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

Optimal. Leaf size=175 \[ -\frac {x \left (-35 x^2 (d+4 f)+59 d+380 f\right )}{3456 \left (x^4-5 x^2+4\right )}+\frac {x \left (-\left (x^2 (5 d+8 f)\right )+17 d+20 f\right )}{144 \left (x^4-5 x^2+4\right )^2}-\frac {(313 d+820 f) \tanh ^{-1}\left (\frac {x}{2}\right )}{20736}+\frac {1}{648} (13 d+25 f) \tanh ^{-1}(x)-\frac {1}{81} e \log \left (1-x^2\right )+\frac {1}{81} e \log \left (4-x^2\right )-\frac {e \left (5-2 x^2\right )}{54 \left (x^4-5 x^2+4\right )}+\frac {e \left (5-2 x^2\right )}{36 \left (x^4-5 x^2+4\right )^2} \]

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Rubi [A]  time = 0.22, antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {1673, 1178, 1166, 207, 12, 1107, 614, 616, 31} \begin {gather*} -\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 (x^2 (-(5 d+8 f))+17 d+20 f\right )}{144 \left (x^4-5 x^2+4\right )^2}-\frac {(313 d+820 f) \tanh ^{-1}\left (\frac {x}{2}\right )}{20736}+\frac {1}{648} (13 d+25 f) \tanh ^{-1}(x)-\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}-\frac {1}{81} e \log \left (1-x^2\right )+\frac {1}{81} e \log \left (4-x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[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 207

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

Rule 614

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 616

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 1107

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 1166

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 1178

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 1673

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*} \int \frac {d+e x+f x^2}{\left (4-5 x^2+x^4\right )^3} \, dx &=\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 \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {Subst}\left (\int \frac {1}{-4+x} \, dx,x,x^2\right )-\frac {1}{81} e \operatorname {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*}

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Mathematica [A]  time = 0.13, size = 161, normalized size = 0.92 \begin {gather*} \frac {\frac {12 \left (d x \left (35 x^2-59\right )+64 e \left (2 x^2-5\right )+20 f x \left (7 x^2-19\right )\right )}{x^4-5 x^2+4}+\frac {288 \left (-5 d x^3+17 d x+e \left (20-8 x^2\right )-8 f x^3+20 f x\right )}{\left (x^4-5 x^2+4\right )^2}-32 \log (1-x) (13 d+16 e+25 f)+\log (2-x) (313 d+512 e+820 f)+32 \log (x+1) (13 d-16 e+25 f)+\log (x+2) (-313 d+512 e-820 f)}{41472} \end {gather*}

Antiderivative was successfully verified.

