3.90.49 \(\int \frac {-192+96 x-60 x^2-104 x^3+13 x^4-32 x^5+4 x^6+\frac {1}{4} e (4+x^2) (-128 x^3+64 x^4-8 x^5)}{256 x^2-128 x^3+80 x^4-32 x^5+4 x^6} \, dx\)

Optimal. Leaf size=31 \[ e+\frac {3}{4 x}-\frac {x^2}{4-x}-\frac {1}{4} e \left (4+x^2\right ) \]

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Rubi [A]  time = 0.21, antiderivative size = 26, normalized size of antiderivative = 0.84, number of steps used = 6, number of rules used = 5, integrand size = 85, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {2055, 1594, 27, 12, 1620} \begin {gather*} -\frac {e x^2}{4}+x-\frac {16}{4-x}+\frac {3}{4 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-192 + 96*x - 60*x^2 - 104*x^3 + 13*x^4 - 32*x^5 + 4*x^6 + (E*(4 + x^2)*(-128*x^3 + 64*x^4 - 8*x^5))/4)/(
256*x^2 - 128*x^3 + 80*x^4 - 32*x^5 + 4*x^6),x]

[Out]

-16/(4 - x) + 3/(4*x) + x - (E*x^2)/4

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 1594

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^
(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] && PosQ[r - p]

Rule 1620

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

Rule 2055

Int[(u_.)*(P_)*(Q_)^(q_), x_Symbol] :> Module[{gcd = PolyGCD[P, Q, x]}, Int[u*gcd^(q + 1)*PolynomialQuotient[P
, gcd, x]*PolynomialQuotient[Q, gcd, x]^q, x] /; NeQ[gcd, 1]] /; ILtQ[q, 0] && PolyQ[P, x] && PolyQ[Q, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-48+24 x-3 x^2+(-32-32 e) x^3+(4+16 e) x^4-2 e x^5}{64 x^2-32 x^3+4 x^4} \, dx\\ &=\int \frac {-48+24 x-3 x^2+(-32-32 e) x^3+(4+16 e) x^4-2 e x^5}{x^2 \left (64-32 x+4 x^2\right )} \, dx\\ &=\int \frac {-48+24 x-3 x^2+(-32-32 e) x^3+(4+16 e) x^4-2 e x^5}{4 (-4+x)^2 x^2} \, dx\\ &=\frac {1}{4} \int \frac {-48+24 x-3 x^2+(-32-32 e) x^3+(4+16 e) x^4-2 e x^5}{(-4+x)^2 x^2} \, dx\\ &=\frac {1}{4} \int \left (4-\frac {64}{(-4+x)^2}-\frac {3}{x^2}-2 e x\right ) \, dx\\ &=-\frac {16}{4-x}+\frac {3}{4 x}+x-\frac {e x^2}{4}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 26, normalized size = 0.84 \begin {gather*} \frac {1}{4} \left (\frac {64}{-4+x}+\frac {3}{x}+4 x-e x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-192 + 96*x - 60*x^2 - 104*x^3 + 13*x^4 - 32*x^5 + 4*x^6 + (E*(4 + x^2)*(-128*x^3 + 64*x^4 - 8*x^5)
)/4)/(256*x^2 - 128*x^3 + 80*x^4 - 32*x^5 + 4*x^6),x]

[Out]

(64/(-4 + x) + 3/x + 4*x - E*x^2)/4

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fricas [A]  time = 0.52, size = 39, normalized size = 1.26 \begin {gather*} \frac {4 \, x^{3} - 16 \, x^{2} - {\left (x^{4} - 4 \, x^{3}\right )} e + 67 \, x - 12}{4 \, {\left (x^{2} - 4 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x^5+64*x^4-128*x^3)*exp(log(1/4*x^2+1)+1)+4*x^6-32*x^5+13*x^4-104*x^3-60*x^2+96*x-192)/(4*x^6-3
2*x^5+80*x^4-128*x^3+256*x^2),x, algorithm="fricas")

[Out]

1/4*(4*x^3 - 16*x^2 - (x^4 - 4*x^3)*e + 67*x - 12)/(x^2 - 4*x)

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giac [A]  time = 0.23, size = 25, normalized size = 0.81 \begin {gather*} -\frac {1}{4} \, x^{2} e + x + \frac {67 \, x - 12}{4 \, {\left (x^{2} - 4 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x^5+64*x^4-128*x^3)*exp(log(1/4*x^2+1)+1)+4*x^6-32*x^5+13*x^4-104*x^3-60*x^2+96*x-192)/(4*x^6-3
2*x^5+80*x^4-128*x^3+256*x^2),x, algorithm="giac")

[Out]

-1/4*x^2*e + x + 1/4*(67*x - 12)/(x^2 - 4*x)

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maple [A]  time = 0.06, size = 22, normalized size = 0.71

method	result	size
default	\(x +\frac {3}{4 x}+\frac {16}{x -4}-\frac {x^{2} {\mathrm e}}{4}\)	\(22\)
risch	\(-\frac {x^{2} {\mathrm e}}{4}+x +\frac {\frac {67 x}{4}-3}{\left (x -4\right ) x}\)	\(24\)
norman	\(\frac {-3+\left (1+{\mathrm e}\right ) x^{3}+\frac {3 x}{4}-\frac {x^{4} {\mathrm e}}{4}}{\left (x -4\right ) x}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-8*x^5+64*x^4-128*x^3)*exp(ln(1/4*x^2+1)+1)+4*x^6-32*x^5+13*x^4-104*x^3-60*x^2+96*x-192)/(4*x^6-32*x^5+8
0*x^4-128*x^3+256*x^2),x,method=_RETURNVERBOSE)

[Out]

x+3/4/x+16/(x-4)-1/4*x^2*exp(1)

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maxima [A]  time = 0.34, size = 25, normalized size = 0.81 \begin {gather*} -\frac {1}{4} \, x^{2} e + x + \frac {67 \, x - 12}{4 \, {\left (x^{2} - 4 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x^5+64*x^4-128*x^3)*exp(log(1/4*x^2+1)+1)+4*x^6-32*x^5+13*x^4-104*x^3-60*x^2+96*x-192)/(4*x^6-3
2*x^5+80*x^4-128*x^3+256*x^2),x, algorithm="maxima")

[Out]

-1/4*x^2*e + x + 1/4*(67*x - 12)/(x^2 - 4*x)

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mupad [B]  time = 0.15, size = 23, normalized size = 0.74 \begin {gather*} x-\frac {x^2\,\mathrm {e}}{4}+\frac {\frac {67\,x}{4}-3}{x\,\left (x-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(log(x^2/4 + 1) + 1)*(128*x^3 - 64*x^4 + 8*x^5) - 96*x + 60*x^2 + 104*x^3 - 13*x^4 + 32*x^5 - 4*x^6 +
192)/(256*x^2 - 128*x^3 + 80*x^4 - 32*x^5 + 4*x^6),x)

[Out]

x - (x^2*exp(1))/4 + ((67*x)/4 - 3)/(x*(x - 4))

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sympy [A]  time = 0.16, size = 22, normalized size = 0.71 \begin {gather*} - \frac {e x^{2}}{4} + x - \frac {12 - 67 x}{4 x^{2} - 16 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x**5+64*x**4-128*x**3)*exp(ln(1/4*x**2+1)+1)+4*x**6-32*x**5+13*x**4-104*x**3-60*x**2+96*x-192)/
(4*x**6-32*x**5+80*x**4-128*x**3+256*x**2),x)

[Out]

-E*x**2/4 + x - (12 - 67*x)/(4*x**2 - 16*x)

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