3.85.77 $\int \frac{-400+16x^2+e^{25}(100-4x^2)}{243(15625+17500x+6150x^2+700x^3+25x^4)}dx$

Optimal. Leaf size=23 $$\frac{(-4+e^{25})x}{6075(x+\frac{1}{4}(5+x{)}^2)}$$

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Rubi [A]  time = 0.05, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 44, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.091, Rules used = {12, 1680, 1814, 8} $$\begin{array}{c}\frac{4(4-e^{25})x}{6075(24-(x+7{)}^2)}\end{array}$$

Antiderivative was successfully verified.

[In]

Int[(-400 + 16*x^2 + E^25*(100 - 4*x^2))/(243*(15625 + 17500*x + 6150*x^2 + 700*x^3 + 25*x^4)),x]

[Out]

(4*(4 - E^25)*x)/(6075*(24 - (7 + x)^2))

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, 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 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]

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

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\frac{1}{243}\int \frac{-400+16x^2+e^{25}(100-4x^2)}{15625+17500x+6150x^2+700x^3+25x^4}dx\\ & =\frac{1}{243}\mathrm{Subst}(\int \frac{4(4-e^{25})(24-14x+x^2)}{25{(24-x^2)}^2}dx,x,7+x)\\ & =\frac{\left(4(4-e^{25})\right)\mathrm{Subst}(\int \frac{24-14x+x^2}{{(24-x^2)}^2}dx,x,7+x)}{6075}\\ & =\frac{4(4-e^{25})x}{6075(24-(7+x{)}^2)}+\frac{(-4+e^{25})\mathrm{Subst}(\int 0dx,x,7+x)}{72900}\\ & =\frac{4(4-e^{25})x}{6075(24-(7+x{)}^2)}\end{array}\end{array}$$

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Mathematica [A]  time = 0.01, size = 20, normalized size = 0.87 $$\begin{array}{c}\frac{4(-4+e^{25})x}{6075(25+14x+x^2)}\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(-400 + 16*x^2 + E^25*(100 - 4*x^2))/(243*(15625 + 17500*x + 6150*x^2 + 700*x^3 + 25*x^4)),x]

[Out]

(4*(-4 + E^25)*x)/(6075*(25 + 14*x + x^2))

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fricas [A]  time = 0.76, size = 20, normalized size = 0.87 $$\begin{array}{c}\frac{4(x{e}^{25}-4x)}{6075(x^2+14x+25)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/243*((-4*x^2+100)*exp(25)+16*x^2-400)/(25*x^4+700*x^3+6150*x^2+17500*x+15625),x, algorithm="fricas
")

[Out]

4/6075*(x*e^25 - 4*x)/(x^2 + 14*x + 25)

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giac [A]  time = 0.20, size = 20, normalized size = 0.87 $$\begin{array}{c}\frac{4(x{e}^{25}-4x)}{6075(x^2+14x+25)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/243*((-4*x^2+100)*exp(25)+16*x^2-400)/(25*x^4+700*x^3+6150*x^2+17500*x+15625),x, algorithm="giac")

[Out]

4/6075*(x*e^25 - 4*x)/(x^2 + 14*x + 25)

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maple [A]  time = 0.04, size = 18, normalized size = 0.78

method	result	size
gosper	$\frac{4x({\mathrm{e}}^{25}-4)}{6075(x^2+14x+25)}$	$18$
norman	$\frac{(-\frac{16}{6075}+\frac{4{\mathrm{e}}^{25}}{6075})x}{x^2+14x+25}$	$19$
default	$\frac{(4{\mathrm{e}}^{25}-16)x}{6075x^2+85050x+151875}$	$20$
risch	$\frac{x(\frac{4{\mathrm{e}}^{25}}{25}-\frac{16}{25})}{243x^2+3402x+6075}$	$20$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/243*((-4*x^2+100)*exp(25)+16*x^2-400)/(25*x^4+700*x^3+6150*x^2+17500*x+15625),x,method=_RETURNVERBOSE)

[Out]

4/6075*x*(exp(25)-4)/(x^2+14*x+25)

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maxima [A]  time = 0.36, size = 17, normalized size = 0.74 $$\begin{array}{c}\frac{4x(e^{25}-4)}{6075(x^2+14x+25)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/243*((-4*x^2+100)*exp(25)+16*x^2-400)/(25*x^4+700*x^3+6150*x^2+17500*x+15625),x, algorithm="maxima
")

[Out]

4/6075*x*(e^25 - 4)/(x^2 + 14*x + 25)

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mupad [B]  time = 0.08, size = 19, normalized size = 0.83 $$\begin{array}{c}\frac{4x({\mathrm{e}}^{25}-4)}{6075(x^2+14x+25)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((exp(25)*(4*x^2 - 100))/243 - (16*x^2)/243 + 400/243)/(17500*x + 6150*x^2 + 700*x^3 + 25*x^4 + 15625),x)

[Out]

(4*x*(exp(25) - 4))/(6075*(14*x + x^2 + 25))

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sympy [A]  time = 0.32, size = 17, normalized size = 0.74 $$\begin{array}{c}\frac{x(-16+4e^{25})}{6075x^2+85050x+151875}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/243*((-4*x**2+100)*exp(25)+16*x**2-400)/(25*x**4+700*x**3+6150*x**2+17500*x+15625),x)

[Out]

x*(-16 + 4*exp(25))/(6075*x**2 + 85050*x + 151875)

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