[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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {d+e x+f x^2}{\left (4-5 x^2+x^4\right )^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 1.41, size = 389, normalized size = 2.22 \begin {gather*} \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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [A]  time = 0.35, size = 157, normalized size = 0.90 \begin {gather*} -\frac {1}{41472} \, {\left (313 \, d + 820 \, f - 512 \, e\right )} \log \left ({\left | x + 2 \right |}\right ) + \frac {1}{1296} \, {\left (13 \, d + 25 \, f - 16 \, e\right )} \log \left ({\left | x + 1 \right |}\right ) - \frac {1}{1296} \, {\left (13 \, d + 25 \, f + 16 \, e\right )} \log \left ({\left | x - 1 \right |}\right ) + \frac {1}{41472} \, {\left (313 \, d + 820 \, f + 512 \, e\right )} \log \left ({\left | x - 2 \right |}\right ) + \frac {35 \, d x^{7} + 140 \, f x^{7} + 128 \, x^{6} e - 234 \, d x^{5} - 1080 \, f x^{5} - 960 \, x^{4} e + 315 \, d x^{3} + 2268 \, f x^{3} + 1920 \, x^{2} e + 172 \, d x - 1040 \, f x - 800 \, e}{3456 \, {\left (x^{4} - 5 \, x^{2} + 4\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.02, size = 278, normalized size = 1.59 \begin {gather*} -\frac {313 d \ln \left (x +2\right )}{41472}+\frac {e \ln \left (x +2\right )}{81}-\frac {e \ln \left (x -1\right )}{81}-\frac {13 d \ln \left (x -1\right )}{1296}-\frac {e \ln \left (x +1\right )}{81}+\frac {13 d \ln \left (x +1\right )}{1296}+\frac {313 d \ln \left (x -2\right )}{41472}+\frac {e \ln \left (x -2\right )}{81}+\frac {205 f \ln \left (x -2\right )}{10368}+\frac {25 f \ln \left (x +1\right )}{1296}-\frac {25 f \ln \left (x -1\right )}{1296}-\frac {205 f \ln \left (x +2\right )}{10368}+\frac {e}{144 x -144}+\frac {d}{432 x +432}+\frac {d}{432 x -432}-\frac {d}{432 \left (x +1\right )^{2}}+\frac {e}{432 \left (x +1\right )^{2}}+\frac {d}{432 \left (x -1\right )^{2}}+\frac {e}{432 \left (x -1\right )^{2}}+\frac {d}{3456 \left (x +2\right )^{2}}-\frac {e}{1728 \left (x +2\right )^{2}}+\frac {f}{864 \left (x +2\right )^{2}}+\frac {f}{432 \left (x -1\right )^{2}}-\frac {f}{432 \left (x +1\right )^{2}}-\frac {f}{864 \left (x -2\right )^{2}}-\frac {d}{3456 \left (x -2\right )^{2}}-\frac {e}{1728 \left (x -2\right )^{2}}+\frac {19 d}{6912 \left (x +2\right )}-\frac {17 e}{3456 \left (x +2\right )}+\frac {19 d}{6912 \left (x -2\right )}+\frac {17 e}{3456 \left (x -2\right )}-\frac {e}{144 \left (x +1\right )}+\frac {5 f}{432 \left (x -1\right )}+\frac {5 f}{576 \left (x +2\right )}+\frac {5 f}{576 \left (x -2\right )}+\frac {5 f}{432 \left (x +1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-313/41472*d*ln(x+2)+1/81*e*ln(x+2)-1/81*e*ln(x-1)-13/1296*d*ln(x-1)-1/81*e*ln(x+1)+13/1296*d*ln(x+1)+313/4147
2*d*ln(x-2)+1/81*e*ln(x-2)+205/10368*f*ln(x-2)+25/1296*f*ln(x+1)-25/1296*f*ln(x-1)-205/10368*f*ln(x+2)-1/432/(
x+1)^2*d+1/432/(x+1)^2*e+1/432/(x-1)^2*d+1/432/(x-1)^2*e+1/3456/(x+2)^2*d-1/1728/(x+2)^2*e+1/864/(x+2)^2*f+1/4
32/(x-1)^2*f-1/432/(x+1)^2*f-1/864/(x-2)^2*f-1/3456/(x-2)^2*d-1/1728/(x-2)^2*e+19/6912/(x+2)*d-17/3456/(x+2)*e
+19/6912/(x-2)*d+17/3456/(x-2)*e+1/432/(x+1)*d-1/144/(x+1)*e+1/432/(x-1)*d+1/144/(x-1)*e+5/432/(x-1)*f+5/576/(
x+2)*f+5/576/(x-2)*f+5/432/(x+1)*f

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maxima [A]  time = 1.10, size = 155, normalized size = 0.89 \begin {gather*} -\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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 0.11, size = 151, normalized size = 0.86 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [B]  time = 124.29, size = 2822, normalized size = 16.13

