3.82 \(\int \frac{(2+x) (d+e x+f x^2+g x^3)}{4-5 x^2+x^4} \, dx\)
Optimal. Leaf size=57 \[ -\frac{1}{2} \log (1-x) (d+e+f+g)+\frac{1}{3} \log (2-x) (d+2 e+4 f+8 g)+\frac{1}{6} \log (x+1) (d-e+f-g)+g x \]
[Out]
g*x - ((d + e + f + g)*Log[1 - x])/2 + ((d + 2*e + 4*f + 8*g)*Log[2 - x])/3 + ((d - e + f - g)*Log[1 + x])/6
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Rubi [A] time = 0.0793655, antiderivative size = 57, normalized size of antiderivative = 1.,
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, 2074} \[ -\frac{1}{2} \log (1-x) (d+e+f+g)+\frac{1}{3} \log (2-x) (d+2 e+4 f+8 g)+\frac{1}{6} \log (x+1) (d-e+f-g)+g x \]
Antiderivative was successfully verified.
[In]
Int[((2 + x)*(d + e*x + f*x^2 + g*x^3))/(4 - 5*x^2 + x^4),x]
[Out]
g*x - ((d + e + f + g)*Log[1 - x])/2 + ((d + 2*e + 4*f + 8*g)*Log[2 - x])/3 + ((d - e + f - g)*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+g x^3\right )}{4-5 x^2+x^4} \, dx &=\int \frac{d+e x+f x^2+g x^3}{2-x-2 x^2+x^3} \, dx\\ &=\int \left (g+\frac{d+2 e+4 f+8 g}{3 (-2+x)}+\frac{-d-e-f-g}{2 (-1+x)}+\frac{d-e+f-g}{6 (1+x)}\right ) \, dx\\ &=g x-\frac{1}{2} (d+e+f+g) \log (1-x)+\frac{1}{3} (d+2 e+4 f+8 g) \log (2-x)+\frac{1}{6} (d-e+f-g) \log (1+x)\\ \end{align*}
Mathematica [A] time = 0.025557, size = 55, normalized size = 0.96 \[ \frac{1}{6} (-3 \log (1-x) (d+e+f+g)+2 \log (2-x) (d+2 e+4 f+8 g)+\log (x+1) (d-e+f-g)+6 g x) \]
Antiderivative was successfully verified.
[In]
Integrate[((2 + x)*(d + e*x + f*x^2 + g*x^3))/(4 - 5*x^2 + x^4),x]
[Out]
(6*g*x - 3*(d + e + f + g)*Log[1 - x] + 2*(d + 2*e + 4*f + 8*g)*Log[2 - x] + (d - e + f - g)*Log[1 + x])/6
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Maple [A] time = 0.007, size = 89, normalized size = 1.6 \begin{align*} gx+{\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 ( 1+x \right ) g}{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{8\,\ln \left ( x-2 \right ) g}{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}}-{\frac{\ln \left ( x-1 \right ) g}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((2+x)*(g*x^3+f*x^2+e*x+d)/(x^4-5*x^2+4),x)
[Out]
g*x+1/6*ln(1+x)*d-1/6*ln(1+x)*e+1/6*ln(1+x)*f-1/6*ln(1+x)*g+1/3*ln(x-2)*d+2/3*ln(x-2)*e+4/3*ln(x-2)*f+8/3*ln(x
-2)*g-1/2*ln(x-1)*d-1/2*ln(x-1)*e-1/2*ln(x-1)*f-1/2*ln(x-1)*g
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Maxima [A] time = 0.959427, size = 63, normalized size = 1.11 \begin{align*} g x + \frac{1}{6} \,{\left (d - e + f - g\right )} \log \left (x + 1\right ) - \frac{1}{2} \,{\left (d + e + f + g\right )} \log \left (x - 1\right ) + \frac{1}{3} \,{\left (d + 2 \, e + 4 \, f + 8 \, g\right )} \log \left (x - 2\right ) \end{align*}
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),x, algorithm="maxima")
[Out]
g*x + 1/6*(d - e + f - g)*log(x + 1) - 1/2*(d + e + f + g)*log(x - 1) + 1/3*(d + 2*e + 4*f + 8*g)*log(x - 2)
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Fricas [A] time = 1.60767, size = 149, normalized size = 2.61 \begin{align*} g x + \frac{1}{6} \,{\left (d - e + f - g\right )} \log \left (x + 1\right ) - \frac{1}{2} \,{\left (d + e + f + g\right )} \log \left (x - 1\right ) + \frac{1}{3} \,{\left (d + 2 \, e + 4 \, f + 8 \, g\right )} \log \left (x - 2\right ) \end{align*}
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),x, algorithm="fricas")
[Out]
g*x + 1/6*(d - e + f - g)*log(x + 1) - 1/2*(d + e + f + g)*log(x - 1) + 1/3*(d + 2*e + 4*f + 8*g)*log(x - 2)
