 630, 31} \[ \int \frac {d+e x+f x^2}{\left (4-5 x^2+x^4\right )^3} \, dx=-\frac {\text {arctanh}\left (\frac {x}{2}\right ) (313 d+820 f)}{20736}+\frac {1}{648} \text {arctanh}(x) (13 d+25 f)-\frac {x \left (-35 x^2 (d+4 f)+59 d+380 f\right )}{3456 \left (x^4-5 x^2+4\right )}+\frac {x \left (-\left (x^2 (5 d+8 f)\right )+17 d+20 f\right )}{144 \left (x^4-5 x^2+4\right )^2}-\frac {1}{81} e \log \left (1-x^2\right )+\frac {1}{81} e \log \left (4-x^2\right )-\frac {e \left (5-2 x^2\right )}{54 \left (x^4-5 x^2+4\right )}+\frac {e \left (5-2 x^2\right )}{36 \left (x^4-5 x^2+4\right )^2} \]

[In]

Int[(d + e*x + f*x^2)/(4 - 5*x^2 + x^4)^3,x]

[Out]

(e*(5 - 2*x^2))/(36*(4 - 5*x^2 + x^4)^2) + (x*(17*d + 20*f - (5*d + 8*f)*x^2))/(144*(4 - 5*x^2 + x^4)^2) - (e*
(5 - 2*x^2))/(54*(4 - 5*x^2 + x^4)) - (x*(59*d + 380*f - 35*(d + 4*f)*x^2))/(3456*(4 - 5*x^2 + x^4)) - ((313*d
+ 820*f)*ArcTanh[x/2])/20736 + ((13*d + 25*f)*ArcTanh[x])/648 - (e*Log[1 - x^2])/81 + (e*Log[4 - x^2])/81

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 628

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1
)*(b^2 - 4*a*c))), x] - Dist[2*c*((2*p + 3)/((p + 1)*(b^2 - 4*a*c))), Int[(a + b*x + c*x^2)^(p + 1), x], x] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2] && IntegerQ[4*p]

Rule 630

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[c/q, Int[1/Simp
[b/2 - q/2 + c*x, x], x], x] - Dist[c/q, Int[1/Simp[b/2 + q/2 + c*x, x], x], x]] /; FreeQ[{a, b, c}, x] && NeQ
[b^2 - 4*a*c, 0] && PosQ[b^2 - 4*a*c] && PerfectSquareQ[b^2 - 4*a*c]

Rule 1121

Int[(x_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[(a + b*x + c*x^2)^p, x],
x, x^2], x] /; FreeQ[{a, b, c, p}, x]

Rule 1180

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rule 1192

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[x*(a*b*e - d*(b^2 - 2*a
*c) - c*(b*d - 2*a*e)*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Dist[1/(2*a*(p + 1)
*(b^2 - 4*a*c)), Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 7)*(d*b - 2*a*e)*c*x^2, x]*(a +
b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e
^2, 0] && LtQ[p, -1] && IntegerQ[2*p]

Rule 1687

Int[(Pq_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], k}, Int[Sum[Coeff[
Pq, x, 2*k]*x^(2*k), {k, 0, q/2}]*(a + b*x^2 + c*x^4)^p, x] + Int[x*Sum[Coeff[Pq, x, 2*k + 1]*x^(2*k), {k, 0,
(q - 1)/2}]*(a + b*x^2 + c*x^4)^p, x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] &&  !PolyQ[Pq, x^2]

