3.1.86 \(\int \frac {(d+e x) (2-x-2 x^2+x^3)}{(4-5 x^2+x^4)^2} \, dx\) [86]

Optimal. Leaf size=71 \[ \frac {d-2 e}{12 (2+x)}-\frac {1}{18} (d+e) \log (1-x)+\frac {1}{48} (d+2 e) \log (2-x)+\frac {1}{6} (d-e) \log (1+x)-\frac {1}{144} (19 d-26 e) \log (2+x) \]

[Out]

1/12*(d-2*e)/(2+x)-1/18*(d+e)*ln(1-x)+1/48*(d+2*e)*ln(2-x)+1/6*(d-e)*ln(1+x)-1/144*(19*d-26*e)*ln(2+x)

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Rubi [A]
time = 0.12, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {1600, 6874} \begin {gather*} \frac {d-2 e}{12 (x+2)}-\frac {1}{18} (d+e) \log (1-x)+\frac {1}{48} (d+2 e) \log (2-x)+\frac {1}{6} (d-e) \log (x+1)-\frac {1}{144} (19 d-26 e) \log (x+2) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(d - 2*e)/(12*(2 + x)) - ((d + e)*Log[1 - x])/18 + ((d + 2*e)*Log[2 - x])/48 + ((d - e)*Log[1 + x])/6 - ((19*d
- 26*e)*Log[2 + x])/144

Rule 1600

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \int \frac {(d+e x) \left (2-x-2 x^2+x^3\right )}{\left (4-5 x^2+x^4\right )^2} \, dx &=\int \frac {d+e x}{(2+x)^2 \left (2-x-2 x^2+x^3\right )} \, dx\\ &=\int \left (\frac {d+2 e}{48 (-2+x)}+\frac {-d-e}{18 (-1+x)}+\frac {d-e}{6 (1+x)}+\frac {-d+2 e}{12 (2+x)^2}+\frac {-19 d+26 e}{144 (2+x)}\right ) \, dx\\ &=\frac {d-2 e}{12 (2+x)}-\frac {1}{18} (d+e) \log (1-x)+\frac {1}{48} (d+2 e) \log (2-x)+\frac {1}{6} (d-e) \log (1+x)-\frac {1}{144} (19 d-26 e) \log (2+x)\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 66, normalized size = 0.93 \begin {gather*} \frac {1}{144} \left (\frac {12 (d-2 e)}{2+x}+24 (d-e) \log (-1-x)-8 (d+e) \log (1-x)+3 (d+2 e) \log (2-x)+(-19 d+26 e) \log (2+x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((12*(d - 2*e))/(2 + x) + 24*(d - e)*Log[-1 - x] - 8*(d + e)*Log[1 - x] + 3*(d + 2*e)*Log[2 - x] + (-19*d + 26
*e)*Log[2 + x])/144

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Maple [A]
time = 0.04, size = 64, normalized size = 0.90

method	result	size
default	\(\left (-\frac {19 d}{144}+\frac {13 e}{72}\right ) \ln \left (x +2\right )-\frac {-\frac {d}{12}+\frac {e}{6}}{x +2}+\left (\frac {d}{48}+\frac {e}{24}\right ) \ln \left (x -2\right )+\left (-\frac {d}{18}-\frac {e}{18}\right ) \ln \left (-1+x \right )+\left (\frac {d}{6}-\frac {e}{6}\right ) \ln \left (1+x \right )\)	\(64\)
risch	\(\frac {d}{12 x +24}-\frac {e}{6 \left (x +2\right )}+\frac {\ln \left (2-x \right ) d}{48}+\frac {\ln \left (2-x \right ) e}{24}-\frac {\ln \left (-1+x \right ) d}{18}-\frac {\ln \left (-1+x \right ) e}{18}+\frac {\ln \left (1+x \right ) d}{6}-\frac {\ln \left (1+x \right ) e}{6}-\frac {19 \ln \left (-x -2\right ) d}{144}+\frac {13 \ln \left (-x -2\right ) e}{72}\)	\(82\)
norman	\(\frac {\left (-\frac {d}{12}+\frac {e}{6}\right ) x +\left (\frac {d}{12}-\frac {e}{6}\right ) x^{3}+\left (-\frac {d}{6}+\frac {e}{3}\right ) x^{2}+\frac {d}{6}-\frac {e}{3}}{x^{4}-5 x^{2}+4}+\left (-\frac {19 d}{144}+\frac {13 e}{72}\right ) \ln \left (x +2\right )+\left (-\frac {d}{18}-\frac {e}{18}\right ) \ln \left (-1+x \right )+\left (\frac {d}{6}-\frac {e}{6}\right ) \ln \left (1+x \right )+\left (\frac {d}{48}+\frac {e}{24}\right ) \ln \left (x -2\right )\)	\(101\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-19/144*d+13/72*e)*ln(x+2)-(-1/12*d+1/6*e)/(x+2)+(1/48*d+1/24*e)*ln(x-2)+(-1/18*d-1/18*e)*ln(-1+x)+(1/6*d-1/6
*e)*ln(1+x)

