\(\int \frac {d+e x+f x^2+g x^3+h x^4}{(4-5 x^2+x^4)^3} \, dx\) [45]

Optimal result

Integrand size = 33, antiderivative size = 224 \[ \int \frac {d+e x+f x^2+g x^3+h x^4}{\left (4-5 x^2+x^4\right )^3} \, dx=\frac {5 e+8 g-(2 e+5 g) x^2}{36 \left (4-5 x^2+x^4\right )^2}+\frac {x \left (17 d+20 f+32 h-(5 d+8 f+20 h) x^2\right )}{144 \left (4-5 x^2+x^4\right )^2}-\frac {(2 e+5 g) \left (5-2 x^2\right )}{108 \left (4-5 x^2+x^4\right )}-\frac {x \left (59 d+380 f+848 h-5 (7 d+28 f+64 h) x^2\right )}{3456 \left (4-5 x^2+x^4\right )}-\frac {(313 d+820 f+1936 h) \text {arctanh}\left (\frac {x}{2}\right )}{20736}+\frac {1}{648} (13 d+25 f+61 h) \text {arctanh}(x)-\frac {1}{162} (2 e+5 g) \log \left (1-x^2\right )+\frac {1}{162} (2 e+5 g) \log \left (4-x^2\right ) \]

[Out]

1/36*(5*e+8*g-(2*e+5*g)*x^2)/(x^4-5*x^2+4)^2+1/144*x*(17*d+20*f+32*h-(5*d+8*f+20*h)*x^2)/(x^4-5*x^2+4)^2-1/108
*(2*e+5*g)*(-2*x^2+5)/(x^4-5*x^2+4)-1/3456*x*(59*d+380*f+848*h-5*(7*d+28*f+64*h)*x^2)/(x^4-5*x^2+4)-1/20736*(3
13*d+820*f+1936*h)*arctanh(1/2*x)+1/648*(13*d+25*f+61*h)*arctanh(x)-1/162*(2*e+5*g)*ln(-x^2+1)+1/162*(2*e+5*g)
*ln(-x^2+4)

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.303, Rules used = {1687, 1692, 1192, 1180, 213, 1261, 652, 628, 630, 31} \[ \int \frac {d+e x+f x^2+g x^3+h x^4}{\left (4-5 x^2+x^4\right )^3} \, dx=-\frac {\text {arctanh}\left (\frac {x}{2}\right ) (313 d+820 f+1936 h)}{20736}+\frac {1}{648} \text {arctanh}(x) (13 d+25 f+61 h)-\frac {x \left (-5 x^2 (7 d+28 f+64 h)+59 d+380 f+848 h\right )}{3456 \left (x^4-5 x^2+4\right )}+\frac {x \left (-\left (x^2 (5 d+8 f+20 h)\right )+17 d+20 f+32 h\right )}{144 \left (x^4-5 x^2+4\right )^2}-\frac {1}{162} (2 e+5 g) \log \left (1-x^2\right )+\frac {1}{162} (2 e+5 g) \log \left (4-x^2\right )-\frac {\left (5-2 x^2\right ) (2 e+5 g)}{108 \left (x^4-5 x^2+4\right )}+\frac {-\left (x^2 (2 e+5 g)\right )+5 e+8 g}{36 \left (x^4-5 x^2+4\right )^2} \]

[In]

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

[Out]

(5*e + 8*g - (2*e + 5*g)*x^2)/(36*(4 - 5*x^2 + x^4)^2) + (x*(17*d + 20*f + 32*h - (5*d + 8*f + 20*h)*x^2))/(14
4*(4 - 5*x^2 + x^4)^2) - ((2*e + 5*g)*(5 - 2*x^2))/(108*(4 - 5*x^2 + x^4)) - (x*(59*d + 380*f + 848*h - 5*(7*d
+ 28*f + 64*h)*x^2))/(3456*(4 - 5*x^2 + x^4)) - ((313*d + 820*f + 1936*h)*ArcTanh[x/2])/20736 + ((13*d + 25*f
+ 61*h)*ArcTanh[x])/648 - ((2*e + 5*g)*Log[1 - x^2])/162 + ((2*e + 5*g)*Log[4 - x^2])/162