Rubi steps \begin{align*} \text {integral}& = \int \frac {e x}{\left (4-5 x^2+x^4\right )^3} \, dx+\int \frac {d+f x^2}{\left (4-5 x^2+x^4\right )^3} \, dx \\ & = \frac {x \left (17 d+20 f-(5 d+8 f) x^2\right )}{144 \left (4-5 x^2+x^4\right )^2}-\frac {1}{144} \int \frac {-19 d+20 f+5 (5 d+8 f) x^2}{\left (4-5 x^2+x^4\right )^2} \, dx+e \int \frac {x}{\left (4-5 x^2+x^4\right )^3} \, dx \\ & = \frac {x \left (17 d+20 f-(5 d+8 f) x^2\right )}{144 \left (4-5 x^2+x^4\right )^2}-\frac {x \left (59 d+380 f-35 (d+4 f) x^2\right )}{3456 \left (4-5 x^2+x^4\right )}+\frac {\int \frac {3 (173 d+260 f)+105 (d+4 f) x^2}{4-5 x^2+x^4} \, dx}{10368}+\frac {1}{2} e \text {Subst}\left (\int \frac {1}{\left (4-5 x+x^2\right )^3} \, dx,x,x^2\right ) \\ & = \frac {e \left (5-2 x^2\right )}{36 \left (4-5 x^2+x^4\right )^2}+\frac {x \left (17 d+20 f-(5 d+8 f) x^2\right )}{144 \left (4-5 x^2+x^4\right )^2}-\frac {x \left (59 d+380 f-35 (d+4 f) x^2\right )}{3456 \left (4-5 x^2+x^4\right )}-\frac {1}{6} e \text {Subst}\left (\int \frac {1}{\left (4-5 x+x^2\right )^2} \, dx,x,x^2\right )+\frac {1}{648} (-13 d-25 f) \int \frac {1}{-1+x^2} \, dx+\frac {(313 d+820 f) \int \frac {1}{-4+x^2} \, dx}{10368} \\ & = \frac {e \left (5-2 x^2\right )}{36 \left (4-5 x^2+x^4\right )^2}+\frac {x \left (17 d+20 f-(5 d+8 f) x^2\right )}{144 \left (4-5 x^2+x^4\right )^2}-\frac {e \left (5-2 x^2\right )}{54 \left (4-5 x^2+x^4\right )}-\frac {x \left (59 d+380 f-35 (d+4 f) x^2\right )}{3456 \left (4-5 x^2+x^4\right )}-\frac {(313 d+820 f) \tanh ^{-1}\left (\frac {x}{2}\right )}{20736}+\frac {1}{648} (13 d+25 f) \tanh ^{-1}(x)+\frac {1}{27} e \text {Subst}\left (\int \frac {1}{4-5 x+x^2} \, dx,x,x^2\right ) \\ & = \frac {e \left (5-2 x^2\right )}{36 \left (4-5 x^2+x^4\right )^2}+\frac {x \left (17 d+20 f-(5 d+8 f) x^2\right )}{144 \left (4-5 x^2+x^4\right )^2}-\frac {e \left (5-2 x^2\right )}{54 \left (4-5 x^2+x^4\right )}-\frac {x \left (59 d+380 f-35 (d+4 f) x^2\right )}{3456 \left (4-5 x^2+x^4\right )}-\frac {(313 d+820 f) \tanh ^{-1}\left (\frac {x}{2}\right )}{20736}+\frac {1}{648} (13 d+25 f) \tanh ^{-1}(x)+\frac {1}{81} e \text {Subst}\left (\int \frac {1}{-4+x} \, dx,x,x^2\right )-\frac {1}{81} e \text {Subst}\left (\int \frac {1}{-1+x} \, dx,x,x^2\right ) \\ & = \frac {e \left (5-2 x^2\right )}{36 \left (4-5 x^2+x^4\right )^2}+\frac {x \left (17 d+20 f-(5 d+8 f) x^2\right )}{144 \left (4-5 x^2+x^4\right )^2}-\frac {e \left (5-2 x^2\right )}{54 \left (4-5 x^2+x^4\right )}-\frac {x \left (59 d+380 f-35 (d+4 f) x^2\right )}{3456 \left (4-5 x^2+x^4\right )}-\frac {(313 d+820 f) \tanh ^{-1}\left (\frac {x}{2}\right )}{20736}+\frac {1}{648} (13 d+25 f) \tanh ^{-1}(x)-\frac {1}{81} e \log \left (1-x^2\right )+\frac {1}{81} e \log \left (4-x^2\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.92 \[ \int \frac {d+e x+f x^2}{\left (4-5 x^2+x^4\right )^3} \, dx=\frac {\frac {288 \left (17 d x+20 f x-5 d x^3-8 f x^3+e \left (20-8 x^2\right )\right )}{\left (4-5 x^2+x^4\right )^2}+\frac {12 \left (64 e \left (-5+2 x^2\right )+20 f x \left (-19+7 x^2\right )+d x \left (-59+35 x^2\right )\right )}{4-5 x^2+x^4}-32 (13 d+16 e+25 f) \log (1-x)+(313 d+512 e+820 f) \log (2-x)+32 (13 d-16 e+25 f) \log (1+x)+(-313 d+512 e-820 f) \log (2+x)}{41472} \]

