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

Optimal. Leaf size=131 \[ -\frac{d-e+f-g+h}{6 (x+1)}-\frac{d-2 e+4 f-8 g+16 h}{12 (x+2)}-\frac{1}{36} \log (1-x) (d+e+f+g+h)+\frac{1}{144} \log (2-x) (d+2 e+4 f+8 g+16 h)-\frac{1}{36} \log (x+1) (7 d-13 e+19 f-25 g+31 h)+\frac{1}{144} \log (x+2) (31 d-50 e+76 f-104 g+112 h) \]

[Out]

-(d - e + f - g + h)/(6*(1 + x)) - (d - 2*e + 4*f - 8*g + 16*h)/(12*(2 + x)) - ((d + e + f + g + h)*Log[1 - x]
)/36 + ((d + 2*e + 4*f + 8*g + 16*h)*Log[2 - x])/144 - ((7*d - 13*e + 19*f - 25*g + 31*h)*Log[1 + x])/36 + ((3
1*d - 50*e + 76*f - 104*g + 112*h)*Log[2 + x])/144

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Rubi [A]  time = 0.28044, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.049, Rules used = {1586, 6728} \[ -\frac{d-e+f-g+h}{6 (x+1)}-\frac{d-2 e+4 f-8 g+16 h}{12 (x+2)}-\frac{1}{36} \log (1-x) (d+e+f+g+h)+\frac{1}{144} \log (2-x) (d+2 e+4 f+8 g+16 h)-\frac{1}{36} \log (x+1) (7 d-13 e+19 f-25 g+31 h)+\frac{1}{144} \log (x+2) (31 d-50 e+76 f-104 g+112 h) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(d - e + f - g + h)/(6*(1 + x)) - (d - 2*e + 4*f - 8*g + 16*h)/(12*(2 + x)) - ((d + e + f + g + h)*Log[1 - x]
)/36 + ((d + 2*e + 4*f + 8*g + 16*h)*Log[2 - x])/144 - ((7*d - 13*e + 19*f - 25*g + 31*h)*Log[1 + x])/36 + ((3
1*d - 50*e + 76*f - 104*g + 112*h)*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 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps

\begin{align*} \int \frac{\left (2-3 x+x^2\right ) \left (d+e x+f x^2+g x^3+h x^4\right )}{\left (4-5 x^2+x^4\right )^2} \, dx &=\int \frac{d+e x+f x^2+g x^3+h x^4}{\left (2-3 x+x^2\right ) \left (2+3 x+x^2\right )^2} \, dx\\ &=\int \left (\frac{d+2 e+4 f+8 g+16 h}{144 (-2+x)}+\frac{-d-e-f-g-h}{36 (-1+x)}+\frac{d-e+f-g+h}{6 (1+x)^2}+\frac{-7 d+13 e-19 f+25 g-31 h}{36 (1+x)}+\frac{d-2 e+4 f-8 g+16 h}{12 (2+x)^2}+\frac{31 d-50 e+76 f-104 g+112 h}{144 (2+x)}\right ) \, dx\\ &=-\frac{d-e+f-g+h}{6 (1+x)}-\frac{d-2 e+4 f-8 g+16 h}{12 (2+x)}-\frac{1}{36} (d+e+f+g+h) \log (1-x)+\frac{1}{144} (d+2 e+4 f+8 g+16 h) \log (2-x)-\frac{1}{36} (7 d-13 e+19 f-25 g+31 h) \log (1+x)+\frac{1}{144} (31 d-50 e+76 f-104 g+112 h) \log (2+x)\\ \end{align*}

Mathematica [A]  time = 0.0694655, size = 136, normalized size = 1.04 \[ \frac{1}{144} \left (-\frac{12 (d (3 x+5)+2 (-e (2 x+3)+3 f x+4 f-5 g x-6 g+9 h x+10 h))}{x^2+3 x+2}-4 \log (1-x) (d+e+f+g+h)+\log (2-x) (d+2 (e+2 f+4 g+8 h))-4 \log (x+1) (7 d-13 e+19 f-25 g+31 h)+\log (x+2) (31 d-50 e+76 f-104 g+112 h)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-12*(d*(5 + 3*x) + 2*(4*f - 6*g + 10*h + 3*f*x - 5*g*x + 9*h*x - e*(3 + 2*x))))/(2 + 3*x + x^2) - 4*(d + e +
f + g + h)*Log[1 - x] + (d + 2*(e + 2*f + 4*g + 8*h))*Log[2 - x] - 4*(7*d - 13*e + 19*f - 25*g + 31*h)*Log[1
+ x] + (31*d - 50*e + 76*f - 104*g + 112*h)*Log[2 + x])/144

