3.89.90 $\int \frac{e^x(-304+152x^2-19x^4+e^2(-304+152x^2-19x^4))+e^{2x}(28-5x^2+x^4+e^2(28-5x^2+x^4))}{5776-2888x^2+361x^4+e^x(1064x-418x^3+38x^5)+e^{2x}(49x^2-14x^4+x^6)}dx$

Optimal. Leaf size=31 $$\frac{1+e^2}{-19e^{-x}+x-x(2-\frac{3}{-4+x^2})}$$

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Rubi [F]  time = 11.21, 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 = {} $$\begin{array}{c}\int \frac{e^x(-304+152x^2-19x^4+e^2(-304+152x^2-19x^4))+e^{2x}(28-5x^2+x^4+e^2(28-5x^2+x^4))}{5776-2888x^2+361x^4+e^x(1064x-418x^3+38x^5)+e^{2x}(49x^2-14x^4+x^6)}dx\end{array}$$

Verification is not applicable to the result.

[In]

Int[(E^x*(-304 + 152*x^2 - 19*x^4 + E^2*(-304 + 152*x^2 - 19*x^4)) + E^(2*x)*(28 - 5*x^2 + x^4 + E^2*(28 - 5*x
^2 + x^4)))/(5776 - 2888*x^2 + 361*x^4 + E^x*(1064*x - 418*x^3 + 38*x^5) + E^(2*x)*(49*x^2 - 14*x^4 + x^6)),x]

[Out]

