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

Optimal. Leaf size=96 \[ -\frac{1}{2} \log (1-x) (d+e+f+g+h+i)+\frac{1}{3} \log (2-x) (d+2 e+4 f+8 g+16 h+32 i)+\frac{1}{6} \log (x+1) (d-e+f-g+h-i)+x (g+2 h+5 i)+\frac{1}{2} x^2 (h+2 i)+\frac{i x^3}{3} \]

[Out]

(g + 2*h + 5*i)*x + ((h + 2*i)*x^2)/2 + (i*x^3)/3 - ((d + e + f + g + h + i)*Log
[1 - x])/2 + ((d + 2*e + 4*f + 8*g + 16*h + 32*i)*Log[2 - x])/3 + ((d - e + f -
g + h - i)*Log[1 + x])/6

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Rubi [A]  time = 0.274685, antiderivative size = 96, 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 \[ -\frac{1}{2} \log (1-x) (d+e+f+g+h+i)+\frac{1}{3} \log (2-x) (d+2 e+4 f+8 g+16 h+32 i)+\frac{1}{6} \log (x+1) (d-e+f-g+h-i)+x (g+2 h+5 i)+\frac{1}{2} x^2 (h+2 i)+\frac{i x^3}{3} \]

Antiderivative was successfully verified.

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

[Out]

(g + 2*h + 5*i)*x + ((h + 2*i)*x^2)/2 + (i*x^3)/3 - ((d + e + f + g + h + i)*Log
[1 - x])/2 + ((d + 2*e + 4*f + 8*g + 16*h + 32*i)*Log[2 - x])/3 + ((d - e + f -
g + h - i)*Log[1 + x])/6

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2+x)*(i*x**5+h*x**4+g*x**3+f*x**2+e*x+d)/(x**4-5*x**2+4),x)

[Out]

Timed out

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Mathematica [A]  time = 0.103942, size = 91, normalized size = 0.95 \[ \frac{1}{6} \left (-3 \log (1-x) (d+e+f+g+h+i)+2 \log (2-x) (d+2 e+4 (f+2 g+4 h+8 i))+\log (x+1) (d-e+f-g+h-i)+6 x (g+2 h+5 i)+3 x^2 (h+2 i)+2 i x^3\right ) \]

Antiderivative was successfully verified.

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

[Out]

(6*(g + 2*h + 5*i)*x + 3*(h + 2*i)*x^2 + 2*i*x^3 - 3*(d + e + f + g + h + i)*Log
[1 - x] + 2*(d + 2*e + 4*(f + 2*g + 4*h + 8*i))*Log[2 - x] + (d - e + f - g + h
- i)*Log[1 + x])/6

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Maple [A]  time = 0.011, size = 156, normalized size = 1.6 \[{\frac{i{x}^{3}}{3}}+{\frac{h{x}^{2}}{2}}+i{x}^{2}+gx+2\,hx+5\,ix-{\frac{\ln \left ( -1+x \right ) d}{2}}-{\frac{\ln \left ( -1+x \right ) e}{2}}-{\frac{\ln \left ( -1+x \right ) f}{2}}-{\frac{\ln \left ( -1+x \right ) g}{2}}-{\frac{\ln \left ( -1+x \right ) h}{2}}-{\frac{\ln \left ( -1+x \right ) i}{2}}+{\frac{\ln \left ( 1+x \right ) d}{6}}-{\frac{\ln \left ( 1+x \right ) e}{6}}+{\frac{\ln \left ( 1+x \right ) f}{6}}-{\frac{\ln \left ( 1+x \right ) g}{6}}+{\frac{\ln \left ( 1+x \right ) h}{6}}-{\frac{\ln \left ( 1+x \right ) i}{6}}+{\frac{\ln \left ( x-2 \right ) d}{3}}+{\frac{2\,\ln \left ( x-2 \right ) e}{3}}+{\frac{4\,\ln \left ( x-2 \right ) f}{3}}+{\frac{8\,\ln \left ( x-2 \right ) g}{3}}+{\frac{16\,\ln \left ( x-2 \right ) h}{3}}+{\frac{32\,\ln \left ( x-2 \right ) i}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*i*x^3+1/2*h*x^2+i*x^2+g*x+2*h*x+5*i*x-1/2*ln(-1+x)*d-1/2*ln(-1+x)*e-1/2*ln(-
1+x)*f-1/2*ln(-1+x)*g-1/2*ln(-1+x)*h-1/2*ln(-1+x)*i+1/6*ln(1+x)*d-1/6*ln(1+x)*e+
1/6*ln(1+x)*f-1/6*ln(1+x)*g+1/6*ln(1+x)*h-1/6*ln(1+x)*i+1/3*ln(x-2)*d+2/3*ln(x-2
)*e+4/3*ln(x-2)*f+8/3*ln(x-2)*g+16/3*ln(x-2)*h+32/3*ln(x-2)*i

