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

Optimal. Leaf size=158 \[ -\frac{d-e+f-g+h}{36 (x+1)}+\frac{d+e+f+g+h}{12 (1-x)}+\frac{d+2 e+4 f+8 g+16 h}{36 (2-x)}+\frac{1}{36} \log (1-x) (2 d+5 e+8 f+11 g+14 h)-\frac{1}{432} \log (2-x) (35 d+58 e+92 f+136 g+176 h)+\frac{1}{108} \log (x+1) (2 d+e-4 f+7 g-10 h)+\frac{1}{144} \log (x+2) (d-2 e+4 f-8 g+16 h) \]

[Out]

(d + e + f + g + h)/(12*(1 - x)) + (d + 2*e + 4*f + 8*g + 16*h)/(36*(2 - x)) - (d - e + f - g + h)/(36*(1 + x)
) + ((2*d + 5*e + 8*f + 11*g + 14*h)*Log[1 - x])/36 - ((35*d + 58*e + 92*f + 136*g + 176*h)*Log[2 - x])/432 +
((2*d + e - 4*f + 7*g - 10*h)*Log[1 + x])/108 + ((d - 2*e + 4*f - 8*g + 16*h)*Log[2 + x])/144

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Rubi [A]  time = 0.28863, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {1586, 6742} \[ -\frac{d-e+f-g+h}{36 (x+1)}+\frac{d+e+f+g+h}{12 (1-x)}+\frac{d+2 e+4 f+8 g+16 h}{36 (2-x)}+\frac{1}{36} \log (1-x) (2 d+5 e+8 f+11 g+14 h)-\frac{1}{432} \log (2-x) (35 d+58 e+92 f+136 g+176 h)+\frac{1}{108} \log (x+1) (2 d+e-4 f+7 g-10 h)+\frac{1}{144} \log (x+2) (d-2 e+4 f-8 g+16 h) \]

Antiderivative was successfully verified.

[In]

Int[((2 + x)*(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)/(12*(1 - x)) + (d + 2*e + 4*f + 8*g + 16*h)/(36*(2 - x)) - (d - e + f - g + h)/(36*(1 + x)
) + ((2*d + 5*e + 8*f + 11*g + 14*h)*Log[1 - x])/36 - ((35*d + 58*e + 92*f + 136*g + 176*h)*Log[2 - x])/432 +
((2*d + e - 4*f + 7*g - 10*h)*Log[1 + x])/108 + ((d - 2*e + 4*f - 8*g + 16*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 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+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}{(2+x) \left (2-x-2 x^2+x^3\right )^2} \, dx\\ &=\int \left (\frac{d+2 e+4 f+8 g+16 h}{36 (-2+x)^2}+\frac{-35 d-58 e-92 f-136 g-176 h}{432 (-2+x)}+\frac{d+e+f+g+h}{12 (-1+x)^2}+\frac{2 d+5 e+8 f+11 g+14 h}{36 (-1+x)}+\frac{d-e+f-g+h}{36 (1+x)^2}+\frac{2 d+e-4 f+7 g-10 h}{108 (1+x)}+\frac{d-2 e+4 f-8 g+16 h}{144 (2+x)}\right ) \, dx\\ &=\frac{d+e+f+g+h}{12 (1-x)}+\frac{d+2 e+4 f+8 g+16 h}{36 (2-x)}-\frac{d-e+f-g+h}{36 (1+x)}+\frac{1}{36} (2 d+5 e+8 f+11 g+14 h) \log (1-x)-\frac{1}{432} (35 d+58 e+92 f+136 g+176 h) \log (2-x)+\frac{1}{108} (2 d+e-4 f+7 g-10 h) \log (1+x)+\frac{1}{144} (d-2 e+4 f-8 g+16 h) \log (2+x)\\ \end{align*}