[In]

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

[Out]

((288*(17*d*x + 20*f*x - 5*d*x^3 - 8*f*x^3 + e*(20 - 8*x^2)))/(4 - 5*x^2 + x^4)^2 + (12*(64*e*(-5 + 2*x^2) + 2
0*f*x*(-19 + 7*x^2) + d*x*(-59 + 35*x^2)))/(4 - 5*x^2 + x^4) - 32*(13*d + 16*e + 25*f)*Log[1 - x] + (313*d + 5
12*e + 820*f)*Log[2 - x] + 32*(13*d - 16*e + 25*f)*Log[1 + x] + (-313*d + 512*e - 820*f)*Log[2 + x])/41472

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.79

method	result	size
norman	\(\frac {\left (-\frac {13 d}{192}-\frac {5 f}{16}\right ) x^{5}+\left (\frac {35 d}{384}+\frac {21 f}{32}\right ) x^{3}+\left (\frac {35 d}{3456}+\frac {35 f}{864}\right ) x^{7}+\left (\frac {43 d}{864}-\frac {65 f}{216}\right ) x +\frac {5 e \,x^{2}}{9}+\frac {e \,x^{6}}{27}-\frac {5 e \,x^{4}}{18}-\frac {25 e}{108}}{\left (x^{4}-5 x^{2}+4\right )^{2}}+\left (-\frac {313 d}{41472}+\frac {e}{81}-\frac {205 f}{10368}\right ) \ln \left (x +2\right )+\left (-\frac {13 d}{1296}-\frac {e}{81}-\frac {25 f}{1296}\right ) \ln \left (x -1\right )+\left (\frac {13 d}{1296}-\frac {e}{81}+\frac {25 f}{1296}\right ) \ln \left (x +1\right )+\left (\frac {313 d}{41472}+\frac {e}{81}+\frac {205 f}{10368}\right ) \ln \left (x -2\right )\)	\(139\)
risch	\(\frac {\left (-\frac {13 d}{192}-\frac {5 f}{16}\right ) x^{5}+\left (\frac {35 d}{384}+\frac {21 f}{32}\right ) x^{3}+\left (\frac {35 d}{3456}+\frac {35 f}{864}\right ) x^{7}+\left (\frac {43 d}{864}-\frac {65 f}{216}\right ) x +\frac {5 e \,x^{2}}{9}+\frac {e \,x^{6}}{27}-\frac {5 e \,x^{4}}{18}-\frac {25 e}{108}}{\left (x^{4}-5 x^{2}+4\right )^{2}}+\frac {313 \ln \left (2-x \right ) d}{41472}+\frac {\ln \left (2-x \right ) e}{81}+\frac {205 \ln \left (2-x \right ) f}{10368}+\frac {13 \ln \left (x +1\right ) d}{1296}-\frac {\ln \left (x +1\right ) e}{81}+\frac {25 \ln \left (x +1\right ) f}{1296}-\frac {13 \ln \left (1-x \right ) d}{1296}-\frac {\ln \left (1-x \right ) e}{81}-\frac {25 \ln \left (1-x \right ) f}{1296}-\frac {313 \ln \left (x +2\right ) d}{41472}+\frac {\ln \left (x +2\right ) e}{81}-\frac {205 \ln \left (x +2\right ) f}{10368}\)	\(175\)
default	\(\left (-\frac {313 d}{41472}+\frac {e}{81}-\frac {205 f}{10368}\right ) \ln \left (x +2\right )-\frac {-\frac {19 d}{6912}+\frac {17 e}{3456}-\frac {5 f}{576}}{x +2}-\frac {-\frac {d}{1728}+\frac {e}{864}-\frac {f}{432}}{2 \left (x +2\right )^{2}}-\frac {-\frac {d}{432}+\frac {e}{144}-\frac {5 f}{432}}{x +1}-\frac {\frac {d}{216}-\frac {e}{216}+\frac {f}{216}}{2 \left (x +1\right )^{2}}+\left (\frac {13 d}{1296}-\frac {e}{81}+\frac {25 f}{1296}\right ) \ln \left (x +1\right )+\left (-\frac {13 d}{1296}-\frac {e}{81}-\frac {25 f}{1296}\right ) \ln \left (x -1\right )-\frac {-\frac {d}{432}-\frac {e}{144}-\frac {5 f}{432}}{x -1}-\frac {-\frac {d}{216}-\frac {e}{216}-\frac {f}{216}}{2 \left (x -1\right )^{2}}-\frac {-\frac {19 d}{6912}-\frac {17 e}{3456}-\frac {5 f}{576}}{x -2}-\frac {\frac {d}{1728}+\frac {e}{864}+\frac {f}{432}}{2 \left (x -2\right )^{2}}+\left (\frac {313 d}{41472}+\frac {e}{81}+\frac {205 f}{10368}\right ) \ln \left (x -2\right )\)	\(198\)
