3.69.58 $\int \frac{-12+6x-2x^6+e^2(-20x^4-40x^5-20x^6)+e^4(-10x^2-40x^3-60x^4-40x^5-10x^6)+e^5(2x+10x^2+20x^3+20x^4+10x^5+2x^6)+e(14-6x+10x^5+10x^6)+e^3(20x^3+60x^4+60x^5+20x^6)}{-x^5+e^2(-10x^3-20x^4-10x^5)+e^4(-5x-20x^2-30x^3-20x^4-5x^5)+e^5(1+5x+10x^2+10x^3+5x^4+x^5)+e(5x^4+5x^5)+e^3(10x^2+30x^3+30x^4+10x^5)}dx$

Optimal. Leaf size=22 $$11+x^2-\frac{3-2x}{(x-e(1+x){)}^4}$$

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Rubi [B]  time = 0.18, antiderivative size = 51, normalized size of antiderivative = 2.32, number of steps used = 2, number of rules used = 1, integrand size = 257, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.004, Rules used = {2074} $$\begin{array}{c}{x}^2-\frac{2}{(1-e)(e-(1-e)x{)}^3}-\frac{3-5e}{(1-e)(e-(1-e)x{)}^4}\end{array}$$

Antiderivative was successfully verified.

[In]

Int[(-12 + 6*x - 2*x^6 + E^2*(-20*x^4 - 40*x^5 - 20*x^6) + E^4*(-10*x^2 - 40*x^3 - 60*x^4 - 40*x^5 - 10*x^6) +
E^5*(2*x + 10*x^2 + 20*x^3 + 20*x^4 + 10*x^5 + 2*x^6) + E*(14 - 6*x + 10*x^5 + 10*x^6) + E^3*(20*x^3 + 60*x^4
+ 60*x^5 + 20*x^6))/(-x^5 + E^2*(-10*x^3 - 20*x^4 - 10*x^5) + E^4*(-5*x - 20*x^2 - 30*x^3 - 20*x^4 - 5*x^5) +
E^5*(1 + 5*x + 10*x^2 + 10*x^3 + 5*x^4 + x^5) + E*(5*x^4 + 5*x^5) + E^3*(10*x^2 + 30*x^3 + 30*x^4 + 10*x^5)),
x]

[Out]

x^2 - (3 - 5*E)/((1 - E)*(E - (1 - E)*x)^4) - 2/((1 - E)*(E - (1 - E)*x)^3)

Rule 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int (2x+\frac{4(-3+5e)}{(e-(1-e)x{)}^5}-\frac{6}{(e-(1-e)x{)}^4})dx\\ & =x^2-\frac{3-5e}{(1-e)(e-(1-e)x{)}^4}-\frac{2}{(1-e)(e-(1-e)x{)}^3}\end{array}\end{array}$$

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Mathematica [B]  time = 0.08, size = 47, normalized size = 2.14 $$\begin{array}{c}-2x+(1+x{)}^2+\frac{3-5e}{(-1+e)(e-x+ex{)}^4}+\frac{2}{(-1+e)(e-x+ex{)}^3}\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(-12 + 6*x - 2*x^6 + E^2*(-20*x^4 - 40*x^5 - 20*x^6) + E^4*(-10*x^2 - 40*x^3 - 60*x^4 - 40*x^5 - 10*
x^6) + E^5*(2*x + 10*x^2 + 20*x^3 + 20*x^4 + 10*x^5 + 2*x^6) + E*(14 - 6*x + 10*x^5 + 10*x^6) + E^3*(20*x^3 +
60*x^4 + 60*x^5 + 20*x^6))/(-x^5 + E^2*(-10*x^3 - 20*x^4 - 10*x^5) + E^4*(-5*x - 20*x^2 - 30*x^3 - 20*x^4 - 5*
x^5) + E^5*(1 + 5*x + 10*x^2 + 10*x^3 + 5*x^4 + x^5) + E*(5*x^4 + 5*x^5) + E^3*(10*x^2 + 30*x^3 + 30*x^4 + 10*
x^5)),x]