Mathematica [A]  time = 0.0998471, size = 169, normalized size = 1.07 \[ \frac{1}{432} \left (\frac{12 \left (d \left (-5 x^2+6 x+5\right )+2 \left (e \left (5-2 x^2\right )+f \left (-4 x^2+3 x+4\right )-5 g x^2+8 g-10 h x^2+3 h x+10 h\right )\right )}{x^3-2 x^2-x+2}+12 \log (1-x) (2 d+5 e+8 f+11 g+14 h)-\log (2-x) (35 d+58 e+92 f+136 g+176 h)+4 \log (x+1) (2 d+e-4 f+7 g-10 h)+3 \log (x+2) (d-2 e+4 f-8 g+16 h)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((12*(d*(5 + 6*x - 5*x^2) + 2*(8*g + 10*h + 3*h*x - 5*g*x^2 - 10*h*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 + 14*h)*Log[1 - x] - (35*d + 58*e + 92*f + 136*g + 176*h)
*Log[2 - x] + 4*(2*d + e - 4*f + 7*g - 10*h)*Log[1 + x] + 3*(d - 2*e + 4*f - 8*g + 16*h)*Log[2 + x])/432

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Maple [A]  time = 0.018, size = 262, normalized size = 1.7 \begin{align*}{\frac{\ln \left ( 2+x \right ) d}{144}}-{\frac{\ln \left ( 2+x \right ) e}{72}}+{\frac{\ln \left ( 1+x \right ) d}{54}}+{\frac{\ln \left ( 1+x \right ) e}{108}}-{\frac{35\,\ln \left ( x-2 \right ) d}{432}}-{\frac{29\,\ln \left ( x-2 \right ) e}{216}}+{\frac{\ln \left ( x-1 \right ) d}{18}}+{\frac{5\,\ln \left ( x-1 \right ) e}{36}}-{\frac{4\,h}{9\,x-18}}-{\frac{h}{12\,x-12}}-{\frac{h}{36+36\,x}}-{\frac{d}{36+36\,x}}+{\frac{e}{36+36\,x}}-{\frac{2\,g}{9\,x-18}}-{\frac{d}{36\,x-72}}-{\frac{e}{18\,x-36}}-{\frac{g}{12\,x-12}}-{\frac{d}{12\,x-12}}-{\frac{e}{12\,x-12}}+{\frac{g}{36+36\,x}}-{\frac{f}{36+36\,x}}-{\frac{f}{9\,x-18}}-{\frac{f}{12\,x-12}}-{\frac{\ln \left ( 2+x \right ) g}{18}}+{\frac{7\,\ln \left ( 1+x \right ) g}{108}}-{\frac{17\,\ln \left ( x-2 \right ) g}{54}}+{\frac{11\,\ln \left ( x-1 \right ) g}{36}}+{\frac{\ln \left ( 2+x \right ) h}{9}}-{\frac{5\,\ln \left ( 1+x \right ) h}{54}}-{\frac{11\,\ln \left ( x-2 \right ) h}{27}}+{\frac{7\,\ln \left ( x-1 \right ) h}{18}}-{\frac{23\,\ln \left ( x-2 \right ) f}{108}}+{\frac{2\,\ln \left ( x-1 \right ) f}{9}}+{\frac{\ln \left ( 2+x \right ) f}{36}}-{\frac{\ln \left ( 1+x \right ) f}{27}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/144*ln(2+x)*d-1/72*ln(2+x)*e+1/54*ln(1+x)*d+1/108*ln(1+x)*e-35/432*ln(x-2)*d-29/216*ln(x-2)*e+1/18*ln(x-1)*d
+5/36*ln(x-1)*e-4/9/(x-2)*h-1/12/(x-1)*h-1/36/(1+x)*h-1/36/(1+x)*d+1/36/(1+x)*e-2/9/(x-2)*g-1/36/(x-2)*d-1/18/
(x-2)*e-1/12/(x-1)*g-1/12/(x-1)*d-1/12/(x-1)*e+1/36/(1+x)*g-1/36/(1+x)*f-1/9/(x-2)*f-1/12/(x-1)*f-1/18*ln(2+x)
*g+7/108*ln(1+x)*g-17/54*ln(x-2)*g+11/36*ln(x-1)*g+1/9*ln(2+x)*h-5/54*ln(1+x)*h-11/27*ln(x-2)*h+7/18*ln(x-1)*h
-23/108*ln(x-2)*f+2/9*ln(x-1)*f+1/36*ln(2+x)*f-1/27*ln(1+x)*f