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(13*d - 16*e + 25*f)*log(x + (-1106258459719280*d**5*e - 13113710954343*d**5*(13*d - 16*e + 25*f) - 1292948240
1572800*d**4*e*f - 107063904267900*d**4*f*(13*d - 16*e + 25*f) - 817263343042560*d**3*e**3 + 153628968222720*d
**3*e**2*(13*d - 16*e + 25*f) - 59478343838144000*d**3*e*f**2 + 9530197557248*d**3*e*(13*d - 16*e + 25*f)**2 -
324891412840800*d**3*f**2*(13*d - 16*e + 25*f) + 88038005760*d**3*(13*d - 16*e + 25*f)**3 - 2885705898393600*
d**2*e**3*f + 1014848673546240*d**2*e**2*f*(13*d - 16*e + 25*f) - 134905286808320000*d**2*e*f**3 + 63469758382
080*d**2*e*f*(13*d - 16*e + 25*f)**2 - 422972724528000*d**2*f**3*(13*d - 16*e + 25*f) + 364616847360*d**2*f*(1
3*d - 16*e + 25*f)**3 + 5035763255214080*d*e**5 + 142661633703936*d*e**4*(13*d - 16*e + 25*f) - 21383148994560
00*d*e**3*f**2 - 19670950215680*d*e**3*(13*d - 16*e + 25*f)**2 + 2257033730457600*d*e**2*f**2*(13*d - 16*e + 2
5*f) - 557272006656*d*e**2*(13*d - 16*e + 25*f)**3 - 151082645593600000*d*e*f**4 + 141056507904000*d*e*f**2*(1
3*d - 16*e + 25*f)**2 - 167683154400000*d*f**4*(13*d - 16*e + 25*f) + 339373670400*d*f**2*(13*d - 16*e + 25*f)
**3 + 10643272556871680*e**5*f + 214404767416320*e**4*f*(13*d - 16*e + 25*f) + 529992253440000*e**3*f**3 - 415
75283425280*e**3*f*(13*d - 16*e + 25*f)**2 + 1671759396864000*e**2*f**3*(13*d - 16*e + 25*f) - 837518622720*e*
*2*f*(13*d - 16*e + 25*f)**3 - 66895452108800000*e*f**5 + 104485486592000*e*f**3*(13*d - 16*e + 25*f)**2 + 510
41923200000*f**5*(13*d - 16*e + 25*f) - 80289792000*f**3*(13*d - 16*e + 25*f)**3)/(22941256248261*d**6 + 19727
1407316645*d**5*f - 2312740746035200*d**4*e**2 + 612862910928900*d**4*f**2 - 20566607354920960*d**3*e**2*f + 7
67363353812000*d**3*f**3 + 4473912813420544*d**2*e**4 - 68552762169753600*d**2*e**2*f**2 + 197499222000000*d**
2*f**4 + 20324472439439360*d*e**4*f - 101559983669248000*d*e**2*f**3 - 182883938400000*d*f**5 + 22539988369408
000*e**4*f**2 - 56422196838400000*e**2*f**4 + 21520080000000*f**6))/1296 - (13*d + 16*e + 25*f)*log(x + (-1106
258459719280*d**5*e + 13113710954343*d**5*(13*d + 16*e + 25*f) - 12929482401572800*d**4*e*f + 107063904267900*
d**4*f*(13*d + 16*e + 25*f) - 817263343042560*d**3*e**3 - 153628968222720*d**3*e**2*(13*d + 16*e + 25*f) - 594
78343838144000*d**3*e*f**2 + 9530197557248*d**3*e*(13*d + 16*e + 25*f)**2 + 324891412840800*d**3*f**2*(13*d +
16*e + 25*f) - 88038005760*d**3*(13*d + 16*e + 25*f)**3 - 2885705898393600*d**2*e**3*f - 1014848673546240*d**2
*e**2*f*(13*d + 16*e + 25*f) - 134905286808320000*d**2*e*f**3 + 63469758382080*d**2*e*f*(13*d + 16*e + 25*f)**
2 + 422972724528000*d**2*f**3*(13*d + 16*e + 25*f) - 364616847360*d**2*f*(13*d + 16*e + 25*f)**3 + 50357632552
