\(\int \frac {d+e x}{(4-5 x^2+x^4)^3} \, dx\) [42]

Optimal result

Integrand size = 18, antiderivative size = 143 \[ \int \frac {d+e x}{\left (4-5 x^2+x^4\right )^3} \, dx=\frac {d x \left (17-5 x^2\right )}{144 \left (4-5 x^2+x^4\right )^2}+\frac {e \left (5-2 x^2\right )}{36 \left (4-5 x^2+x^4\right )^2}-\frac {d x \left (59-35 x^2\right )}{3456 \left (4-5 x^2+x^4\right )}-\frac {e \left (5-2 x^2\right )}{54 \left (4-5 x^2+x^4\right )}-\frac {313 d \text {arctanh}\left (\frac {x}{2}\right )}{20736}+\frac {13}{648} d \text {arctanh}(x)-\frac {1}{81} e \log \left (1-x^2\right )+\frac {1}{81} e \log \left (4-x^2\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {1687, 12, 1106, 1192, 1180, 213, 1121, 628, 630, 31} \[ \int \frac {d+e x}{\left (4-5 x^2+x^4\right )^3} \, dx=-\frac {313 d \text {arctanh}\left (\frac {x}{2}\right )}{20736}+\frac {13}{648} d \text {arctanh}(x)-\frac {d x \left (59-35 x^2\right )}{3456 \left (x^4-5 x^2+4\right )}+\frac {d x \left (17-5 x^2\right )}{144 \left (x^4-5 x^2+4\right )^2}-\frac {1}{81} e \log \left (1-x^2\right )+\frac {1}{81} e \log \left (4-x^2\right )-\frac {e \left (5-2 x^2\right )}{54 \left (x^4-5 x^2+4\right )}+\frac {e \left (5-2 x^2\right )}{36 \left (x^4-5 x^2+4\right )^2} \]

[In]

Int[(d + e*x)/(4 - 5*x^2 + x^4)^3,x]

[Out]

(d*x*(17 - 5*x^2))/(144*(4 - 5*x^2 + x^4)^2) + (e*(5 - 2*x^2))/(36*(4 - 5*x^2 + x^4)^2) - (d*x*(59 - 35*x^2))/
(3456*(4 - 5*x^2 + x^4)) - (e*(5 - 2*x^2))/(54*(4 - 5*x^2 + x^4)) - (313*d*ArcTanh[x/2])/20736 + (13*d*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 1106

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

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 {d}{\left (4-5 x^2+x^4\right )^3} \, dx+\int \frac {e x}{\left (4-5 x^2+x^4\right )^3} \, dx \\ & = d \int \frac {1}{\left (4-5 x^2+x^4\right )^3} \, dx+e \int \frac {x}{\left (4-5 x^2+x^4\right )^3} \, dx \\ & = \frac {d x \left (17-5 x^2\right )}{144 \left (4-5 x^2+x^4\right )^2}-\frac {1}{144} d \int \frac {-19+25 x^2}{\left (4-5 x^2+x^4\right )^2} \, dx+\frac {1}{2} e \text {Subst}\left (\int \frac {1}{\left (4-5 x+x^2\right )^3} \, dx,x,x^2\right ) \\ & = \frac {d x \left (17-5 x^2\right )}{144 \left (4-5 x^2+x^4\right )^2}+\frac {e \left (5-2 x^2\right )}{36 \left (4-5 x^2+x^4\right )^2}-\frac {d x \left (59-35 x^2\right )}{3456 \left (4-5 x^2+x^4\right )}+\frac {d \int \frac {519+105 x^2}{4-5 x^2+x^4} \, dx}{10368}-\frac {1}{6} e \text {Subst}\left (\int \frac {1}{\left (4-5 x+x^2\right )^2} \, dx,x,x^2\right ) \\ & = \frac {d x \left (17-5 x^2\right )}{144 \left (4-5 x^2+x^4\right )^2}+\frac {e \left (5-2 x^2\right )}{36 \left (4-5 x^2+x^4\right )^2}-\frac {d x \left (59-35 x^2\right )}{3456 \left (4-5 x^2+x^4\right )}-\frac {e \left (5-2 x^2\right )}{54 \left (4-5 x^2+x^4\right )}-\frac {1}{648} (13 d) \int \frac {1}{-1+x^2} \, dx+\frac {(313 d) \int \frac {1}{-4+x^2} \, dx}{10368}+\frac {1}{27} e \text {Subst}\left (\int \frac {1}{4-5 x+x^2} \, dx,x,x^2\right ) \\ & = \frac {d x \left (17-5 x^2\right )}{144 \left (4-5 x^2+x^4\right )^2}+\frac {e \left (5-2 x^2\right )}{36 \left (4-5 x^2+x^4\right )^2}-\frac {d x \left (59-35 x^2\right )}{3456 \left (4-5 x^2+x^4\right )}-\frac {e \left (5-2 x^2\right )}{54 \left (4-5 x^2+x^4\right )}-\frac {313 d \tanh ^{-1}\left (\frac {x}{2}\right )}{20736}+\frac {13}{648} d \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 {d x \left (17-5 x^2\right )}{144 \left (4-5 x^2+x^4\right )^2}+\frac {e \left (5-2 x^2\right )}{36 \left (4-5 x^2+x^4\right )^2}-\frac {d x \left (59-35 x^2\right )}{3456 \left (4-5 x^2+x^4\right )}-\frac {e \left (5-2 x^2\right )}{54 \left (4-5 x^2+x^4\right )}-\frac {313 d \tanh ^{-1}\left (\frac {x}{2}\right )}{20736}+\frac {13}{648} d \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 = 0.07 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.90 \[ \int \frac {d+e x}{\left (4-5 x^2+x^4\right )^3} \, dx=\frac {\frac {288 \left (e \left (20-8 x^2\right )+d x \left (17-5 x^2\right )\right )}{\left (4-5 x^2+x^4\right )^2}+\frac {12 \left (64 e \left (-5+2 x^2\right )+d x \left (-59+35 x^2\right )\right )}{4-5 x^2+x^4}-32 (13 d+16 e) \log (1-x)+(313 d+512 e) \log (2-x)+32 (13 d-16 e) \log (1+x)+(-313 d+512 e) \log (2+x)}{41472} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.75

