3.13.51 \(\int \frac {-60 x-21 x^2+5 x^3+135 x^4+135 x^5+99 x^6+59 x^7+18 x^8+2 x^9+e^{2 x} (135 x^2+135 x^3+99 x^4+59 x^5+18 x^6+2 x^7)+e^x (45+34 x-2 x^2-271 x^3-270 x^4-198 x^5-118 x^6-36 x^7-4 x^8)}{54 x^6+54 x^7+18 x^8+2 x^9+e^{2 x} (54 x^4+54 x^5+18 x^6+2 x^7)+e^x (-108 x^5-108 x^6-36 x^7-4 x^8)} \, dx\)

Optimal. Leaf size=37 \[ 4+x+\frac {(5-x) \left (-x+\frac {1}{x (3+x)^2 \left (-e^x+x\right )}\right )}{2 x^2} \]

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Rubi [F]  time = 3.13, 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 {-60 x-21 x^2+5 x^3+135 x^4+135 x^5+99 x^6+59 x^7+18 x^8+2 x^9+e^{2 x} \left (135 x^2+135 x^3+99 x^4+59 x^5+18 x^6+2 x^7\right )+e^x \left (45+34 x-2 x^2-271 x^3-270 x^4-198 x^5-118 x^6-36 x^7-4 x^8\right )}{54 x^6+54 x^7+18 x^8+2 x^9+e^{2 x} \left (54 x^4+54 x^5+18 x^6+2 x^7\right )+e^x \left (-108 x^5-108 x^6-36 x^7-4 x^8\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-60*x - 21*x^2 + 5*x^3 + 135*x^4 + 135*x^5 + 99*x^6 + 59*x^7 + 18*x^8 + 2*x^9 + E^(2*x)*(135*x^2 + 135*x^
3 + 99*x^4 + 59*x^5 + 18*x^6 + 2*x^7) + E^x*(45 + 34*x - 2*x^2 - 271*x^3 - 270*x^4 - 198*x^5 - 118*x^6 - 36*x^
7 - 4*x^8))/(54*x^6 + 54*x^7 + 18*x^8 + 2*x^9 + E^(2*x)*(54*x^4 + 54*x^5 + 18*x^6 + 2*x^7) + E^x*(-108*x^5 - 1
08*x^6 - 36*x^7 - 4*x^8)),x]

[Out]

-5/(2*x) + x + (5*Defer[Int][1/((E^x - x)*x^4), x])/6 - (5*Defer[Int][1/((E^x - x)^2*x^3), x])/18 - (11*Defer[
Int][1/((E^x - x)*x^3), x])/54 + (14*Defer[Int][1/((E^x - x)^2*x^2), x])/27 - Defer[Int][1/((E^x - x)*x^2), x]
/9 - (10*Defer[Int][1/((E^x - x)^2*x), x])/27 + (7*Defer[Int][1/((E^x - x)*x), x])/54 - (8*Defer[Int][1/((E^x
- x)*(3 + x)^3), x])/27 + (16*Defer[Int][1/((E^x - x)^2*(3 + x)^2), x])/27 - (5*Defer[Int][1/((E^x - x)*(3 + x
)^2), x])/18 + (10*Defer[Int][1/((E^x - x)^2*(3 + x)), x])/27 - (7*Defer[Int][1/((E^x - x)*(3 + x)), x])/54