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Maxima [A]
time = 0.29, size = 62, normalized size = 0.87 \begin {gather*} -\frac {1}{144} \, {\left (19 \, d - 26 \, e\right )} \log \left (x + 2\right ) + \frac {1}{6} \, {\left (d - e\right )} \log \left (x + 1\right ) - \frac {1}{18} \, {\left (d + e\right )} \log \left (x - 1\right ) + \frac {1}{48} \, {\left (d + 2 \, e\right )} \log \left (x - 2\right ) + \frac {d - 2 \, e}{12 \, {\left (x + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/144*(19*d - 26*e)*log(x + 2) + 1/6*(d - e)*log(x + 1) - 1/18*(d + e)*log(x - 1) + 1/48*(d + 2*e)*log(x - 2)
+ 1/12*(d - 2*e)/(x + 2)

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Fricas [A]
time = 0.41, size = 93, normalized size = 1.31 \begin {gather*} -\frac {{\left ({\left (19 \, d - 26 \, e\right )} x + 38 \, d - 52 \, e\right )} \log \left (x + 2\right ) - 24 \, {\left ({\left (d - e\right )} x + 2 \, d - 2 \, e\right )} \log \left (x + 1\right ) + 8 \, {\left ({\left (d + e\right )} x + 2 \, d + 2 \, e\right )} \log \left (x - 1\right ) - 3 \, {\left ({\left (d + 2 \, e\right )} x + 2 \, d + 4 \, e\right )} \log \left (x - 2\right ) - 12 \, d + 24 \, e}{144 \, {\left (x + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/144*(((19*d - 26*e)*x + 38*d - 52*e)*log(x + 2) - 24*((d - e)*x + 2*d - 2*e)*log(x + 1) + 8*((d + e)*x + 2*
d + 2*e)*log(x - 1) - 3*((d + 2*e)*x + 2*d + 4*e)*log(x - 2) - 12*d + 24*e)/(x + 2)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1188 vs. \(2 (60) = 120\).
time = 7.55, size = 1188, normalized size = 16.73 \begin {gather*} \frac {d - 2 e}{12 x + 24} + \frac {\left (d - e\right ) \log {\left (x + \frac {- 1534775 d^{6} + 8032360 d^{5} e - 984027 d^{5} \left (d - e\right ) - 12991180 d^{4} e^{2} + 11797266 d^{4} e \left (d - e\right ) + 3567168 d^{4} \left (d - e\right )^{2} + 1075200 d^{3} e^{3} - 32721528 d^{3} e^{2} \left (d - e\right ) - 8725248 d^{3} e \left (d - e\right )^{2} - 247104 d^{3} \left (d - e\right )^{3} + 16959280 d^{2} e^{4} + 38977296 d^{2} e^{3} \left (d - e\right ) - 2820096 d^{2} e^{2} \left (d - e\right )^{2} - 10357632 d^{2} e \left (d - e\right )^{3} - 15836800 d e^{5} - 21294960 d e^{4} \left (d - e\right ) + 15436800 d e^{3} \left (d - e\right )^{2} + 16277760 d e^{2} \left (d - e\right )^{3} + 4283840 e^{6} + 3876000 e^{5} \left (d - e\right ) - 6865920 e^{4} \left (d - e\right )^{2} - 4078080 e^{3} \left (d - e\right )^{3}}{801262 d^{6} - 4662251 d^{5} e + 7296938 d^{4} e^{2} + 1388616 d^{3} e^{3} - 12447440 d^{2} e^{4} + 9990800 d e^{5} - 2380000 e^{6}} \right )}}{6} - \frac {\left (d + e\right ) \log {\left (x + \frac {- 1534775 d^{6} + 8032360 d^{5} e + 328009 d^{5} \left (d + e\right ) - 12991180 d^{4} e^{2} - 3932422 d^{4} e \left (d + e\right ) + 396352 d^{4} \left (d + e\right )^{2} + 1075200 d^{3} e^{3} + 10907176 