3.32.80 \(\int \frac {e^{5+x+\frac {e^{5+x} (244-40 x+4 x^2)}{36 x+9 e^{2 x} x+36 x^2}} (-976-976 x+992 x^2-144 x^3+16 x^4+e^{2 x} (-244-244 x+44 x^2-4 x^3))}{144 x^2+9 e^{4 x} x^2+288 x^3+144 x^4+e^{2 x} (72 x^2+72 x^3)} \, dx\)

Optimal. Leaf size=38 \[ e^{\frac {e^{5+x} \left (4+\frac {1}{9} (5-x)^2\right )}{x+x \left (\frac {e^{2 x}}{4}+x\right )}} \]

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Rubi [F]  time = 21.86, 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 {\exp \left (5+x+\frac {e^{5+x} \left (244-40 x+4 x^2\right )}{36 x+9 e^{2 x} x+36 x^2}\right ) \left (-976-976 x+992 x^2-144 x^3+16 x^4+e^{2 x} \left (-244-244 x+44 x^2-4 x^3\right )\right )}{144 x^2+9 e^{4 x} x^2+288 x^3+144 x^4+e^{2 x} \left (72 x^2+72 x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(5 + x + (E^(5 + x)*(244 - 40*x + 4*x^2))/(36*x + 9*E^(2*x)*x + 36*x^2))*(-976 - 976*x + 992*x^2 - 144*
x^3 + 16*x^4 + E^(2*x)*(-244 - 244*x + 44*x^2 - 4*x^3)))/(144*x^2 + 9*E^(4*x)*x^2 + 288*x^3 + 144*x^4 + E^(2*x
)*(72*x^2 + 72*x^3)),x]

[Out]

(1792*Defer[Int][E^(5 + x + (E^(5 + x)*(244 - 40*x + 4*x^2))/(36*x + 9*E^(2*x)*x + 36*x^2))/(4 + E^(2*x) + 4*x
)^2, x])/9 + (976*Defer[Int][E^(5 + x + (E^(5 + x)*(244 - 40*x + 4*x^2))/(36*x + 9*E^(2*x)*x + 36*x^2))/(x*(4
+ E^(2*x) + 4*x)^2), x])/9 - (304*Defer[Int][(E^(5 + x + (E^(5 + x)*(244 - 40*x + 4*x^2))/(36*x + 9*E^(2*x)*x
+ 36*x^2))*x)/(4 + E^(2*x) + 4*x)^2, x])/9 + (32*Defer[Int][(E^(5 + x + (E^(5 + x)*(244 - 40*x + 4*x^2))/(36*x
+ 9*E^(2*x)*x + 36*x^2))*x^2)/(4 + E^(2*x) + 4*x)^2, x])/9 + (44*Defer[Int][E^(5 + x + (E^(5 + x)*(244 - 40*x
+ 4*x^2))/(36*x + 9*E^(2*x)*x + 36*x^2))/(4 + E^(2*x) + 4*x), x])/9 - (244*Defer[Int][E^(5 + x + (E^(5 + x)*(
244 - 40*x + 4*x^2))/(36*x + 9*E^(2*x)*x + 36*x^2))/(x^2*(4 + E^(2*x) + 4*x)), x])/9 - (244*Defer[Int][E^(5 +
x + (E^(5 + x)*(244 - 40*x + 4*x^2))/(36*x + 9*E^(2*x)*x + 36*x^2))/(x*(4 + E^(2*x) + 4*x)), x])/9 - (4*Defer[
Int][(E^(5 + x + (E^(5 + x)*(244 - 40*x + 4*x^2))/(36*x + 9*E^(2*x)*x + 36*x^2))*x)/(4 + E^(2*x) + 4*x), x])/9

