3.89.62 \(\int \frac {192+88 x+9 x^2+e (-96-48 x-6 x^2)}{128 x^4+64 x^5+8 x^6} \, dx\)

Optimal. Leaf size=22 \[ 1+\frac {-2+e+\frac {x}{2 (4+x)}}{4 x^3} \]

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Rubi [A]  time = 0.07, antiderivative size = 36, normalized size of antiderivative = 1.64, number of steps used = 5, number of rules used = 4, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.098, Rules used = {1594, 27, 12, 1820} \begin {gather*} -\frac {2-e}{4 x^3}+\frac {1}{32 x^2}+\frac {1}{128 (x+4)}-\frac {1}{128 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(192 + 88*x + 9*x^2 + E*(-96 - 48*x - 6*x^2))/(128*x^4 + 64*x^5 + 8*x^6),x]

[Out]

-1/4*(2 - E)/x^3 + 1/(32*x^2) - 1/(128*x) + 1/(128*(4 + x))

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 1820

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a +
b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {192+88 x+9 x^2+e \left (-96-48 x-6 x^2\right )}{x^4 \left (128+64 x+8 x^2\right )} \, dx\\ &=\int \frac {192+88 x+9 x^2+e \left (-96-48 x-6 x^2\right )}{8 x^4 (4+x)^2} \, dx\\ &=\frac {1}{8} \int \frac {192+88 x+9 x^2+e \left (-96-48 x-6 x^2\right )}{x^4 (4+x)^2} \, dx\\ &=\frac {1}{8} \int \left (-\frac {6 (-2+e)}{x^4}-\frac {1}{2 x^3}+\frac {1}{16 x^2}-\frac {1}{16 (4+x)^2}\right ) \, dx\\ &=-\frac {2-e}{4 x^3}+\frac {1}{32 x^2}-\frac {1}{128 x}+\frac {1}{128 (4+x)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 23, normalized size = 1.05 \begin {gather*} \frac {-16-3 x+2 e (4+x)}{8 x^3 (4+x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(192 + 88*x + 9*x^2 + E*(-96 - 48*x - 6*x^2))/(128*x^4 + 64*x^5 + 8*x^6),x]

[Out]

(-16 - 3*x + 2*E*(4 + x))/(8*x^3*(4 + x))

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fricas [A]  time = 0.47, size = 25, normalized size = 1.14 \begin {gather*} \frac {2 \, {\left (x + 4\right )} e - 3 \, x - 16}{8 \, {\left (x^{4} + 4 \, x^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x^2-48*x-96)*exp(1)+9*x^2+88*x+192)/(8*x^6+64*x^5+128*x^4),x, algorithm="fricas")

[Out]

1/8*(2*(x + 4)*e - 3*x - 16)/(x^4 + 4*x^3)

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giac [A]  time = 0.16, size = 25, normalized size = 1.14 \begin {gather*} \frac {1}{128 \, {\left (x + 4\right )}} - \frac {x^{2} - 4 \, x - 32 \, e + 64}{128 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x^2-48*x-96)*exp(1)+9*x^2+88*x+192)/(8*x^6+64*x^5+128*x^4),x, algorithm="giac")

[Out]

1/128/(x + 4) - 1/128*(x^2 - 4*x - 32*e + 64)/x^3

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maple [A]  time = 0.05, size = 22, normalized size = 1.00

method	result	size
norman	\(\frac {\left (\frac {{\mathrm e}}{4}-\frac {3}{8}\right ) x -2+{\mathrm e}}{x^{3} \left (4+x \right )}\)	\(22\)
risch	\(\frac {\left (\frac {{\mathrm e}}{4}-\frac {3}{8}\right ) x -2+{\mathrm e}}{x^{3} \left (4+x \right )}\)	\(22\)
gosper	\(\frac {2 x \,{\mathrm e}+8 \,{\mathrm e}-3 x -16}{8 x^{3} \left (4+x \right )}\)	\(25\)
default	\(-\frac {-6 \,{\mathrm e}+12}{24 x^{3}}+\frac {1}{32 x^{2}}-\frac {1}{128 x}+\frac {1}{512+128 x}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-6*x^2-48*x-96)*exp(1)+9*x^2+88*x+192)/(8*x^6+64*x^5+128*x^4),x,method=_RETURNVERBOSE)

[Out]

((1/4*exp(1)-3/8)*x-2+exp(1))/x^3/(4+x)

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maxima [A]  time = 0.37, size = 27, normalized size = 1.23 \begin {gather*} \frac {x {\left (2 \, e - 3\right )} + 8 \, e - 16}{8 \, {\left (x^{4} + 4 \, x^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x^2-48*x-96)*exp(1)+9*x^2+88*x+192)/(8*x^6+64*x^5+128*x^4),x, algorithm="maxima")

[Out]

1/8*(x*(2*e - 3) + 8*e - 16)/(x^4 + 4*x^3)

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mupad [B]  time = 0.10, size = 24, normalized size = 1.09 \begin {gather*} \frac {\mathrm {e}+x\,\left (\frac {\mathrm {e}}{4}-\frac {3}{8}\right )-2}{x^4+4\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((88*x - exp(1)*(48*x + 6*x^2 + 96) + 9*x^2 + 192)/(128*x^4 + 64*x^5 + 8*x^6),x)

[Out]

(exp(1) + x*(exp(1)/4 - 3/8) - 2)/(4*x^3 + x^4)

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sympy [A]  time = 0.44, size = 24, normalized size = 1.09 \begin {gather*} \frac {x \left (-3 + 2 e\right ) - 16 + 8 e}{8 x^{4} + 32 x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x**2-48*x-96)*exp(1)+9*x**2+88*x+192)/(8*x**6+64*x**5+128*x**4),x)

[Out]

(x*(-3 + 2*E) - 16 + 8*E)/(8*x**4 + 32*x**3)

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