3.17.33 $\int \frac{e^{12+4x}(5+10x)+e^{6+2x}(-10x+10x^2)}{x^4+e^{6+2x}(18x^2-2x^3-2x^4)+e^{12+4x}(81-18x-17x^2+2x^3+x^4)}dx$

Optimal. Leaf size=25 $$\frac{5}{9-x-x^2+e^{-6-2x}{x}^2}$$

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Rubi [F]  time = 9.51, 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 = {} $$\begin{array}{c}\int \frac{e^{12+4x}(5+10x)+e^{6+2x}(-10x+10x^2)}{x^4+e^{6+2x}(18x^2-2x^3-2x^4)+e^{12+4x}(81-18x-17x^2+2x^3+x^4)}dx\end{array}$$

Verification is not applicable to the result.

[In]

Int[(E^(12 + 4*x)*(5 + 10*x) + E^(6 + 2*x)*(-10*x + 10*x^2))/(x^4 + E^(6 + 2*x)*(18*x^2 - 2*x^3 - 2*x^4) + E^(
12 + 4*x)*(81 - 18*x - 17*x^2 + 2*x^3 + x^4)),x]

[Out]

-5*Defer[Int][E^(6 + 2*x)/(x^2 - E^(6 + 2*x)*(-9 + x + x^2))^2, x] + (90*Defer[Int][E^(6 + 2*x)/((-1 + Sqrt[37
] - 2*x)*(x^2 - E^(6 + 2*x)*(-9 + x + x^2))^2), x])/Sqrt[37] + 10*Defer[Int][(E^(6 + 2*x)*x^2)/(x^2 - E^(6 + 2
*x)*(-9 + x + x^2))^2, x] + (95*(37 - Sqrt[37])*Defer[Int][E^(6 + 2*x)/((1 - Sqrt[37] + 2*x)*(x^2 - E^(6 + 2*x
)*(-9 + x + x^2))^2), x])/37 + (90*Defer[Int][E^(6 + 2*x)/((1 + Sqrt[37] + 2*x)*(x^2 - E^(6 + 2*x)*(-9 + x + x
^2))^2), x])/Sqrt[37] + (95*(37 + Sqrt[37])*Defer[Int][E^(6 + 2*x)/((1 + Sqrt[37] + 2*x)*(x^2 - E^(6 + 2*x)*(-
9 + x + x^2))^2), x])/37 + (10*(37 + Sqrt[37])*Defer[Int][E^(6 + 2*x)/((-1 - Sqrt[37] - 2*x)*(x^2 - E^(6 + 2*x
)*(-9 + x + x^2))), x])/37 + (10*Defer[Int][E^(6 + 2*x)/((-1 + Sqrt[37] - 2*x)*(x^2 - E^(6 + 2*x)*(-9 + x + x^
2))), x])/Sqrt[37] + (10*(37 - Sqrt[37])*Defer[Int][E^(6 + 2*x)/((-1 + Sqrt[37] - 2*x)*(x^2 - E^(6 + 2*x)*(-9
+ x + x^2))), x])/37 + (10*Defer[Int][E^(6 + 2*x)/((1 + Sqrt[37] + 2*x)*(x^2 - E^(6 + 2*x)*(-9 + x + x^2))), x
])/Sqrt[37]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{5e^{6+2x}(2(-1+x)x+e^{6+2x}(1+2x))}{{(x^2-e^{6+2x}(-9+x+x^2))}^2}dx\\ & =5\int \frac{e^{6+2x}(2(-1+x)x+e^{6+2x}(1+2x))}{{(x^2-e^{6+2x}(-9+x+x^2))}^2}dx\\ & =5\int (\frac{e^{6+2x}(1+2x)}{(-9+x+x^2)(-9e^{6+2x}+e^{6+2x}x-x^2+e^{6+2x}{x}^2)}+\frac{e^{6+2x}x(18-19x+2x^2+2x^3)}{(-9+x+x^2){(-9e^{6+2x}+e^{6+2x}x-x^2+e^{6+2x}{x}^2)}^2})dx\\ & =5\int \frac{e^{6+2x}(1+2x)}{(-9+x+x^2)(-9e^{6+2x}+e^{6+2x}x-x^2+e^{6+2x}{x}^2)}dx+5\int \frac{e^{6+2x}x(18-19x+2x^2+2x^3)}{(-9+x+x^2){(-9e^{6+2x}+e^{6+2x}x-x^2+e^{6+2x}{x}^2)}^2}dx\\ & =5\int \frac{e^{6+2x}(1+2x)}{(9-x-x^2)(x^2-e^{6+2x}(-9+x+x^2))}dx+5\int (-\frac{e^{6+2x}}{{(-9e^{6+2x}+e^{6+2x}x-x^2+e^{6+2x}{x}^2)}^2}+\frac{2e^{6+2x}{x}^2}{{(-9e^{6+2x}+e^{6+2x}x-x^2+e^{6+2x}{x}^2)}^2}+\frac{e^{6+2x}(-9+19x)}{(-9+x+x^2){(-9e^{6+2x}+e^{6+2x}x-x^2+e^{6+2x}{x}^2)}^2})dx\\ & =-(5\int \frac{e^{6+2x}}{{(-9e^{6+2x}+e^{6+2x}x-x^2+e^{6+2x}{x}^2)}^2}dx)+5\int \frac{e^{6+2x}(-9+19x)}{(-9+x+x^2){(-9e^{6+2x}+e^{6+2x}x-x^2+e^{6+2x}{x}^2)}^2}dx+5\int (\frac{e^{6+2x}}{(-9+x+x^2)(-9e^{6+2x}+e^{6+2x}x-x^2+e^{6+2x}{x}^2)}+\frac{2e^{6+2x}x}{(-9+x+x^2)(-9e^{6+2x}+e^{6+2x}x-x^2+e^{6+2x}{x}^2)})dx+10\int \frac{e^{6+2x}{x}^2}{{(-9e^{6+2x}+e^{6+2x}x-x^2+e^{6+2x}{x}^2)}^2}dx\\ & =5\int \frac{e^{6+2x}}{(-9+x+x^2)(-9e^{6+2x}+e^{6+2x}x-x^2+e^{6+2x}{x}^2)}dx-5\int \frac{e^{6+2x}}{{(x^2-e^{6+2x}(-9+x+x^2))}^2}dx+5\int (-\frac{9e^{6+2x}}{(-9+x+x^2){(-9e^{6+2x}+e^{6+2x}x-x^2+e^{6+2x}{x}^2)}^2}+\frac{19e^{6+2x}x}{(-9+x+x^2){(-9e^{6+2x}+e^{6+2x}x-x^2+e^{6+2x}{x}^2)}^2})dx+10\int \frac{e^{6+2x}x}{(-9+x+x^2)(-9e^{6+2x}+e^{6+2x}x-x^2+e^{6+2x}{x}^2)}dx+10\int \frac{e^{6+2x}{x}^2}{{(x^2-e^{6+2x}(-9+x+x^2))}^2}dx\\ & =-(5\int \frac{e^{6+2x}}{{(x^2-e^{6+2x}(-9+x+x^2))}^2}dx)+5\int \frac{e^{6+2x}}{(9-x-x^2)(x^2-e^{6+2x}(-9+x+x^2))}dx+10\int \frac{e^{6+2x}{x}^2}{{(x^2-e^{6+2x}(-9+x+x^2))}^2}dx+10\int \frac{e^{6+2x}x}{(9-x-x^2)(x^2-e^{6+2x}(-9+x+x^2))}dx-45\int \frac{e^{6+2x}}{(-9+x+x^2){(-9e^{6+2x}+e^{6+2x}x-x^2+e^{6+2x}{x}^2)}^2}dx+95\int \frac{e^{6+2x}x}{(-9+x+x^2){(-9e^{6+2x}+e^{6+2x}x-x^2+e^{6+2x}{x}^2)}^2}dx\\ & =-(5\int \frac{e^{6+2x}}{{(x^2-e^{6+2x}(-9+x+x^2))}^2}dx)+5\int (\frac{2e^{6+2x}}{\sqrt{37}(-1+\sqrt{37}-2x)(x^2-e^{6+2x}(-9+x+x^2))}+\frac{2e^{6+2x}}{\sqrt{37}(1+\sqrt{37}+2x)(x^2-e^{6+2x}(-9+x+x^2))})dx+10\int \frac{e^{6+2x}{x}^2}{{(x^2-e^{6+2x}(-9+x+x^2))}^2}dx+10\int (\frac{(1+\frac{1}{\sqrt{37}})e^{6+2x}}{(-1-\sqrt{37}-2x)(x^2-e^{6+2x}(-9+x+x^2))}+\frac{(1-\frac{1}{\sqrt{37}})e^{6+2x}}{(-1+\sqrt{37}-2x)(x^2-e^{6+2x}(-9+x+x^2))})dx-45\int \frac{e^{6+2x}}{(-9+x+x^2){(x^2-e^{6+2x}(-9+x+x^2))}^2}dx+95\int \frac{e^{6+2x}x}{(-9+x+x^2){(x^2-e^{6+2x}(-9+x+x^2))}^2}dx\\ & =-(5\int \frac{e^{6+2x}}{{(x^2-e^{6+2x}(-9+x+x^2))}^2}dx)+10\int \frac{e^{6+2x}{x}^2}{{(x^2-e^{6+2x}(-9+x+x^2))}^2}dx-45\int (-\frac{2e^{6+2x}}{\sqrt{37}(-1+\sqrt{37}-2x){(x^2-e^{6+2x}(-9+x+x^2))}^2}-\frac{2e^{6+2x}}{\sqrt{37}(1+\sqrt{37}+2x){(x^2-e^{6+2x}(-9+x+x^2))}^2})dx+95\int (\frac{(1-\frac{1}{\sqrt{37}})e^{6+2x}}{(1-\sqrt{37}+2x){(x^2-e^{6+2x}(-9+x+x^2))}^2}+\frac{(1+\frac{1}{\sqrt{37}})e^{6+2x}}{(1+\sqrt{37}+2x){(x^2-e^{6+2x}(-9+x+x^2))}^2})dx+\frac{10\int \frac{e^{6+2x}}{(-1+\sqrt{37}-2x)(x^2-e^{6+2x}(-9+x+x^2))}dx}{\sqrt{37}}+\frac{10\int \frac{e^{6+2x}}{(1+\sqrt{37}+2x)(x^2-e^{6+2x}(-9+x+x^2))}dx}{\sqrt{37}}+\frac{1}{37}\left(10(37-\sqrt{37})\right)\int \frac{e^{6+2x}}{(-1+\sqrt{37}-2x)(x^2-e^{6+2x}(-9+x+x^2))}dx+\frac{1}{37}\left(10(37+\sqrt{37})\right)\int \frac{e^{6+2x}}{(-1-\sqrt{37}-2x)(x^2-e^{6+2x}(-9+x+x^2))}dx\\ & =-(5\int \frac{e^{6+2x}}{{(x^2-e^{6+2x}(-9+x+x^2))}^2}dx)+10\int \frac{e^{6+2x}{x}^2}{{(x^2-e^{6+2x}(-9+x+x^2))}^2}dx+\frac{10\int \frac{e^{6+2x}}{(-1+\sqrt{37}-2x)(x^2-e^{6+2x}(-9+x+x^2))}dx}{\sqrt{37}}+\frac{10\int \frac{e^{6+2x}}{(1+\sqrt{37}+2x)(x^2-e^{6+2x}(-9+x+x^2))}dx}{\sqrt{37}}+\frac{90\int \frac{e^{6+2x}}{(-1+\sqrt{37}-2x){(x^2-e^{6+2x}(-9+x+x^2))}^2}dx}{\sqrt{37}}+\frac{90\int \frac{e^{6+2x}}{(1+\sqrt{37}+2x){(x^2-e^{6+2x}(-9+x+x^2))}^2}dx}{\sqrt{37}}+\frac{1}{37}\left(10(37-\sqrt{37})\right)\int \frac{e^{6+2x}}{(-1+\sqrt{37}-2x)(x^2-e^{6+2x}(-9+x+x^2))}dx+\frac{1}{37}\left(95(37-\sqrt{37})\right)\int \frac{e^{6+2x}}{(1-\sqrt{37}+2x){(x^2-e^{6+2x}(-9+x+x^2))}^2}dx+\frac{1}{37}\left(10(37+\sqrt{37})\right)\int \frac{e^{6+2x}}{(-1-\sqrt{37}-2x)(x^2-e^{6+2x}(-9+x+x^2))}dx+\frac{1}{37}\left(95(37+\sqrt{37})\right)\int \frac{e^{6+2x}}{(1+\sqrt{37}+2x){(x^2-e^{6+2x}(-9+x+x^2))}^2}dx\end{array}\end{array}$$

