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

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

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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 {gather*} \int \frac {e^x \left (-304+152 x^2-19 x^4+e^2 \left (-304+152 x^2-19 x^4\right )\right )+e^{2 x} \left (28-5 x^2+x^4+e^2 \left (28-5 x^2+x^4\right )\right )}{5776-2888 x^2+361 x^4+e^x \left (1064 x-418 x^3+38 x^5\right )+e^{2 x} \left (49 x^2-14 x^4+x^6\right )} \, dx \end {gather*}

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

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

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 {gather*} -\frac {{\left (x^{2} + {\left (x^{2} - 4\right )} e^{2} - 4\right )} e^{x}}{19 \, x^{2} + {\left (x^{3} - 7 \, x\right )} e^{x} - 76} \end {gather*}

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 {gather*} -\frac {x^{2} e^{\left (x + 2\right )} + x^{2} e^{x} - 4 \, e^{\left (x + 2\right )} - 4 \, e^{x}}{x^{3} e^{x} + 19 \, x^{2} - 7 \, x e^{x} - 76} \end {gather*}

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	\(\frac {\left (4 \,{\mathrm e}^{2}+4\right ) {\mathrm e}^{x}+\left (-{\mathrm e}^{2}-1\right ) x^{2} {\mathrm e}^{x}}{{\mathrm e}^{x} x^{3}-7 \,{\mathrm e}^{x} x +19 x^{2}-76}\)	\(44\)
risch	\(\frac {\left (-{\mathrm e}^{2}-1\right ) x^{2}+4+4 \,{\mathrm e}^{2}}{\left (x^{2}-7\right ) x}+\frac {19 x^{4} {\mathrm e}^{2}+19 x^{4}-152 x^{2} {\mathrm e}^{2}-152 x^{2}+304 \,{\mathrm e}^{2}+304}{\left (x^{2}-7\right ) x \left ({\mathrm e}^{x} x^{3}-7 \,{\mathrm e}^{x} x +19 x^{2}-76\right )}\)	\(88\)

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 {gather*} -\frac {{\left (x^{2} {\left (e^{2} + 1\right )} - 4 \, e^{2} - 4\right )} e^{x}}{19 \, x^{2} + {\left (x^{3} - 7 \, x\right )} e^{x} - 76} \end {gather*}

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 {gather*} \int \frac {{\mathrm {e}}^{2\,x}\,\left ({\mathrm {e}}^2\,\left (x^4-5\,x^2+28\right )-5\,x^2+x^4+28\right )-{\mathrm {e}}^x\,\left ({\mathrm {e}}^2\,\left (19\,x^4-152\,x^2+304\right )-152\,x^2+19\,x^4+304\right )}{{\mathrm {e}}^{2\,x}\,\left (x^6-14\,x^4+49\,x^2\right )-2888\,x^2+361\,x^4+{\mathrm {e}}^x\,\left (38\,x^5-418\,x^3+1064\,x\right )+5776} \,d x \end {gather*}

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 {gather*} \frac {19 x^{4} + 19 x^{4} e^{2} - 152 x^{2} e^{2} - 152 x^{2} + 304 + 304 e^{2}}{19 x^{5} - 209 x^{3} + 532 x + \left (x^{6} - 14 x^{4} + 49 x^{2}\right ) e^{x}} + \frac {x^{2} \left (- e^{2} - 1\right ) + 4 + 4 e^{2}}{x^{3} - 7 x} \end {gather*}

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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