d^{3} e^{2} \left (d + e\right ) - 969472 d^{3} e \left (d + e\right )^{2} + 9152 d^{3} \left (d + e\right )^{3} + 16959280 d^{2} e^{4} - 12992432 d^{2} e^{3} \left (d + e\right ) - 313344 d^{2} e^{2} \left (d + e\right )^{2} + 383616 d^{2} e \left (d + e\right )^{3} - 15836800 d e^{5} + 7098320 d e^{4} \left (d + e\right ) + 1715200 d e^{3} \left (d + e\right )^{2} - 602880 d e^{2} \left (d + e\right )^{3} + 4283840 e^{6} - 1292000 e^{5} \left (d + e\right ) - 762880 e^{4} \left (d + e\right )^{2} + 151040 e^{3} \left (d + e\right )^{3}}{801262 d^{6} - 4662251 d^{5} e + 7296938 d^{4} e^{2} + 1388616 d^{3} e^{3} - 12447440 d^{2} e^{4} + 9990800 d e^{5} - 2380000 e^{6}} \right )}}{18} + \frac {\left (d + 2 e\right ) \log {\left (x + \frac {- 1534775 d^{6} + 8032360 d^{5} e - \frac {984027 d^{5} \left (d + 2 e\right )}{8} - 12991180 d^{4} e^{2} + \frac {5898633 d^{4} e \left (d + 2 e\right )}{4} + 55737 d^{4} \left (d + 2 e\right )^{2} + 1075200 d^{3} e^{3} - 4090191 d^{3} e^{2} \left (d + 2 e\right ) - 136332 d^{3} e \left (d + 2 e\right )^{2} - \frac {3861 d^{3} \left (d + 2 e\right )^{3}}{8} + 16959280 d^{2} e^{4} + 4872162 d^{2} e^{3} \left (d + 2 e\right ) - 44064 d^{2} e^{2} \left (d + 2 e\right )^{2} - \frac {80919 d^{2} e \left (d + 2 e\right )^{3}}{4} - 15836800 d e^{5} - 2661870 d e^{4} \left (d + 2 e\right ) + 241200 d e^{3} \left (d + 2 e\right )^{2} + \frac {63585 d e^{2} \left (d + 2 e\right )^{3}}{2} + 4283840 e^{6} + 484500 e^{5} \left (d + 2 e\right ) - 107280 e^{4} \left (d + 2 e\right )^{2} - 7965 e^{3} \left (d + 2 e\right )^{3}}{801262 d^{6} - 4662251 d^{5} e + 7296938 d^{4} e^{2} + 1388616 d^{3} e^{3} - 12447440 d^{2} e^{4} + 9990800 d e^{5} - 2380000 e^{6}} \right )}}{48} - \frac {\left (19 d - 26 e\right ) \log {\left (x + \frac {- 1534775 d^{6} + 8032360 d^{5} e + \frac {328009 d^{5} \cdot \left (19 d - 26 e\right )}{8} - 12991180 d^{4} e^{2} - \frac {1966211 d^{4} e \left (19 d - 26 e\right )}{4} + 6193 d^{4} \left (19 d - 26 e\right )^{2} + 1075200 d^{3} e^{3} + 1363397 d^{3} e^{2} \cdot \left (19 d - 26 e\right ) - 15148 d^{3} e \left (19 d - 26 e\right )^{2} + \frac {143 d^{3} \left (19 d - 26 e\right )^{3}}{8} + 16959280 d^{2} e^{4} - 1624054 d^{2} e^{3} \cdot \left (19 d - 26 e\right ) - 4896 d^{2} e^{2} \left (19 d - 26 e\right )^{2} + \frac {2997 d^{2} e \left (19 d - 26 e\right )^{3}}{4} - 15836800 d e^{5} + 887290 d e^{4} \cdot \left (19 d - 26 e\right ) + 26800 d e^{3} \left (19 d - 26 e\right )^{2} - \frac {2355 d e^{2} \left (19 d - 26 e\right )^{3}}{2} + 4283840 e^{6} - 161500 e^{5} \cdot \left (19 d - 26 e\right ) - 11920 e^{4} \left (19 d - 26 e\right )^{2} + 295 e^{3} \left (19 d - 26 e\right )^{3}}{801262 d^{6} - 4662251 d^{5} e + 7296938 d^{4} e^{2} + 1388616 d^{3} e^{3} - 12447440 d^{2} e^{4} + 9990800 d e^{5} - 2380000 e^{6}} \right )}}{144} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(d - 2*e)/(12*x + 24) + (d - e)*log(x + (-1534775*d**6 + 8032360*d**5*e - 984027*d**5*(d - e) - 12991180*d**4*