parallelrisch	\(\frac {-12960 f \,x^{5}+1536 e \,x^{6}-11520 e \,x^{4}-9600 e +27216 f \,x^{3}+420 d \,x^{7}+1680 f \,x^{7}+2064 d x +5008 \ln \left (x -2\right ) d +8192 \ln \left (x -2\right ) e -6656 \ln \left (x -1\right ) d -8192 \ln \left (x -1\right ) e -4160 \ln \left (x +1\right ) x^{6} d +16896 \ln \left (x -2\right ) x^{4} e -13120 \ln \left (x +2\right ) f +12800 \ln \left (x +1\right ) f +23040 e \,x^{2}-20480 \ln \left (x -2\right ) x^{2} e +16640 \ln \left (x -1\right ) x^{2} d +20480 \ln \left (x -1\right ) x^{2} e -16640 \ln \left (x +1\right ) x^{2} d +20480 \ln \left (x +1\right ) x^{2} e +12520 \ln \left (x +2\right ) x^{2} d -20480 \ln \left (x +2\right ) x^{2} e +10329 \ln \left (x -2\right ) x^{4} d +820 \ln \left (x -2\right ) x^{8} f -800 \ln \left (x -1\right ) x^{8} f +800 \ln \left (x +1\right ) x^{8} f -820 \ln \left (x +2\right ) x^{8} f -8200 \ln \left (x -2\right ) x^{6} f +8000 \ln \left (x -1\right ) x^{6} f -8000 \ln \left (x +1\right ) x^{6} f +8200 \ln \left (x +2\right ) x^{6} f -5008 \ln \left (x +2\right ) d +5120 \ln \left (x +1\right ) x^{6} e +3130 \ln \left (x +2\right ) x^{6} d -5120 \ln \left (x +2\right ) x^{6} e +8192 \ln \left (x +2\right ) e +6656 \ln \left (x +1\right ) d -8192 \ln \left (x +1\right ) e -2808 x^{5} d +3780 x^{3} d -512 \ln \left (x +1\right ) x^{8} e -26400 \ln \left (x -1\right ) x^{4} f +26400 \ln \left (x +1\right ) x^{4} f -27060 \ln \left (x +2\right ) x^{4} f +512 \ln \left (x -2\right ) x^{8} e -416 \ln \left (x -1\right ) x^{8} d -512 \ln \left (x -1\right ) x^{8} e +4160 \ln \left (x -1\right ) x^{6} d +5120 \ln \left (x -1\right ) x^{6} e -32800 \ln \left (x -2\right ) x^{2} f +32000 \ln \left (x -1\right ) x^{2} f -32000 \ln \left (x +1\right ) x^{2} f +32800 \ln \left (x +2\right ) x^{2} f +27060 \ln \left (x -2\right ) x^{4} f -313 \ln \left (x +2\right ) x^{8} d +512 \ln \left (x +2\right ) x^{8} e -3130 \ln \left (x -2\right ) x^{6} d +416 \ln \left (x +1\right ) x^{8} d -5120 \ln \left (x -2\right ) x^{6} e +13120 \ln \left (x -2\right ) f -12800 \ln \left (x -1\right ) f +313 \ln \left (x -2\right ) x^{8} d -13728 \ln \left (x -1\right ) x^{4} d -16896 \ln \left (x -1\right ) x^{4} e +13728 \ln \left (x +1\right ) x^{4} d -16896 \ln \left (x +1\right ) x^{4} e -10329 \ln \left (x +2\right ) x^{4} d +16896 \ln \left (x +2\right ) x^{4} e -12520 \ln \left (x -2\right ) x^{2} d -12480 f x}{41472 \left (x^{4}-5 x^{2}+4\right )^{2}}\)	\(645\)