method	result	size
norman	\(\frac {\frac {5}{9} e \,x^{2}+\frac {1}{27} e \,x^{6}-\frac {5}{18} e \,x^{4}+\frac {43}{864} d x +\frac {35}{3456} d \,x^{7}+\frac {35}{384} x^{3} d -\frac {13}{192} x^{5} d -\frac {25}{108} e}{\left (x^{4}-5 x^{2}+4\right )^{2}}+\left (-\frac {313 d}{41472}+\frac {e}{81}\right ) \ln \left (x +2\right )+\left (-\frac {13 d}{1296}-\frac {e}{81}\right ) \ln \left (x -1\right )+\left (\frac {13 d}{1296}-\frac {e}{81}\right ) \ln \left (x +1\right )+\left (\frac {313 d}{41472}+\frac {e}{81}\right ) \ln \left (x -2\right )\)	\(107\)
risch	\(\frac {\frac {5}{9} e \,x^{2}+\frac {1}{27} e \,x^{6}-\frac {5}{18} e \,x^{4}+\frac {43}{864} d x +\frac {35}{3456} d \,x^{7}+\frac {35}{384} x^{3} d -\frac {13}{192} x^{5} d -\frac {25}{108} e}{\left (x^{4}-5 x^{2}+4\right )^{2}}+\frac {13 \ln \left (x +1\right ) d}{1296}-\frac {\ln \left (x +1\right ) e}{81}-\frac {313 \ln \left (x +2\right ) d}{41472}+\frac {\ln \left (x +2\right ) e}{81}+\frac {313 \ln \left (2-x \right ) d}{41472}+\frac {\ln \left (2-x \right ) e}{81}-\frac {13 \ln \left (1-x \right ) d}{1296}-\frac {\ln \left (1-x \right ) e}{81}\)	\(123\)
default	\(\left (-\frac {313 d}{41472}+\frac {e}{81}\right ) \ln \left (x +2\right )-\frac {-\frac {19 d}{6912}+\frac {17 e}{3456}}{x +2}-\frac {-\frac {d}{1728}+\frac {e}{864}}{2 \left (x +2\right )^{2}}-\frac {-\frac {d}{432}+\frac {e}{144}}{x +1}-\frac {\frac {d}{216}-\frac {e}{216}}{2 \left (x +1\right )^{2}}+\left (\frac {13 d}{1296}-\frac {e}{81}\right ) \ln \left (x +1\right )+\left (-\frac {13 d}{1296}-\frac {e}{81}\right ) \ln \left (x -1\right )-\frac {-\frac {d}{432}-\frac {e}{144}}{x -1}-\frac {-\frac {d}{216}-\frac {e}{216}}{2 \left (x -1\right )^{2}}-\frac {-\frac {19 d}{6912}-\frac {17 e}{3456}}{x -2}-\frac {\frac {d}{1728}+\frac {e}{864}}{2 \left (x -2\right )^{2}}+\left (\frac {313 d}{41472}+\frac {e}{81}\right ) \ln \left (x -2\right )\)	\(162\)
parallelrisch	\(\frac {1536 e \,x^{6}-11520 e \,x^{4}-9600 e +420 d \,x^{7}+2064 d x +5008 \ln \left (x -2\right ) d +8192 \ln \left (x -2\right ) e -6656 \ln \left (x -1\right ) d -8192 \ln \left (x -1\right ) e -4160 \ln \left (x +1\right ) x^{6} d +16896 \ln \left (x -2\right ) x^{4} e +23040 e \,x^{2}-20480 \ln \left (x -2\right ) x^{2} e +16640 \ln \left (x -1\right ) x^{2} d +20480 \ln \left (x -1\right ) x^{2} e -16640 \ln \left (x +1\right ) x^{2} d +20480 \ln \left (x +1\right ) x^{2} e +12520 \ln \left (x +2\right ) x^{2} d -20480 \ln \left (x +2\right ) x^{2} e +10329 \ln \left (x -2\right ) x^{4} d -5008 \ln \left (x +2\right ) d +5120 \ln \left (x +1\right ) x^{6} e +3130 \ln \left (x +2\right ) x^{6} d -5120 \ln \left (x +2\right ) x^{6} e +8192 \ln \left (x +2\right ) e +6656 \ln \left (x +1\right ) d -8192 \ln \left (x +1\right ) e -2808 x^{5} d +3780 x^{3} d -512 \ln \left (x +1\right ) x^{8} e +512 \ln \left (x -2\right ) x^{8} e -416 \ln \left (x -1\right ) x^{8} d -512 \ln \left (x -1\right ) x^{8} e +4160 \ln \left (x -1\right ) x^{6} d +5120 \ln \left (x -1\right ) x^{6} e -313 \ln \left (x +2\right ) x^{8} d +512 \ln \left (x +2\right ) x^{8} e -3130 \ln \left (x -2\right ) x^{6} d +416 \ln \left (x +1\right ) x^{8} d -5120 \ln \left (x -2\right ) x^{6} e +313 \ln \left (x -2\right ) x^{8} d -13728 \ln \left (x -1\right ) x^{4} d -16896 \ln \left (x -1\right ) x^{4} e +13728 \ln \left (x +1\right ) x^{4} d -16896 \ln \left (x +1\right ) x^{4} e -10329 \ln \left (x +2\right ) x^{4} d +16896 \ln \left (x +2\right ) x^{4} e -12520 \ln \left (x -2\right ) x^{2} d}{41472 \left (x^{4}-5 x^{2}+4\right )^{2}}\)	\(435\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 307 vs. \(2 (125) = 250\).