-304*(1 + E^2)*Defer[Int][E^x/(-76 - 7*E^x*x + 19*x^2 + E^x*x^3)^2, x] + 171*(1 + E^2)*Defer[Int][E^x/((Sqrt[7
] - x)*(-76 - 7*E^x*x + 19*x^2 + E^x*x^3)^2), x] - 304*(1 + E^2)*Defer[Int][E^x/(x*(-76 - 7*E^x*x + 19*x^2 + E
^x*x^3)^2), x] + 38*(1 + E^2)*Defer[Int][(E^x*x)/(-76 - 7*E^x*x + 19*x^2 + E^x*x^3)^2, x] + 152*(1 + E^2)*Defe
r[Int][(E^x*x^2)/(-76 - 7*E^x*x + 19*x^2 + E^x*x^3)^2, x] - 19*(1 + E^2)*Defer[Int][(E^x*x^3)/(-76 - 7*E^x*x +
19*x^2 + E^x*x^3)^2, x] - 19*(1 + E^2)*Defer[Int][(E^x*x^4)/(-76 - 7*E^x*x + 19*x^2 + E^x*x^3)^2, x] - 171*(1
+ E^2)*Defer[Int][E^x/((Sqrt[7] + x)*(-76 - 7*E^x*x + 19*x^2 + E^x*x^3)^2), x] - 3*(1 + E^2)*Defer[Int][E^x/(
(Sqrt[7] - x)*(-76 - 7*E^x*x + 19*x^2 + E^x*x^3)), x] - 4*(1 + E^2)*Defer[Int][E^x/(x*(-76 - 7*E^x*x + 19*x^2
+ E^x*x^3)), x] + (1 + E^2)*Defer[Int][(E^x*x)/(-76 - 7*E^x*x + 19*x^2 + E^x*x^3), x] + 3*(1 + E^2)*Defer[Int]
[E^x/((Sqrt[7] + x)*(-76 - 7*E^x*x + 19*x^2 + E^x*x^3)), x]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{e^x(1+e^2)(-19{(-4+x^2)}^2+e^x(28-5x^2+x^4))}{{(e^{x}x(-7+x^2)+19(-4+x^2))}^2}dx\\ & =(1+e^2)\int \frac{e^x(-19{(-4+x^2)}^2+e^x(28-5x^2+x^4))}{{(e^{x}x(-7+x^2)+19(-4+x^2))}^2}dx\\ & =(1+e^2)\int (\frac{e^x(28-5x^2+x^4)}{x(-7+x^2)(-76-7e^{x}x+19x^2+e^x{x}^3)}-\frac{19e^x(-112-112x+48x^2+72x^3-9x^4-15x^5+x^6+x^7)}{x(-7+x^2){(-76-7e^{x}x+19x^2+e^x{x}^3)}^2})dx\\ & =(1+e^2)\int \frac{e^x(28-5x^2+x^4)}{x(-7+x^2)(-76-7e^{x}x+19x^2+e^x{x}^3)}dx-\left(19(1+e^2)\right)\int \frac{e^x(-112-112x+48x^2+72x^3-9x^4-15x^5+x^6+x^7)}{x(-7+x^2){(-76-7e^{x}x+19x^2+e^x{x}^3)}^2}dx\\ & =(1+e^2)\int (-\frac{4e^x}{x(-76-7e^{x}x+19x^2+e^x{x}^3)}+\frac{e^{x}x}{-76-7e^{x}x+19x^2+e^x{x}^3}+\frac{6e^{x}x}{(-7+x^2)(-76-7e^{x}x+19x^2+e^x{x}^3)})dx-\left(19(1+e^2)\right)\int (\frac{16e^x}{{(-76-7e^{x}x+19x^2+e^x{x}^3)}^2}+\frac{16e^x}{x{(-76-7e^{x}x+19x^2+e^x{x}^3)}^2}-\frac{2e^{x}x}{{(-76-7e^{x}x+19x^2+e^x{x}^3)}^2}-\frac{8e^x{x}^2}{{(-76-7e^{x}x+19x^2+e^x{x}^3)}^2}+\frac{e^x{x}^3}{{(-76-7e^{x}x+19x^2+e^x{x}^3)}^2}+\frac{e^x{x}^4}{{(-76-7e^{x}x+19x^2+e^x{x}^3)}^2}+\frac{18e^{x}x}{(-7+x^2){(-76-7e^{x}x+19x^2+e^x{x}^3)}^2})dx\\ & =(1+e^2)\int \frac{e^{x}x}{-76-7e^{x}x+19x^2+e^x{x}^3}dx-\left(4(1+e^2)\right)\int \frac{e^x}{x(-76-7e^{x}x+19x^2+e^x{x}^3)}dx+\left(6(1+e^2)\right)\int \frac{e^{x}x}{(-7+x^2)(-76-7e^{x}x+19x^2+e^x{x}^3)}dx-\left(19(1+e^2)\right)\int \frac{e^x{x}^3}{{(-76-7e^{x}x+19x^2+e^x{x}^3)}^2}dx-\left(19(1+e^2)\right)\int \frac{e^x{x}^4}{{(-76-7e^{x}x+19x^2+e^x{x}^3)}^2}dx+\left(38(1+e^2)\right)\int \frac{e^{x}x}{{(-76-7e^{x}x+19x^2+e^x{x}^3)}^2}dx+\left(152(1+e^2)\right)\int \frac{e^x{x}^2}{{(-76-7e^{x}x+19x^2+e^x{x}^3)}^2}dx-\left(304(1+e^2)\right)\int \frac{e^x}{{(-76-7e^{x}x+19x^2+e^x{x}^3)}^2}dx-\left(304(1+e^2)\right)\int \frac{e^x}{x{(-76-7e^{x}x+19x^2+e^x{x}^3)}^2}dx-\left(342(1+e^2)\right)\int \frac{e^{x}x}{(-7+x^2){(-76-7e^{x}x+19x^2+e^x{x}^3)}^2}dx\\ & =(1+e^2)\int \frac{e^{x}x}{-76-7e^{x}x+19x^2+e^x{x}^3}dx-\left(4(1+e^2)\right)\int \frac{e^x}{x(-76-7e^{x}x+19x^2+e^x{x}^3)}dx+\left(6(1+e^2)\right)\int (-\frac{e^x}{2(\sqrt{7}-x)(-76-7e^{x}x+19x^2+e^x{x}^3)}+\frac{e^x}{2(\sqrt{7}+x)(-76-7e^{x}x+19x^2+e^x{x}^3)})dx-\left(19(1+e^2)\right)\int \frac{e^x{x}^3}{{(-76-7e^{x}x+19x^2+e^x{x}^3)}^2}dx-\left(19(1+e^2)\right)\int \frac{e^x{x}^4}{{(-76-7e^{x}x+19x^2+e^x{x}^3)}^2}dx+\left(38(1+e^2)\right)\int \frac{e^{x}x}{{(-76-7e^{x}x+19x^2+e^x{x}^3)}^2}dx+\left(152(1+e^2)\right)\int \frac{e^x{x}^2}{{(-76-7e^{x}x+19x^2+e^x{x}^3)}^2}dx-\left(304(1+e^2)\right)\int \frac{e^x}{{(-76-7e^{x}x+19x^2+e^x{x}^3)}^2}dx-\left(304(1+e^2)\right)\int \frac{e^x}{x{(-76-7e^{x}x+19x^2+e^x{x}^3)}^2}dx-\left(342(1+e^2)\right)\int (-\frac{e^x}{2(\sqrt{7}-x){(-76-7e^{x}x+19x^2+e^x{x}^3)}^2}+\frac{e^x}{2(\sqrt{7}+x){(-76-7e^{x}x+19x^2+e^x{x}^3)}^2})dx\\ & =(1+e^2)\int \frac{e^{x}x}{-76-7e^{x}x+19x^2+e^x{x}^3}dx-\left(3(1+e^2)\right)\int \frac{e^x}{(\sqrt{7}-x)(-76-7e^{x}x+19x^2+e^x{x}^3)}dx+\left(3(1+e^2)\right)\int \frac{e^x}{(\sqrt{7}+x)(-76-7e^{x}x+19x^2+e^x{x}^3)}dx-\left(4(1+e^2)\right)\int \frac{e^x}{x(-76-7e^{x}x+19x^2+e^x{x}^3)}dx-\left(19(1+e^2)\right)\int \frac{e^x{x}^3}{{(-76-7e^{x}x+19x^2+e^x{x}^3)}^2}dx-\left(19(1+e^2)\right)\int \frac{e^x{x}^4}{{(-76-7e^{x}x+19x^2+e^x{x}^3)}^2}dx+\left(38(1+e^2)\right)\int \frac{e^{x}x}{{(-76-7e^{x}x+19x^2+e^x{x}^3)}^2}dx+\left(152(1+e^2)\right)\int \frac{e^x{x}^2}{{(-76-7e^{x}x+19x^2+e^x{x}^3)}^2}dx+\left(171(1+e^2)\right)\int \frac{e^x}{(\sqrt{7}-x){(-76-7e^{x}x+19x^2+e^x{x}^3)}^2}dx-\left(171(1+e^2)\right)\int \frac{e^x}{(\sqrt{7}+x){(-76-7e^{x}x+19x^2+e^x{x}^3)}^2}dx-\left(304(1+e^2)\right)\int \frac{e^x}{{(-76-7e^{x}x+19x^2+e^x{x}^3)}^2}dx-\left(304(1+e^2)\right)\int \frac{e^x}{x{(-76-7e^{x}x+19x^2+e^x{x}^3)}^2}dx\end{array}\end{array}$$

