3.2.1 \(\int \frac {(2+x) (d+e x+f x^2+g x^3+h x^4)}{(4-5 x^2+x^4)^2} \, dx\) [101]

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

[Out]

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

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Rubi [A]
time = 0.19, antiderivative size = 158, normalized size of antiderivative = 1.00, 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 = {1600, 6874} \begin {gather*} -\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) \end {gather*}

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 1600

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 6874

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*}

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Mathematica [A]
time = 0.06, size = 169, normalized size = 1.07 \begin {gather*} \frac {1}{432} \left (\frac {12 \left (d \left (5+6 x-5 x^2\right )+2 \left (8 g+10 h+3 h x-5 g x^2-10 h x^2+f \left (4+3 x-4 x^2\right )+e \left (5-2 x^2\right )\right )\right )}{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)\right ) \end {gather*}

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 = 0.06, size = 155, normalized size = 0.98

method	result	size
default	\(\left (\frac {d}{144}-\frac {e}{72}+\frac {f}{36}-\frac {g}{18}+\frac {h}{9}\right ) \ln \left (x +2\right )+\left (-\frac {35 d}{432}-\frac {29 e}{216}-\frac {23 f}{108}-\frac {17 g}{54}-\frac {11 h}{27}\right ) \ln \left (x -2\right )-\frac {\frac {d}{36}+\frac {e}{18}+\frac {f}{9}+\frac {2 g}{9}+\frac {4 h}{9}}{x -2}-\frac {\frac {d}{12}+\frac {e}{12}+\frac {f}{12}+\frac {g}{12}+\frac {h}{12}}{-1+x}+\left (\frac {d}{18}+\frac {5 e}{36}+\frac {2 f}{9}+\frac {11 g}{36}+\frac {7 h}{18}\right ) \ln \left (-1+x \right )-\frac {\frac {d}{36}-\frac {e}{36}+\frac {f}{36}-\frac {g}{36}+\frac {h}{36}}{1+x}+\left (\frac {d}{54}+\frac {e}{108}-\frac {f}{27}+\frac {7 g}{108}-\frac {5 h}{54}\right ) \ln \left (1+x \right )\)	\(155\)
norman	\(\frac {\left (-\frac {5 d}{36}-\frac {e}{9}-\frac {2 f}{9}-\frac {5 g}{18}-\frac {5 h}{9}\right ) x^{3}+\left (\frac {17 d}{36}+\frac {5 e}{18}+\frac {5 f}{9}+\frac {4 g}{9}+\frac {8 h}{9}\right ) x +\left (-\frac {d}{9}-\frac {2 e}{9}-\frac {5 f}{18}-\frac {5 g}{9}-\frac {17 h}{18}\right ) x^{2}+\frac {5 d}{18}+\frac {5 e}{9}+\frac {10 h}{9}+\frac {4 f}{9}+\frac {8 g}{9}}{x^{4}-5 x^{2}+4}+\left (-\frac {35 d}{432}-\frac {29 e}{216}-\frac {23 f}{108}-\frac {17 g}{54}-\frac {11 h}{27}\right ) \ln \left (x -2\right )+\left (\frac {d}{18}+\frac {5 e}{36}+\frac {2 f}{9}+\frac {11 g}{36}+\frac {7 h}{18}\right ) \ln \left (-1+x \right )+\left (\frac {d}{54}+\frac {e}{108}-\frac {f}{27}+\frac {7 g}{108}-\frac {5 h}{54}\right ) \ln \left (1+x \right )+\left (\frac {d}{144}-\frac {e}{72}+\frac {f}{36}-\frac {g}{18}+\frac {h}{9}\right ) \ln \left (x +2\right )\)	\(173\)
risch	\(\frac {\left (-\frac {5 d}{36}-\frac {e}{9}-\frac {2 f}{9}-\frac {5 g}{18}-\frac {5 h}{9}\right ) x^{2}+\left (\frac {h}{6}+\frac {f}{6}+\frac {d}{6}\right ) x +\frac {5 d}{36}+\frac {5 e}{18}+\frac {2 f}{9}+\frac {4 g}{9}+\frac {5 h}{9}}{x^{3}-2 x^{2}-x +2}+\frac {\ln \left (-1-x \right ) d}{54}+\frac {\ln \left (-1-x \right ) e}{108}-\frac {\ln \left (-1-x \right ) f}{27}+\frac {7 \ln \left (-1-x \right ) g}{108}-\frac {5 \ln \left (-1-x \right ) h}{54}+\frac {\ln \left (x +2\right ) d}{144}-\frac {\ln \left (x +2\right ) e}{72}+\frac {\ln \left (x +2\right ) f}{36}-\frac {\ln \left (x +2\right ) g}{18}+\frac {\ln \left (x +2\right ) h}{9}+\frac {\ln \left (-1+x \right ) d}{18}+\frac {5 \ln \left (-1+x \right ) e}{36}+\frac {2 \ln \left (-1+x \right ) f}{9}+\frac {11 \ln \left (-1+x \right ) g}{36}+\frac {7 \ln \left (-1+x \right ) h}{18}-\frac {35 \ln \left (2-x \right ) d}{432}-\frac {29 \ln \left (2-x \right ) e}{216}-\frac {23 \ln \left (2-x \right ) f}{108}-\frac {17 \ln \left (2-x \right ) g}{54}-\frac {11 \ln \left (2-x \right ) h}{27}\)	\(226\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 151, normalized size = 0.96 \begin {gather*} \frac {1}{144} \, {\left (d + 4 \, f - 8 \, g + 16 \, h - 2 \, e\right )} \log \left (x + 2\right ) + \frac {1}{108} \, {\left (2 \, d - 4 \, f + 7 \, g - 10 \, h + e\right )} \log \left (x + 1\right ) + \frac {1}{36} \, {\left (2 \, d + 8 \, f + 11 \, g + 14 \, h + 5 \, e\right )} \log \left (x - 1\right ) - \frac {1}{432} \, {\left (35 \, d + 92 \, f + 136 \, g + 176 \, h + 58 \, e\right )} \log \left (x - 2\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^{3} - 2 \, x^{2} - x + 2\right )}} \end {gather*}