Time = 0.29 (sec) , antiderivative size = 307, normalized size of antiderivative = 2.15 \[ \int \frac {d+e x}{\left (4-5 x^2+x^4\right )^3} \, dx=\frac {420 \, d x^{7} + 1536 \, e x^{6} - 2808 \, d x^{5} - 11520 \, e x^{4} + 3780 \, d x^{3} + 23040 \, e x^{2} + 2064 \, d x - {\left ({\left (313 \, d - 512 \, e\right )} x^{8} - 10 \, {\left (313 \, d - 512 \, e\right )} x^{6} + 33 \, {\left (313 \, d - 512 \, e\right )} x^{4} - 40 \, {\left (313 \, d - 512 \, e\right )} x^{2} + 5008 \, d - 8192 \, e\right )} \log \left (x + 2\right ) + 32 \, {\left ({\left (13 \, d - 16 \, e\right )} x^{8} - 10 \, {\left (13 \, d - 16 \, e\right )} x^{6} + 33 \, {\left (13 \, d - 16 \, e\right )} x^{4} - 40 \, {\left (13 \, d - 16 \, e\right )} x^{2} + 208 \, d - 256 \, e\right )} \log \left (x + 1\right ) - 32 \, {\left ({\left (13 \, d + 16 \, e\right )} x^{8} - 10 \, {\left (13 \, d + 16 \, e\right )} x^{6} + 33 \, {\left (13 \, d + 16 \, e\right )} x^{4} - 40 \, {\left (13 \, d + 16 \, e\right )} x^{2} + 208 \, d + 256 \, e\right )} \log \left (x - 1\right ) + {\left ({\left (313 \, d + 512 \, e\right )} x^{8} - 10 \, {\left (313 \, d + 512 \, e\right )} x^{6} + 33 \, {\left (313 \, d + 512 \, e\right )} x^{4} - 40 \, {\left (313 \, d + 512 \, e\right )} x^{2} + 5008 \, d + 8192 \, e\right )} \log \left (x - 2\right ) - 9600 \, e}{41472 \, {\left (x^{8} - 10 \, x^{6} + 33 \, x^{4} - 40 \, x^{2} + 16\right )}} \]