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{2 x} x^2 (3+x)^3 \left (5+2 x^2\right )+x \left (-60-21 x+5 x^2+135 x^3+135 x^4+99 x^5+59 x^6+18 x^7+2 x^8\right )-e^x \left (-45-34 x+2 x^2+271 x^3+270 x^4+198 x^5+118 x^6+36 x^7+4 x^8\right )}{2 \left (e^x-x\right )^2 x^4 (3+x)^3} \, dx\\ &=\frac {1}{2} \int \frac {e^{2 x} x^2 (3+x)^3 \left (5+2 x^2\right )+x \left (-60-21 x+5 x^2+135 x^3+135 x^4+99 x^5+59 x^6+18 x^7+2 x^8\right )-e^x \left (-45-34 x+2 x^2+271 x^3+270 x^4+198 x^5+118 x^6+36 x^7+4 x^8\right )}{\left (e^x-x\right )^2 x^4 (3+x)^3} \, dx\\ &=\frac {1}{2} \int \left (-\frac {5-6 x+x^2}{\left (e^x-x\right )^2 x^3 (3+x)^2}+\frac {5+2 x^2}{x^2}-\frac {-45-34 x+2 x^2+x^3}{\left (e^x-x\right ) x^4 (3+x)^3}\right ) \, dx\\ &=-\left (\frac {1}{2} \int \frac {5-6 x+x^2}{\left (e^x-x\right )^2 x^3 (3+x)^2} \, dx\right )+\frac {1}{2} \int \frac {5+2 x^2}{x^2} \, dx-\frac {1}{2} \int \frac {-45-34 x+2 x^2+x^3}{\left (e^x-x\right ) x^4 (3+x)^3} \, dx\\ &=\frac {1}{2} \int \left (2+\frac {5}{x^2}\right ) \, dx-\frac {1}{2} \int \left (\frac {5}{9 \left (e^x-x\right )^2 x^3}-\frac {28}{27 \left (e^x-x\right )^2 x^2}+\frac {20}{27 \left (e^x-x\right )^2 x}-\frac {32}{27 \left (e^x-x\right )^2 (3+x)^2}-\frac {20}{27 \left (e^x-x\right )^2 (3+x)}\right ) \, dx-\frac {1}{2} \int \left (-\frac {5}{3 \left (e^x-x\right ) x^4}+\frac {11}{27 \left (e^x-x\right ) x^3}+\frac {2}{9 \left (e^x-x\right ) x^2}-\frac {7}{27 \left (e^x-x\right ) x}+\frac {16}{27 \left (e^x-x\right ) (3+x)^3}+\frac {5}{9 \left (e^x-x\right ) (3+x)^2}+\frac {7}{27 \left (e^x-x\right ) (3+x)}\right ) \, dx\\ &=-\frac {5}{2 x}+x-\frac {1}{9} \int \frac {1}{\left (e^x-x\right ) x^2} \, dx+\frac {7}{54} \int \frac {1}{\left (e^x-x\right ) x} \, dx-\frac {7}{54} \int \frac {1}{\left (e^x-x\right ) (3+x)} \, dx-\frac {11}{54} \int \frac {1}{\left (e^x-x\right ) x^3} \, dx-\frac {5}{18} \int \frac {1}{\left (e^x-x\right )^2 x^3} \, dx-\frac {5}{18} \int \frac {1}{\left (e^x-x\right ) (3+x)^2} \, dx-\frac {8}{27} \int \frac {1}{\left (e^x-x\right ) (3+x)^3} \, dx-\frac {10}{27} \int \frac {1}{\left (e^x-x\right )^2 x} \, dx+\frac {10}{27} \int \frac {1}{\left (e^x-x\right )^2 (3+x)} \, dx+\frac {14}{27} \int \frac {1}{\left (e^x-x\right )^2 x^2} \, dx+\frac {16}{27} \int \frac {1}{\left (e^x-x\right )^2 (3+x)^2} \, dx+\frac {5}{6} \int \frac {1}{\left (e^x-x\right ) x^4} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 34, normalized size = 0.92 \begin {gather*} \frac {1}{2} \left (-\frac {5}{x}+2 x+\frac {-5+x}{\left (e^x-x\right ) x^3 (3+x)^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-60*x - 21*x^2 + 5*x^3 + 135*x^4 + 135*x^5 + 99*x^6 + 59*x^7 + 18*x^8 + 2*x^9 + E^(2*x)*(135*x^2 +
135*x^3 + 99*x^4 + 59*x^5 + 18*x^6 + 2*x^7) + E^x*(45 + 34*x - 2*x^2 - 271*x^3 - 270*x^4 - 198*x^5 - 118*x^6 -
36*x^7 - 4*x^8))/(54*x^6 + 54*x^7 + 18*x^8 + 2*x^9 + E^(2*x)*(54*x^4 + 54*x^5 + 18*x^6 + 2*x^7) + E^x*(-108*x
^5 - 108*x^6 - 36*x^7 - 4*x^8)),x]

[Out]