[In]

int((f*x^2+e*x+d)/(x^4-5*x^2+4)^3,x,method=_RETURNVERBOSE)

[Out]

((-13/192*d-5/16*f)*x^5+(35/384*d+21/32*f)*x^3+(35/3456*d+35/864*f)*x^7+(43/864*d-65/216*f)*x+5/9*e*x^2+1/27*e
*x^6-5/18*e*x^4-25/108*e)/(x^4-5*x^2+4)^2+(-313/41472*d+1/81*e-205/10368*f)*ln(x+2)+(-13/1296*d-1/81*e-25/1296
*f)*ln(x-1)+(13/1296*d-1/81*e+25/1296*f)*ln(x+1)+(313/41472*d+1/81*e+205/10368*f)*ln(x-2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 389 vs. \(2 (156) = 312\).

Time = 0.32 (sec) , antiderivative size = 389, normalized size of antiderivative = 2.22 \[ \int \frac {d+e x+f x^2}{\left (4-5 x^2+x^4\right )^3} \, dx=\frac {420 \, {\left (d + 4 \, f\right )} x^{7} + 1536 \, e x^{6} - 216 \, {\left (13 \, d + 60 \, f\right )} x^{5} - 11520 \, e x^{4} + 756 \, {\left (5 \, d + 36 \, f\right )} x^{3} + 23040 \, e x^{2} + 48 \, {\left (43 \, d - 260 \, f\right )} x - {\left ({\left (313 \, d - 512 \, e + 820 \, f\right )} x^{8} - 10 \, {\left (313 \, d - 512 \, e + 820 \, f\right )} x^{6} + 33 \, {\left (313 \, d - 512 \, e + 820 \, f\right )} x^{4} - 40 \, {\left (313 \, d - 512 \, e + 820 \, f\right )} x^{2} + 5008 \, d - 8192 \, e + 13120 \, f\right )} \log \left (x + 2\right ) + 32 \, {\left ({\left (13 \, d - 16 \, e + 25 \, f\right )} x^{8} - 10 \, {\left (13 \, d - 16 \, e + 25 \, f\right )} x^{6} + 33 \, {\left (13 \, d - 16 \, e + 25 \, f\right )} x^{4} - 40 \, {\left (13 \, d - 16 \, e + 25 \, f\right )} x^{2} + 208 \, d - 256 \, e + 400 \, f\right )} \log \left (x + 1\right ) - 32 \, {\left ({\left (13 \, d + 16 \, e + 25 \, f\right )} x^{8} - 10 \, {\left (13 \, d + 16 \, e + 25 \, f\right )} x^{6} + 33 \, {\left (13 \, d + 16 \, e + 25 \, f\right )} x^{4} - 40 \, {\left (13 \, d + 16 \, e + 25 \, f\right )} x^{2} + 208 \, d + 256 \, e + 400 \, f\right )} \log \left (x - 1\right ) + {\left ({\left (313 \, d + 512 \, e + 820 \, f\right )} x^{8} - 10 \, {\left (313 \, d + 512 \, e + 820 \, f\right )} x^{6} + 33 \, {\left (313 \, d + 512 \, e + 820 \, f\right )} x^{4} - 40 \, {\left (313 \, d + 512 \, e + 820 \, f\right )} x^{2} + 5008 \, d + 8192 \, e + 13120 \, f\right )} \log \left (x - 2\right ) - 9600 \, e}{41472 \, {\left (x^{8} - 10 \, x^{6} + 33 \, x^{4} - 40 \, x^{2} + 16\right )}} \]