[In]

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

[Out]

1/41472*(420*d*x^7 + 1536*e*x^6 - 2808*d*x^5 - 11520*e*x^4 + 3780*d*x^3 + 23040*e*x^2 + 2064*d*x - ((313*d - 5
12*e)*x^8 - 10*(313*d - 512*e)*x^6 + 33*(313*d - 512*e)*x^4 - 40*(313*d - 512*e)*x^2 + 5008*d - 8192*e)*log(x
+ 2) + 32*((13*d - 16*e)*x^8 - 10*(13*d - 16*e)*x^6 + 33*(13*d - 16*e)*x^4 - 40*(13*d - 16*e)*x^2 + 208*d - 25
6*e)*log(x + 1) - 32*((13*d + 16*e)*x^8 - 10*(13*d + 16*e)*x^6 + 33*(13*d + 16*e)*x^4 - 40*(13*d + 16*e)*x^2 +
208*d + 256*e)*log(x - 1) + ((313*d + 512*e)*x^8 - 10*(313*d + 512*e)*x^6 + 33*(313*d + 512*e)*x^4 - 40*(313*
d + 512*e)*x^2 + 5008*d + 8192*e)*log(x - 2) - 9600*e)/(x^8 - 10*x^6 + 33*x^4 - 40*x^2 + 16)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 668 vs. \(2 (126) = 252\).

Time = 2.23 (sec) , antiderivative size = 668, normalized size of antiderivative = 4.67 \[ \int \frac {d+e x}{\left (4-5 x^2+x^4\right )^3} \, dx=\frac {\left (13 d - 16 e\right ) \log {\left (x + \frac {- 1106258459719280 d^{4} e - 13113710954343 d^{4} \cdot \left (13 d - 16 e\right ) - 817263343042560 d^{2} e^{3} + 153628968222720 d^{2} e^{2} \cdot \left (13 d - 16 e\right ) + 9530197557248 d^{2} e \left (13 d - 16 e\right )^{2} + 88038005760 d^{2} \left (13 d - 16 e\right )^{3} + 5035763255214080 e^{5} + 142661633703936 e^{4} \cdot \left (13 d - 16 e\right ) - 19670950215680 e^{3} \left (13 d - 16 e\right )^{2} - 557272006656 e^{2} \left (13 d - 16 e\right )^{3}}{22941256248261 d^{5} - 2312740746035200 d^{3} e^{2} + 4473912813420544 d e^{4}} \right )}}{1296} - \frac {\left (13 d + 16 e\right ) \log {\left (x + \frac {- 1106258459719280 d^{4} e + 13113710954343 d^{4} \cdot \left (13 d + 16 e\right ) - 817263343042560 d^{2} e^{3} - 153628968222720 d^{2} e^{2} \cdot \left (13 d + 16 e\right ) + 9530197557248 d^{2} e \left (13 d + 16 e\right )^{2} - 88038005760 d^{2} \left (13 d + 16 e\right )^{3} + 5035763255214080 e^{5} - 142661633703936 e^{4} \cdot \left (13 d + 16 e\right ) - 19670950215680 e^{3} \left (13 d + 16 e\right )^{2} + 557272006656 e^{2} \left (13 d + 16 e\right )^{3}}{22941256248261 d^{5} - 2312740746035200 d^{3} e^{2} + 4473912813420544 d e^{4}} \right )}}{1296} - \frac {\left (313 d - 512 e\right ) \log {\left (x + \frac {- 1106258459719280 d^{4} e + \frac {13113710954343 d^{4} \cdot \left (313 d - 512 e\right )}{32} - 817263343042560 d^{2} e^{3} - 4800905256960 d^{2} e^{2} \cdot \left (313 d - 512 e\right ) + 9306833552 d^{2} e \left (313 d - 512 e\right )^{2} - \frac {85974615 d^{2} \left (313 d - 512 e\right )^{3}}{32} + 5035763255214080 e^{5} - 4458176053248 e^{4} \cdot \left (313 d - 512 e\right ) - 19209912320 e^{3} \left (313 d - 512 e\right )^{2} + 17006592 e^{2} \left (313 d - 512 e\right )^{3}}{22941256248261 d^{5} - 2312740746035200 d^{3} e^{2} + 4473912813420544 d e^{4}} \right )}}{41472} + \frac {\left (313 d + 512 e\right ) \log {\left (x + \frac {- 1106258459719280 d^{4} e - \frac {13113710954343 d^{4} \cdot \left (313 d + 512 e\right )}{32} - 817263343042560 d^{2} e^{3} + 4800905256960 d^{2} e^{2} \cdot \left (313 d + 512 e\right ) + 9306833552 d^{2} e \left (313 d + 512 e\right )^{2} + \frac {85974615 d^{2} \left (313 d + 512 e\right )^{3}}{32} + 5035763255214080 e^{5} + 4458176053248 e^{4} \cdot \left (313 d + 512 e\right ) - 19209912320 e^{3} \left (313 d + 512 e\right )^{2} - 17006592 e^{2} \left (313 d + 512 e\right )^{3}}{22941256248261 d^{5} - 2312740746035200 d^{3} e^{2} + 4473912813420544 d e^{4}} \right )}}{41472} + \frac {35 d x^{7} - 234 d x^{5} + 315 d x^{3} + 172 d x + 128 e x^{6} - 960 e x^{4} + 1920 e x^{2} - 800 e}{3456 x^{8} - 34560 x^{6} + 114048 x^{4} - 138240 x^{2} + 55296} \]