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Maple [A]  time = 0.015, size = 222, normalized size = 1.7 \begin{align*}{\frac{31\,\ln \left ( 2+x \right ) d}{144}}-{\frac{25\,\ln \left ( 2+x \right ) e}{72}}-{\frac{7\,\ln \left ( 1+x \right ) d}{36}}+{\frac{13\,\ln \left ( 1+x \right ) e}{36}}+{\frac{\ln \left ( x-2 \right ) d}{144}}+{\frac{\ln \left ( x-2 \right ) e}{72}}-{\frac{\ln \left ( x-1 \right ) d}{36}}-{\frac{\ln \left ( x-1 \right ) e}{36}}-{\frac{h}{6+6\,x}}-{\frac{4\,h}{6+3\,x}}+{\frac{2\,g}{6+3\,x}}-{\frac{d}{6+6\,x}}+{\frac{e}{6+6\,x}}-{\frac{d}{24+12\,x}}+{\frac{e}{12+6\,x}}+{\frac{g}{6+6\,x}}-{\frac{f}{6+6\,x}}-{\frac{f}{6+3\,x}}-{\frac{13\,\ln \left ( 2+x \right ) g}{18}}+{\frac{25\,\ln \left ( 1+x \right ) g}{36}}+{\frac{\ln \left ( x-2 \right ) g}{18}}-{\frac{\ln \left ( x-1 \right ) g}{36}}+{\frac{7\,\ln \left ( 2+x \right ) h}{9}}-{\frac{31\,\ln \left ( 1+x \right ) h}{36}}+{\frac{\ln \left ( x-2 \right ) h}{9}}-{\frac{\ln \left ( x-1 \right ) h}{36}}+{\frac{\ln \left ( x-2 \right ) f}{36}}-{\frac{\ln \left ( x-1 \right ) f}{36}}+{\frac{19\,\ln \left ( 2+x \right ) f}{36}}-{\frac{19\,\ln \left ( 1+x \right ) f}{36}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

31/144*ln(2+x)*d-25/72*ln(2+x)*e-7/36*ln(1+x)*d+13/36*ln(1+x)*e+1/144*ln(x-2)*d+1/72*ln(x-2)*e-1/36*ln(x-1)*d-
1/36*ln(x-1)*e-1/6/(1+x)*h-4/3/(2+x)*h+2/3/(2+x)*g-1/6/(1+x)*d+1/6/(1+x)*e-1/12/(2+x)*d+1/6/(2+x)*e+1/6/(1+x)*
g-1/6/(1+x)*f-1/3/(2+x)*f-13/18*ln(2+x)*g+25/36*ln(1+x)*g+1/18*ln(x-2)*g-1/36*ln(x-1)*g+7/9*ln(2+x)*h-31/36*ln
(1+x)*h+1/9*ln(x-2)*h-1/36*ln(x-1)*h+1/36*ln(x-2)*f-1/36*ln(x-1)*f+19/36*ln(2+x)*f-19/36*ln(1+x)*f

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Maxima [A]  time = 0.957816, size = 166, normalized size = 1.27 \begin{align*} \frac{1}{144} \,{\left (31 \, d - 50 \, e + 76 \, f - 104 \, g + 112 \, h\right )} \log \left (x + 2\right ) - \frac{1}{36} \,{\left (7 \, d - 13 \, e + 19 \, f - 25 \, g + 31 \, h\right )} \log \left (x + 1\right ) - \frac{1}{36} \,{\left (d + e + f + g + h\right )} \log \left (x - 1\right ) + \frac{1}{144} \,{\left (d + 2 \, e + 4 \, f + 8 \, g + 16 \, h\right )} \log \left (x - 2\right ) - \frac{{\left (3 \, d - 4 \, e + 6 \, f - 10 \, g + 18 \, h\right )} x + 5 \, d - 6 \, e + 8 \, f - 12 \, g + 20 \, h}{12 \,{\left (x^{2} + 3 \, x + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/144*(31*d - 50*e + 76*f - 104*g + 112*h)*log(x + 2) - 1/36*(7*d - 13*e + 19*f - 25*g + 31*h)*log(x + 1) - 1/
36*(d + e + f + g + h)*log(x - 1) + 1/144*(d + 2*e + 4*f + 8*g + 16*h)*log(x - 2) - 1/12*((3*d - 4*e + 6*f - 1
0*g + 18*h)*x + 5*d - 6*e + 8*f - 12*g + 20*h)/(x^2 + 3*x + 2)