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 652

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

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 1261

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

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]

Rule 1692

Int[(Pq_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{d = Coeff[PolynomialRemainder[Pq, a +
b*x^2 + c*x^4, x], x, 0], e = Coeff[PolynomialRemainder[Pq, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 +
c*x^4)^(p + 1)*((a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Dist[1/(2*
a*(p + 1)*(b^2 - 4*a*c)), Int[(a + b*x^2 + c*x^4)^(p + 1)*ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuot
ient[Pq, a + b*x^2 + c*x^4, x] + b^2*d*(2*p + 3) - 2*a*c*d*(4*p + 5) - a*b*e + c*(4*p + 7)*(b*d - 2*a*e)*x^2,
x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x^2] && Expon[Pq, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1
]

Rubi steps \begin{align*} \text {integral}& = \int \frac {x \left (e+g x^2\right )}{\left (4-5 x^2+x^4\right )^3} \, dx+\int \frac {d+f x^2+h x^4}{\left (4-5 x^2+x^4\right )^3} \, dx \\ & = \frac {x \left (17 d+20 f+32 h-(5 d+8 f+20 h) x^2\right )}{144 \left (4-5 x^2+x^4\right )^2}-\frac {1}{144} \int \frac {-19 d+20 f+32 h+5 (5 d+8 f+20 h) x^2}{\left (4-5 x^2+x^4\right )^2} \, dx+\frac {1}{2} \text {Subst}\left (\int \frac {e+g x}{\left (4-5 x+x^2\right )^3} \, dx,x,x^2\right ) \\ & = \frac {5 e+8 g-(2 e+5 g) x^2}{36 \left (4-5 x^2+x^4\right )^2}+\frac {x \left (17 d+20 f+32 h-(5 d+8 f+20 h) x^2\right )}{144 \left (4-5 x^2+x^4\right )^2}-\frac {x \left (59 d+380 f+848 h-5 (7 d+28 f+64 h) x^2\right )}{3456 \left (4-5 x^2+x^4\right )}+\frac {\int \frac {3 (173 d+260 f+656 h)+15 (7 d+28 f+64 h) x^2}{4-5 x^2+x^4} \, dx}{10368}+\frac {1}{12} (-2 e-5 g) \text {Subst}\left (\int \frac {1}{\left (4-5 x+x^2\right )^2} \, dx,x,x^2\right ) \\ & = \frac {5 e+8 g-(2 e+5 g) x^2}{36 \left (4-5 x^2+x^4\right )^2}+\frac {x \left (17 d+20 f+32 h-(5 d+8 f+20 h) x^2\right )}{144 \left (4-5 x^2+x^4\right )^2}-\frac {(2 e+5 g) \left (5-2 x^2\right )}{108 \left (4-5 x^2+x^4\right )}-\frac {x \left (59 d+380 f+848 h-5 (7 d+28 f+64 h) x^2\right )}{3456 \left (4-5 x^2+x^4\right )}+\frac {1}{54} (2 e+5 g) \text {Subst}\left (\int \frac {1}{4-5 x+x^2} \, dx,x,x^2\right )+\frac {1}{648} (-13 d-25 f-61 h) \int \frac {1}{-1+x^2} \, dx+\frac {(313 d+820 f+1936 h) \int \frac {1}{-4+x^2} \, dx}{10368} \\ & = \frac {5 e+8 g-(2 e+5 g) x^2}{36 \left (4-5 x^2+x^4\right )^2}+\frac {x \left (17 d+20 f+32 h-(5 d+8 f+20 h) x^2\right )}{144 \left (4-5 x^2+x^4\right )^2}-\frac {(2 e+5 g) \left (5-2 x^2\right )}{108 \left (4-5 x^2+x^4\right )}-\frac {x \left (59 d+380 f+848 h-5 (7 d+28 f+64 h) x^2\right )}{3456 \left (4-5 x^2+x^4\right )}-\frac {(313 d+820 f+1936 h) \tanh ^{-1}\left (\frac {x}{2}\right )}{20736}+\frac {1}{648} (13 d+25 f+61 h) \tanh ^{-1}(x)+\frac {1}{162} (-2 e-5 g) \text {Subst}\left (\int \frac {1}{-1+x} \, dx,x,x^2\right )+\frac {1}{162} (2 e+5 g) \text {Subst}\left (\int \frac {1}{-4+x} \, dx,x,x^2\right ) \\ & = \frac {5 e+8 g-(2 e+5 g) x^2}{36 \left (4-5 x^2+x^4\right )^2}+\frac {x \left (17 d+20 f+32 h-(5 d+8 f+20 h) x^2\right )}{144 \left (4-5 x^2+x^4\right )^2}-\frac {(2 e+5 g) \left (5-2 x^2\right )}{108 \left (4-5 x^2+x^4\right )}-\frac {x \left (59 d+380 f+848 h-5 (7 d+28 f+64 h) x^2\right )}{3456 \left (4-5 x^2+x^4\right )}-\frac {(313 d+820 f+1936 h) \tanh ^{-1}\left (\frac {x}{2}\right )}{20736}+\frac {1}{648} (13 d+25 f+61 h) \tanh ^{-1}(x)-\frac {1}{162} (2 e+5 g) \log \left (1-x^2\right )+\frac {1}{162} (2 e+5 g) \log \left (4-x^2\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.03 \[ \int \frac {d+e x+f x^2+g x^3+h x^4}{\left (4-5 x^2+x^4\right )^3} \, dx=\frac {20 e+32 g+17 d x+20 f x+32 h x-8 e x^2-20 g x^2-5 d x^3-8 f x^3-20 h x^3}{144 \left (4-5 x^2+x^4\right )^2}+\frac {-320 e-800 g-59 d x-380 f x-848 h x+128 e x^2+320 g x^2+35 d x^3+140 f x^3+320 h x^3}{3456 \left (4-5 x^2+x^4\right )}+\frac {(-13 d-16 e-25 f-40 g-61 h) \log (1-x)}{1296}+\frac {(313 d+512 e+820 f+1280 g+1936 h) \log (2-x)}{41472}+\frac {(13 d-16 e+25 f-40 g+61 h) \log (1+x)}{1296}+\frac {(-313 d+512 e-820 f+1280 g-1936 h) \log (2+x)}{41472} \]

