3.14.34 $\int \frac{132-25x^2+e^{-10+4x}{x}^2+x^4+e^{-5+2x}x(24+2x-2x^2)}{144-24x^2+e^{-10+4x}{x}^2+x^4+e^{-5+2x}x(24-2x^2)}dx$

Optimal. Leaf size=24 $$2+x-\frac{x}{12+e^{-5+2x}x-x^2}$$

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Rubi [F]  time = 1.08, 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 = {} $$\begin{array}{c}\int \frac{132-25x^2+e^{-10+4x}{x}^2+x^4+e^{-5+2x}x(24+2x-2x^2)}{144-24x^2+e^{-10+4x}{x}^2+x^4+e^{-5+2x}x(24-2x^2)}dx\end{array}$$

Verification is not applicable to the result.

[In]

Int[(132 - 25*x^2 + E^(-10 + 4*x)*x^2 + x^4 + E^(-5 + 2*x)*x*(24 + 2*x - 2*x^2))/(144 - 24*x^2 + E^(-10 + 4*x)
*x^2 + x^4 + E^(-5 + 2*x)*x*(24 - 2*x^2)),x]

[Out]

x - 12*E^10*Defer[Int][(-12*E^5 - E^(2*x)*x + E^5*x^2)^(-2), x] - 24*E^10*Defer[Int][x/(-12*E^5 - E^(2*x)*x +
E^5*x^2)^2, x] - E^10*Defer[Int][x^2/(-12*E^5 - E^(2*x)*x + E^5*x^2)^2, x] + 2*E^10*Defer[Int][x^3/(-12*E^5 -
E^(2*x)*x + E^5*x^2)^2, x] - 2*E^5*Defer[Int][x/(-12*E^5 - E^(2*x)*x + E^5*x^2), x]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{e^{10}(132-25x^2+e^{-10+4x}{x}^2+x^4+e^{-5+2x}x(24+2x-2x^2))}{{(12e^5+e^{2x}x-e^5{x}^2)}^2}dx\\ & =e^{10}\int \frac{132-25x^2+e^{-10+4x}{x}^2+x^4+e^{-5+2x}x(24+2x-2x^2)}{{(12e^5+e^{2x}x-e^5{x}^2)}^2}dx\\ & =e^{10}\int (\frac{1}{e^{10}}-\frac{2x}{e^5(-12e^5-e^{2x}x+e^5{x}^2)}+\frac{-12-24x-x^2+2x^3}{{(-12e^5-e^{2x}x+e^5{x}^2)}^2})dx\\ & =x-\left(2e^5\right)\int \frac{x}{-12e^5-e^{2x}x+e^5{x}^2}dx+e^{10}\int \frac{-12-24x-x^2+2x^3}{{(-12e^5-e^{2x}x+e^5{x}^2)}^2}dx\\ & =x-\left(2e^5\right)\int \frac{x}{-12e^5-e^{2x}x+e^5{x}^2}dx+e^{10}\int (-\frac{12}{{(-12e^5-e^{2x}x+e^5{x}^2)}^2}-\frac{24x}{{(-12e^5-e^{2x}x+e^5{x}^2)}^2}-\frac{x^2}{{(-12e^5-e^{2x}x+e^5{x}^2)}^2}+\frac{2x^3}{{(-12e^5-e^{2x}x+e^5{x}^2)}^2})dx\\ & =x-\left(2e^5\right)\int \frac{x}{-12e^5-e^{2x}x+e^5{x}^2}dx-e^{10}\int \frac{x^2}{{(-12e^5-e^{2x}x+e^5{x}^2)}^2}dx+\left(2e^{10}\right)\int \frac{x^3}{{(-12e^5-e^{2x}x+e^5{x}^2)}^2}dx-\left(12e^{10}\right)\int \frac{1}{{(-12e^5-e^{2x}x+e^5{x}^2)}^2}dx-\left(24e^{10}\right)\int \frac{x}{{(-12e^5-e^{2x}x+e^5{x}^2)}^2}dx\end{array}\end{array}$$

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Mathematica [A]  time = 0.32, size = 27, normalized size = 1.12 $$\begin{array}{c}x+\frac{e^{5}x}{-e^{2x}x+e^5(-12+x^2)}\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(132 - 25*x^2 + E^(-10 + 4*x)*x^2 + x^4 + E^(-5 + 2*x)*x*(24 + 2*x - 2*x^2))/(144 - 24*x^2 + E^(-10
+ 4*x)*x^2 + x^4 + E^(-5 + 2*x)*x*(24 - 2*x^2)),x]