14080*d*e**5 - 142661633703936*d*e**4*(13*d + 16*e + 25*f) - 2138314899456000*d*e**3*f**2 - 19670950215680*d*e
**3*(13*d + 16*e + 25*f)**2 - 2257033730457600*d*e**2*f**2*(13*d + 16*e + 25*f) + 557272006656*d*e**2*(13*d +
16*e + 25*f)**3 - 151082645593600000*d*e*f**4 + 141056507904000*d*e*f**2*(13*d + 16*e + 25*f)**2 + 16768315440
0000*d*f**4*(13*d + 16*e + 25*f) - 339373670400*d*f**2*(13*d + 16*e + 25*f)**3 + 10643272556871680*e**5*f - 21
4404767416320*e**4*f*(13*d + 16*e + 25*f) + 529992253440000*e**3*f**3 - 41575283425280*e**3*f*(13*d + 16*e + 2
5*f)**2 - 1671759396864000*e**2*f**3*(13*d + 16*e + 25*f) + 837518622720*e**2*f*(13*d + 16*e + 25*f)**3 - 6689
5452108800000*e*f**5 + 104485486592000*e*f**3*(13*d + 16*e + 25*f)**2 - 51041923200000*f**5*(13*d + 16*e + 25*
f) + 80289792000*f**3*(13*d + 16*e + 25*f)**3)/(22941256248261*d**6 + 197271407316645*d**5*f - 231274074603520
0*d**4*e**2 + 612862910928900*d**4*f**2 - 20566607354920960*d**3*e**2*f + 767363353812000*d**3*f**3 + 44739128
13420544*d**2*e**4 - 68552762169753600*d**2*e**2*f**2 + 197499222000000*d**2*f**4 + 20324472439439360*d*e**4*f
- 101559983669248000*d*e**2*f**3 - 182883938400000*d*f**5 + 22539988369408000*e**4*f**2 - 56422196838400000*e
**2*f**4 + 21520080000000*f**6))/1296 - (313*d - 512*e + 820*f)*log(x + (-1106258459719280*d**5*e + 1311371095
4343*d**5*(313*d - 512*e + 820*f)/32 - 12929482401572800*d**4*e*f + 26765976066975*d**4*f*(313*d - 512*e + 820
*f)/8 - 817263343042560*d**3*e**3 - 4800905256960*d**3*e**2*(313*d - 512*e + 820*f) - 59478343838144000*d**3*e
*f**2 + 9306833552*d**3*e*(313*d - 512*e + 820*f)**2 + 10152856651275*d**3*f**2*(313*d - 512*e + 820*f) - 8597
4615*d**3*(313*d - 512*e + 820*f)**3/32 - 2885705898393600*d**2*e**3*f - 31714021048320*d**2*e**2*f*(313*d - 5
12*e + 820*f) - 134905286808320000*d**2*e*f**3 + 61982185920*d**2*e*f*(313*d - 512*e + 820*f)**2 + 13217897641
500*d**2*f**3*(313*d - 512*e + 820*f) - 89017785*d**2*f*(313*d - 512*e + 820*f)**3/8 + 5035763255214080*d*e**5
- 4458176053248*d*e**4*(313*d - 512*e + 820*f) - 2138314899456000*d*e**3*f**2 - 19209912320*d*e**3*(313*d - 5
12*e + 820*f)**2 - 70532304076800*d*e**2*f**2*(313*d - 512*e + 820*f) + 17006592*d*e**2*(313*d - 512*e + 820*f
)**3 - 151082645593600000*d*e*f**4 + 137750496000*d*e*f**2*(313*d - 512*e + 820*f)**2 + 5240098575000*d*f**4*(
313*d - 512*e + 820*f) - 20713725*d*f**2*(313*d - 512*e + 820*f)**3/2 + 10643272556871680*e**5*f - 67001489817
60*e**4*f*(313*d - 512*e + 820*f) + 529992253440000*e**3*f**3 - 40600862720*e**3*f*(313*d - 512*e + 820*f)**2
- 52242481152000*e**2*f**3*(313*d - 512*e + 820*f) + 25559040*e**2*f*(313*d - 512*e + 820*f)**3 - 668954521088
00000*e*f**5 + 102036608000*e*f**3*(313*d - 512*e + 820*f)**2 - 1595060100000*f**5*(313*d - 512*e + 820*f) + 2