e**2 + 11797266*d**4*e*(d - e) + 3567168*d**4*(d - e)**2 + 1075200*d**3*e**3 - 32721528*d**3*e**2*(d - e) - 87
25248*d**3*e*(d - e)**2 - 247104*d**3*(d - e)**3 + 16959280*d**2*e**4 + 38977296*d**2*e**3*(d - e) - 2820096*d
**2*e**2*(d - e)**2 - 10357632*d**2*e*(d - e)**3 - 15836800*d*e**5 - 21294960*d*e**4*(d - e) + 15436800*d*e**3
*(d - e)**2 + 16277760*d*e**2*(d - e)**3 + 4283840*e**6 + 3876000*e**5*(d - e) - 6865920*e**4*(d - e)**2 - 407
8080*e**3*(d - e)**3)/(801262*d**6 - 4662251*d**5*e + 7296938*d**4*e**2 + 1388616*d**3*e**3 - 12447440*d**2*e*
*4 + 9990800*d*e**5 - 2380000*e**6))/6 - (d + e)*log(x + (-1534775*d**6 + 8032360*d**5*e + 328009*d**5*(d + e)
- 12991180*d**4*e**2 - 3932422*d**4*e*(d + e) + 396352*d**4*(d + e)**2 + 1075200*d**3*e**3 + 10907176*d**3*e*
*2*(d + e) - 969472*d**3*e*(d + e)**2 + 9152*d**3*(d + e)**3 + 16959280*d**2*e**4 - 12992432*d**2*e**3*(d + e)
- 313344*d**2*e**2*(d + e)**2 + 383616*d**2*e*(d + e)**3 - 15836800*d*e**5 + 7098320*d*e**4*(d + e) + 1715200
*d*e**3*(d + e)**2 - 602880*d*e**2*(d + e)**3 + 4283840*e**6 - 1292000*e**5*(d + e) - 762880*e**4*(d + e)**2 +
151040*e**3*(d + e)**3)/(801262*d**6 - 4662251*d**5*e + 7296938*d**4*e**2 + 1388616*d**3*e**3 - 12447440*d**2
*e**4 + 9990800*d*e**5 - 2380000*e**6))/18 + (d + 2*e)*log(x + (-1534775*d**6 + 8032360*d**5*e - 984027*d**5*(
d + 2*e)/8 - 12991180*d**4*e**2 + 5898633*d**4*e*(d + 2*e)/4 + 55737*d**4*(d + 2*e)**2 + 1075200*d**3*e**3 - 4
090191*d**3*e**2*(d + 2*e) - 136332*d**3*e*(d + 2*e)**2 - 3861*d**3*(d + 2*e)**3/8 + 16959280*d**2*e**4 + 4872
162*d**2*e**3*(d + 2*e) - 44064*d**2*e**2*(d + 2*e)**2 - 80919*d**2*e*(d + 2*e)**3/4 - 15836800*d*e**5 - 26618
70*d*e**4*(d + 2*e) + 241200*d*e**3*(d + 2*e)**2 + 63585*d*e**2*(d + 2*e)**3/2 + 4283840*e**6 + 484500*e**5*(d
+ 2*e) - 107280*e**4*(d + 2*e)**2 - 7965*e**3*(d + 2*e)**3)/(801262*d**6 - 4662251*d**5*e + 7296938*d**4*e**2
+ 1388616*d**3*e**3 - 12447440*d**2*e**4 + 9990800*d*e**5 - 2380000*e**6))/48 - (19*d - 26*e)*log(x + (-15347
75*d**6 + 8032360*d**5*e + 328009*d**5*(19*d - 26*e)/8 - 12991180*d**4*e**2 - 1966211*d**4*e*(19*d - 26*e)/4 +
6193*d**4*(19*d - 26*e)**2 + 1075200*d**3*e**3 + 1363397*d**3*e**2*(19*d - 26*e) - 15148*d**3*e*(19*d - 26*e)
**2 + 143*d**3*(19*d - 26*e)**3/8 + 16959280*d**2*e**4 - 1624054*d**2*e**3*(19*d - 26*e) - 4896*d**2*e**2*(19*
d - 26*e)**2 + 2997*d**2*e*(19*d - 26*e)**3/4 - 15836800*d*e**5 + 887290*d*e**4*(19*d - 26*e) + 26800*d*e**3*(
19*d - 26*e)**2 - 2355*d*e**2*(19*d - 26*e)**3/2 + 4283840*e**6 - 161500*e**5*(19*d - 26*e) - 11920*e**4*(19*d
- 26*e)**2 + 295*e**3*(19*d - 26*e)**3)/(801262*d**6 - 4662251*d**5*e + 7296938*d**4*e**2 + 1388616*d**3*e**3
- 12447440*d**2*e**4 + 9990800*d*e**5 - 2380000*e**6))/144

