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

Optimal. Leaf size=141 \[ -\frac {d-e+f-g}{36 (x+1)}+\frac {d+e+f+g}{12 (1-x)}+\frac {d+2 e+4 f+8 g}{36 (2-x)}+\frac {1}{36} \log (1-x) (2 d+5 e+8 f+11 g)-\frac {1}{432} \log (2-x) (35 d+58 e+92 f+136 g)+\frac {1}{108} \log (x+1) (2 d+e-4 f+7 g)+\frac {1}{144} \log (x+2) (d-2 e+4 f-8 g) \]

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Rubi [A]  time = 0.25, antiderivative size = 141, 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 = {1586, 6742} \begin {gather*} -\frac {d-e+f-g}{36 (x+1)}+\frac {d+e+f+g}{12 (1-x)}+\frac {d+2 e+4 f+8 g}{36 (2-x)}+\frac {1}{36} \log (1-x) (2 d+5 e+8 f+11 g)-\frac {1}{432} \log (2-x) (35 d+58 e+92 f+136 g)+\frac {1}{108} \log (x+1) (2 d+e-4 f+7 g)+\frac {1}{144} \log (x+2) (d-2 e+4 f-8 g) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(d + e + f + g)/(12*(1 - x)) + (d + 2*e + 4*f + 8*g)/(36*(2 - x)) - (d - e + f - g)/(36*(1 + x)) + ((2*d + 5*e
+ 8*f + 11*g)*Log[1 - x])/36 - ((35*d + 58*e + 92*f + 136*g)*Log[2 - x])/432 + ((2*d + e - 4*f + 7*g)*Log[1 +
x])/108 + ((d - 2*e + 4*f - 8*g)*Log[2 + x])/144

Rule 1586

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 6742

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

Rubi steps

\begin {align*} \int \frac {(2+x) \left (d+e x+f x^2+g x^3\right )}{\left (4-5 x^2+x^4\right )^2} \, dx &=\int \frac {d+e x+f x^2+g x^3}{(2+x) \left (2-x-2 x^2+x^3\right )^2} \, dx\\ &=\int \left (\frac {d+2 e+4 f+8 g}{36 (-2+x)^2}+\frac {-35 d-58 e-92 f-136 g}{432 (-2+x)}+\frac {d+e+f+g}{12 (-1+x)^2}+\frac {2 d+5 e+8 f+11 g}{36 (-1+x)}+\frac {d-e+f-g}{36 (1+x)^2}+\frac {2 d+e-4 f+7 g}{108 (1+x)}+\frac {d-2 e+4 f-8 g}{144 (2+x)}\right ) \, dx\\ &=\frac {d+e+f+g}{12 (1-x)}+\frac {d+2 e+4 f+8 g}{36 (2-x)}-\frac {d-e+f-g}{36 (1+x)}+\frac {1}{36} (2 d+5 e+8 f+11 g) \log (1-x)-\frac {1}{432} (35 d+58 e+92 f+136 g) \log (2-x)+\frac {1}{108} (2 d+e-4 f+7 g) \log (1+x)+\frac {1}{144} (d-2 e+4 f-8 g) \log (2+x)\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 144, normalized size = 1.02 \begin {gather*} \frac {1}{432} \left (\frac {12 \left (d \left (-5 x^2+6 x+5\right )+2 \left (e \left (5-2 x^2\right )+f \left (-4 x^2+3 x+4\right )+g \left (8-5 x^2\right )\right )\right )}{x^3-2 x^2-x+2}+12 \log (1-x) (2 d+5 e+8 f+11 g)-\log (2-x) (35 d+58 e+92 f+136 g)+4 \log (x+1) (2 d+e-4 f+7 g)+3 \log (x+2) (d-2 e+4 f-8 g)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((12*(d*(5 + 6*x - 5*x^2) + 2*(g*(8 - 5*x^2) + f*(4 + 3*x - 4*x^2) + e*(5 - 2*x^2))))/(2 - x - 2*x^2 + x^3) +
12*(2*d + 5*e + 8*f + 11*g)*Log[1 - x] - (35*d + 58*e + 92*f + 136*g)*Log[2 - x] + 4*(2*d + e - 4*f + 7*g)*Log
[1 + x] + 3*(d - 2*e + 4*f - 8*g)*Log[2 + x])/432

