3.71.100 \(\int \frac {e^4 (72-72 x)}{169-312 x+300 x^2-144 x^3+36 x^4} \, dx\)

Optimal. Leaf size=16 \[ \frac {e^4}{\frac {13}{6}-2 x+x^2} \]

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Rubi [A]  time = 0.02, antiderivative size = 18, normalized size of antiderivative = 1.12, number of steps used = 4, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {12, 1680, 261} \begin {gather*} \frac {6 e^4}{6 (1-x)^2+7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^4*(72 - 72*x))/(169 - 312*x + 300*x^2 - 144*x^3 + 36*x^4),x]

[Out]

(6*E^4)/(7 + 6*(1 - x)^2)

Rule 12

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

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 1680

Int[(Pq_)*(Q4_)^(p_), x_Symbol] :> With[{a = Coeff[Q4, x, 0], b = Coeff[Q4, x, 1], c = Coeff[Q4, x, 2], d = Co
eff[Q4, x, 3], e = Coeff[Q4, x, 4]}, Subst[Int[SimplifyIntegrand[(Pq /. x -> -(d/(4*e)) + x)*(a + d^4/(256*e^3
) - (b*d)/(8*e) + (c - (3*d^2)/(8*e))*x^2 + e*x^4)^p, x], x], x, d/(4*e) + x] /; EqQ[d^3 - 4*c*d*e + 8*b*e^2,
0] && NeQ[d, 0]] /; FreeQ[p, x] && PolyQ[Pq, x] && PolyQ[Q4, x, 4] &&  !IGtQ[p, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=e^4 \int \frac {72-72 x}{169-312 x+300 x^2-144 x^3+36 x^4} \, dx\\ &=e^4 \operatorname {Subst}\left (\int -\frac {72 x}{\left (7+6 x^2\right )^2} \, dx,x,-1+x\right )\\ &=-\left (\left (72 e^4\right ) \operatorname {Subst}\left (\int \frac {x}{\left (7+6 x^2\right )^2} \, dx,x,-1+x\right )\right )\\ &=\frac {6 e^4}{7+6 (1-x)^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 17, normalized size = 1.06 \begin {gather*} \frac {6 e^4}{13-12 x+6 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^4*(72 - 72*x))/(169 - 312*x + 300*x^2 - 144*x^3 + 36*x^4),x]

[Out]

(6*E^4)/(13 - 12*x + 6*x^2)

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fricas [A]  time = 0.80, size = 16, normalized size = 1.00 \begin {gather*} \frac {6 \, e^{4}}{6 \, x^{2} - 12 \, x + 13} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-72*x+72)*exp(1)^4/(36*x^4-144*x^3+300*x^2-312*x+169),x, algorithm="fricas")

[Out]

6*e^4/(6*x^2 - 12*x + 13)

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giac [A]  time = 0.22, size = 16, normalized size = 1.00 \begin {gather*} \frac {6 \, e^{4}}{6 \, x^{2} - 12 \, x + 13} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-72*x+72)*exp(1)^4/(36*x^4-144*x^3+300*x^2-312*x+169),x, algorithm="giac")

[Out]

6*e^4/(6*x^2 - 12*x + 13)

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maple [A]  time = 0.02, size = 14, normalized size = 0.88

method	result	size
risch	\(\frac {{\mathrm e}^{4}}{\frac {13}{6}+x^{2}-2 x}\)	\(14\)
gosper	\(\frac {6 \,{\mathrm e}^{4}}{6 x^{2}-12 x +13}\)	\(19\)
default	\(\frac {6 \,{\mathrm e}^{4}}{6 x^{2}-12 x +13}\)	\(19\)
norman	\(\frac {6 \,{\mathrm e}^{4}}{6 x^{2}-12 x +13}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-72*x+72)*exp(1)^4/(36*x^4-144*x^3+300*x^2-312*x+169),x,method=_RETURNVERBOSE)

[Out]

exp(4)/(13/6+x^2-2*x)

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maxima [A]  time = 0.38, size = 16, normalized size = 1.00 \begin {gather*} \frac {6 \, e^{4}}{6 \, x^{2} - 12 \, x + 13} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-72*x+72)*exp(1)^4/(36*x^4-144*x^3+300*x^2-312*x+169),x, algorithm="maxima")

[Out]

6*e^4/(6*x^2 - 12*x + 13)

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mupad [B]  time = 4.15, size = 16, normalized size = 1.00 \begin {gather*} \frac {6\,{\mathrm {e}}^4}{6\,x^2-12\,x+13} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(4)*(72*x - 72))/(300*x^2 - 312*x - 144*x^3 + 36*x^4 + 169),x)

[Out]

(6*exp(4))/(6*x^2 - 12*x + 13)

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sympy [A]  time = 0.19, size = 14, normalized size = 0.88 \begin {gather*} \frac {6 e^{4}}{6 x^{2} - 12 x + 13} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-72*x+72)*exp(1)**4/(36*x**4-144*x**3+300*x**2-312*x+169),x)

[Out]

6*exp(4)/(6*x**2 - 12*x + 13)

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