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Mathematica [A]  time = 5.05, size = 35, normalized size = 1.13 $$\begin{array}{c}-\frac{e^x(1+e^2)(-4+x^2)}{e^{x}x(-7+x^2)+19(-4+x^2)}\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(E^x*(-304 + 152*x^2 - 19*x^4 + E^2*(-304 + 152*x^2 - 19*x^4)) + E^(2*x)*(28 - 5*x^2 + x^4 + E^2*(28
- 5*x^2 + x^4)))/(5776 - 2888*x^2 + 361*x^4 + E^x*(1064*x - 418*x^3 + 38*x^5) + E^(2*x)*(49*x^2 - 14*x^4 + x^
6)),x]

[Out]

-((E^x*(1 + E^2)*(-4 + x^2))/(E^x*x*(-7 + x^2) + 19*(-4 + x^2)))

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fricas [A]  time = 0.58, size = 36, normalized size = 1.16 $$\begin{array}{c}-\frac{(x^2+(x^2-4)e^2-4)e^x}{19x^2+(x^3-7x)e^x-76}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^4-5*x^2+28)*exp(2)+x^4-5*x^2+28)*exp(x)^2+((-19*x^4+152*x^2-304)*exp(2)-19*x^4+152*x^2-304)*exp
(x))/((x^6-14*x^4+49*x^2)*exp(x)^2+(38*x^5-418*x^3+1064*x)*exp(x)+361*x^4-2888*x^2+5776),x, algorithm="fricas"
)

[Out]

-(x^2 + (x^2 - 4)*e^2 - 4)*e^x/(19*x^2 + (x^3 - 7*x)*e^x - 76)

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giac [A]  time = 0.23, size = 47, normalized size = 1.52 $$\begin{array}{c}-\frac{x^2{e}^{(x+2)}+x^2{e}^x-4e^{(x+2)}-4e^x}{x^3{e}^x+19x^2-7x{e}^x-76}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^4-5*x^2+28)*exp(2)+x^4-5*x^2+28)*exp(x)^2+((-19*x^4+152*x^2-304)*exp(2)-19*x^4+152*x^2-304)*exp
(x))/((x^6-14*x^4+49*x^2)*exp(x)^2+(38*x^5-418*x^3+1064*x)*exp(x)+361*x^4-2888*x^2+5776),x, algorithm="giac")

[Out]

-(x^2*e^(x + 2) + x^2*e^x - 4*e^(x + 2) - 4*e^x)/(x^3*e^x + 19*x^2 - 7*x*e^x - 76)