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Sympy [B] time = 26.0929, size = 1389, normalized size = 24.37 \begin{align*} \text{result too large to display} \end{align*}
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),x)
[Out]
g*x + (d - e + f - g)*log(x + (26*d**3 + 66*d**2*e + 132*d**2*f + 174*d**2*g - 9*d**2*(d - e + f - g) + 78*d*e
**2 + 276*d*e*f + 444*d*e*g - 12*d*e*(d - e + f - g) + 222*d*f**2 + 636*d*f*g + 6*d*f*(d - e + f - g) + 510*d*
g**2 + 36*d*g*(d - e + f - g) - 7*d*(d - e + f - g)**2 + 46*e**3 + 204*e**2*f + 390*e**2*g + 3*e**2*(d - e + f
- g) + 282*e*f**2 + 984*e*f*g + 36*e*f*(d - e + f - g) + 930*e*g**2 + 102*e*g*(d - e + f - g) - 8*e*(d - e +
f - g)**2 + 116*f**3 + 534*f**2*g + 51*f**2*(d - e + f - g) + 924*f*g**2 + 228*f*g*(d - e + f - g) - 13*f*(d -
e + f - g)**2 + 586*g**3 + 243*g**2*(d - e + f - g) - 20*g*(d - e + f - g)**2)/(10*d**3 + 69*d**2*e + 102*d**
2*f + 213*d**2*g + 102*d*e**2 + 318*d*e*f + 564*d*e*g + 246*d*f**2 + 894*d*f*g + 750*d*g**2 + 35*e**3 + 174*e*
*2*f + 249*e**2*g + 285*e*f**2 + 852*e*f*g + 537*e*g**2 + 154*f**3 + 717*f**2*g + 966*f*g**2 + 323*g**3))/6 -
(d + e + f + g)*log(x + (26*d**3 + 66*d**2*e + 132*d**2*f + 174*d**2*g + 27*d**2*(d + e + f + g) + 78*d*e**2 +
276*d*e*f + 444*d*e*g + 36*d*e*(d + e + f + g) + 222*d*f**2 + 636*d*f*g - 18*d*f*(d + e + f + g) + 510*d*g**2
- 108*d*g*(d + e + f + g) - 63*d*(d + e + f + g)**2 + 46*e**3 + 204*e**2*f + 390*e**2*g - 9*e**2*(d + e + f +
g) + 282*e*f**2 + 984*e*f*g - 108*e*f*(d + e + f + g) + 930*e*g**2 - 306*e*g*(d + e + f + g) - 72*e*(d + e +
f + g)**2 + 116*f**3 + 534*f**2*g - 153*f**2*(d + e + f + g) + 924*f*g**2 - 684*f*g*(d + e + f + g) - 117*f*(d
+ e + f + g)**2 + 586*g**3 - 729*g**2*(d + e + f + g) - 180*g*(d + e + f + g)**2)/(10*d**3 + 69*d**2*e + 102*
d**2*f + 213*d**2*g + 102*d*e**2 + 318*d*e*f + 564*d*e*g + 246*d*f**2 + 894*d*f*g + 750*d*g**2 + 35*e**3 + 174
*e**2*f + 249*e**2*g + 285*e*f**2 + 852*e*f*g + 537*e*g**2 + 154*f**3 + 717*f**2*g + 966*f*g**2 + 323*g**3))/2
+ (d + 2*e + 4*f + 8*g)*log(x + (26*d**3 + 66*d**2*e + 132*d**2*f + 174*d**2*g - 18*d**2*(d + 2*e + 4*f + 8*g
) + 78*d*e**2 + 276*d*e*f + 444*d*e*g - 24*d*e*(d + 2*e + 4*f + 8*g) + 222*d*f**2 + 636*d*f*g + 12*d*f*(d + 2*
e + 4*f + 8*g) + 510*d*g**2 + 72*d*g*(d + 2*e + 4*f + 8*g) - 28*d*(d + 2*e + 4*f + 8*g)**2 + 46*e**3 + 204*e**
2*f + 390*e**2*g + 6*e**2*(d + 2*e + 4*f + 8*g) + 282*e*f**2 + 984*e*f*g + 72*e*f*(d + 2*e + 4*f + 8*g) + 930*
e*g**2 + 204*e*g*(d + 2*e + 4*f + 8*g) - 32*e*(d + 2*e + 4*f + 8*g)**2 + 116*f**3 + 534*f**2*g + 102*f**2*(d +
2*e + 4*f + 8*g) + 924*f*g**2 + 456*f*g*(d + 2*e + 4*f + 8*g) - 52*f*(d + 2*e + 4*f + 8*g)**2 + 586*g**3 + 48
6*g**2*(d + 2*e + 4*f + 8*g) - 80*g*(d + 2*e + 4*f + 8*g)**2)/(10*d**3 + 69*d**2*e + 102*d**2*f + 213*d**2*g +
102*d*e**2 + 318*d*e*f + 564*d*e*g + 246*d*f**2 + 894*d*f*g + 750*d*g**2 + 35*e**3 + 174*e**2*f + 249*e**2*g
+ 285*e*f**2 + 852*e*f*g + 537*e*g**2 + 154*f**3 + 717*f**2*g + 966*f*g**2 + 323*g**3))/3
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Giac [A] time = 1.08989, size = 72, normalized size = 1.26 \begin{align*} g x + \frac{1}{6} \,{\left (d + f - g - e\right )} \log \left ({\left | x + 1 \right |}\right ) - \frac{1}{2} \,{\left (d + f + g + e\right )} \log \left ({\left | x - 1 \right |}\right ) + \frac{1}{3} \,{\left (d + 4 \, f + 8 \, g + 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)*(g*x^3+f*x^2+e*x+d)/(x^4-5*x^2+4),x, algorithm="giac")
[Out]
g*x + 1/6*(d + f - g - e)*log(abs(x + 1)) - 1/2*(d + f + g + e)*log(abs(x - 1)) + 1/3*(d + 4*f + 8*g + 2*e)*lo
g(abs(x - 2))