[Out]

-2*x + (1 + x)^2 + (3 - 5*E)/((-1 + E)*(E - x + E*x)^4) + 2/((-1 + E)*(E - x + E*x)^3)

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fricas [B]  time = 0.53, size = 155, normalized size = 7.05 $$\begin{array}{c}\frac{x^6+(x^6+4x^5+6x^4+4x^3+x^2)e^4-4(x^6+3x^5+3x^4+x^3)e^3+6(x^6+2x^5+x^4)e^2-4(x^6+x^5)e+2x-3}{x^4+(x^4+4x^3+6x^2+4x+1)e^4-4(x^4+3x^3+3x^2+x)e^3+6(x^4+2x^3+x^2)e^2-4(x^4+x^3)e}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^6+10*x^5+20*x^4+20*x^3+10*x^2+2*x)*exp(1)^5+(-10*x^6-40*x^5-60*x^4-40*x^3-10*x^2)*exp(1)^4+(20
*x^6+60*x^5+60*x^4+20*x^3)*exp(1)^3+(-20*x^6-40*x^5-20*x^4)*exp(1)^2+(10*x^6+10*x^5-6*x+14)*exp(1)-2*x^6+6*x-1
2)/((x^5+5*x^4+10*x^3+10*x^2+5*x+1)*exp(1)^5+(-5*x^5-20*x^4-30*x^3-20*x^2-5*x)*exp(1)^4+(10*x^5+30*x^4+30*x^3+
10*x^2)*exp(1)^3+(-10*x^5-20*x^4-10*x^3)*exp(1)^2+(5*x^5+5*x^4)*exp(1)-x^5),x, algorithm="fricas")

[Out]

(x^6 + (x^6 + 4*x^5 + 6*x^4 + 4*x^3 + x^2)*e^4 - 4*(x^6 + 3*x^5 + 3*x^4 + x^3)*e^3 + 6*(x^6 + 2*x^5 + x^4)*e^2
- 4*(x^6 + x^5)*e + 2*x - 3)/(x^4 + (x^4 + 4*x^3 + 6*x^2 + 4*x + 1)*e^4 - 4*(x^4 + 3*x^3 + 3*x^2 + x)*e^3 + 6
*(x^4 + 2*x^3 + x^2)*e^2 - 4*(x^4 + x^3)*e)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 $$\begin{array}{c}\text{Timed out}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^6+10*x^5+20*x^4+20*x^3+10*x^2+2*x)*exp(1)^5+(-10*x^6-40*x^5-60*x^4-40*x^3-10*x^2)*exp(1)^4+(20
*x^6+60*x^5+60*x^4+20*x^3)*exp(1)^3+(-20*x^6-40*x^5-20*x^4)*exp(1)^2+(10*x^6+10*x^5-6*x+14)*exp(1)-2*x^6+6*x-1
2)/((x^5+5*x^4+10*x^3+10*x^2+5*x+1)*exp(1)^5+(-5*x^5-20*x^4-30*x^3-20*x^2-5*x)*exp(1)^4+(10*x^5+30*x^4+30*x^3+
10*x^2)*exp(1)^3+(-10*x^5-20*x^4-10*x^3)*exp(1)^2+(5*x^5+5*x^4)*exp(1)-x^5),x, algorithm="giac")

[Out]

Timed out

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maple [B]  time = 0.38, size = 105, normalized size = 4.77