[In]

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

[Out]

(20*e + 32*g + 17*d*x + 20*f*x + 32*h*x - 8*e*x^2 - 20*g*x^2 - 5*d*x^3 - 8*f*x^3 - 20*h*x^3)/(144*(4 - 5*x^2 +
x^4)^2) + (-320*e - 800*g - 59*d*x - 380*f*x - 848*h*x + 128*e*x^2 + 320*g*x^2 + 35*d*x^3 + 140*f*x^3 + 320*h
*x^3)/(3456*(4 - 5*x^2 + x^4)) + ((-13*d - 16*e - 25*f - 40*g - 61*h)*Log[1 - x])/1296 + ((313*d + 512*e + 820
*f + 1280*g + 1936*h)*Log[2 - x])/41472 + ((13*d - 16*e + 25*f - 40*g + 61*h)*Log[1 + x])/1296 + ((-313*d + 51
2*e - 820*f + 1280*g - 1936*h)*Log[2 + x])/41472

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.86

method	result	size
norman	\(\frac {\left (-\frac {13 d}{192}-\frac {5 f}{16}-\frac {17 h}{24}\right ) x^{5}+\left (\frac {35 d}{384}+\frac {21 f}{32}+\frac {35 h}{24}\right ) x^{3}+\left (\frac {35 d}{3456}+\frac {35 f}{864}+\frac {5 h}{54}\right ) x^{7}+\left (\frac {43 d}{864}-\frac {65 f}{216}-\frac {41 h}{54}\right ) x +\left (-\frac {5 e}{18}-\frac {25 g}{36}\right ) x^{4}+\left (\frac {5 e}{9}+\frac {25 g}{18}\right ) x^{2}+\left (\frac {e}{27}+\frac {5 g}{54}\right ) x^{6}-\frac {25 e}{108}-\frac {19 g}{27}}{\left (x^{4}-5 x^{2}+4\right )^{2}}+\left (-\frac {313 d}{41472}+\frac {e}{81}-\frac {205 f}{10368}+\frac {5 g}{162}-\frac {121 h}{2592}\right ) \ln \left (x +2\right )+\left (-\frac {13 d}{1296}-\frac {e}{81}-\frac {25 f}{1296}-\frac {5 g}{162}-\frac {61 h}{1296}\right ) \ln \left (x -1\right )+\left (\frac {13 d}{1296}-\frac {e}{81}+\frac {25 f}{1296}-\frac {5 g}{162}+\frac {61 h}{1296}\right ) \ln \left (x +1\right )+\left (\frac {313 d}{41472}+\frac {e}{81}+\frac {205 f}{10368}+\frac {5 g}{162}+\frac {121 h}{2592}\right ) \ln \left (x -2\right )\)	\(193\)
risch	\(\frac {\left (-\frac {13 d}{192}-\frac {5 f}{16}-\frac {17 h}{24}\right ) x^{5}+\left (\frac {35 d}{384}+\frac {21 f}{32}+\frac {35 h}{24}\right ) x^{3}+\left (\frac {35 d}{3456}+\frac {35 f}{864}+\frac {5 h}{54}\right ) x^{7}+\left (\frac {43 d}{864}-\frac {65 f}{216}-\frac {41 h}{54}\right ) x +\left (-\frac {5 e}{18}-\frac {25 g}{36}\right ) x^{4}+\left (\frac {5 e}{9}+\frac {25 g}{18}\right ) x^{2}+\left (\frac {e}{27}+\frac {5 g}{54}\right ) x^{6}-\frac {25 e}{108}-\frac {19 g}{27}}{\left (x^{4}-5 x^{2}+4\right )^{2}}-\frac {13 \ln \left (1-x \right ) d}{1296}-\frac {\ln \left (1-x \right ) e}{81}-\frac {25 \ln \left (1-x \right ) f}{1296}-\frac {5 \ln \left (1-x \right ) g}{162}-\frac {61 \ln \left (1-x \right ) h}{1296}-\frac {313 \ln \left (x +2\right ) d}{41472}+\frac {\ln \left (x +2\right ) e}{81}-\frac {205 \ln \left (x +2\right ) f}{10368}+\frac {5 \ln \left (x +2\right ) g}{162}-\frac {121 \ln \left (x +2\right ) h}{2592}+\frac {13 \ln \left (x +1\right ) d}{1296}-\frac {\ln \left (x +1\right ) e}{81}+\frac {25 \ln \left (x +1\right ) f}{1296}-\frac {5 \ln \left (x +1\right ) g}{162}+\frac {61 \ln \left (x +1\right ) h}{1296}+\frac {313 \ln \left (2-x \right ) d}{41472}+\frac {\ln \left (2-x \right ) e}{81}+\frac {205 \ln \left (2-x \right ) f}{10368}+\frac {5 \ln \left (2-x \right ) g}{162}+\frac {121 \ln \left (2-x \right ) h}{2592}\)	\(269\)
default	\(\left (-\frac {313 d}{41472}+\frac {e}{81}-\frac {205 f}{10368}+\frac {5 g}{162}-\frac {121 h}{2592}\right ) \ln \left (x +2\right )-\frac {-\frac {19 d}{6912}+\frac {17 e}{3456}-\frac {5 f}{576}+\frac {13 g}{864}-\frac {11 h}{432}}{x +2}-\frac {-\frac {d}{1728}+\frac {e}{864}-\frac {f}{432}+\frac {g}{216}-\frac {h}{108}}{2 \left (x +2\right )^{2}}-\frac {-\frac {d}{432}+\frac {e}{144}-\frac {5 f}{432}+\frac {7 g}{432}-\frac {h}{48}}{x +1}-\frac {\frac {d}{216}-\frac {e}{216}+\frac {f}{216}-\frac {g}{216}+\frac {h}{216}}{2 \left (x +1\right )^{2}}+\left (\frac {13 d}{1296}-\frac {e}{81}+\frac {25 f}{1296}-\frac {5 g}{162}+\frac {61 h}{1296}\right ) \ln \left (x +1\right )+\left (-\frac {13 d}{1296}-\frac {e}{81}-\frac {25 f}{1296}-\frac {5 g}{162}-\frac {61 h}{1296}\right ) \ln \left (x -1\right )-\frac {-\frac {d}{432}-\frac {e}{144}-\frac {5 f}{432}-\frac {7 g}{432}-\frac {h}{48}}{x -1}-\frac {-\frac {d}{216}-\frac {e}{216}-\frac {f}{216}-\frac {g}{216}-\frac {h}{216}}{2 \left (x -1\right )^{2}}-\frac {-\frac {19 d}{6912}-\frac {17 e}{3456}-\frac {5 f}{576}-\frac {13 g}{864}-\frac {11 h}{432}}{x -2}-\frac {\frac {d}{1728}+\frac {e}{864}+\frac {f}{432}+\frac {g}{216}+\frac {h}{108}}{2 \left (x -2\right )^{2}}+\left (\frac {313 d}{41472}+\frac {e}{81}+\frac {205 f}{10368}+\frac {5 g}{162}+\frac {121 h}{2592}\right ) \ln \left (x -2\right )\)	\(270\)
parallelrisch	\(\text {Expression too large to display}\)	\(1064\)