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 376 vs. \(2 (140) = 280\).
time = 3.60, size = 376, normalized size = 2.38 \begin {gather*} -\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 {gather*}

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)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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 = 3.46, size = 155, normalized size = 0.98 \begin {gather*} \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 {gather*}

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))

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Mupad [B]
time = 1.39, size = 152, normalized size = 0.96 \begin {gather*} \ln \left (x-1\right )\,\left (\frac {d}{18}+\frac {5\,e}{36}+\frac {2\,f}{9}+\frac {11\,g}{36}+\frac {7\,h}{18}\right )-\frac {\left (-\frac {5\,d}{36}-\frac {e}{9}-\frac {2\,f}{9}-\frac {5\,g}{18}-\frac {5\,h}{9}\right )\,x^2+\left (\frac {d}{6}+\frac {f}{6}+\frac {h}{6}\right )\,x+\frac {5\,d}{36}+\frac {5\,e}{18}+\frac {2\,f}{9}+\frac {4\,g}{9}+\frac {5\,h}{9}}{-x^3+2\,x^2+x-2}+\ln \left (x+2\right )\,\left (\frac {d}{144}-\frac {e}{72}+\frac {f}{36}-\frac {g}{18}+\frac {h}{9}\right )+\ln \left (x+1\right )\,\left (\frac {d}{54}+\frac {e}{108}-\frac {f}{27}+\frac {7\,g}{108}-\frac {5\,h}{54}\right )-\ln \left (x-2\right )\,\left (\frac {35\,d}{432}+\frac {29\,e}{216}+\frac {23\,f}{108}+\frac {17\,g}{54}+\frac {11\,h}{27}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x - 1)*(d/18 + (5*e)/36 + (2*f)/9 + (11*g)/36 + (7*h)/18) - ((5*d)/36 + (5*e)/18 + (2*f)/9 + (4*g)/9 + (5*
h)/9 - x^2*((5*d)/36 + e/9 + (2*f)/9 + (5*g)/18 + (5*h)/9) + x*(d/6 + f/6 + h/6))/(x + 2*x^2 - x^3 - 2) + log(
x + 2)*(d/144 - e/72 + f/36 - g/18 + h/9) + log(x + 1)*(d/54 + e/108 - f/27 + (7*g)/108 - (5*h)/54) - log(x -
2)*((35*d)/432 + (29*e)/216 + (23*f)/108 + (17*g)/54 + (11*h)/27)

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