3.33.82 $\int \frac{-405-774x+36e^{2}x-432x^2-60x^3+9x^4+2x^5}{450-60x^2+2x^4}dx$

Optimal. Leaf size=33 $$\frac{3-e^2+x+(4+x{)}^2}{2-\frac{9}{3+\frac{1}{4}(-9+x^2)}}$$

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Rubi [A]  time = 0.04, antiderivative size = 34, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 4, integrand size = 46, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.087, Rules used = {6, 28, 1814, 1586} $$\begin{array}{c}\frac{x^2}{2}-\frac{9(9x-e^2+34)}{15-x^2}+\frac{9x}{2}\end{array}$$

Antiderivative was successfully verified.

[In]

Int[(-405 - 774*x + 36*E^2*x - 432*x^2 - 60*x^3 + 9*x^4 + 2*x^5)/(450 - 60*x^2 + 2*x^4),x]

[Out]

(9*x)/2 + x^2/2 - (9*(34 - E^2 + 9*x))/(15 - x^2)

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 1586

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 1814

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x^2, x], f = Coeff[P
olynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[((a
*g - b*f*x)*(a + b*x^2)^(p + 1))/(2*a*b*(p + 1)), x] + Dist[1/(2*a*(p + 1)), Int[(a + b*x^2)^(p + 1)*ExpandToS
um[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{-405+(-774+36e^2)x-432x^2-60x^3+9x^4+2x^5}{450-60x^2+2x^4}dx\\ & =2\int \frac{-405+(-774+36e^2)x-432x^2-60x^3+9x^4+2x^5}{{(-30+2x^2)}^2}dx\\ & =-\frac{9(34-e^2+9x)}{15-x^2}+\frac{1}{30}\int \frac{-4050-900x+270x^2+60x^3}{-30+2x^2}dx\\ & =-\frac{9(34-e^2+9x)}{15-x^2}+\frac{1}{30}\int (135+30x)dx\\ & =\frac{9x}{2}+\frac{x^2}{2}-\frac{9(34-e^2+9x)}{15-x^2}\end{array}\end{array}$$

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Mathematica [A]  time = 0.02, size = 28, normalized size = 0.85 $$\begin{array}{c}\frac{1}{2}(9x+x^2-\frac{18(-34+e^2-9x)}{-15+x^2})\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(-405 - 774*x + 36*E^2*x - 432*x^2 - 60*x^3 + 9*x^4 + 2*x^5)/(450 - 60*x^2 + 2*x^4),x]

[Out]

(9*x + x^2 - (18*(-34 + E^2 - 9*x))/(-15 + x^2))/2

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fricas [A]  time = 0.72, size = 31, normalized size = 0.94 $$\begin{array}{c}\frac{x^4+9x^3-15x^2+27x-18e^2+612}{2(x^2-15)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((36*exp(2)*x+2*x^5+9*x^4-60*x^3-432*x^2-774*x-405)/(2*x^4-60*x^2+450),x, algorithm="fricas")

[Out]

1/2*(x^4 + 9*x^3 - 15*x^2 + 27*x - 18*e^2 + 612)/(x^2 - 15)

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giac [A]  time = 0.38, size = 27, normalized size = 0.82 $$\begin{array}{c}\frac{1}{2}{x}^2+\frac{9}{2}x+\frac{9(9x-e^2+34)}{x^2-15}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((36*exp(2)*x+2*x^5+9*x^4-60*x^3-432*x^2-774*x-405)/(2*x^4-60*x^2+450),x, algorithm="giac")

[Out]

1/2*x^2 + 9/2*x + 9*(9*x - e^2 + 34)/(x^2 - 15)

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maple [A]  time = 0.04, size = 27, normalized size = 0.82

method	result	size
risch	$\frac{x^2}{2}+\frac{9x}{2}+\frac{81x+306-9{\mathrm{e}}^2}{x^2-15}$	$27$
default	$\frac{x^2}{2}+\frac{9x}{2}+\frac{81x+306-9{\mathrm{e}}^2}{x^2-15}$	$28$
norman	$\frac{\frac{27x}{2}+\frac{9x^3}{2}+\frac{x^4}{2}+\frac{387}{2}-9{\mathrm{e}}^2}{x^2-15}$	$28$
gosper	$-\frac{-x^4-9x^3+18{\mathrm{e}}^2-27x-387}{2(x^2-15)}$	$29$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((36*exp(2)*x+2*x^5+9*x^4-60*x^3-432*x^2-774*x-405)/(2*x^4-60*x^2+450),x,method=_RETURNVERBOSE)

[Out]

1/2*x^2+9/2*x+(81*x+306-9*exp(2))/(x^2-15)

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maxima [A]  time = 0.41, size = 27, normalized size = 0.82 $$\begin{array}{c}\frac{1}{2}{x}^2+\frac{9}{2}x+\frac{9(9x-e^2+34)}{x^2-15}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((36*exp(2)*x+2*x^5+9*x^4-60*x^3-432*x^2-774*x-405)/(2*x^4-60*x^2+450),x, algorithm="maxima")

[Out]

1/2*x^2 + 9/2*x + 9*(9*x - e^2 + 34)/(x^2 - 15)

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mupad [B]  time = 0.08, size = 26, normalized size = 0.79 $$\begin{array}{c}\frac{9x}{2}+\frac{x^2}{2}+\frac{81x-9{\mathrm{e}}^2+306}{x^2-15}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(774*x - 36*x*exp(2) + 432*x^2 + 60*x^3 - 9*x^4 - 2*x^5 + 405)/(2*x^4 - 60*x^2 + 450),x)

[Out]

(9*x)/2 + x^2/2 + (81*x - 9*exp(2) + 306)/(x^2 - 15)

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sympy [A]  time = 0.31, size = 24, normalized size = 0.73 $$\begin{array}{c}\frac{x^2}{2}+\frac{9x}{2}+\frac{81x-9e^2+306}{x^2-15}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((36*exp(2)*x+2*x**5+9*x**4-60*x**3-432*x**2-774*x-405)/(2*x**4-60*x**2+450),x)

[Out]

x**2/2 + 9*x/2 + (81*x - 9*exp(2) + 306)/(x**2 - 15)

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