[In]

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

[Out]

((-13/192*d-5/16*f-17/24*h)*x^5+(35/384*d+21/32*f+35/24*h)*x^3+(35/3456*d+35/864*f+5/54*h)*x^7+(43/864*d-65/21
6*f-41/54*h)*x+(-5/18*e-25/36*g)*x^4+(5/9*e+25/18*g)*x^2+(1/27*e+5/54*g)*x^6-25/108*e-19/27*g)/(x^4-5*x^2+4)^2
+(-313/41472*d+1/81*e-205/10368*f+5/162*g-121/2592*h)*ln(x+2)+(-13/1296*d-1/81*e-25/1296*f-5/162*g-61/1296*h)*
ln(x-1)+(13/1296*d-1/81*e+25/1296*f-5/162*g+61/1296*h)*ln(x+1)+(313/41472*d+1/81*e+205/10368*f+5/162*g+121/259
2*h)*ln(x-2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 544 vs. \(2 (204) = 408\).

Time = 1.45 (sec) , antiderivative size = 544, normalized size of antiderivative = 2.43 \[ \int \frac {d+e x+f x^2+g x^3+h x^4}{\left (4-5 x^2+x^4\right )^3} \, dx=\frac {60 \, {\left (7 \, d + 28 \, f + 64 \, h\right )} x^{7} + 768 \, {\left (2 \, e + 5 \, g\right )} x^{6} - 216 \, {\left (13 \, d + 60 \, f + 136 \, h\right )} x^{5} - 5760 \, {\left (2 \, e + 5 \, g\right )} x^{4} + 756 \, {\left (5 \, d + 36 \, f + 80 \, h\right )} x^{3} + 11520 \, {\left (2 \, e + 5 \, g\right )} x^{2} + 48 \, {\left (43 \, d - 260 \, f - 656 \, h\right )} x - {\left ({\left (313 \, d - 512 \, e + 820 \, f - 1280 \, g + 1936 \, h\right )} x^{8} - 10 \, {\left (313 \, d - 512 \, e + 820 \, f - 1280 \, g + 1936 \, h\right )} x^{6} + 33 \, {\left (313 \, d - 512 \, e + 820 \, f - 1280 \, g + 1936 \, h\right )} x^{4} - 40 \, {\left (313 \, d - 512 \, e + 820 \, f - 1280 \, g + 1936 \, h\right )} x^{2} + 5008 \, d - 8192 \, e + 13120 \, f - 20480 \, g + 30976 \, h\right )} \log \left (x + 2\right ) + 32 \, {\left ({\left (13 \, d - 16 \, e + 25 \, f - 40 \, g + 61 \, h\right )} x^{8} - 10 \, {\left (13 \, d - 16 \, e + 25 \, f - 40 \, g + 61 \, h\right )} x^{6} + 33 \, {\left (13 \, d - 16 \, e + 25 \, f - 40 \, g + 61 \, h\right )} x^{4} - 40 \, {\left (13 \, d - 16 \, e + 25 \, f - 40 \, g + 61 \, h\right )} x^{2} + 208 \, d - 256 \, e + 400 \, f - 640 \, g + 976 \, h\right )} \log \left (x + 1\right ) - 32 \, {\left ({\left (13 \, d + 16 \, e + 25 \, f + 40 \, g + 61 \, h\right )} x^{8} - 10 \, {\left (13 \, d + 16 \, e + 25 \, f + 40 \, g + 61 \, h\right )} x^{6} + 33 \, {\left (13 \, d + 16 \, e + 25 \, f + 40 \, g + 61 \, h\right )} x^{4} - 40 \, {\left (13 \, d + 16 \, e + 25 \, f + 40 \, g + 61 \, h\right )} x^{2} + 208 \, d + 256 \, e + 400 \, f + 640 \, g + 976 \, h\right )} \log \left (x - 1\right ) + {\left ({\left (313 \, d + 512 \, e + 820 \, f + 1280 \, g + 1936 \, h\right )} x^{8} - 10 \, {\left (313 \, d + 512 \, e + 820 \, f + 1280 \, g + 1936 \, h\right )} x^{6} + 33 \, {\left (313 \, d + 512 \, e + 820 \, f + 1280 \, g + 1936 \, h\right )} x^{4} - 40 \, {\left (313 \, d + 512 \, e + 820 \, f + 1280 \, g + 1936 \, h\right )} x^{2} + 5008 \, d + 8192 \, e + 13120 \, f + 20480 \, g + 30976 \, h\right )} \log \left (x - 2\right ) - 9600 \, e - 29184 \, g}{41472 \, {\left (x^{8} - 10 \, x^{6} + 33 \, x^{4} - 40 \, x^{2} + 16\right )}} \]