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 3.68, size = 321, normalized size = 2.28 \begin {gather*} -\frac {12 \, {\left (5 \, d + 4 \, e + 8 \, f + 10 \, g\right )} x^{2} - 72 \, {\left (d + f\right )} x - 3 \, {\left ({\left (d - 2 \, e + 4 \, f - 8 \, g\right )} x^{3} - 2 \, {\left (d - 2 \, e + 4 \, f - 8 \, g\right )} x^{2} - {\left (d - 2 \, e + 4 \, f - 8 \, g\right )} x + 2 \, d - 4 \, e + 8 \, f - 16 \, g\right )} \log \left (x + 2\right ) - 4 \, {\left ({\left (2 \, d + e - 4 \, f + 7 \, g\right )} x^{3} - 2 \, {\left (2 \, d + e - 4 \, f + 7 \, g\right )} x^{2} - {\left (2 \, d + e - 4 \, f + 7 \, g\right )} x + 4 \, d + 2 \, e - 8 \, f + 14 \, g\right )} \log \left (x + 1\right ) - 12 \, {\left ({\left (2 \, d + 5 \, e + 8 \, f + 11 \, g\right )} x^{3} - 2 \, {\left (2 \, d + 5 \, e + 8 \, f + 11 \, g\right )} x^{2} - {\left (2 \, d + 5 \, e + 8 \, f + 11 \, g\right )} x + 4 \, d + 10 \, e + 16 \, f + 22 \, g\right )} \log \left (x - 1\right ) + {\left ({\left (35 \, d + 58 \, e + 92 \, f + 136 \, g\right )} x^{3} - 2 \, {\left (35 \, d + 58 \, e + 92 \, f + 136 \, g\right )} x^{2} - {\left (35 \, d + 58 \, e + 92 \, f + 136 \, g\right )} x + 70 \, d + 116 \, e + 184 \, f + 272 \, g\right )} \log \left (x - 2\right ) - 60 \, d - 120 \, e - 96 \, f - 192 \, g}{432 \, {\left (x^{3} - 2 \, x^{2} - x + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/432*(12*(5*d + 4*e + 8*f + 10*g)*x^2 - 72*(d + f)*x - 3*((d - 2*e + 4*f - 8*g)*x^3 - 2*(d - 2*e + 4*f - 8*g
)*x^2 - (d - 2*e + 4*f - 8*g)*x + 2*d - 4*e + 8*f - 16*g)*log(x + 2) - 4*((2*d + e - 4*f + 7*g)*x^3 - 2*(2*d +
e - 4*f + 7*g)*x^2 - (2*d + e - 4*f + 7*g)*x + 4*d + 2*e - 8*f + 14*g)*log(x + 1) - 12*((2*d + 5*e + 8*f + 11
*g)*x^3 - 2*(2*d + 5*e + 8*f + 11*g)*x^2 - (2*d + 5*e + 8*f + 11*g)*x + 4*d + 10*e + 16*f + 22*g)*log(x - 1) +
((35*d + 58*e + 92*f + 136*g)*x^3 - 2*(35*d + 58*e + 92*f + 136*g)*x^2 - (35*d + 58*e + 92*f + 136*g)*x + 70*
d + 116*e + 184*f + 272*g)*log(x - 2) - 60*d - 120*e - 96*f - 192*g)/(x^3 - 2*x^2 - x + 2)

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giac [A]  time = 0.32, size = 136, normalized size = 0.96 \begin {gather*} \frac {1}{144} \, {\left (d + 4 \, f - 8 \, g - 2 \, e\right )} \log \left ({\left | x + 2 \right |}\right ) + \frac {1}{108} \, {\left (2 \, d - 4 \, f + 7 \, g + e\right )} \log \left ({\left | x + 1 \right |}\right ) + \frac {1}{36} \, {\left (2 \, d + 8 \, f + 11 \, g + 5 \, e\right )} \log \left ({\left | x - 1 \right |}\right ) - \frac {1}{432} \, {\left (35 \, d + 92 \, f + 136 \, g + 58 \, e\right )} \log \left ({\left | x - 2 \right |}\right ) - \frac {{\left (5 \, d + 8 \, f + 10 \, g + 4 \, e\right )} x^{2} - 6 \, {\left (d + f\right )} x - 5 \, d - 8 \, f - 16 \, g - 10 \, e}{36 \, {\left (x + 1\right )} {\left (x - 1\right )} {\left (x - 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/144*(d + 4*f - 8*g - 2*e)*log(abs(x + 2)) + 1/108*(2*d - 4*f + 7*g + e)*log(abs(x + 1)) + 1/36*(2*d + 8*f +
11*g + 5*e)*log(abs(x - 1)) - 1/432*(35*d + 92*f + 136*g + 58*e)*log(abs(x - 2)) - 1/36*((5*d + 8*f + 10*g + 4
*e)*x^2 - 6*(d + f)*x - 5*d - 8*f - 16*g - 10*e)/((x + 1)*(x - 1)*(x - 2))