[Out]

x + (E^5*x)/(-(E^(2*x)*x) + E^5*(-12 + x^2))

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fricas [A]  time = 0.98, size = 36, normalized size = 1.50 $$\begin{array}{c}\frac{x^3-x{e}^{(2x+\log (x)-5)}-11x}{x^2-e^{(2x+\log (x)-5)}-12}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(2*x+log(x)-5)^2+(-2*x^2+2*x+24)*exp(2*x+log(x)-5)+x^4-25*x^2+132)/(exp(2*x+log(x)-5)^2+(-2*x^2+
24)*exp(2*x+log(x)-5)+x^4-24*x^2+144),x, algorithm="fricas")

[Out]

(x^3 - x*e^(2*x + log(x) - 5) - 11*x)/(x^2 - e^(2*x + log(x) - 5) - 12)

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giac [A]  time = 0.41, size = 42, normalized size = 1.75 $$\begin{array}{c}\frac{x^3{e}^5-x^2{e}^{\left(2x\right)}-11x{e}^5}{x^2{e}^5-x{e}^{\left(2x\right)}-12e^5}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(2*x+log(x)-5)^2+(-2*x^2+2*x+24)*exp(2*x+log(x)-5)+x^4-25*x^2+132)/(exp(2*x+log(x)-5)^2+(-2*x^2+
24)*exp(2*x+log(x)-5)+x^4-24*x^2+144),x, algorithm="giac")

[Out]

(x^3*e^5 - x^2*e^(2*x) - 11*x*e^5)/(x^2*e^5 - x*e^(2*x) - 12*e^5)

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maple [A]  time = 0.16, size = 21, normalized size = 0.88

method	result	size
risch	$x+\frac{x}{x^2-x{\mathrm{e}}^{2x-5}-12}$	$21$
norman	$\frac{x^3-11x-x{\mathrm{e}}^{2x+\ln (x)-5}}{x^2-{\mathrm{e}}^{2x+\ln (x)-5}-12}$	$37$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(2*x+ln(x)-5)^2+(-2*x^2+2*x+24)*exp(2*x+ln(x)-5)+x^4-25*x^2+132)/(exp(2*x+ln(x)-5)^2+(-2*x^2+24)*exp(2
*x+ln(x)-5)+x^4-24*x^2+144),x,method=_RETURNVERBOSE)

[Out]

x+x/(x^2-x*exp(2*x-5)-12)

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maxima [A]  time = 0.47, size = 42, normalized size = 1.75 $$\begin{array}{c}\frac{x^3{e}^5-x^2{e}^{\left(2x\right)}-11x{e}^5}{x^2{e}^5-x{e}^{\left(2x\right)}-12e^5}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(2*x+log(x)-5)^2+(-2*x^2+2*x+24)*exp(2*x+log(x)-5)+x^4-25*x^2+132)/(exp(2*x+log(x)-5)^2+(-2*x^2+
24)*exp(2*x+log(x)-5)+x^4-24*x^2+144),x, algorithm="maxima")

[Out]

(x^3*e^5 - x^2*e^(2*x) - 11*x*e^5)/(x^2*e^5 - x*e^(2*x) - 12*e^5)

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mupad [B]  time = 1.04, size = 22, normalized size = 0.92 $$\begin{array}{c}x-\frac{x}{x{\mathrm{e}}^{2x-5}-x^2+12}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(4*x + 2*log(x) - 10) + exp(2*x + log(x) - 5)*(2*x - 2*x^2 + 24) - 25*x^2 + x^4 + 132)/(exp(4*x + 2*lo
g(x) - 10) - exp(2*x + log(x) - 5)*(2*x^2 - 24) - 24*x^2 + x^4 + 144),x)

[Out]

x - x/(x*exp(2*x - 5) - x^2 + 12)

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sympy [A]  time = 0.16, size = 15, normalized size = 0.62 $$\begin{array}{c}x-\frac{x}{-x^2+x{e}^{2x-5}+12}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(2*x+ln(x)-5)**2+(-2*x**2+2*x+24)*exp(2*x+ln(x)-5)+x**4-25*x**2+132)/(exp(2*x+ln(x)-5)**2+(-2*x*
*2+24)*exp(2*x+ln(x)-5)+x**4-24*x**2+144),x)

[Out]

x - x/(-x**2 + x*exp(2*x - 5) + 12)

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