3.69.71 $\int \frac{-75+140x+20x^3+e^{2x}(-3x^2+9x^4+6x^5)+e^x(30x-29x^2-61x^3-4x^4-x^5+x^6)}{25+e^x(-10x-10x^2)+e^{2x}(x^2+2x^3+x^4)}dx$

Optimal. Leaf size=32 $$3-(1-x)(3x+\frac{x}{-e^x+\frac{5}{x+x^2}})$$

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Rubi [F]  time = 1.30, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.000, Rules used = {} $$\begin{array}{c}\int \frac{-75+140x+20x^3+e^{2x}(-3x^2+9x^4+6x^5)+e^x(30x-29x^2-61x^3-4x^4-x^5+x^6)}{25+e^x(-10x-10x^2)+e^{2x}(x^2+2x^3+x^4)}dx\end{array}$$

Verification is not applicable to the result.

[In]

Int[(-75 + 140*x + 20*x^3 + E^(2*x)*(-3*x^2 + 9*x^4 + 6*x^5) + E^x*(30*x - 29*x^2 - 61*x^3 - 4*x^4 - x^5 + x^6
))/(25 + E^x*(-10*x - 10*x^2) + E^(2*x)*(x^2 + 2*x^3 + x^4)),x]

[Out]

(3*(1 - 2*x)^2)/4 - 5*Defer[Int][x/(-5 + E^x*x + E^x*x^2)^2, x] - 10*Defer[Int][x^2/(-5 + E^x*x + E^x*x^2)^2,
x] + 10*Defer[Int][x^3/(-5 + E^x*x + E^x*x^2)^2, x] + 5*Defer[Int][x^4/(-5 + E^x*x + E^x*x^2)^2, x] + Defer[In
t][x/(-5 + E^x*x + E^x*x^2), x] - 2*Defer[Int][x^2/(-5 + E^x*x + E^x*x^2), x] - 2*Defer[Int][x^3/(-5 + E^x*x +
E^x*x^2), x] + Defer[Int][x^4/(-5 + E^x*x + E^x*x^2), x]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{-75+140x+20x^3+e^{2x}(-3x^2+9x^4+6x^5)+e^x(30x-29x^2-61x^3-4x^4-x^5+x^6)}{{(5-e^{x}x-e^x{x}^2)}^2}dx\\ & =\int (3(-1+2x)+\frac{x(1-2x-2x^2+x^3)}{-5+e^{x}x+e^x{x}^2}+\frac{5x(-1-2x+2x^2+x^3)}{{(-5+e^{x}x+e^x{x}^2)}^2})dx\\ & =\frac{3}{4}(1-2x{)}^2+5\int \frac{x(-1-2x+2x^2+x^3)}{{(-5+e^{x}x+e^x{x}^2)}^2}dx+\int \frac{x(1-2x-2x^2+x^3)}{-5+e^{x}x+e^x{x}^2}dx\\ & =\frac{3}{4}(1-2x{)}^2+5\int (-\frac{x}{{(-5+e^{x}x+e^x{x}^2)}^2}-\frac{2x^2}{{(-5+e^{x}x+e^x{x}^2)}^2}+\frac{2x^3}{{(-5+e^{x}x+e^x{x}^2)}^2}+\frac{x^4}{{(-5+e^{x}x+e^x{x}^2)}^2})dx+\int (\frac{x}{-5+e^{x}x+e^x{x}^2}-\frac{2x^2}{-5+e^{x}x+e^x{x}^2}-\frac{2x^3}{-5+e^{x}x+e^x{x}^2}+\frac{x^4}{-5+e^{x}x+e^x{x}^2})dx\\ & =\frac{3}{4}(1-2x{)}^2-2\int \frac{x^2}{-5+e^{x}x+e^x{x}^2}dx-2\int \frac{x^3}{-5+e^{x}x+e^x{x}^2}dx-5\int \frac{x}{{(-5+e^{x}x+e^x{x}^2)}^2}dx+5\int \frac{x^4}{{(-5+e^{x}x+e^x{x}^2)}^2}dx-10\int \frac{x^2}{{(-5+e^{x}x+e^x{x}^2)}^2}dx+10\int \frac{x^3}{{(-5+e^{x}x+e^x{x}^2)}^2}dx+\int \frac{x}{-5+e^{x}x+e^x{x}^2}dx+\int \frac{x^4}{-5+e^{x}x+e^x{x}^2}dx\end{array}\end{array}$$