[In]

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

[Out]

(13*d - 16*e)*log(x + (-1106258459719280*d**4*e - 13113710954343*d**4*(13*d - 16*e) - 817263343042560*d**2*e**
3 + 153628968222720*d**2*e**2*(13*d - 16*e) + 9530197557248*d**2*e*(13*d - 16*e)**2 + 88038005760*d**2*(13*d -
16*e)**3 + 5035763255214080*e**5 + 142661633703936*e**4*(13*d - 16*e) - 19670950215680*e**3*(13*d - 16*e)**2
- 557272006656*e**2*(13*d - 16*e)**3)/(22941256248261*d**5 - 2312740746035200*d**3*e**2 + 4473912813420544*d*e
**4))/1296 - (13*d + 16*e)*log(x + (-1106258459719280*d**4*e + 13113710954343*d**4*(13*d + 16*e) - 81726334304
2560*d**2*e**3 - 153628968222720*d**2*e**2*(13*d + 16*e) + 9530197557248*d**2*e*(13*d + 16*e)**2 - 88038005760
*d**2*(13*d + 16*e)**3 + 5035763255214080*e**5 - 142661633703936*e**4*(13*d + 16*e) - 19670950215680*e**3*(13*
d + 16*e)**2 + 557272006656*e**2*(13*d + 16*e)**3)/(22941256248261*d**5 - 2312740746035200*d**3*e**2 + 4473912
813420544*d*e**4))/1296 - (313*d - 512*e)*log(x + (-1106258459719280*d**4*e + 13113710954343*d**4*(313*d - 512
*e)/32 - 817263343042560*d**2*e**3 - 4800905256960*d**2*e**2*(313*d - 512*e) + 9306833552*d**2*e*(313*d - 512*
e)**2 - 85974615*d**2*(313*d - 512*e)**3/32 + 5035763255214080*e**5 - 4458176053248*e**4*(313*d - 512*e) - 192
09912320*e**3*(313*d - 512*e)**2 + 17006592*e**2*(313*d - 512*e)**3)/(22941256248261*d**5 - 2312740746035200*d
**3*e**2 + 4473912813420544*d*e**4))/41472 + (313*d + 512*e)*log(x + (-1106258459719280*d**4*e - 1311371095434
3*d**4*(313*d + 512*e)/32 - 817263343042560*d**2*e**3 + 4800905256960*d**2*e**2*(313*d + 512*e) + 9306833552*d
**2*e*(313*d + 512*e)**2 + 85974615*d**2*(313*d + 512*e)**3/32 + 5035763255214080*e**5 + 4458176053248*e**4*(3
13*d + 512*e) - 19209912320*e**3*(313*d + 512*e)**2 - 17006592*e**2*(313*d + 512*e)**3)/(22941256248261*d**5 -
2312740746035200*d**3*e**2 + 4473912813420544*d*e**4))/41472 + (35*d*x**7 - 234*d*x**5 + 315*d*x**3 + 172*d*x
+ 128*e*x**6 - 960*e*x**4 + 1920*e*x**2 - 800*e)/(3456*x**8 - 34560*x**6 + 114048*x**4 - 138240*x**2 + 55296)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.85 \[ \int \frac {d+e x}{\left (4-5 x^2+x^4\right )^3} \, dx=-\frac {1}{41472} \, {\left (313 \, d - 512 \, e\right )} \log \left (x + 2\right ) + \frac {1}{1296} \, {\left (13 \, d - 16 \, e\right )} \log \left (x + 1\right ) - \frac {1}{1296} \, {\left (13 \, d + 16 \, e\right )} \log \left (x - 1\right ) + \frac {1}{41472} \, {\left (313 \, d + 512 \, e\right )} \log \left (x - 2\right ) + \frac {35 \, d x^{7} + 128 \, e x^{6} - 234 \, d x^{5} - 960 \, e x^{4} + 315 \, d x^{3} + 1920 \, e x^{2} + 172 \, d x - 800 \, e}{3456 \, {\left (x^{8} - 10 \, x^{6} + 33 \, x^{4} - 40 \, x^{2} + 16\right )}} \]