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Maxima [A]  time = 0.975518, size = 196, normalized size = 1.24 \begin{align*} \frac{1}{144} \,{\left (d - 2 \, e + 4 \, f - 8 \, g + 16 \, h\right )} \log \left (x + 2\right ) + \frac{1}{108} \,{\left (2 \, d + e - 4 \, f + 7 \, g - 10 \, h\right )} \log \left (x + 1\right ) + \frac{1}{36} \,{\left (2 \, d + 5 \, e + 8 \, f + 11 \, g + 14 \, h\right )} \log \left (x - 1\right ) - \frac{1}{432} \,{\left (35 \, d + 58 \, e + 92 \, f + 136 \, g + 176 \, h\right )} \log \left (x - 2\right ) - \frac{{\left (5 \, d + 4 \, e + 8 \, f + 10 \, g + 20 \, h\right )} x^{2} - 6 \,{\left (d + f + h\right )} x - 5 \, d - 10 \, e - 8 \, f - 16 \, g - 20 \, h}{36 \,{\left (x^{3} - 2 \, x^{2} - x + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+x)*(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*(d - 2*e + 4*f - 8*g + 16*h)*log(x + 2) + 1/108*(2*d + e - 4*f + 7*g - 10*h)*log(x + 1) + 1/36*(2*d + 5*
e + 8*f + 11*g + 14*h)*log(x - 1) - 1/432*(35*d + 58*e + 92*f + 136*g + 176*h)*log(x - 2) - 1/36*((5*d + 4*e +
8*f + 10*g + 20*h)*x^2 - 6*(d + f + h)*x - 5*d - 10*e - 8*f - 16*g - 20*h)/(x^3 - 2*x^2 - x + 2)