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maple [A]  time = 0.02, size = 210, normalized size = 1.49 \begin {gather*} \frac {11 g \ln \left (x -1\right )}{36}-\frac {g \ln \left (x +2\right )}{18}-\frac {17 g \ln \left (x -2\right )}{54}+\frac {7 g \ln \left (x +1\right )}{108}+\frac {d \ln \left (x +2\right )}{144}-\frac {e \ln \left (x +2\right )}{72}+\frac {5 e \ln \left (x -1\right )}{36}+\frac {d \ln \left (x -1\right )}{18}+\frac {e \ln \left (x +1\right )}{108}+\frac {d \ln \left (x +1\right )}{54}-\frac {35 d \ln \left (x -2\right )}{432}-\frac {29 e \ln \left (x -2\right )}{216}-\frac {23 f \ln \left (x -2\right )}{108}-\frac {f \ln \left (x +1\right )}{27}+\frac {2 f \ln \left (x -1\right )}{9}+\frac {f \ln \left (x +2\right )}{36}+\frac {g}{36 x +36}+\frac {e}{36 x +36}-\frac {g}{12 \left (x -1\right )}-\frac {2 g}{9 \left (x -2\right )}-\frac {d}{36 \left (x -2\right )}-\frac {e}{18 \left (x -2\right )}-\frac {d}{36 \left (x +1\right )}-\frac {d}{12 \left (x -1\right )}-\frac {e}{12 \left (x -1\right )}-\frac {f}{12 \left (x -1\right )}-\frac {f}{9 \left (x -2\right )}-\frac {f}{36 \left (x +1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

11/36*g*ln(x-1)-1/18*g*ln(x+2)-17/54*g*ln(x-2)+7/108*g*ln(x+1)+1/144*d*ln(x+2)-1/72*e*ln(x+2)+5/36*e*ln(x-1)+1
/18*d*ln(x-1)+1/108*e*ln(x+1)+1/54*d*ln(x+1)-35/432*d*ln(x-2)-29/216*e*ln(x-2)-23/108*f*ln(x-2)-1/27*f*ln(x+1)
+2/9*f*ln(x-1)+1/36*f*ln(x+2)+1/36/(x+1)*g-1/12/(x-1)*g-2/9/(x-2)*g-1/36/(x-2)*d-1/18/(x-2)*e-1/36/(x+1)*d+1/3
6/(x+1)*e-1/12/(x-1)*d-1/12/(x-1)*e-1/12/(x-1)*f-1/9/(x-2)*f-1/36/(x+1)*f

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maxima [A]  time = 0.45, size = 126, normalized size = 0.89 \begin {gather*} \frac {1}{144} \, {\left (d - 2 \, e + 4 \, f - 8 \, g\right )} \log \left (x + 2\right ) + \frac {1}{108} \, {\left (2 \, d + e - 4 \, f + 7 \, g\right )} \log \left (x + 1\right ) + \frac {1}{36} \, {\left (2 \, d + 5 \, e + 8 \, f + 11 \, g\right )} \log \left (x - 1\right ) - \frac {1}{432} \, {\left (35 \, d + 58 \, e + 92 \, f + 136 \, g\right )} \log \left (x - 2\right ) - \frac {{\left (5 \, d + 4 \, e + 8 \, f + 10 \, g\right )} x^{2} - 6 \, {\left (d + f\right )} x - 5 \, d - 10 \, e - 8 \, f - 16 \, g}{36 \, {\left (x^{3} - 2 \, x^{2} - x + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/144*(d - 2*e + 4*f - 8*g)*log(x + 2) + 1/108*(2*d + e - 4*f + 7*g)*log(x + 1) + 1/36*(2*d + 5*e + 8*f + 11*g
)*log(x - 1) - 1/432*(35*d + 58*e + 92*f + 136*g)*log(x - 2) - 1/36*((5*d + 4*e + 8*f + 10*g)*x^2 - 6*(d + f)*
x - 5*d - 10*e - 8*f - 16*g)/(x^3 - 2*x^2 - x + 2)

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mupad [B]  time = 0.88, size = 131, normalized size = 0.93 \begin {gather*} \ln \left (x-1\right )\,\left (\frac {d}{18}+\frac {5\,e}{36}+\frac {2\,f}{9}+\frac {11\,g}{36}\right )+\ln \left (x+2\right )\,\left (\frac {d}{144}-\frac {e}{72}+\frac {f}{36}-\frac {g}{18}\right )+\ln \left (x+1\right )\,\left (\frac {d}{54}+\frac {e}{108}-\frac {f}{27}+\frac {7\,g}{108}\right )-\ln \left (x-2\right )\,\left (\frac {35\,d}{432}+\frac {29\,e}{216}+\frac {23\,f}{108}+\frac {17\,g}{54}\right )-\frac {\left (-\frac {5\,d}{36}-\frac {e}{9}-\frac {2\,f}{9}-\frac {5\,g}{18}\right )\,x^2+\left (\frac {d}{6}+\frac {f}{6}\right )\,x+\frac {5\,d}{36}+\frac {5\,e}{18}+\frac {2\,f}{9}+\frac {4\,g}{9}}{-x^3+2\,x^2+x-2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x - 1)*(d/18 + (5*e)/36 + (2*f)/9 + (11*g)/36) + log(x + 2)*(d/144 - e/72 + f/36 - g/18) + log(x + 1)*(d/5
4 + e/108 - f/27 + (7*g)/108) - log(x - 2)*((35*d)/432 + (29*e)/216 + (23*f)/108 + (17*g)/54) - ((5*d)/36 + (5
*e)/18 + (2*f)/9 + (4*g)/9 - x^2*((5*d)/36 + e/9 + (2*f)/9 + (5*g)/18) + x*(d/6 + f/6))/(x + 2*x^2 - x^3 - 2)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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