[In]

integrate((f*x^2+e*x+d)/(x^4-5*x^2+4)^3,x, algorithm="fricas")

[Out]

1/41472*(420*(d + 4*f)*x^7 + 1536*e*x^6 - 216*(13*d + 60*f)*x^5 - 11520*e*x^4 + 756*(5*d + 36*f)*x^3 + 23040*e
*x^2 + 48*(43*d - 260*f)*x - ((313*d - 512*e + 820*f)*x^8 - 10*(313*d - 512*e + 820*f)*x^6 + 33*(313*d - 512*e
+ 820*f)*x^4 - 40*(313*d - 512*e + 820*f)*x^2 + 5008*d - 8192*e + 13120*f)*log(x + 2) + 32*((13*d - 16*e + 25
*f)*x^8 - 10*(13*d - 16*e + 25*f)*x^6 + 33*(13*d - 16*e + 25*f)*x^4 - 40*(13*d - 16*e + 25*f)*x^2 + 208*d - 25
6*e + 400*f)*log(x + 1) - 32*((13*d + 16*e + 25*f)*x^8 - 10*(13*d + 16*e + 25*f)*x^6 + 33*(13*d + 16*e + 25*f)
*x^4 - 40*(13*d + 16*e + 25*f)*x^2 + 208*d + 256*e + 400*f)*log(x - 1) + ((313*d + 512*e + 820*f)*x^8 - 10*(31
3*d + 512*e + 820*f)*x^6 + 33*(313*d + 512*e + 820*f)*x^4 - 40*(313*d + 512*e + 820*f)*x^2 + 5008*d + 8192*e +
13120*f)*log(x - 2) - 9600*e)/(x^8 - 10*x^6 + 33*x^4 - 40*x^2 + 16)

Sympy [F(-1)]

Timed out. \[ \int \frac {d+e x+f x^2}{\left (4-5 x^2+x^4\right )^3} \, dx=\text {Timed out} \]

[In]

integrate((f*x**2+e*x+d)/(x**4-5*x**2+4)**3,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.89 \[ \int \frac {d+e x+f x^2}{\left (4-5 x^2+x^4\right )^3} \, dx=-\frac {1}{41472} \, {\left (313 \, d - 512 \, e + 820 \, f\right )} \log \left (x + 2\right ) + \frac {1}{1296} \, {\left (13 \, d - 16 \, e + 25 \, f\right )} \log \left (x + 1\right ) - \frac {1}{1296} \, {\left (13 \, d + 16 \, e + 25 \, f\right )} \log \left (x - 1\right ) + \frac {1}{41472} \, {\left (313 \, d + 512 \, e + 820 \, f\right )} \log \left (x - 2\right ) + \frac {35 \, {\left (d + 4 \, f\right )} x^{7} + 128 \, e x^{6} - 18 \, {\left (13 \, d + 60 \, f\right )} x^{5} - 960 \, e x^{4} + 63 \, {\left (5 \, d + 36 \, f\right )} x^{3} + 1920 \, e x^{2} + 4 \, {\left (43 \, d - 260 \, f\right )} x - 800 \, e}{3456 \, {\left (x^{8} - 10 \, x^{6} + 33 \, x^{4} - 40 \, x^{2} + 16\right )}} \]

[In]

integrate((f*x^2+e*x+d)/(x^4-5*x^2+4)^3,x, algorithm="maxima")

[Out]

-1/41472*(313*d - 512*e + 820*f)*log(x + 2) + 1/1296*(13*d - 16*e + 25*f)*log(x + 1) - 1/1296*(13*d + 16*e + 2
5*f)*log(x - 1) + 1/41472*(313*d + 512*e + 820*f)*log(x - 2) + 1/3456*(35*(d + 4*f)*x^7 + 128*e*x^6 - 18*(13*d
+ 60*f)*x^5 - 960*e*x^4 + 63*(5*d + 36*f)*x^3 + 1920*e*x^2 + 4*(43*d - 260*f)*x - 800*e)/(x^8 - 10*x^6 + 33*x
^4 - 40*x^2 + 16)

