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

Optimal. Leaf size=47 \[ -\frac{1}{2} \log (1-x) (d+e+f)+\frac{1}{3} \log (2-x) (d+2 e+4 f)+\frac{1}{6} \log (x+1) (d-e+f) \]

[Out]

-((d + e + f)*Log[1 - x])/2 + ((d + 2*e + 4*f)*Log[2 - x])/3 + ((d - e + f)*Log[1 + x])/6

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Rubi [A]  time = 0.0638697, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {1586, 2074} \[ -\frac{1}{2} \log (1-x) (d+e+f)+\frac{1}{3} \log (2-x) (d+2 e+4 f)+\frac{1}{6} \log (x+1) (d-e+f) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((d + e + f)*Log[1 - x])/2 + ((d + 2*e + 4*f)*Log[2 - x])/3 + ((d - e + f)*Log[1 + x])/6

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 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps

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

Mathematica [A]  time = 0.0204724, size = 44, normalized size = 0.94 \[ \frac{1}{6} (-3 \log (1-x) (d+e+f)+2 \log (2-x) (d+2 e+4 f)+\log (x+1) (d-e+f)) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*(d + e + f)*Log[1 - x] + 2*(d + 2*e + 4*f)*Log[2 - x] + (d - e + f)*Log[1 + x])/6

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Maple [A]  time = 0.007, size = 65, normalized size = 1.4 \begin{align*}{\frac{\ln \left ( 1+x \right ) d}{6}}-{\frac{\ln \left ( 1+x \right ) e}{6}}+{\frac{\ln \left ( 1+x \right ) f}{6}}+{\frac{\ln \left ( x-2 \right ) d}{3}}+{\frac{2\,\ln \left ( x-2 \right ) e}{3}}+{\frac{4\,\ln \left ( x-2 \right ) f}{3}}-{\frac{\ln \left ( x-1 \right ) d}{2}}-{\frac{\ln \left ( x-1 \right ) e}{2}}-{\frac{\ln \left ( x-1 \right ) f}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*ln(1+x)*d-1/6*ln(1+x)*e+1/6*ln(1+x)*f+1/3*ln(x-2)*d+2/3*ln(x-2)*e+4/3*ln(x-2)*f-1/2*ln(x-1)*d-1/2*ln(x-1)*
e-1/2*ln(x-1)*f

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Maxima [A]  time = 0.944502, size = 50, normalized size = 1.06 \begin{align*} \frac{1}{6} \,{\left (d - e + f\right )} \log \left (x + 1\right ) - \frac{1}{2} \,{\left (d + e + f\right )} \log \left (x - 1\right ) + \frac{1}{3} \,{\left (d + 2 \, e + 4 \, f\right )} \log \left (x - 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(d - e + f)*log(x + 1) - 1/2*(d + e + f)*log(x - 1) + 1/3*(d + 2*e + 4*f)*log(x - 2)

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Fricas [A]  time = 1.62067, size = 122, normalized size = 2.6 \begin{align*} \frac{1}{6} \,{\left (d - e + f\right )} \log \left (x + 1\right ) - \frac{1}{2} \,{\left (d + e + f\right )} \log \left (x - 1\right ) + \frac{1}{3} \,{\left (d + 2 \, e + 4 \, f\right )} \log \left (x - 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(d - e + f)*log(x + 1) - 1/2*(d + e + f)*log(x - 1) + 1/3*(d + 2*e + 4*f)*log(x - 2)