[In]

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

[Out]

1/41472*(60*(7*d + 28*f + 64*h)*x^7 + 768*(2*e + 5*g)*x^6 - 216*(13*d + 60*f + 136*h)*x^5 - 5760*(2*e + 5*g)*x
^4 + 756*(5*d + 36*f + 80*h)*x^3 + 11520*(2*e + 5*g)*x^2 + 48*(43*d - 260*f - 656*h)*x - ((313*d - 512*e + 820
*f - 1280*g + 1936*h)*x^8 - 10*(313*d - 512*e + 820*f - 1280*g + 1936*h)*x^6 + 33*(313*d - 512*e + 820*f - 128
0*g + 1936*h)*x^4 - 40*(313*d - 512*e + 820*f - 1280*g + 1936*h)*x^2 + 5008*d - 8192*e + 13120*f - 20480*g + 3
0976*h)*log(x + 2) + 32*((13*d - 16*e + 25*f - 40*g + 61*h)*x^8 - 10*(13*d - 16*e + 25*f - 40*g + 61*h)*x^6 +
33*(13*d - 16*e + 25*f - 40*g + 61*h)*x^4 - 40*(13*d - 16*e + 25*f - 40*g + 61*h)*x^2 + 208*d - 256*e + 400*f
- 640*g + 976*h)*log(x + 1) - 32*((13*d + 16*e + 25*f + 40*g + 61*h)*x^8 - 10*(13*d + 16*e + 25*f + 40*g + 61*
h)*x^6 + 33*(13*d + 16*e + 25*f + 40*g + 61*h)*x^4 - 40*(13*d + 16*e + 25*f + 40*g + 61*h)*x^2 + 208*d + 256*e
+ 400*f + 640*g + 976*h)*log(x - 1) + ((313*d + 512*e + 820*f + 1280*g + 1936*h)*x^8 - 10*(313*d + 512*e + 82
0*f + 1280*g + 1936*h)*x^6 + 33*(313*d + 512*e + 820*f + 1280*g + 1936*h)*x^4 - 40*(313*d + 512*e + 820*f + 12
80*g + 1936*h)*x^2 + 5008*d + 8192*e + 13120*f + 20480*g + 30976*h)*log(x - 2) - 9600*e - 29184*g)/(x^8 - 10*x
^6 + 33*x^4 - 40*x^2 + 16)

Sympy [F(-1)]