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Mathematica [A]  time = 0.07, size = 27, normalized size = 0.84 $$\begin{array}{c}x(-3+3x+\frac{x-x^3}{-5+e^{x}x(1+x)})\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(-75 + 140*x + 20*x^3 + E^(2*x)*(-3*x^2 + 9*x^4 + 6*x^5) + E^x*(30*x - 29*x^2 - 61*x^3 - 4*x^4 - x^5
+ x^6))/(25 + E^x*(-10*x - 10*x^2) + E^(2*x)*(x^2 + 2*x^3 + x^4)),x]

[Out]

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

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fricas [A]  time = 1.46, size = 39, normalized size = 1.22 $$\begin{array}{c}-\frac{x^4+14x^2-3(x^4-x^2)e^x-15x}{(x^2+x)e^x-5}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x^5+9*x^4-3*x^2)*exp(x)^2+(x^6-x^5-4*x^4-61*x^3-29*x^2+30*x)*exp(x)+20*x^3+140*x-75)/((x^4+2*x^3
+x^2)*exp(x)^2+(-10*x^2-10*x)*exp(x)+25),x, algorithm="fricas")

[Out]

-(x^4 + 14*x^2 - 3*(x^4 - x^2)*e^x - 15*x)/((x^2 + x)*e^x - 5)

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giac [A]  time = 0.14, size = 43, normalized size = 1.34 $$\begin{array}{c}\frac{3x^4{e}^x-x^4-3x^2{e}^x-14x^2+15x}{x^2{e}^x+x{e}^x-5}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x^5+9*x^4-3*x^2)*exp(x)^2+(x^6-x^5-4*x^4-61*x^3-29*x^2+30*x)*exp(x)+20*x^3+140*x-75)/((x^4+2*x^3
+x^2)*exp(x)^2+(-10*x^2-10*x)*exp(x)+25),x, algorithm="giac")

[Out]

(3*x^4*e^x - x^4 - 3*x^2*e^x - 14*x^2 + 15*x)/(x^2*e^x + x*e^x - 5)

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maple [A]  time = 0.06, size = 35, normalized size = 1.09

method	result	size
risch	$3x^2-3x-\frac{(x-1)(x+1)x^2}{{\mathrm{e}}^x{x}^2+{\mathrm{e}}^{x}x-5}$	$35$
norman	$\frac{3{\mathrm{e}}^{x}x+15x-14x^2-x^4+3{\mathrm{e}}^x{x}^4-15}{{\mathrm{e}}^x{x}^2+{\mathrm{e}}^{x}x-5}$	$43$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((6*x^5+9*x^4-3*x^2)*exp(x)^2+(x^6-x^5-4*x^4-61*x^3-29*x^2+30*x)*exp(x)+20*x^3+140*x-75)/((x^4+2*x^3+x^2)*
exp(x)^2+(-10*x^2-10*x)*exp(x)+25),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.41, size = 39, normalized size = 1.22 $$\begin{array}{c}-\frac{x^4+14x^2-3(x^4-x^2)e^x-15x}{(x^2+x)e^x-5}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x^5+9*x^4-3*x^2)*exp(x)^2+(x^6-x^5-4*x^4-61*x^3-29*x^2+30*x)*exp(x)+20*x^3+140*x-75)/((x^4+2*x^3
+x^2)*exp(x)^2+(-10*x^2-10*x)*exp(x)+25),x, algorithm="maxima")

[Out]

-(x^4 + 14*x^2 - 3*(x^4 - x^2)*e^x - 15*x)/((x^2 + x)*e^x - 5)

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mupad [B]  time = 4.18, size = 38, normalized size = 1.19 $$\begin{array}{c}-\frac{x(x-1)(x-3x^2{\mathrm{e}}^x-3x{\mathrm{e}}^x+x^2+15)}{x^2{\mathrm{e}}^x+x{\mathrm{e}}^x-5}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((140*x - exp(x)*(29*x^2 - 30*x + 61*x^3 + 4*x^4 + x^5 - x^6) + exp(2*x)*(9*x^4 - 3*x^2 + 6*x^5) + 20*x^3 -
75)/(exp(2*x)*(x^2 + 2*x^3 + x^4) - exp(x)*(10*x + 10*x^2) + 25),x)

[Out]

-(x*(x - 1)*(x - 3*x^2*exp(x) - 3*x*exp(x) + x^2 + 15))/(x^2*exp(x) + x*exp(x) - 5)

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sympy [A]  time = 0.20, size = 24, normalized size = 0.75 $$\begin{array}{c}3x^2-3x+\frac{-x^4+x^2}{(x^2+x)e^x-5}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x**5+9*x**4-3*x**2)*exp(x)**2+(x**6-x**5-4*x**4-61*x**3-29*x**2+30*x)*exp(x)+20*x**3+140*x-75)/(
(x**4+2*x**3+x**2)*exp(x)**2+(-10*x**2-10*x)*exp(x)+25),x)

[Out]

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

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