method	result	size
risch	$x^2+\frac{2x-3}{x^4{\mathrm{e}}^4+4x^3{\mathrm{e}}^4-4x^4{\mathrm{e}}^3+6x^2{\mathrm{e}}^4-12x^3{\mathrm{e}}^3+6x^4{\mathrm{e}}^2+4x{\mathrm{e}}^4-12x^2{\mathrm{e}}^3+12x^3{\mathrm{e}}^2-4x^4\mathrm{e}+{\mathrm{e}}^4-4x{\mathrm{e}}^3+6x^2{\mathrm{e}}^2-4x^3\mathrm{e}+x^4}$	$105$
norman	$\frac{({\mathrm{e}}^4-4{\mathrm{e}}^3+6{\mathrm{e}}^2-4\mathrm{e}+1)x^6+{\mathrm{e}}^{-2}({\mathrm{e}}^6+18{\mathrm{e}}^2-36\mathrm{e}+18)x^2+(6{\mathrm{e}}^8-12{\mathrm{e}}^7+6{\mathrm{e}}^6+3+3{\mathrm{e}}^4-12{\mathrm{e}}^3+18{\mathrm{e}}^2-12\mathrm{e}){\mathrm{e}}^{-4}{x}^4+2{\mathrm{e}}^{-1}(7\mathrm{e}-6)x+4({\mathrm{e}}^7-{\mathrm{e}}^6+3{\mathrm{e}}^3-9{\mathrm{e}}^2+9\mathrm{e}-3){\mathrm{e}}^{-3}{x}^3+4\mathrm{e}({\mathrm{e}}^3-3{\mathrm{e}}^2+3\mathrm{e}-1)x^5}{{(x\mathrm{e}+\mathrm{e}-x)}^4}$	$188$
default	$x^2+\frac{2(\sum _{\mathit{\text{\_R}}=\mathrm{RootOf}(({\mathrm{e}}^5-10{\mathrm{e}}^2+5\mathrm{e}-5{\mathrm{e}}^4+10{\mathrm{e}}^3-1){\mathit{\text{\_Z}}}^5+(5{\mathrm{e}}^5-20{\mathrm{e}}^2+5\mathrm{e}-20{\mathrm{e}}^4+30{\mathrm{e}}^3){\mathit{\text{\_Z}}}^4+(10{\mathrm{e}}^5-10{\mathrm{e}}^2-30{\mathrm{e}}^4+30{\mathrm{e}}^3){\mathit{\text{\_Z}}}^3+(10{\mathrm{e}}^5-20{\mathrm{e}}^4+10{\mathrm{e}}^3){\mathit{\text{\_Z}}}^2+(5{\mathrm{e}}^5-5{\mathrm{e}}^4)\mathit{\text{\_Z}}+{\mathrm{e}}^5)}\frac{(-6+3(1-\mathrm{e})\mathit{\text{\_R}}+7\mathrm{e})\ln (x-\mathit{\text{\_R}})}{{\mathit{\text{\_R}}}^4{\mathrm{e}}^5+4{\mathit{\text{\_R}}}^3{\mathrm{e}}^5-5{\mathit{\text{\_R}}}^4{\mathrm{e}}^4+6{\mathit{\text{\_R}}}^2{\mathrm{e}}^5-16{\mathit{\text{\_R}}}^3{\mathrm{e}}^4+10{\mathit{\text{\_R}}}^4{\mathrm{e}}^3+4\mathit{\text{\_R}}{\mathrm{e}}^5-18{\mathit{\text{\_R}}}^2{\mathrm{e}}^4+24{\mathit{\text{\_R}}}^3{\mathrm{e}}^3-10{\mathit{\text{\_R}}}^4{\mathrm{e}}^2+{\mathrm{e}}^5-8\mathit{\text{\_R}}{\mathrm{e}}^4+18{\mathit{\text{\_R}}}^2{\mathrm{e}}^3-16{\mathit{\text{\_R}}}^3{\mathrm{e}}^2+5{\mathit{\text{\_R}}}^4\mathrm{e}-{\mathrm{e}}^4+4\mathit{\text{\_R}}{\mathrm{e}}^3-6{\mathit{\text{\_R}}}^2{\mathrm{e}}^2+4{\mathit{\text{\_R}}}^3\mathrm{e}-{\mathit{\text{\_R}}}^4})}{5}$	$260$
gosper	$\frac{x(18{\mathrm{e}}^{2}x+x{\mathrm{e}}^8+6x^3{\mathrm{e}}^6+4x^2{\mathrm{e}}^8+18x{\mathrm{e}}^4+14{\mathrm{e}}^4-12{\mathrm{e}}^3+3x^3+3x^3{\mathrm{e}}^4+12x^2{\mathrm{e}}^4-4x^5{\mathrm{e}}^5+18x^3{\mathrm{e}}^2+36x^2{\mathrm{e}}^2-4x^4{\mathrm{e}}^5-36x{\mathrm{e}}^3-12x^2\mathrm{e}-12x^3\mathrm{e}-12x^3{\mathrm{e}}^3-36x^2{\mathrm{e}}^3+4x^4{\mathrm{e}}^8+6x^3{\mathrm{e}}^8-4x^2{\mathrm{e}}^7+x^5{\mathrm{e}}^4+6{\mathrm{e}}^6{x}^5+12{\mathrm{e}}^6{x}^4+{\mathrm{e}}^8{x}^5-4{\mathrm{e}}^7{x}^5-12{\mathrm{e}}^7{x}^4-12{\mathrm{e}}^7{x}^3){\mathrm{e}}^{-4}}{x^4{\mathrm{e}}^4+4x^3{\mathrm{e}}^4-4x^4{\mathrm{e}}^3+6x^2{\mathrm{e}}^4-12x^3{\mathrm{e}}^3+6x^4{\mathrm{e}}^2+4x{\mathrm{e}}^4-12x^2{\mathrm{e}}^3+12x^3{\mathrm{e}}^2-4x^4\mathrm{e}+{\mathrm{e}}^4-4x{\mathrm{e}}^3+6x^2{\mathrm{e}}^2-4x^3\mathrm{e}+x^4}$	$362$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^6+10*x^5+20*x^4+20*x^3+10*x^2+2*x)*exp(1)^5+(-10*x^6-40*x^5-60*x^4-40*x^3-10*x^2)*exp(1)^4+(20*x^6+6
0*x^5+60*x^4+20*x^3)*exp(1)^3+(-20*x^6-40*x^5-20*x^4)*exp(1)^2+(10*x^6+10*x^5-6*x+14)*exp(1)-2*x^6+6*x-12)/((x
^5+5*x^4+10*x^3+10*x^2+5*x+1)*exp(1)^5+(-5*x^5-20*x^4-30*x^3-20*x^2-5*x)*exp(1)^4+(10*x^5+30*x^4+30*x^3+10*x^2
)*exp(1)^3+(-10*x^5-20*x^4-10*x^3)*exp(1)^2+(5*x^5+5*x^4)*exp(1)-x^5),x,method=_RETURNVERBOSE)