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

[In]

integrate((h*x**4+g*x**3+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 = 214, normalized size of antiderivative = 0.96 \[ \int \frac {d+e x+f x^2+g x^3+h x^4}{\left (4-5 x^2+x^4\right )^3} \, dx=-\frac {1}{41472} \, {\left (313 \, d - 512 \, e + 820 \, f - 1280 \, g + 1936 \, h\right )} \log \left (x + 2\right ) + \frac {1}{1296} \, {\left (13 \, d - 16 \, e + 25 \, f - 40 \, g + 61 \, h\right )} \log \left (x + 1\right ) - \frac {1}{1296} \, {\left (13 \, d + 16 \, e + 25 \, f + 40 \, g + 61 \, h\right )} \log \left (x - 1\right ) + \frac {1}{41472} \, {\left (313 \, d + 512 \, e + 820 \, f + 1280 \, g + 1936 \, h\right )} \log \left (x - 2\right ) + \frac {5 \, {\left (7 \, d + 28 \, f + 64 \, h\right )} x^{7} + 64 \, {\left (2 \, e + 5 \, g\right )} x^{6} - 18 \, {\left (13 \, d + 60 \, f + 136 \, h\right )} x^{5} - 480 \, {\left (2 \, e + 5 \, g\right )} x^{4} + 63 \, {\left (5 \, d + 36 \, f + 80 \, h\right )} x^{3} + 960 \, {\left (2 \, e + 5 \, g\right )} x^{2} + 4 \, {\left (43 \, d - 260 \, f - 656 \, h\right )} x - 800 \, e - 2432 \, g}{3456 \, {\left (x^{8} - 10 \, x^{6} + 33 \, x^{4} - 40 \, x^{2} + 16\right )}} \]

[In]

integrate((h*x^4+g*x^3+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 - 1280*g + 1936*h)*log(x + 2) + 1/1296*(13*d - 16*e + 25*f - 40*g + 61*h)*log(
x + 1) - 1/1296*(13*d + 16*e + 25*f + 40*g + 61*h)*log(x - 1) + 1/41472*(313*d + 512*e + 820*f + 1280*g + 1936
*h)*log(x - 2) + 1/3456*(5*(7*d + 28*f + 64*h)*x^7 + 64*(2*e + 5*g)*x^6 - 18*(13*d + 60*f + 136*h)*x^5 - 480*(
2*e + 5*g)*x^4 + 63*(5*d + 36*f + 80*h)*x^3 + 960*(2*e + 5*g)*x^2 + 4*(43*d - 260*f - 656*h)*x - 800*e - 2432*
g)/(x^8 - 10*x^6 + 33*x^4 - 40*x^2 + 16)

Giac [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.96 \[ \int \frac {d+e x+f x^2+g x^3+h x^4}{\left (4-5 x^2+x^4\right )^3} \, dx=-\frac {1}{41472} \, {\left (313 \, d - 512 \, e + 820 \, f - 1280 \, g + 1936 \, h\right )} \log \left ({\left | x + 2 \right |}\right ) + \frac {1}{1296} \, {\left (13 \, d - 16 \, e + 25 \, f - 40 \, g + 61 \, h\right )} \log \left ({\left | x + 1 \right |}\right ) - \frac {1}{1296} \, {\left (13 \, d + 16 \, e + 25 \, f + 40 \, g + 61 \, h\right )} \log \left ({\left | x - 1 \right |}\right ) + \frac {1}{41472} \, {\left (313 \, d + 512 \, e + 820 \, f + 1280 \, g + 1936 \, h\right )} \log \left ({\left | x - 2 \right |}\right ) + \frac {35 \, d x^{7} + 140 \, f x^{7} + 320 \, h x^{7} + 128 \, e x^{6} + 320 \, g x^{6} - 234 \, d x^{5} - 1080 \, f x^{5} - 2448 \, h x^{5} - 960 \, e x^{4} - 2400 \, g x^{4} + 315 \, d x^{3} + 2268 \, f x^{3} + 5040 \, h x^{3} + 1920 \, e x^{2} + 4800 \, g x^{2} + 172 \, d x - 1040 \, f x - 2624 \, h x - 800 \, e - 2432 \, g}{3456 \, {\left (x^{4} - 5 \, x^{2} + 4\right )}^{2}} \]