450250*f**3*(313*d - 512*e + 820*f)**3)/(22941256248261*d**6 + 197271407316645*d**5*f - 2312740746035200*d**4*
e**2 + 612862910928900*d**4*f**2 - 20566607354920960*d**3*e**2*f + 767363353812000*d**3*f**3 + 447391281342054
4*d**2*e**4 - 68552762169753600*d**2*e**2*f**2 + 197499222000000*d**2*f**4 + 20324472439439360*d*e**4*f - 1015
59983669248000*d*e**2*f**3 - 182883938400000*d*f**5 + 22539988369408000*e**4*f**2 - 56422196838400000*e**2*f**
4 + 21520080000000*f**6))/41472 + (313*d + 512*e + 820*f)*log(x + (-1106258459719280*d**5*e - 13113710954343*d
**5*(313*d + 512*e + 820*f)/32 - 12929482401572800*d**4*e*f - 26765976066975*d**4*f*(313*d + 512*e + 820*f)/8
- 817263343042560*d**3*e**3 + 4800905256960*d**3*e**2*(313*d + 512*e + 820*f) - 59478343838144000*d**3*e*f**2
+ 9306833552*d**3*e*(313*d + 512*e + 820*f)**2 - 10152856651275*d**3*f**2*(313*d + 512*e + 820*f) + 85974615*d
**3*(313*d + 512*e + 820*f)**3/32 - 2885705898393600*d**2*e**3*f + 31714021048320*d**2*e**2*f*(313*d + 512*e +
820*f) - 134905286808320000*d**2*e*f**3 + 61982185920*d**2*e*f*(313*d + 512*e + 820*f)**2 - 13217897641500*d*
*2*f**3*(313*d + 512*e + 820*f) + 89017785*d**2*f*(313*d + 512*e + 820*f)**3/8 + 5035763255214080*d*e**5 + 445
8176053248*d*e**4*(313*d + 512*e + 820*f) - 2138314899456000*d*e**3*f**2 - 19209912320*d*e**3*(313*d + 512*e +
820*f)**2 + 70532304076800*d*e**2*f**2*(313*d + 512*e + 820*f) - 17006592*d*e**2*(313*d + 512*e + 820*f)**3 -
151082645593600000*d*e*f**4 + 137750496000*d*e*f**2*(313*d + 512*e + 820*f)**2 - 5240098575000*d*f**4*(313*d
+ 512*e + 820*f) + 20713725*d*f**2*(313*d + 512*e + 820*f)**3/2 + 10643272556871680*e**5*f + 6700148981760*e**
4*f*(313*d + 512*e + 820*f) + 529992253440000*e**3*f**3 - 40600862720*e**3*f*(313*d + 512*e + 820*f)**2 + 5224
2481152000*e**2*f**3*(313*d + 512*e + 820*f) - 25559040*e**2*f*(313*d + 512*e + 820*f)**3 - 66895452108800000*
e*f**5 + 102036608000*e*f**3*(313*d + 512*e + 820*f)**2 + 1595060100000*f**5*(313*d + 512*e + 820*f) - 2450250
*f**3*(313*d + 512*e + 820*f)**3)/(22941256248261*d**6 + 197271407316645*d**5*f - 2312740746035200*d**4*e**2 +
612862910928900*d**4*f**2 - 20566607354920960*d**3*e**2*f + 767363353812000*d**3*f**3 + 4473912813420544*d**2
*e**4 - 68552762169753600*d**2*e**2*f**2 + 197499222000000*d**2*f**4 + 20324472439439360*d*e**4*f - 1015599836
69248000*d*e**2*f**3 - 182883938400000*d*f**5 + 22539988369408000*e**4*f**2 - 56422196838400000*e**2*f**4 + 21
520080000000*f**6))/41472 + (128*e*x**6 - 960*e*x**4 + 1920*e*x**2 - 800*e + x**7*(35*d + 140*f) + x**5*(-234*
d - 1080*f) + x**3*(315*d + 2268*f) + x*(172*d - 1040*f))/(3456*x**8 - 34560*x**6 + 114048*x**4 - 138240*x**2
+ 55296)

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