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (5+x+\frac {e^{5+x} \left (244-40 x+4 x^2\right )}{36 x+9 e^{2 x} x+36 x^2}\right ) \left (-976-976 x+992 x^2-144 x^3+16 x^4+e^{2 x} \left (-244-244 x+44 x^2-4 x^3\right )\right )}{9 x^2 \left (4+e^{2 x}+4 x\right )^2} \, dx\\ &=\frac {1}{9} \int \frac {\exp \left (5+x+\frac {e^{5+x} \left (244-40 x+4 x^2\right )}{36 x+9 e^{2 x} x+36 x^2}\right ) \left (-976-976 x+992 x^2-144 x^3+16 x^4+e^{2 x} \left (-244-244 x+44 x^2-4 x^3\right )\right )}{x^2 \left (4+e^{2 x}+4 x\right )^2} \, dx\\ &=\frac {1}{9} \int \left (-\frac {4 \exp \left (5+x+\frac {e^{5+x} \left (244-40 x+4 x^2\right )}{36 x+9 e^{2 x} x+36 x^2}\right ) \left (61+61 x-11 x^2+x^3\right )}{x^2 \left (4+e^{2 x}+4 x\right )}+\frac {16 \exp \left (5+x+\frac {e^{5+x} \left (244-40 x+4 x^2\right )}{36 x+9 e^{2 x} x+36 x^2}\right ) \left (61+112 x-19 x^2+2 x^3\right )}{x \left (4+e^{2 x}+4 x\right )^2}\right ) \, dx\\ &=-\left (\frac {4}{9} \int \frac {\exp \left (5+x+\frac {e^{5+x} \left (244-40 x+4 x^2\right )}{36 x+9 e^{2 x} x+36 x^2}\right ) \left (61+61 x-11 x^2+x^3\right )}{x^2 \left (4+e^{2 x}+4 x\right )} \, dx\right )+\frac {16}{9} \int \frac {\exp \left (5+x+\frac {e^{5+x} \left (244-40 x+4 x^2\right )}{36 x+9 e^{2 x} x+36 x^2}\right ) \left (61+112 x-19 x^2+2 x^3\right )}{x \left (4+e^{2 x}+4 x\right )^2} \, dx\\ &=-\left (\frac {4}{9} \int \left (-\frac {11 \exp \left (5+x+\frac {e^{5+x} \left (244-40 x+4 x^2\right )}{36 x+9 e^{2 x} x+36 x^2}\right )}{4+e^{2 x}+4 x}+\frac {61 \exp \left (5+x+\frac {e^{5+x} \left (244-40 x+4 x^2\right )}{36 x+9 e^{2 x} x+36 x^2}\right )}{x^2 \left (4+e^{2 x}+4 x\right )}+\frac {61 \exp \left (5+x+\frac {e^{5+x} \left (244-40 x+4 x^2\right )}{36 x+9 e^{2 x} x+36 x^2}\right )}{x \left (4+e^{2 x}+4 x\right )}+\frac {\exp \left (5+x+\frac {e^{5+x} \left (244-40 x+4 x^2\right )}{36 x+9 e^{2 x} x+36 x^2}\right ) x}{4+e^{2 x}+4 x}\right ) \, dx\right )+\frac {16}{9} \int \left (\frac {112 \exp \left (5+x+\frac {e^{5+x} \left (244-40 x+4 x^2\right )}{36 x+9 e^{2 x} x+36 x^2}\right )}{\left (4+e^{2 x}+4 x\right )^2}+\frac {61 \exp \left (5+x+\frac {e^{5+x} \left (244-40 x+4 x^2\right )}{36 x+9 e^{2 x} x+36 x^2}\right )}{x \left (4+e^{2 x}+4 x\right )^2}-\frac {19 \exp \left (5+x+\frac {e^{5+x} \left (244-40 x+4 x^2\right )}{36 x+9 e^{2 x} x+36 x^2}\right ) x}{\left (4+e^{2 x}+4 x\right )^2}+\frac {2 \exp \left (5+x+\frac {e^{5+x} \left (244-40 x+4 x^2\right )}{36 x+9 e^{2 x} x+36 x^2}\right ) x^2}{\left (4+e^{2 x}+4 x\right )^2}\right ) \, dx\\ &=-\left (\frac {4}{9} \int \frac {\exp \left (5+x+\frac {e^{5+x} \left (244-40 x+4 x^2\right )}{36 x+9 e^{2 x} x+36 x^2}\right ) x}{4+e^{2 x}+4 x} \, dx\right )+\frac {32}{9} \int \frac {\exp \left (5+x+\frac {e^{5+x} \left (244-40 x+4 x^2\right )}{36 x+9 e^{2 x} x+36 x^2}\right ) x^2}{\left (4+e^{2 x}+4 x\right )^2} \, dx+\frac {44}{9} \int \frac {\exp \left (5+x+\frac {e^{5+x} \left (244-40 x+4 x^2\right )}{36 x+9 e^{2 x} x+36 x^2}\right )}{4+e^{2 x}+4 x} \, dx-\frac {244}{9} \int \frac {\exp \left (5+x+\frac {e^{5+x} \left (244-40 x+4 x^2\right )}{36 x+9 e^{2 x} x+36 x^2}\right )}{x^2 \left (4+e^{2 x}+4 x\right )} \, dx-\frac {244}{9} \int \frac {\exp \left (5+x+\frac {e^{5+x} \left (244-40 x+4 x^2\right )}{36 x+9 e^{2 x} x+36 x^2}\right )}{x \left (4+e^{2 x}+4 x\right )} \, dx-\frac {304}{9} \int \frac {\exp \left (5+x+\frac {e^{5+x} \left (244-40 x+4 x^2\right )}{36 x+9 e^{2 x} x+36 x^2}\right ) x}{\left (4+e^{2 x}+4 x\right )^2} \, dx+\frac {976}{9} \int \frac {\exp \left (5+x+\frac {e^{5+x} \left (244-40 x+4 x^2\right )}{36 x+9 e^{2 x} x+36 x^2}\right )}{x \left (4+e^{2 x}+4 x\right )^2} \, dx+\frac {1792}{9} \int \frac {\exp \left (5+x+\frac {e^{5+x} \left (244-40 x+4 x^2\right )}{36 x+9 e^{2 x} x+36 x^2}\right )}{\left (4+e^{2 x}+4 x\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.15, size = 34, normalized size = 0.89 \begin {gather*} e^{\frac {4 e^{5+x} \left (61-10 x+x^2\right )}{9 x \left (4+e^{2 x}+4 x\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(5 + x + (E^(5 + x)*(244 - 40*x + 4*x^2))/(36*x + 9*E^(2*x)*x + 36*x^2))*(-976 - 976*x + 992*x^2
- 144*x^3 + 16*x^4 + E^(2*x)*(-244 - 244*x + 44*x^2 - 4*x^3)))/(144*x^2 + 9*E^(4*x)*x^2 + 288*x^3 + 144*x^4 +
E^(2*x)*(72*x^2 + 72*x^3)),x]

