3.47.82 \(\int \frac {e^2 (144-288 x+172 x^2-40 x^4+16 x^5-2 x^6)}{1296-3744 x+5296 x^2-4680 x^3+2792 x^4-1144 x^5+313 x^6-52 x^7+4 x^8} \, dx\)

Optimal. Leaf size=29 \[ \frac {e^2 x}{-x+x^2+\left (3-x-\frac {x}{-2+x}\right )^2} \]

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Rubi [F]  time = 0.57, 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^2 \left (144-288 x+172 x^2-40 x^4+16 x^5-2 x^6\right )}{1296-3744 x+5296 x^2-4680 x^3+2792 x^4-1144 x^5+313 x^6-52 x^7+4 x^8} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^2*(144 - 288*x + 172*x^2 - 40*x^4 + 16*x^5 - 2*x^6))/(1296 - 3744*x + 5296*x^2 - 4680*x^3 + 2792*x^4 -
1144*x^5 + 313*x^6 - 52*x^7 + 4*x^8),x]

[Out]

(21*E^2)/(32*(36 - 52*x + 36*x^2 - 13*x^3 + 2*x^4)) - (1353*E^2*Defer[Int][(36 - 52*x + 36*x^2 - 13*x^3 + 2*x^
4)^(-2), x])/8 + (433*E^2*Defer[Int][x/(36 - 52*x + 36*x^2 - 13*x^3 + 2*x^4)^2, x])/4 - (595*E^2*Defer[Int][x^
2/(36 - 52*x + 36*x^2 - 13*x^3 + 2*x^4)^2, x])/32 + (31*E^2*Defer[Int][(36 - 52*x + 36*x^2 - 13*x^3 + 2*x^4)^(
-1), x])/4 + (3*E^2*Defer[Int][x/(36 - 52*x + 36*x^2 - 13*x^3 + 2*x^4), x])/2 - E^2*Defer[Int][x^2/(36 - 52*x
+ 36*x^2 - 13*x^3 + 2*x^4), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=e^2 \int \frac {144-288 x+172 x^2-40 x^4+16 x^5-2 x^6}{1296-3744 x+5296 x^2-4680 x^3+2792 x^4-1144 x^5+313 x^6-52 x^7+4 x^8} \, dx\\ &=e^2 \int \left (\frac {-540+244 x+28 x^2-21 x^3}{4 \left (36-52 x+36 x^2-13 x^3+2 x^4\right )^2}+\frac {31+6 x-4 x^2}{4 \left (36-52 x+36 x^2-13 x^3+2 x^4\right )}\right ) \, dx\\ &=\frac {1}{4} e^2 \int \frac {-540+244 x+28 x^2-21 x^3}{\left (36-52 x+36 x^2-13 x^3+2 x^4\right )^2} \, dx+\frac {1}{4} e^2 \int \frac {31+6 x-4 x^2}{36-52 x+36 x^2-13 x^3+2 x^4} \, dx\\ &=\frac {21 e^2}{32 \left (36-52 x+36 x^2-13 x^3+2 x^4\right )}+\frac {1}{32} e^2 \int \frac {-5412+3464 x-595 x^2}{\left (36-52 x+36 x^2-13 x^3+2 x^4\right )^2} \, dx+\frac {1}{4} e^2 \int \left (\frac {31}{36-52 x+36 x^2-13 x^3+2 x^4}+\frac {6 x}{36-52 x+36 x^2-13 x^3+2 x^4}-\frac {4 x^2}{36-52 x+36 x^2-13 x^3+2 x^4}\right ) \, dx\\ &=\frac {21 e^2}{32 \left (36-52 x+36 x^2-13 x^3+2 x^4\right )}+\frac {1}{32} e^2 \int \left (-\frac {5412}{\left (36-52 x+36 x^2-13 x^3+2 x^4\right )^2}+\frac {3464 x}{\left (36-52 x+36 x^2-13 x^3+2 x^4\right )^2}-\frac {595 x^2}{\left (36-52 x+36 x^2-13 x^3+2 x^4\right )^2}\right ) \, dx-e^2 \int \frac {x^2}{36-52 x+36 x^2-13 x^3+2 x^4} \, dx+\frac {1}{2} \left (3 e^2\right ) \int \frac {x}{36-52 x+36 x^2-13 x^3+2 x^4} \, dx+\frac {1}{4} \left (31 e^2\right ) \int \frac {1}{36-52 x+36 x^2-13 x^3+2 x^4} \, dx\\ &=\frac {21 e^2}{32 \left (36-52 x+36 x^2-13 x^3+2 x^4\right )}-e^2 \int \frac {x^2}{36-52 x+36 x^2-13 x^3+2 x^4} \, dx+\frac {1}{2} \left (3 e^2\right ) \int \frac {x}{36-52 x+36 x^2-13 x^3+2 x^4} \, dx+\frac {1}{4} \left (31 e^2\right ) \int \frac {1}{36-52 x+36 x^2-13 x^3+2 x^4} \, dx-\frac {1}{32} \left (595 e^2\right ) \int \frac {x^2}{\left (36-52 x+36 x^2-13 x^3+2 x^4\right )^2} \, dx+\frac {1}{4} \left (433 e^2\right ) \int \frac {x}{\left (36-52 x+36 x^2-13 x^3+2 x^4\right )^2} \, dx-\frac {1}{8} \left (1353 e^2\right ) \int \frac {1}{\left (36-52 x+36 x^2-13 x^3+2 x^4\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 33, normalized size = 1.14 \begin {gather*} \frac {2 e^2 (-2+x)^2 x}{72-104 x+72 x^2-26 x^3+4 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^2*(144 - 288*x + 172*x^2 - 40*x^4 + 16*x^5 - 2*x^6))/(1296 - 3744*x + 5296*x^2 - 4680*x^3 + 2792*
x^4 - 1144*x^5 + 313*x^6 - 52*x^7 + 4*x^8),x]