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.85 \[ \int \frac {d+e x+f x^2}{\left (4-5 x^2+x^4\right )^3} \, dx=-\frac {1}{41472} \, {\left (313 \, d - 512 \, e + 820 \, f\right )} \log \left ({\left | x + 2 \right |}\right ) + \frac {1}{1296} \, {\left (13 \, d - 16 \, e + 25 \, f\right )} \log \left ({\left | x + 1 \right |}\right ) - \frac {1}{1296} \, {\left (13 \, d + 16 \, e + 25 \, f\right )} \log \left ({\left | x - 1 \right |}\right ) + \frac {1}{41472} \, {\left (313 \, d + 512 \, e + 820 \, f\right )} \log \left ({\left | x - 2 \right |}\right ) + \frac {35 \, d x^{7} + 140 \, f x^{7} + 128 \, e x^{6} - 234 \, d x^{5} - 1080 \, f x^{5} - 960 \, e x^{4} + 315 \, d x^{3} + 2268 \, f x^{3} + 1920 \, e x^{2} + 172 \, d x - 1040 \, f x - 800 \, e}{3456 \, {\left (x^{4} - 5 \, x^{2} + 4\right )}^{2}} \]

[In]

integrate((f*x^2+e*x+d)/(x^4-5*x^2+4)^3,x, algorithm="giac")

[Out]

-1/41472*(313*d - 512*e + 820*f)*log(abs(x + 2)) + 1/1296*(13*d - 16*e + 25*f)*log(abs(x + 1)) - 1/1296*(13*d
+ 16*e + 25*f)*log(abs(x - 1)) + 1/41472*(313*d + 512*e + 820*f)*log(abs(x - 2)) + 1/3456*(35*d*x^7 + 140*f*x^
7 + 128*e*x^6 - 234*d*x^5 - 1080*f*x^5 - 960*e*x^4 + 315*d*x^3 + 2268*f*x^3 + 1920*e*x^2 + 172*d*x - 1040*f*x
- 800*e)/(x^4 - 5*x^2 + 4)^2

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.86 \[ \int \frac {d+e x+f x^2}{\left (4-5 x^2+x^4\right )^3} \, dx=\ln \left (x+1\right )\,\left (\frac {13\,d}{1296}-\frac {e}{81}+\frac {25\,f}{1296}\right )-\ln \left (x-1\right )\,\left (\frac {13\,d}{1296}+\frac {e}{81}+\frac {25\,f}{1296}\right )+\ln \left (x-2\right )\,\left (\frac {313\,d}{41472}+\frac {e}{81}+\frac {205\,f}{10368}\right )-\ln \left (x+2\right )\,\left (\frac {313\,d}{41472}-\frac {e}{81}+\frac {205\,f}{10368}\right )+\frac {\left (\frac {35\,d}{3456}+\frac {35\,f}{864}\right )\,x^7+\frac {e\,x^6}{27}+\left (-\frac {13\,d}{192}-\frac {5\,f}{16}\right )\,x^5-\frac {5\,e\,x^4}{18}+\left (\frac {35\,d}{384}+\frac {21\,f}{32}\right )\,x^3+\frac {5\,e\,x^2}{9}+\left (\frac {43\,d}{864}-\frac {65\,f}{216}\right )\,x-\frac {25\,e}{108}}{x^8-10\,x^6+33\,x^4-40\,x^2+16} \]

[In]

int((d + e*x + f*x^2)/(x^4 - 5*x^2 + 4)^3,x)

[Out]

log(x + 1)*((13*d)/1296 - e/81 + (25*f)/1296) - log(x - 1)*((13*d)/1296 + e/81 + (25*f)/1296) + log(x - 2)*((3
13*d)/41472 + e/81 + (205*f)/10368) - log(x + 2)*((313*d)/41472 - e/81 + (205*f)/10368) + (x^3*((35*d)/384 + (
21*f)/32) - x^5*((13*d)/192 + (5*f)/16) - (25*e)/108 + x^7*((35*d)/3456 + (35*f)/864) + (5*e*x^2)/9 - (5*e*x^4
)/18 + (e*x^6)/27 + x*((43*d)/864 - (65*f)/216))/(33*x^4 - 40*x^2 - 10*x^6 + x^8 + 16)