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Fricas [B]  time = 34.3086, size = 1049, normalized size = 6.64 \begin{align*} -\frac{12 \,{\left (5 \, d + 4 \, e + 8 \, f + 10 \, g + 20 \, h\right )} x^{2} - 72 \,{\left (d + f + h\right )} x - 3 \,{\left ({\left (d - 2 \, e + 4 \, f - 8 \, g + 16 \, h\right )} x^{3} - 2 \,{\left (d - 2 \, e + 4 \, f - 8 \, g + 16 \, h\right )} x^{2} -{\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 ) - 4 \,{\left ({\left (2 \, d + e - 4 \, f + 7 \, g - 10 \, h\right )} x^{3} - 2 \,{\left (2 \, d + e - 4 \, f + 7 \, g - 10 \, h\right )} x^{2} -{\left (2 \, d + e - 4 \, f + 7 \, g - 10 \, h\right )} x + 4 \, d + 2 \, e - 8 \, f + 14 \, g - 20 \, h\right )} \log \left (x + 1\right ) - 12 \,{\left ({\left (2 \, d + 5 \, e + 8 \, f + 11 \, g + 14 \, h\right )} x^{3} - 2 \,{\left (2 \, d + 5 \, e + 8 \, f + 11 \, g + 14 \, h\right )} x^{2} -{\left (2 \, d + 5 \, e + 8 \, f + 11 \, g + 14 \, h\right )} x + 4 \, d + 10 \, e + 16 \, f + 22 \, g + 28 \, h\right )} \log \left (x - 1\right ) +{\left ({\left (35 \, d + 58 \, e + 92 \, f + 136 \, g + 176 \, h\right )} x^{3} - 2 \,{\left (35 \, d + 58 \, e + 92 \, f + 136 \, g + 176 \, h\right )} x^{2} -{\left (35 \, d + 58 \, e + 92 \, f + 136 \, g + 176 \, h\right )} x + 70 \, d + 116 \, e + 184 \, f + 272 \, g + 352 \, h\right )} \log \left (x - 2\right ) - 60 \, d - 120 \, e - 96 \, f - 192 \, g - 240 \, h}{432 \,{\left (x^{3} - 2 \, x^{2} - x + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+x)*(h*x^4+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 + 20*h)*x^2 - 72*(d + f + h)*x - 3*((d - 2*e + 4*f - 8*g + 16*h)*x^3 - 2*(d
- 2*e + 4*f - 8*g + 16*h)*x^2 - (d - 2*e + 4*f - 8*g + 16*h)*x + 2*d - 4*e + 8*f - 16*g + 32*h)*log(x + 2) -
4*((2*d + e - 4*f + 7*g - 10*h)*x^3 - 2*(2*d + e - 4*f + 7*g - 10*h)*x^2 - (2*d + e - 4*f + 7*g - 10*h)*x + 4*
d + 2*e - 8*f + 14*g - 20*h)*log(x + 1) - 12*((2*d + 5*e + 8*f + 11*g + 14*h)*x^3 - 2*(2*d + 5*e + 8*f + 11*g
+ 14*h)*x^2 - (2*d + 5*e + 8*f + 11*g + 14*h)*x + 4*d + 10*e + 16*f + 22*g + 28*h)*log(x - 1) + ((35*d + 58*e
+ 92*f + 136*g + 176*h)*x^3 - 2*(35*d + 58*e + 92*f + 136*g + 176*h)*x^2 - (35*d + 58*e + 92*f + 136*g + 176*h
)*x + 70*d + 116*e + 184*f + 272*g + 352*h)*log(x - 2) - 60*d - 120*e - 96*f - 192*g - 240*h)/(x^3 - 2*x^2 - 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((2+x)*(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 = 1.08478, size = 209, normalized size = 1.32 \begin{align*} \frac{1}{144} \,{\left (d + 4 \, f - 8 \, g + 16 \, h - 2 \, e\right )} \log \left ({\left | x + 2 \right |}\right ) + \frac{1}{108} \,{\left (2 \, d - 4 \, f + 7 \, g - 10 \, h + e\right )} \log \left ({\left | x + 1 \right |}\right ) + \frac{1}{36} \,{\left (2 \, d + 8 \, f + 11 \, g + 14 \, h + 5 \, e\right )} \log \left ({\left | x - 1 \right |}\right ) - \frac{1}{432} \,{\left (35 \, d + 92 \, f + 136 \, g + 176 \, h + 58 \, e\right )} \log \left ({\left | x - 2 \right |}\right ) - \frac{{\left (5 \, d + 8 \, f + 10 \, g + 20 \, h + 4 \, e\right )} x^{2} - 6 \,{\left (d + f + h\right )} x - 5 \, d - 8 \, f - 16 \, g - 20 \, h - 10 \, e}{36 \,{\left (x + 1\right )}{\left (x - 1\right )}{\left (x - 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+x)*(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*(d + 4*f - 8*g + 16*h - 2*e)*log(abs(x + 2)) + 1/108*(2*d - 4*f + 7*g - 10*h + e)*log(abs(x + 1)) + 1/36
*(2*d + 8*f + 11*g + 14*h + 5*e)*log(abs(x - 1)) - 1/432*(35*d + 92*f + 136*g + 176*h + 58*e)*log(abs(x - 2))
- 1/36*((5*d + 8*f + 10*g + 20*h + 4*e)*x^2 - 6*(d + f + h)*x - 5*d - 8*f - 16*g - 20*h - 10*e)/((x + 1)*(x -
1)*(x - 2))