[In]

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

[Out]

-1/41472*(313*d - 512*e)*log(x + 2) + 1/1296*(13*d - 16*e)*log(x + 1) - 1/1296*(13*d + 16*e)*log(x - 1) + 1/41
472*(313*d + 512*e)*log(x - 2) + 1/3456*(35*d*x^7 + 128*e*x^6 - 234*d*x^5 - 960*e*x^4 + 315*d*x^3 + 1920*e*x^2
+ 172*d*x - 800*e)/(x^8 - 10*x^6 + 33*x^4 - 40*x^2 + 16)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.80 \[ \int \frac {d+e x}{\left (4-5 x^2+x^4\right )^3} \, dx=-\frac {1}{41472} \, {\left (313 \, d - 512 \, e\right )} \log \left ({\left | x + 2 \right |}\right ) + \frac {1}{1296} \, {\left (13 \, d - 16 \, e\right )} \log \left ({\left | x + 1 \right |}\right ) - \frac {1}{1296} \, {\left (13 \, d + 16 \, e\right )} \log \left ({\left | x - 1 \right |}\right ) + \frac {1}{41472} \, {\left (313 \, d + 512 \, e\right )} \log \left ({\left | x - 2 \right |}\right ) + \frac {35 \, d x^{7} + 128 \, e x^{6} - 234 \, d x^{5} - 960 \, e x^{4} + 315 \, d x^{3} + 1920 \, e x^{2} + 172 \, d x - 800 \, e}{3456 \, {\left (x^{4} - 5 \, x^{2} + 4\right )}^{2}} \]

[In]

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

[Out]

-1/41472*(313*d - 512*e)*log(abs(x + 2)) + 1/1296*(13*d - 16*e)*log(abs(x + 1)) - 1/1296*(13*d + 16*e)*log(abs
(x - 1)) + 1/41472*(313*d + 512*e)*log(abs(x - 2)) + 1/3456*(35*d*x^7 + 128*e*x^6 - 234*d*x^5 - 960*e*x^4 + 31
5*d*x^3 + 1920*e*x^2 + 172*d*x - 800*e)/(x^4 - 5*x^2 + 4)^2

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.83 \[ \int \frac {d+e x}{\left (4-5 x^2+x^4\right )^3} \, dx=\ln \left (x+1\right )\,\left (\frac {13\,d}{1296}-\frac {e}{81}\right )-\ln \left (x-1\right )\,\left (\frac {13\,d}{1296}+\frac {e}{81}\right )+\ln \left (x-2\right )\,\left (\frac {313\,d}{41472}+\frac {e}{81}\right )-\ln \left (x+2\right )\,\left (\frac {313\,d}{41472}-\frac {e}{81}\right )+\frac {\frac {35\,d\,x^7}{3456}+\frac {e\,x^6}{27}-\frac {13\,d\,x^5}{192}-\frac {5\,e\,x^4}{18}+\frac {35\,d\,x^3}{384}+\frac {5\,e\,x^2}{9}+\frac {43\,d\,x}{864}-\frac {25\,e}{108}}{x^8-10\,x^6+33\,x^4-40\,x^2+16} \]

[In]

int((d + e*x)/(x^4 - 5*x^2 + 4)^3,x)

[Out]

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