3.33.16 $\int \frac{e^5(-4x+2x^2)}{45-90x-135x^2+180x^3+180x^4}dx$

Optimal. Leaf size=26 $$\frac{e^{5}x}{3(5-\frac{5}{2x})(3+3x)}$$

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Rubi [A]  time = 0.07, antiderivative size = 25, normalized size of antiderivative = 0.96, number of steps used = 5, number of rules used = 4, integrand size = 35, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.114, Rules used = {12, 1593, 1680, 776} $$\begin{array}{c}-\frac{8e^5(1-x)}{45(9-(4x+1{)}^2)}\end{array}$$

Antiderivative was successfully verified.

[In]

Int[(E^5*(-4*x + 2*x^2))/(45 - 90*x - 135*x^2 + 180*x^3 + 180*x^4),x]

[Out]

(-8*E^5*(1 - x))/(45*(9 - (1 + 4*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 776

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(((e*f + d*g)*(2*p
+ 3) + 2*e*g*(p + 1)*x)*(a + c*x^2)^(p + 1))/(2*c*(p + 1)*(2*p + 3)), x] /; FreeQ[{a, c, d, e, f, g, p}, x] &
& EqQ[a*e*g - c*d*f*(2*p + 3), 0] && NeQ[p, -1]

Rule 1593

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

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

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =e^5\int \frac{-4x+2x^2}{45-90x-135x^2+180x^3+180x^4}dx\\ & =e^5\int \frac{x(-4+2x)}{45-90x-135x^2+180x^3+180x^4}dx\\ & =e^5\mathrm{Subst}(\int \frac{8(1-4x)(9-4x)}{45{(9-16x^2)}^2}dx,x,\frac{1}{4}+x)\\ & =\frac{1}{45}\left(8e^5\right)\mathrm{Subst}(\int \frac{(1-4x)(9-4x)}{{(9-16x^2)}^2}dx,x,\frac{1}{4}+x)\\ & =-\frac{8e^5(1-x)}{45(9-(1+4x{)}^2)}\end{array}\end{array}$$

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Mathematica [A]  time = 0.01, size = 22, normalized size = 0.85 $$\begin{array}{c}\frac{e^5(1-x)}{45(-1+x+2x^2)}\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(E^5*(-4*x + 2*x^2))/(45 - 90*x - 135*x^2 + 180*x^3 + 180*x^4),x]

[Out]

(E^5*(1 - x))/(45*(-1 + x + 2*x^2))

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fricas [A]  time = 0.61, size = 17, normalized size = 0.65 $$\begin{array}{c}-\frac{(x-1)e^5}{45(2x^2+x-1)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2-4*x)*exp(5)/(180*x^4+180*x^3-135*x^2-90*x+45),x, algorithm="fricas")

[Out]

-1/45*(x - 1)*e^5/(2*x^2 + x - 1)

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giac [A]  time = 0.34, size = 17, normalized size = 0.65 $$\begin{array}{c}-\frac{(x-1)e^5}{45(2x^2+x-1)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2-4*x)*exp(5)/(180*x^4+180*x^3-135*x^2-90*x+45),x, algorithm="giac")

[Out]

-1/45*(x - 1)*e^5/(2*x^2 + x - 1)

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maple [A]  time = 0.03, size = 18, normalized size = 0.69

method	result	size
gosper	$-\frac{(x-1){\mathrm{e}}^5}{45(2x^2+x-1)}$	$18$
norman	$\frac{2{\mathrm{e}}^5{x}^2}{45(2x^2+x-1)}$	$18$
risch	$\frac{{\mathrm{e}}^5(-\frac{x}{90}+\frac{1}{90})}{x^2+\frac{1}{2}x-\frac{1}{2}}$	$19$
default	$\frac{2{\mathrm{e}}^5(\frac{1}{12x-6}-\frac{1}{3(x+1)})}{45}$	$22$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^2-4*x)*exp(5)/(180*x^4+180*x^3-135*x^2-90*x+45),x,method=_RETURNVERBOSE)

[Out]

-1/45*(x-1)*exp(5)/(2*x^2+x-1)

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maxima [A]  time = 0.44, size = 17, normalized size = 0.65 $$\begin{array}{c}-\frac{(x-1)e^5}{45(2x^2+x-1)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2-4*x)*exp(5)/(180*x^4+180*x^3-135*x^2-90*x+45),x, algorithm="maxima")

[Out]

-1/45*(x - 1)*e^5/(2*x^2 + x - 1)

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mupad [B]  time = 0.07, size = 22, normalized size = 0.85 $$\begin{array}{c}\frac{{\mathrm{e}}^5}{135(2x-1)}-\frac{2{\mathrm{e}}^5}{135(x+1)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(5)*(4*x - 2*x^2))/(180*x^3 - 135*x^2 - 90*x + 180*x^4 + 45),x)

[Out]

exp(5)/(135*(2*x - 1)) - (2*exp(5))/(135*(x + 1))

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sympy [A]  time = 0.18, size = 17, normalized size = 0.65 $$\begin{array}{c}\frac{-x{e}^5+e^5}{90x^2+45x-45}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x**2-4*x)*exp(5)/(180*x**4+180*x**3-135*x**2-90*x+45),x)

[Out]

(-x*exp(5) + exp(5))/(90*x**2 + 45*x - 45)

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