[Out]

x^2+(2*x-3)/(x^4*exp(4)+4*x^3*exp(4)-4*x^4*exp(3)+6*x^2*exp(4)-12*x^3*exp(3)+6*x^4*exp(2)+4*x*exp(4)-12*x^2*ex
p(3)+12*x^3*exp(2)-4*x^4*exp(1)+exp(4)-4*x*exp(3)+6*x^2*exp(2)-4*x^3*exp(1)+x^4)

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maxima [B]  time = 0.37, size = 79, normalized size = 3.59 $$\begin{array}{c}{x}^2+\frac{2x-3}{x^4(e^4-4e^3+6e^2-4e+1)+4x^3(e^4-3e^3+3e^2-e)+6x^2(e^4-2e^3+e^2)+4x(e^4-e^3)+e^4}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^6+10*x^5+20*x^4+20*x^3+10*x^2+2*x)*exp(1)^5+(-10*x^6-40*x^5-60*x^4-40*x^3-10*x^2)*exp(1)^4+(20
*x^6+60*x^5+60*x^4+20*x^3)*exp(1)^3+(-20*x^6-40*x^5-20*x^4)*exp(1)^2+(10*x^6+10*x^5-6*x+14)*exp(1)-2*x^6+6*x-1
2)/((x^5+5*x^4+10*x^3+10*x^2+5*x+1)*exp(1)^5+(-5*x^5-20*x^4-30*x^3-20*x^2-5*x)*exp(1)^4+(10*x^5+30*x^4+30*x^3+
10*x^2)*exp(1)^3+(-10*x^5-20*x^4-10*x^3)*exp(1)^2+(5*x^5+5*x^4)*exp(1)-x^5),x, algorithm="maxima")