[In]

integrate((h*x^4+g*x^3+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 - 1280*g + 1936*h)*log(abs(x + 2)) + 1/1296*(13*d - 16*e + 25*f - 40*g + 61*h)
*log(abs(x + 1)) - 1/1296*(13*d + 16*e + 25*f + 40*g + 61*h)*log(abs(x - 1)) + 1/41472*(313*d + 512*e + 820*f
+ 1280*g + 1936*h)*log(abs(x - 2)) + 1/3456*(35*d*x^7 + 140*f*x^7 + 320*h*x^7 + 128*e*x^6 + 320*g*x^6 - 234*d*
x^5 - 1080*f*x^5 - 2448*h*x^5 - 960*e*x^4 - 2400*g*x^4 + 315*d*x^3 + 2268*f*x^3 + 5040*h*x^3 + 1920*e*x^2 + 48
00*g*x^2 + 172*d*x - 1040*f*x - 2624*h*x - 800*e - 2432*g)/(x^4 - 5*x^2 + 4)^2

Mupad [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.93 \[ \int \frac {d+e x+f x^2+g x^3+h x^4}{\left (4-5 x^2+x^4\right )^3} \, dx=\ln \left (x+1\right )\,\left (\frac {13\,d}{1296}-\frac {e}{81}+\frac {25\,f}{1296}-\frac {5\,g}{162}+\frac {61\,h}{1296}\right )-\ln \left (x-1\right )\,\left (\frac {13\,d}{1296}+\frac {e}{81}+\frac {25\,f}{1296}+\frac {5\,g}{162}+\frac {61\,h}{1296}\right )-\frac {\left (-\frac {35\,d}{3456}-\frac {35\,f}{864}-\frac {5\,h}{54}\right )\,x^7+\left (-\frac {e}{27}-\frac {5\,g}{54}\right )\,x^6+\left (\frac {13\,d}{192}+\frac {5\,f}{16}+\frac {17\,h}{24}\right )\,x^5+\left (\frac {5\,e}{18}+\frac {25\,g}{36}\right )\,x^4+\left (-\frac {35\,d}{384}-\frac {21\,f}{32}-\frac {35\,h}{24}\right )\,x^3+\left (-\frac {5\,e}{9}-\frac {25\,g}{18}\right )\,x^2+\left (\frac {65\,f}{216}-\frac {43\,d}{864}+\frac {41\,h}{54}\right )\,x+\frac {25\,e}{108}+\frac {19\,g}{27}}{x^8-10\,x^6+33\,x^4-40\,x^2+16}+\ln \left (x-2\right )\,\left (\frac {313\,d}{41472}+\frac {e}{81}+\frac {205\,f}{10368}+\frac {5\,g}{162}+\frac {121\,h}{2592}\right )-\ln \left (x+2\right )\,\left (\frac {313\,d}{41472}-\frac {e}{81}+\frac {205\,f}{10368}-\frac {5\,g}{162}+\frac {121\,h}{2592}\right ) \]

[In]

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

[Out]

log(x + 1)*((13*d)/1296 - e/81 + (25*f)/1296 - (5*g)/162 + (61*h)/1296) - log(x - 1)*((13*d)/1296 + e/81 + (25
*f)/1296 + (5*g)/162 + (61*h)/1296) - ((25*e)/108 + (19*g)/27 - x^2*((5*e)/9 + (25*g)/18) + x^4*((5*e)/18 + (2
5*g)/36) - x^6*(e/27 + (5*g)/54) + x*((65*f)/216 - (43*d)/864 + (41*h)/54) + x^5*((13*d)/192 + (5*f)/16 + (17*
h)/24) - x^3*((35*d)/384 + (21*f)/32 + (35*h)/24) - x^7*((35*d)/3456 + (35*f)/864 + (5*h)/54))/(33*x^4 - 40*x^
2 - 10*x^6 + x^8 + 16) + log(x - 2)*((313*d)/41472 + e/81 + (205*f)/10368 + (5*g)/162 + (121*h)/2592) - log(x
+ 2)*((313*d)/41472 - e/81 + (205*f)/10368 - (5*g)/162 + (121*h)/2592)