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Giac [A]
time = 3.77, size = 66, normalized size = 0.93 \begin {gather*} -\frac {1}{144} \, {\left (19 \, d - 26 \, e\right )} \log \left ({\left | x + 2 \right |}\right ) + \frac {1}{6} \, {\left (d - e\right )} \log \left ({\left | x + 1 \right |}\right ) - \frac {1}{18} \, {\left (d + e\right )} \log \left ({\left | x - 1 \right |}\right ) + \frac {1}{48} \, {\left (d + 2 \, e\right )} \log \left ({\left | x - 2 \right |}\right ) + \frac {d - 2 \, e}{12 \, {\left (x + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/144*(19*d - 26*e)*log(abs(x + 2)) + 1/6*(d - e)*log(abs(x + 1)) - 1/18*(d + e)*log(abs(x - 1)) + 1/48*(d +
2*e)*log(abs(x - 2)) + 1/12*(d - 2*e)/(x + 2)

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Mupad [B]
time = 0.81, size = 64, normalized size = 0.90 \begin {gather*} \frac {\frac {d}{12}-\frac {e}{6}}{x+2}+\ln \left (x+1\right )\,\left (\frac {d}{6}-\frac {e}{6}\right )-\ln \left (x-1\right )\,\left (\frac {d}{18}+\frac {e}{18}\right )+\ln \left (x-2\right )\,\left (\frac {d}{48}+\frac {e}{24}\right )-\ln \left (x+2\right )\,\left (\frac {19\,d}{144}-\frac {13\,e}{72}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(d/12 - e/6)/(x + 2) + log(x + 1)*(d/6 - e/6) - log(x - 1)*(d/18 + e/18) + log(x - 2)*(d/48 + e/24) - log(x +
2)*((19*d)/144 - (13*e)/72)

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