[Out]

E^((4*E^(5 + x)*(61 - 10*x + x^2))/(9*x*(4 + E^(2*x) + 4*x)))

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fricas [B]  time = 0.73, size = 74, normalized size = 1.95 \begin {gather*} e^{\left (-x + \frac {36 \, {\left (x^{3} + 6 \, x^{2} + 5 \, x\right )} e^{10} + 9 \, {\left (x^{2} + 5 \, x\right )} e^{\left (2 \, x + 10\right )} + 4 \, {\left (x^{2} - 10 \, x + 61\right )} e^{\left (x + 15\right )}}{9 \, {\left (4 \, {\left (x^{2} + x\right )} e^{10} + x e^{\left (2 \, x + 10\right )}\right )}} - 5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^3+44*x^2-244*x-244)*exp(x)^2+16*x^4-144*x^3+992*x^2-976*x-976)*exp(5+x)*exp((4*x^2-40*x+244)*
exp(5+x)/(9*x*exp(x)^2+36*x^2+36*x))/(9*x^2*exp(x)^4+(72*x^3+72*x^2)*exp(x)^2+144*x^4+288*x^3+144*x^2),x, algo
rithm="fricas")

[Out]

e^(-x + 1/9*(36*(x^3 + 6*x^2 + 5*x)*e^10 + 9*(x^2 + 5*x)*e^(2*x + 10) + 4*(x^2 - 10*x + 61)*e^(x + 15))/(4*(x^
2 + x)*e^10 + x*e^(2*x + 10)) - 5)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^3+44*x^2-244*x-244)*exp(x)^2+16*x^4-144*x^3+992*x^2-976*x-976)*exp(5+x)*exp((4*x^2-40*x+244)*
exp(5+x)/(9*x*exp(x)^2+36*x^2+36*x))/(9*x^2*exp(x)^4+(72*x^3+72*x^2)*exp(x)^2+144*x^4+288*x^3+144*x^2),x, algo
rithm="giac")

[Out]

Timed out

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maple [A]  time = 0.20, size = 30, normalized size = 0.79

method	result	size
risch	\({\mathrm e}^{\frac {4 \left (x^{2}-10 x +61\right ) {\mathrm e}^{5+x}}{9 x \left ({\mathrm e}^{2 x}+4 x +4\right )}}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-4*x^3+44*x^2-244*x-244)*exp(x)^2+16*x^4-144*x^3+992*x^2-976*x-976)*exp(5+x)*exp((4*x^2-40*x+244)*exp(5+
x)/(9*x*exp(x)^2+36*x^2+36*x))/(9*x^2*exp(x)^4+(72*x^3+72*x^2)*exp(x)^2+144*x^4+288*x^3+144*x^2),x,method=_RET
URNVERBOSE)