[Out]

x^2 + (2*x - 3)/(x^4*(e^4 - 4*e^3 + 6*e^2 - 4*e + 1) + 4*x^3*(e^4 - 3*e^3 + 3*e^2 - e) + 6*x^2*(e^4 - 2*e^3 +
e^2) + 4*x*(e^4 - e^3) + e^4)

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mupad [B]  time = 4.43, size = 48, normalized size = 2.18 $$\begin{array}{c}\frac{2}{{(\mathrm{e}+x(\mathrm{e}-1))}^3(\mathrm{e}-1)}+x^2-\frac{5\mathrm{e}-3}{{(\mathrm{e}+x(\mathrm{e}-1))}^4(\mathrm{e}-1)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((6*x + exp(5)*(2*x + 10*x^2 + 20*x^3 + 20*x^4 + 10*x^5 + 2*x^6) - exp(4)*(10*x^2 + 40*x^3 + 60*x^4 + 40*x^
5 + 10*x^6) + exp(1)*(10*x^5 - 6*x + 10*x^6 + 14) - exp(2)*(20*x^4 + 40*x^5 + 20*x^6) - 2*x^6 + exp(3)*(20*x^3
+ 60*x^4 + 60*x^5 + 20*x^6) - 12)/(exp(5)*(5*x + 10*x^2 + 10*x^3 + 5*x^4 + x^5 + 1) + exp(1)*(5*x^4 + 5*x^5)
- exp(2)*(10*x^3 + 20*x^4 + 10*x^5) - exp(4)*(5*x + 20*x^2 + 30*x^3 + 20*x^4 + 5*x^5) - x^5 + exp(3)*(10*x^2 +
30*x^3 + 30*x^4 + 10*x^5)),x)

[Out]

2/((exp(1) + x*(exp(1) - 1))^3*(exp(1) - 1)) + x^2 - (5*exp(1) - 3)/((exp(1) + x*(exp(1) - 1))^4*(exp(1) - 1))

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sympy [B]  time = 4.74, size = 88, normalized size = 4.00 $$\begin{array}{c}{x}^2+\frac{2x-3}{x^4(-4e^3-4e+1+6e^2+e^4)+x^3(-12e^3-4e+12e^2+4e^4)+x^2(-12e^3+6e^2+6e^4)+x(-4e^3+4e^4)+e^4}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**6+10*x**5+20*x**4+20*x**3+10*x**2+2*x)*exp(1)**5+(-10*x**6-40*x**5-60*x**4-40*x**3-10*x**2)*e
xp(1)**4+(20*x**6+60*x**5+60*x**4+20*x**3)*exp(1)**3+(-20*x**6-40*x**5-20*x**4)*exp(1)**2+(10*x**6+10*x**5-6*x
+14)*exp(1)-2*x**6+6*x-12)/((x**5+5*x**4+10*x**3+10*x**2+5*x+1)*exp(1)**5+(-5*x**5-20*x**4-30*x**3-20*x**2-5*x
)*exp(1)**4+(10*x**5+30*x**4+30*x**3+10*x**2)*exp(1)**3+(-10*x**5-20*x**4-10*x**3)*exp(1)**2+(5*x**5+5*x**4)*e
xp(1)-x**5),x)

[Out]

x**2 + (2*x - 3)/(x**4*(-4*exp(3) - 4*E + 1 + 6*exp(2) + exp(4)) + x**3*(-12*exp(3) - 4*E + 12*exp(2) + 4*exp(
4)) + x**2*(-12*exp(3) + 6*exp(2) + 6*exp(4)) + x*(-4*exp(3) + 4*exp(4)) + exp(4))

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