(-5/x + 2*x + (-5 + x)/((E^x - x)*x^3*(3 + x)^2))/2

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fricas [B]  time = 0.61, size = 96, normalized size = 2.59 \begin {gather*} \frac {2 \, x^{7} + 12 \, x^{6} + 13 \, x^{5} - 30 \, x^{4} - 45 \, x^{3} - {\left (2 \, x^{6} + 12 \, x^{5} + 13 \, x^{4} - 30 \, x^{3} - 45 \, x^{2}\right )} e^{x} - x + 5}{2 \, {\left (x^{6} + 6 \, x^{5} + 9 \, x^{4} - {\left (x^{5} + 6 \, x^{4} + 9 \, x^{3}\right )} e^{x}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^7+18*x^6+59*x^5+99*x^4+135*x^3+135*x^2)*exp(x)^2+(-4*x^8-36*x^7-118*x^6-198*x^5-270*x^4-271*x^
3-2*x^2+34*x+45)*exp(x)+2*x^9+18*x^8+59*x^7+99*x^6+135*x^5+135*x^4+5*x^3-21*x^2-60*x)/((2*x^7+18*x^6+54*x^5+54
*x^4)*exp(x)^2+(-4*x^8-36*x^7-108*x^6-108*x^5)*exp(x)+2*x^9+18*x^8+54*x^7+54*x^6),x, algorithm="fricas")

[Out]

1/2*(2*x^7 + 12*x^6 + 13*x^5 - 30*x^4 - 45*x^3 - (2*x^6 + 12*x^5 + 13*x^4 - 30*x^3 - 45*x^2)*e^x - x + 5)/(x^6
+ 6*x^5 + 9*x^4 - (x^5 + 6*x^4 + 9*x^3)*e^x)

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giac [B]  time = 0.39, size = 104, normalized size = 2.81 \begin {gather*} \frac {2 \, x^{7} - 2 \, x^{6} e^{x} + 12 \, x^{6} - 12 \, x^{5} e^{x} + 13 \, x^{5} - 13 \, x^{4} e^{x} - 30 \, x^{4} + 30 \, x^{3} e^{x} - 45 \, x^{3} + 45 \, x^{2} e^{x} - x + 5}{2 \, {\left (x^{6} - x^{5} e^{x} + 6 \, x^{5} - 6 \, x^{4} e^{x} + 9 \, x^{4} - 9 \, x^{3} e^{x}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^7+18*x^6+59*x^5+99*x^4+135*x^3+135*x^2)*exp(x)^2+(-4*x^8-36*x^7-118*x^6-198*x^5-270*x^4-271*x^
3-2*x^2+34*x+45)*exp(x)+2*x^9+18*x^8+59*x^7+99*x^6+135*x^5+135*x^4+5*x^3-21*x^2-60*x)/((2*x^7+18*x^6+54*x^5+54
*x^4)*exp(x)^2+(-4*x^8-36*x^7-108*x^6-108*x^5)*exp(x)+2*x^9+18*x^8+54*x^7+54*x^6),x, algorithm="giac")

[Out]

1/2*(2*x^7 - 2*x^6*e^x + 12*x^6 - 12*x^5*e^x + 13*x^5 - 13*x^4*e^x - 30*x^4 + 30*x^3*e^x - 45*x^3 + 45*x^2*e^x
- x + 5)/(x^6 - x^5*e^x + 6*x^5 - 6*x^4*e^x + 9*x^4 - 9*x^3*e^x)

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maple [A]  time = 0.10, size = 34, normalized size = 0.92

method	result	size
risch	\(x -\frac {5}{2 x}-\frac {x -5}{2 x^{3} \left (x^{2}+6 x +9\right ) \left (x -{\mathrm e}^{x}\right )}\)	\(34\)
norman	\(\frac {\frac {5}{2}+x^{7}-69 x^{4}-\frac {59 x^{5}}{2}+69 \,{\mathrm e}^{x} x^{3}+\frac {59 \,{\mathrm e}^{x} x^{4}}{2}-\frac {x}{2}-\frac {45 x^{3}}{2}-x^{6} {\mathrm e}^{x}+\frac {45 \,{\mathrm e}^{x} x^{2}}{2}}{x^{3} \left (x -{\mathrm e}^{x}\right ) \left (3+x \right )^{2}}\)	\(69\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^7+18*x^6+59*x^5+99*x^4+135*x^3+135*x^2)*exp(x)^2+(-4*x^8-36*x^7-118*x^6-198*x^5-270*x^4-271*x^3-2*x^
2+34*x+45)*exp(x)+2*x^9+18*x^8+59*x^7+99*x^6+135*x^5+135*x^4+5*x^3-21*x^2-60*x)/((2*x^7+18*x^6+54*x^5+54*x^4)*
exp(x)^2+(-4*x^8-36*x^7-108*x^6-108*x^5)*exp(x)+2*x^9+18*x^8+54*x^7+54*x^6),x,method=_RETURNVERBOSE)