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Mathematica [A]  time = 1.62, size = 31, normalized size = 1.24 $$\begin{array}{c}-\frac{5e^{6+2x}}{-x^2+e^{6+2x}(-9+x+x^2)}\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(E^(12 + 4*x)*(5 + 10*x) + E^(6 + 2*x)*(-10*x + 10*x^2))/(x^4 + E^(6 + 2*x)*(18*x^2 - 2*x^3 - 2*x^4)
+ E^(12 + 4*x)*(81 - 18*x - 17*x^2 + 2*x^3 + x^4)),x]

[Out]

(-5*E^(6 + 2*x))/(-x^2 + E^(6 + 2*x)*(-9 + x + x^2))

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fricas [A]  time = 0.84, size = 28, normalized size = 1.12 $$\begin{array}{c}\frac{5e^{(2x+6)}}{x^2-(x^2+x-9)e^{(2x+6)}}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x+5)*exp(3)^4*exp(x)^4+(10*x^2-10*x)*exp(3)^2*exp(x)^2)/((x^4+2*x^3-17*x^2-18*x+81)*exp(3)^4*ex
p(x)^4+(-2*x^4-2*x^3+18*x^2)*exp(3)^2*exp(x)^2+x^4),x, algorithm="fricas")

[Out]

5*e^(2*x + 6)/(x^2 - (x^2 + x - 9)*e^(2*x + 6))