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maple [A]  time = 0.18, size = 44, normalized size = 1.42

method	result	size
norman	$\frac{(4{\mathrm{e}}^2+4){\mathrm{e}}^x+(-{\mathrm{e}}^2-1)x^2{\mathrm{e}}^x}{{\mathrm{e}}^x{x}^3-7{\mathrm{e}}^{x}x+19x^2-76}$	$44$
risch	$\frac{(-{\mathrm{e}}^2-1)x^2+4+4{\mathrm{e}}^2}{(x^2-7)x}+\frac{19x^4{\mathrm{e}}^2+19x^4-152x^2{\mathrm{e}}^2-152x^2+304{\mathrm{e}}^2+304}{(x^2-7)x({\mathrm{e}}^x{x}^3-7{\mathrm{e}}^{x}x+19x^2-76)}$	$88$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((x^4-5*x^2+28)*exp(2)+x^4-5*x^2+28)*exp(x)^2+((-19*x^4+152*x^2-304)*exp(2)-19*x^4+152*x^2-304)*exp(x))/(
(x^6-14*x^4+49*x^2)*exp(x)^2+(38*x^5-418*x^3+1064*x)*exp(x)+361*x^4-2888*x^2+5776),x,method=_RETURNVERBOSE)

[Out]

((4*exp(2)+4)*exp(x)+(-exp(2)-1)*x^2*exp(x))/(exp(x)*x^3-7*exp(x)*x+19*x^2-76)

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maxima [A]  time = 0.42, size = 37, normalized size = 1.19 $$\begin{array}{c}-\frac{(x^2(e^2+1)-4e^2-4)e^x}{19x^2+(x^3-7x)e^x-76}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^4-5*x^2+28)*exp(2)+x^4-5*x^2+28)*exp(x)^2+((-19*x^4+152*x^2-304)*exp(2)-19*x^4+152*x^2-304)*exp
(x))/((x^6-14*x^4+49*x^2)*exp(x)^2+(38*x^5-418*x^3+1064*x)*exp(x)+361*x^4-2888*x^2+5776),x, algorithm="maxima"
)

[Out]

-(x^2*(e^2 + 1) - 4*e^2 - 4)*e^x/(19*x^2 + (x^3 - 7*x)*e^x - 76)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 $$\begin{array}{c}\int \frac{{\mathrm{e}}^{2x}({\mathrm{e}}^2(x^4-5x^2+28)-5x^2+x^4+28)-{\mathrm{e}}^x({\mathrm{e}}^2(19x^4-152x^2+304)-152x^2+19x^4+304)}{{\mathrm{e}}^{2x}(x^6-14x^4+49x^2)-2888x^2+361x^4+{\mathrm{e}}^x(38x^5-418x^3+1064x)+5776}dx\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(2*x)*(exp(2)*(x^4 - 5*x^2 + 28) - 5*x^2 + x^4 + 28) - exp(x)*(exp(2)*(19*x^4 - 152*x^2 + 304) - 152*x
^2 + 19*x^4 + 304))/(exp(2*x)*(49*x^2 - 14*x^4 + x^6) - 2888*x^2 + 361*x^4 + exp(x)*(1064*x - 418*x^3 + 38*x^5
) + 5776),x)

[Out]

int((exp(2*x)*(exp(2)*(x^4 - 5*x^2 + 28) - 5*x^2 + x^4 + 28) - exp(x)*(exp(2)*(19*x^4 - 152*x^2 + 304) - 152*x
^2 + 19*x^4 + 304))/(exp(2*x)*(49*x^2 - 14*x^4 + x^6) - 2888*x^2 + 361*x^4 + exp(x)*(1064*x - 418*x^3 + 38*x^5
) + 5776), x)

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sympy [B]  time = 0.64, size = 87, normalized size = 2.81 $$\begin{array}{c}\frac{19x^4+19x^4{e}^2-152x^2{e}^2-152x^2+304+304e^2}{19x^5-209x^3+532x+(x^6-14x^4+49x^2)e^x}+\frac{x^2(-e^2-1)+4+4e^2}{x^3-7x}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x**4-5*x**2+28)*exp(2)+x**4-5*x**2+28)*exp(x)**2+((-19*x**4+152*x**2-304)*exp(2)-19*x**4+152*x**2
-304)*exp(x))/((x**6-14*x**4+49*x**2)*exp(x)**2+(38*x**5-418*x**3+1064*x)*exp(x)+361*x**4-2888*x**2+5776),x)

[Out]

(19*x**4 + 19*x**4*exp(2) - 152*x**2*exp(2) - 152*x**2 + 304 + 304*exp(2))/(19*x**5 - 209*x**3 + 532*x + (x**6
- 14*x**4 + 49*x**2)*exp(x)) + (x**2*(-exp(2) - 1) + 4 + 4*exp(2))/(x**3 - 7*x)

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