[Out]

x-5/2/x-1/2*(x-5)/x^3/(x^2+6*x+9)/(x-exp(x))

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maxima [B]  time = 0.65, size = 96, normalized size = 2.59 \begin {gather*} \frac {2 \, x^{7} + 12 \, x^{6} + 13 \, x^{5} - 30 \, x^{4} - 45 \, x^{3} - {\left (2 \, x^{6} + 12 \, x^{5} + 13 \, x^{4} - 30 \, x^{3} - 45 \, x^{2}\right )} e^{x} - x + 5}{2 \, {\left (x^{6} + 6 \, x^{5} + 9 \, x^{4} - {\left (x^{5} + 6 \, x^{4} + 9 \, x^{3}\right )} e^{x}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^7+18*x^6+59*x^5+99*x^4+135*x^3+135*x^2)*exp(x)^2+(-4*x^8-36*x^7-118*x^6-198*x^5-270*x^4-271*x^
3-2*x^2+34*x+45)*exp(x)+2*x^9+18*x^8+59*x^7+99*x^6+135*x^5+135*x^4+5*x^3-21*x^2-60*x)/((2*x^7+18*x^6+54*x^5+54
*x^4)*exp(x)^2+(-4*x^8-36*x^7-108*x^6-108*x^5)*exp(x)+2*x^9+18*x^8+54*x^7+54*x^6),x, algorithm="maxima")

[Out]

1/2*(2*x^7 + 12*x^6 + 13*x^5 - 30*x^4 - 45*x^3 - (2*x^6 + 12*x^5 + 13*x^4 - 30*x^3 - 45*x^2)*e^x - x + 5)/(x^6
+ 6*x^5 + 9*x^4 - (x^5 + 6*x^4 + 9*x^3)*e^x)

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mupad [B]  time = 1.15, size = 45, normalized size = 1.22 \begin {gather*} x-\frac {5}{2\,x}+\frac {-x^3+3\,x^2+13\,x-15}{2\,x^3\,\left (x-{\mathrm {e}}^x\right )\,\left (x-1\right )\,{\left (x+3\right )}^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(2*x)*(135*x^2 + 135*x^3 + 99*x^4 + 59*x^5 + 18*x^6 + 2*x^7) - 60*x - 21*x^2 + 5*x^3 + 135*x^4 + 135*x
^5 + 99*x^6 + 59*x^7 + 18*x^8 + 2*x^9 - exp(x)*(2*x^2 - 34*x + 271*x^3 + 270*x^4 + 198*x^5 + 118*x^6 + 36*x^7
+ 4*x^8 - 45))/(exp(2*x)*(54*x^4 + 54*x^5 + 18*x^6 + 2*x^7) - exp(x)*(108*x^5 + 108*x^6 + 36*x^7 + 4*x^8) + 54
*x^6 + 54*x^7 + 18*x^8 + 2*x^9),x)

[Out]

x - 5/(2*x) + (13*x + 3*x^2 - x^3 - 15)/(2*x^3*(x - exp(x))*(x - 1)*(x + 3)^3)

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sympy [A]  time = 0.24, size = 42, normalized size = 1.14 \begin {gather*} x + \frac {x - 5}{- 2 x^{6} - 12 x^{5} - 18 x^{4} + \left (2 x^{5} + 12 x^{4} + 18 x^{3}\right ) e^{x}} - \frac {5}{2 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**7+18*x**6+59*x**5+99*x**4+135*x**3+135*x**2)*exp(x)**2+(-4*x**8-36*x**7-118*x**6-198*x**5-270
*x**4-271*x**3-2*x**2+34*x+45)*exp(x)+2*x**9+18*x**8+59*x**7+99*x**6+135*x**5+135*x**4+5*x**3-21*x**2-60*x)/((
2*x**7+18*x**6+54*x**5+54*x**4)*exp(x)**2+(-4*x**8-36*x**7-108*x**6-108*x**5)*exp(x)+2*x**9+18*x**8+54*x**7+54
*x**6),x)

[Out]

x + (x - 5)/(-2*x**6 - 12*x**5 - 18*x**4 + (2*x**5 + 12*x**4 + 18*x**3)*exp(x)) - 5/(2*x)

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