[Out]

(2*E^2*(-2 + x)^2*x)/(72 - 104*x + 72*x^2 - 26*x^3 + 4*x^4)

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fricas [A]  time = 0.57, size = 37, normalized size = 1.28 \begin {gather*} \frac {{\left (x^{3} - 4 \, x^{2} + 4 \, x\right )} e^{2}}{2 \, x^{4} - 13 \, x^{3} + 36 \, x^{2} - 52 \, x + 36} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^6+16*x^5-40*x^4+172*x^2-288*x+144)*exp(2)/(4*x^8-52*x^7+313*x^6-1144*x^5+2792*x^4-4680*x^3+529
6*x^2-3744*x+1296),x, algorithm="fricas")

[Out]

(x^3 - 4*x^2 + 4*x)*e^2/(2*x^4 - 13*x^3 + 36*x^2 - 52*x + 36)

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giac [A]  time = 0.14, size = 37, normalized size = 1.28 \begin {gather*} \frac {{\left (x^{3} - 4 \, x^{2} + 4 \, x\right )} e^{2}}{2 \, x^{4} - 13 \, x^{3} + 36 \, x^{2} - 52 \, x + 36} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^6+16*x^5-40*x^4+172*x^2-288*x+144)*exp(2)/(4*x^8-52*x^7+313*x^6-1144*x^5+2792*x^4-4680*x^3+529
6*x^2-3744*x+1296),x, algorithm="giac")

[Out]

(x^3 - 4*x^2 + 4*x)*e^2/(2*x^4 - 13*x^3 + 36*x^2 - 52*x + 36)

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maple [A]  time = 0.06, size = 32, normalized size = 1.10

method	result	size
gosper	\(\frac {x \left (x -2\right )^{2} {\mathrm e}^{2}}{2 x^{4}-13 x^{3}+36 x^{2}-52 x +36}\)	\(32\)
default	\(-\frac {2 \,{\mathrm e}^{2} \left (-\frac {1}{4} x^{3}+x^{2}-x \right )}{x^{4}-\frac {13}{2} x^{3}+18 x^{2}-26 x +18}\)	\(37\)
risch	\(\frac {{\mathrm e}^{2} \left (\frac {1}{2} x^{3}-2 x^{2}+2 x \right )}{x^{4}-\frac {13}{2} x^{3}+18 x^{2}-26 x +18}\)	\(38\)
norman	\(\frac {4 \,{\mathrm e}^{2} x -4 x^{2} {\mathrm e}^{2}+x^{3} {\mathrm e}^{2}}{2 x^{4}-13 x^{3}+36 x^{2}-52 x +36}\)	\(43\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x^6+16*x^5-40*x^4+172*x^2-288*x+144)*exp(2)/(4*x^8-52*x^7+313*x^6-1144*x^5+2792*x^4-4680*x^3+5296*x^2-
3744*x+1296),x,method=_RETURNVERBOSE)

[Out]

x*(x-2)^2*exp(2)/(2*x^4-13*x^3+36*x^2-52*x+36)

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maxima [A]  time = 0.36, size = 37, normalized size = 1.28 \begin {gather*} \frac {{\left (x^{3} - 4 \, x^{2} + 4 \, x\right )} e^{2}}{2 \, x^{4} - 13 \, x^{3} + 36 \, x^{2} - 52 \, x + 36} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^6+16*x^5-40*x^4+172*x^2-288*x+144)*exp(2)/(4*x^8-52*x^7+313*x^6-1144*x^5+2792*x^4-4680*x^3+529
6*x^2-3744*x+1296),x, algorithm="maxima")

[Out]

(x^3 - 4*x^2 + 4*x)*e^2/(2*x^4 - 13*x^3 + 36*x^2 - 52*x + 36)

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mupad [B]  time = 3.13, size = 31, normalized size = 1.07 \begin {gather*} \frac {x\,{\mathrm {e}}^2\,{\left (x-2\right )}^2}{2\,x^4-13\,x^3+36\,x^2-52\,x+36} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(2)*(288*x - 172*x^2 + 40*x^4 - 16*x^5 + 2*x^6 - 144))/(5296*x^2 - 3744*x - 4680*x^3 + 2792*x^4 - 114
4*x^5 + 313*x^6 - 52*x^7 + 4*x^8 + 1296),x)

[Out]

(x*exp(2)*(x - 2)^2)/(36*x^2 - 52*x - 13*x^3 + 2*x^4 + 36)

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sympy [B]  time = 0.43, size = 42, normalized size = 1.45 \begin {gather*} - \frac {- x^{3} e^{2} + 4 x^{2} e^{2} - 4 x e^{2}}{2 x^{4} - 13 x^{3} + 36 x^{2} - 52 x + 36} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x**6+16*x**5-40*x**4+172*x**2-288*x+144)*exp(2)/(4*x**8-52*x**7+313*x**6-1144*x**5+2792*x**4-468
0*x**3+5296*x**2-3744*x+1296),x)

[Out]

-(-x**3*exp(2) + 4*x**2*exp(2) - 4*x*exp(2))/(2*x**4 - 13*x**3 + 36*x**2 - 52*x + 36)

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