3.41.32 $\int \frac{1600x^4+1250x^5+300x^6+25x^7+e^{\frac{6}{25x^3}}(-1152-864x-216x^2-18x^3)}{1600x^4+1200x^5+300x^6+25x^7}dx$

Optimal. Leaf size=24 $$1+e^{\frac{6}{25x^3}}+x+\frac{x^2}{4(4+x{)}^2}$$

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Rubi [A]  time = 0.16, antiderivative size = 25, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 3, integrand size = 70, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.043, Rules used = {6688, 2209, 1850} $$\begin{array}{c}{e}^{\frac{6}{25x^3}}+x-\frac{2}{x+4}+\frac{4}{(x+4{)}^2}\end{array}$$

Antiderivative was successfully verified.

[In]

Int[(1600*x^4 + 1250*x^5 + 300*x^6 + 25*x^7 + E^(6/(25*x^3))*(-1152 - 864*x - 216*x^2 - 18*x^3))/(1600*x^4 + 1
200*x^5 + 300*x^6 + 25*x^7),x]

[Out]

E^(6/(25*x^3)) + x + 4/(4 + x)^2 - 2/(4 + x)

Rule 1850

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

Rule 2209

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^n*
F^(a + b*(c + d*x)^n))/(b*f*n*(c + d*x)^n*Log[F]), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int (-\frac{18e^{\frac{6}{25x^3}}}{25x^4}+\frac{64+50x+12x^2+x^3}{(4+x{)}^3})dx\\ & =-(\frac{18}{25}\int \frac{e^{\frac{6}{25x^3}}}{x^4}dx)+\int \frac{64+50x+12x^2+x^3}{(4+x{)}^3}dx\\ & =e^{\frac{6}{25x^3}}+\int (1-\frac{8}{(4+x{)}^3}+\frac{2}{(4+x{)}^2})dx\\ & =e^{\frac{6}{25x^3}}+x+\frac{4}{(4+x{)}^2}-\frac{2}{4+x}\end{array}\end{array}$$

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Mathematica [A]  time = 0.03, size = 25, normalized size = 1.04 $$\begin{array}{c}{e}^{\frac{6}{25x^3}}+x+\frac{4}{(4+x{)}^2}-\frac{2}{4+x}\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(1600*x^4 + 1250*x^5 + 300*x^6 + 25*x^7 + E^(6/(25*x^3))*(-1152 - 864*x - 216*x^2 - 18*x^3))/(1600*x
^4 + 1200*x^5 + 300*x^6 + 25*x^7),x]

[Out]

E^(6/(25*x^3)) + x + 4/(4 + x)^2 - 2/(4 + x)

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fricas [B]  time = 0.67, size = 39, normalized size = 1.62 $$\begin{array}{c}\frac{x^3+8x^2+(x^2+8x+16)e^{\left(\frac{6}{25x^3}\right)}+14x-4}{x^2+8x+16}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-18*x^3-216*x^2-864*x-1152)*exp(6/25/x^3)+25*x^7+300*x^6+1250*x^5+1600*x^4)/(25*x^7+300*x^6+1200*x
^5+1600*x^4),x, algorithm="fricas")

[Out]

(x^3 + 8*x^2 + (x^2 + 8*x + 16)*e^(6/25/x^3) + 14*x - 4)/(x^2 + 8*x + 16)

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giac [A]  time = 0.15, size = 23, normalized size = 0.96 $$\begin{array}{c}x-\frac{2(x+2)}{x^2+8x+16}+e^{\left(\frac{6}{25x^3}\right)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-18*x^3-216*x^2-864*x-1152)*exp(6/25/x^3)+25*x^7+300*x^6+1250*x^5+1600*x^4)/(25*x^7+300*x^6+1200*x
^5+1600*x^4),x, algorithm="giac")

[Out]

x - 2*(x + 2)/(x^2 + 8*x + 16) + e^(6/25/x^3)

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maple [A]  time = 0.29, size = 25, normalized size = 1.04

method	result	size
risch	$x+\frac{-2x-4}{x^2+8x+16}+{\mathrm{e}}^{\frac{6}{25x^3}}$	$25$
norman	$\frac{x^6-132x^3-50x^4+{\mathrm{e}}^{\frac{6}{25x^3}}{x}^5+16{\mathrm{e}}^{\frac{6}{25x^3}}{x}^3+8{\mathrm{e}}^{\frac{6}{25x^3}}{x}^4}{x^3{(4+x)}^2}$	$56$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-18*x^3-216*x^2-864*x-1152)*exp(6/25/x^3)+25*x^7+300*x^6+1250*x^5+1600*x^4)/(25*x^7+300*x^6+1200*x^5+160
0*x^4),x,method=_RETURNVERBOSE)

[Out]

x+(-2*x-4)/(x^2+8*x+16)+exp(6/25/x^3)

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maxima [B]  time = 0.40, size = 67, normalized size = 2.79 $$\begin{array}{c}x-\frac{16(3x+10)}{x^2+8x+16}+\frac{96(x+3)}{x^2+8x+16}-\frac{50(x+2)}{x^2+8x+16}-\frac{32}{x^2+8x+16}+e^{\left(\frac{6}{25x^3}\right)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-18*x^3-216*x^2-864*x-1152)*exp(6/25/x^3)+25*x^7+300*x^6+1250*x^5+1600*x^4)/(25*x^7+300*x^6+1200*x
^5+1600*x^4),x, algorithm="maxima")

[Out]

x - 16*(3*x + 10)/(x^2 + 8*x + 16) + 96*(x + 3)/(x^2 + 8*x + 16) - 50*(x + 2)/(x^2 + 8*x + 16) - 32/(x^2 + 8*x
+ 16) + e^(6/25/x^3)

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mupad [B]  time = 3.08, size = 20, normalized size = 0.83 $$\begin{array}{c}x+{\mathrm{e}}^{\frac{6}{25x^3}}-\frac{2x+4}{{(x+4)}^2}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1600*x^4 - exp(6/(25*x^3))*(864*x + 216*x^2 + 18*x^3 + 1152) + 1250*x^5 + 300*x^6 + 25*x^7)/(1600*x^4 + 1
200*x^5 + 300*x^6 + 25*x^7),x)

[Out]

x + exp(6/(25*x^3)) - (2*x + 4)/(x + 4)^2

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sympy [A]  time = 0.19, size = 24, normalized size = 1.00 $$\begin{array}{c}x+\frac{-2x-4}{x^2+8x+16}+e^{\frac{6}{25x^3}}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-18*x**3-216*x**2-864*x-1152)*exp(6/25/x**3)+25*x**7+300*x**6+1250*x**5+1600*x**4)/(25*x**7+300*x*
*6+1200*x**5+1600*x**4),x)

[Out]

x + (-2*x - 4)/(x**2 + 8*x + 16) + exp(6/(25*x**3))

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