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Maxima [A]  time = 0.704455, size = 111, normalized size = 1.16 \[ \frac{1}{3} \, i x^{3} + \frac{1}{2} \,{\left (h + 2 \, i\right )} x^{2} +{\left (g + 2 \, h + 5 \, i\right )} x + \frac{1}{6} \,{\left (d - e + f - g + h - i\right )} \log \left (x + 1\right ) - \frac{1}{2} \,{\left (d + e + f + g + h + i\right )} \log \left (x - 1\right ) + \frac{1}{3} \,{\left (d + 2 \, e + 4 \, f + 8 \, g + 16 \, h + 32 \, i\right )} \log \left (x - 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*i*x^3 + 1/2*(h + 2*i)*x^2 + (g + 2*h + 5*i)*x + 1/6*(d - e + f - g + h - i)*
log(x + 1) - 1/2*(d + e + f + g + h + i)*log(x - 1) + 1/3*(d + 2*e + 4*f + 8*g +
16*h + 32*i)*log(x - 2)

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Fricas [A]  time = 0.32984, size = 111, normalized size = 1.16 \[ \frac{1}{3} \, i x^{3} + \frac{1}{2} \,{\left (h + 2 \, i\right )} x^{2} +{\left (g + 2 \, h + 5 \, i\right )} x + \frac{1}{6} \,{\left (d - e + f - g + h - i\right )} \log \left (x + 1\right ) - \frac{1}{2} \,{\left (d + e + f + g + h + i\right )} \log \left (x - 1\right ) + \frac{1}{3} \,{\left (d + 2 \, e + 4 \, f + 8 \, g + 16 \, h + 32 \, i\right )} \log \left (x - 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*i*x^3 + 1/2*(h + 2*i)*x^2 + (g + 2*h + 5*i)*x + 1/6*(d - e + f - g + h - i)*
log(x + 1) - 1/2*(d + e + f + g + h + i)*log(x - 1) + 1/3*(d + 2*e + 4*f + 8*g +
16*h + 32*i)*log(x - 2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.283505, size = 122, normalized size = 1.27 \[ \frac{1}{3} \, i x^{3} + \frac{1}{2} \, h x^{2} + i x^{2} + g x + 2 \, h x + 5 \, i x + \frac{1}{6} \,{\left (d + f - g + h - i - e\right )}{\rm ln}\left ({\left | x + 1 \right |}\right ) - \frac{1}{2} \,{\left (d + f + g + h + i + e\right )}{\rm ln}\left ({\left | x - 1 \right |}\right ) + \frac{1}{3} \,{\left (d + 4 \, f + 8 \, g + 16 \, h + 32 \, i + 2 \, e\right )}{\rm ln}\left ({\left | x - 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*i*x^3 + 1/2*h*x^2 + i*x^2 + g*x + 2*h*x + 5*i*x + 1/6*(d + f - g + h - i - e
)*ln(abs(x + 1)) - 1/2*(d + f + g + h + i + e)*ln(abs(x - 1)) + 1/3*(d + 4*f + 8
*g + 16*h + 32*i + 2*e)*ln(abs(x - 2))