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Sympy [B]  time = 5.15638, size = 716, normalized size = 15.23 \begin{align*} \frac{\left (d - e + f\right ) \log{\left (x + \frac{26 d^{3} + 66 d^{2} e + 132 d^{2} f - 9 d^{2} \left (d - e + f\right ) + 78 d e^{2} + 276 d e f - 12 d e \left (d - e + f\right ) + 222 d f^{2} + 6 d f \left (d - e + f\right ) - 7 d \left (d - e + f\right )^{2} + 46 e^{3} + 204 e^{2} f + 3 e^{2} \left (d - e + f\right ) + 282 e f^{2} + 36 e f \left (d - e + f\right ) - 8 e \left (d - e + f\right )^{2} + 116 f^{3} + 51 f^{2} \left (d - e + f\right ) - 13 f \left (d - e + f\right )^{2}}{10 d^{3} + 69 d^{2} e + 102 d^{2} f + 102 d e^{2} + 318 d e f + 246 d f^{2} + 35 e^{3} + 174 e^{2} f + 285 e f^{2} + 154 f^{3}} \right )}}{6} - \frac{\left (d + e + f\right ) \log{\left (x + \frac{26 d^{3} + 66 d^{2} e + 132 d^{2} f + 27 d^{2} \left (d + e + f\right ) + 78 d e^{2} + 276 d e f + 36 d e \left (d + e + f\right ) + 222 d f^{2} - 18 d f \left (d + e + f\right ) - 63 d \left (d + e + f\right )^{2} + 46 e^{3} + 204 e^{2} f - 9 e^{2} \left (d + e + f\right ) + 282 e f^{2} - 108 e f \left (d + e + f\right ) - 72 e \left (d + e + f\right )^{2} + 116 f^{3} - 153 f^{2} \left (d + e + f\right ) - 117 f \left (d + e + f\right )^{2}}{10 d^{3} + 69 d^{2} e + 102 d^{2} f + 102 d e^{2} + 318 d e f + 246 d f^{2} + 35 e^{3} + 174 e^{2} f + 285 e f^{2} + 154 f^{3}} \right )}}{2} + \frac{\left (d + 2 e + 4 f\right ) \log{\left (x + \frac{26 d^{3} + 66 d^{2} e + 132 d^{2} f - 18 d^{2} \left (d + 2 e + 4 f\right ) + 78 d e^{2} + 276 d e f - 24 d e \left (d + 2 e + 4 f\right ) + 222 d f^{2} + 12 d f \left (d + 2 e + 4 f\right ) - 28 d \left (d + 2 e + 4 f\right )^{2} + 46 e^{3} + 204 e^{2} f + 6 e^{2} \left (d + 2 e + 4 f\right ) + 282 e f^{2} + 72 e f \left (d + 2 e + 4 f\right ) - 32 e \left (d + 2 e + 4 f\right )^{2} + 116 f^{3} + 102 f^{2} \left (d + 2 e + 4 f\right ) - 52 f \left (d + 2 e + 4 f\right )^{2}}{10 d^{3} + 69 d^{2} e + 102 d^{2} f + 102 d e^{2} + 318 d e f + 246 d f^{2} + 35 e^{3} + 174 e^{2} f + 285 e f^{2} + 154 f^{3}} \right )}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(d - e + f)*log(x + (26*d**3 + 66*d**2*e + 132*d**2*f - 9*d**2*(d - e + f) + 78*d*e**2 + 276*d*e*f - 12*d*e*(d
- e + f) + 222*d*f**2 + 6*d*f*(d - e + f) - 7*d*(d - e + f)**2 + 46*e**3 + 204*e**2*f + 3*e**2*(d - e + f) +
282*e*f**2 + 36*e*f*(d - e + f) - 8*e*(d - e + f)**2 + 116*f**3 + 51*f**2*(d - e + f) - 13*f*(d - e + f)**2)/(
10*d**3 + 69*d**2*e + 102*d**2*f + 102*d*e**2 + 318*d*e*f + 246*d*f**2 + 35*e**3 + 174*e**2*f + 285*e*f**2 + 1
54*f**3))/6 - (d + e + f)*log(x + (26*d**3 + 66*d**2*e + 132*d**2*f + 27*d**2*(d + e + f) + 78*d*e**2 + 276*d*
e*f + 36*d*e*(d + e + f) + 222*d*f**2 - 18*d*f*(d + e + f) - 63*d*(d + e + f)**2 + 46*e**3 + 204*e**2*f - 9*e*
*2*(d + e + f) + 282*e*f**2 - 108*e*f*(d + e + f) - 72*e*(d + e + f)**2 + 116*f**3 - 153*f**2*(d + e + f) - 11
7*f*(d + e + f)**2)/(10*d**3 + 69*d**2*e + 102*d**2*f + 102*d*e**2 + 318*d*e*f + 246*d*f**2 + 35*e**3 + 174*e*
*2*f + 285*e*f**2 + 154*f**3))/2 + (d + 2*e + 4*f)*log(x + (26*d**3 + 66*d**2*e + 132*d**2*f - 18*d**2*(d + 2*
e + 4*f) + 78*d*e**2 + 276*d*e*f - 24*d*e*(d + 2*e + 4*f) + 222*d*f**2 + 12*d*f*(d + 2*e + 4*f) - 28*d*(d + 2*
e + 4*f)**2 + 46*e**3 + 204*e**2*f + 6*e**2*(d + 2*e + 4*f) + 282*e*f**2 + 72*e*f*(d + 2*e + 4*f) - 32*e*(d +
2*e + 4*f)**2 + 116*f**3 + 102*f**2*(d + 2*e + 4*f) - 52*f*(d + 2*e + 4*f)**2)/(10*d**3 + 69*d**2*e + 102*d**2
*f + 102*d*e**2 + 318*d*e*f + 246*d*f**2 + 35*e**3 + 174*e**2*f + 285*e*f**2 + 154*f**3))/3

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Giac [A]  time = 1.0817, size = 58, normalized size = 1.23 \begin{align*} \frac{1}{6} \,{\left (d + f - e\right )} \log \left ({\left | x + 1 \right |}\right ) - \frac{1}{2} \,{\left (d + f + e\right )} \log \left ({\left | x - 1 \right |}\right ) + \frac{1}{3} \,{\left (d + 4 \, f + 2 \, e\right )} \log \left ({\left | x - 2 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(d + f - e)*log(abs(x + 1)) - 1/2*(d + f + e)*log(abs(x - 1)) + 1/3*(d + 4*f + 2*e)*log(abs(x - 2))