[Out]

exp(4/9*(x^2-10*x+61)*exp(5+x)/x/(exp(2*x)+4*x+4))

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maxima [B]  time = 1.51, size = 88, normalized size = 2.32 \begin {gather*} e^{\left (-\frac {e^{\left (3 \, x + 5\right )}}{9 \, {\left (4 \, x + e^{\left (2 \, x\right )} + 4\right )}} - \frac {976 \, e^{\left (x + 5\right )}}{9 \, {\left (4 \, {\left (x + 2\right )} e^{\left (2 \, x\right )} + 16 \, x + e^{\left (4 \, x\right )} + 16\right )}} + \frac {244 \, e^{\left (x + 5\right )}}{9 \, {\left (x e^{\left (2 \, x\right )} + 4 \, x\right )}} - \frac {44 \, e^{\left (x + 5\right )}}{9 \, {\left (4 \, x + e^{\left (2 \, x\right )} + 4\right )}} + \frac {1}{9} \, e^{\left (x + 5\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^3+44*x^2-244*x-244)*exp(x)^2+16*x^4-144*x^3+992*x^2-976*x-976)*exp(5+x)*exp((4*x^2-40*x+244)*
exp(5+x)/(9*x*exp(x)^2+36*x^2+36*x))/(9*x^2*exp(x)^4+(72*x^3+72*x^2)*exp(x)^2+144*x^4+288*x^3+144*x^2),x, algo
rithm="maxima")

[Out]

e^(-1/9*e^(3*x + 5)/(4*x + e^(2*x) + 4) - 976/9*e^(x + 5)/(4*(x + 2)*e^(2*x) + 16*x + e^(4*x) + 16) + 244/9*e^
(x + 5)/(x*e^(2*x) + 4*x) - 44/9*e^(x + 5)/(4*x + e^(2*x) + 4) + 1/9*e^(x + 5))

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mupad [B]  time = 2.65, size = 67, normalized size = 1.76 \begin {gather*} {\mathrm {e}}^{-\frac {40\,{\mathrm {e}}^5\,{\mathrm {e}}^x}{36\,x+9\,{\mathrm {e}}^{2\,x}+36}}\,{\mathrm {e}}^{\frac {4\,x\,{\mathrm {e}}^5\,{\mathrm {e}}^x}{36\,x+9\,{\mathrm {e}}^{2\,x}+36}}\,{\mathrm {e}}^{\frac {244\,{\mathrm {e}}^5\,{\mathrm {e}}^x}{36\,x+9\,x\,{\mathrm {e}}^{2\,x}+36\,x^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(x + 5)*exp((exp(x + 5)*(4*x^2 - 40*x + 244))/(36*x + 9*x*exp(2*x) + 36*x^2))*(976*x + exp(2*x)*(244*
x - 44*x^2 + 4*x^3 + 244) - 992*x^2 + 144*x^3 - 16*x^4 + 976))/(exp(2*x)*(72*x^2 + 72*x^3) + 9*x^2*exp(4*x) +
144*x^2 + 288*x^3 + 144*x^4),x)

[Out]

exp(-(40*exp(5)*exp(x))/(36*x + 9*exp(2*x) + 36))*exp((4*x*exp(5)*exp(x))/(36*x + 9*exp(2*x) + 36))*exp((244*e
xp(5)*exp(x))/(36*x + 9*x*exp(2*x) + 36*x^2))

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sympy [A]  time = 0.76, size = 39, normalized size = 1.03 \begin {gather*} e^{\frac {\left (4 x^{2} - 40 x + 244\right ) e^{5} \sqrt {e^{2 x}}}{36 x^{2} + 9 x e^{2 x} + 36 x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x**3+44*x**2-244*x-244)*exp(x)**2+16*x**4-144*x**3+992*x**2-976*x-976)*exp(5+x)*exp((4*x**2-40*
x+244)*exp(5+x)/(9*x*exp(x)**2+36*x**2+36*x))/(9*x**2*exp(x)**4+(72*x**3+72*x**2)*exp(x)**2+144*x**4+288*x**3+
144*x**2),x)

[Out]

exp((4*x**2 - 40*x + 244)*exp(5)*sqrt(exp(2*x))/(36*x**2 + 9*x*exp(2*x) + 36*x))

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