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Fricas [B]  time = 34.4963, size = 783, normalized size = 5.98 \begin{align*} -\frac{12 \,{\left (3 \, d - 4 \, e + 6 \, f - 10 \, g + 18 \, h\right )} x -{\left ({\left (31 \, d - 50 \, e + 76 \, f - 104 \, g + 112 \, h\right )} x^{2} + 3 \,{\left (31 \, d - 50 \, e + 76 \, f - 104 \, g + 112 \, h\right )} x + 62 \, d - 100 \, e + 152 \, f - 208 \, g + 224 \, h\right )} \log \left (x + 2\right ) + 4 \,{\left ({\left (7 \, d - 13 \, e + 19 \, f - 25 \, g + 31 \, h\right )} x^{2} + 3 \,{\left (7 \, d - 13 \, e + 19 \, f - 25 \, g + 31 \, h\right )} x + 14 \, d - 26 \, e + 38 \, f - 50 \, g + 62 \, h\right )} \log \left (x + 1\right ) + 4 \,{\left ({\left (d + e + f + g + h\right )} x^{2} + 3 \,{\left (d + e + f + g + h\right )} x + 2 \, d + 2 \, e + 2 \, f + 2 \, g + 2 \, h\right )} \log \left (x - 1\right ) -{\left ({\left (d + 2 \, e + 4 \, f + 8 \, g + 16 \, h\right )} x^{2} + 3 \,{\left (d + 2 \, e + 4 \, f + 8 \, g + 16 \, h\right )} x + 2 \, d + 4 \, e + 8 \, f + 16 \, g + 32 \, h\right )} \log \left (x - 2\right ) + 60 \, d - 72 \, e + 96 \, f - 144 \, g + 240 \, h}{144 \,{\left (x^{2} + 3 \, x + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/144*(12*(3*d - 4*e + 6*f - 10*g + 18*h)*x - ((31*d - 50*e + 76*f - 104*g + 112*h)*x^2 + 3*(31*d - 50*e + 76
*f - 104*g + 112*h)*x + 62*d - 100*e + 152*f - 208*g + 224*h)*log(x + 2) + 4*((7*d - 13*e + 19*f - 25*g + 31*h
)*x^2 + 3*(7*d - 13*e + 19*f - 25*g + 31*h)*x + 14*d - 26*e + 38*f - 50*g + 62*h)*log(x + 1) + 4*((d + e + f +
g + h)*x^2 + 3*(d + e + f + g + h)*x + 2*d + 2*e + 2*f + 2*g + 2*h)*log(x - 1) - ((d + 2*e + 4*f + 8*g + 16*h
)*x^2 + 3*(d + 2*e + 4*f + 8*g + 16*h)*x + 2*d + 4*e + 8*f + 16*g + 32*h)*log(x - 2) + 60*d - 72*e + 96*f - 14
4*g + 240*h)/(x^2 + 3*x + 2)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.08553, size = 180, normalized size = 1.37 \begin{align*} \frac{1}{144} \,{\left (31 \, d + 76 \, f - 104 \, g + 112 \, h - 50 \, e\right )} \log \left ({\left | x + 2 \right |}\right ) - \frac{1}{36} \,{\left (7 \, d + 19 \, f - 25 \, g + 31 \, h - 13 \, e\right )} \log \left ({\left | x + 1 \right |}\right ) - \frac{1}{36} \,{\left (d + f + g + h + e\right )} \log \left ({\left | x - 1 \right |}\right ) + \frac{1}{144} \,{\left (d + 4 \, f + 8 \, g + 16 \, h + 2 \, e\right )} \log \left ({\left | x - 2 \right |}\right ) - \frac{{\left (3 \, d + 6 \, f - 10 \, g + 18 \, h - 4 \, e\right )} x + 5 \, d + 8 \, f - 12 \, g + 20 \, h - 6 \, e}{12 \,{\left (x + 2\right )}{\left (x + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/144*(31*d + 76*f - 104*g + 112*h - 50*e)*log(abs(x + 2)) - 1/36*(7*d + 19*f - 25*g + 31*h - 13*e)*log(abs(x
+ 1)) - 1/36*(d + f + g + h + e)*log(abs(x - 1)) + 1/144*(d + 4*f + 8*g + 16*h + 2*e)*log(abs(x - 2)) - 1/12*(
(3*d + 6*f - 10*g + 18*h - 4*e)*x + 5*d + 8*f - 12*g + 20*h - 6*e)/((x + 2)*(x + 1))