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giac [A]  time = 0.53, size = 42, normalized size = 1.68 $$\begin{array}{c}-\frac{5e^{(2x+6)}}{x^2{e}^{(2x+6)}-x^2+x{e}^{(2x+6)}-9e^{(2x+6)}}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x+5)*exp(3)^4*exp(x)^4+(10*x^2-10*x)*exp(3)^2*exp(x)^2)/((x^4+2*x^3-17*x^2-18*x+81)*exp(3)^4*ex
p(x)^4+(-2*x^4-2*x^3+18*x^2)*exp(3)^2*exp(x)^2+x^4),x, algorithm="giac")

[Out]

-5*e^(2*x + 6)/(x^2*e^(2*x + 6) - x^2 + x*e^(2*x + 6) - 9*e^(2*x + 6))

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maple [A]  time = 0.35, size = 51, normalized size = 2.04

method	result	size
norman	$-\frac{5{\mathrm{e}}^6{\mathrm{e}}^{2x}}{x^2{\mathrm{e}}^6{\mathrm{e}}^{2x}+{\mathrm{e}}^6{\mathrm{e}}^{2x}x-9{\mathrm{e}}^6{\mathrm{e}}^{2x}-x^2}$	$51$
risch	$-\frac{5}{x^2+x-9}-\frac{5x^2}{(x^2+x-9)(x^2{\mathrm{e}}^{2x+6}+x{\mathrm{e}}^{2x+6}-9{\mathrm{e}}^{2x+6}-x^2)}$	$59$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((10*x+5)*exp(3)^4*exp(x)^4+(10*x^2-10*x)*exp(3)^2*exp(x)^2)/((x^4+2*x^3-17*x^2-18*x+81)*exp(3)^4*exp(x)^4
+(-2*x^4-2*x^3+18*x^2)*exp(3)^2*exp(x)^2+x^4),x,method=_RETURNVERBOSE)

[Out]

-5*exp(3)^2*exp(x)^2/(x^2*exp(3)^2*exp(x)^2+exp(3)^2*exp(x)^2*x-9*exp(3)^2*exp(x)^2-x^2)

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maxima [A]  time = 0.45, size = 35, normalized size = 1.40 $$\begin{array}{c}\frac{5e^{(2x+6)}}{x^2-(x^2{e}^6+x{e}^6-9e^6)e^{\left(2x\right)}}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x+5)*exp(3)^4*exp(x)^4+(10*x^2-10*x)*exp(3)^2*exp(x)^2)/((x^4+2*x^3-17*x^2-18*x+81)*exp(3)^4*ex
p(x)^4+(-2*x^4-2*x^3+18*x^2)*exp(3)^2*exp(x)^2+x^4),x, algorithm="maxima")

[Out]

5*e^(2*x + 6)/(x^2 - (x^2*e^6 + x*e^6 - 9*e^6)*e^(2*x))

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mupad [B]  time = 0.19, size = 57, normalized size = 2.28 $$\begin{array}{c}\frac{5x({\mathrm{e}}^{2x+6}-x+x{\mathrm{e}}^{2x+6})}{9(9{\mathrm{e}}^{2x+6}-x{\mathrm{e}}^{2x+6}-x^2{\mathrm{e}}^{2x+6}+x^2)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(4*x)*exp(12)*(10*x + 5) - exp(2*x)*exp(6)*(10*x - 10*x^2))/(x^4 - exp(2*x)*exp(6)*(2*x^3 - 18*x^2 + 2
*x^4) + exp(4*x)*exp(12)*(2*x^3 - 17*x^2 - 18*x + x^4 + 81)),x)

[Out]

(5*x*(exp(2*x + 6) - x + x*exp(2*x + 6)))/(9*(9*exp(2*x + 6) - x*exp(2*x + 6) - x^2*exp(2*x + 6) + x^2))

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sympy [B]  time = 0.34, size = 66, normalized size = 2.64 $$\begin{array}{c}-\frac{5x^2}{-x^4-x^3+9x^2+(x^4{e}^6+2x^3{e}^6-17x^2{e}^6-18x{e}^6+81e^6)e^{2x}}-\frac{5}{x^2+x-9}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x+5)*exp(3)**4*exp(x)**4+(10*x**2-10*x)*exp(3)**2*exp(x)**2)/((x**4+2*x**3-17*x**2-18*x+81)*exp
(3)**4*exp(x)**4+(-2*x**4-2*x**3+18*x**2)*exp(3)**2*exp(x)**2+x**4),x)

[Out]

-5*x**2/(-x**4 - x**3 + 9*x**2 + (x**4*exp(6) + 2*x**3*exp(6) - 17*x**2*exp(6) - 18*x*exp(6) + 81*exp(6))*exp(
2*x)) - 5/(x**2 + x - 9)

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