3.57.100 \(\int \frac {e^4 (152-6 x)+50 x+e^2 (-3800+148 x)+e^{2 x} (-6 e^4+e^2 (-6+6 x))+e^x (2 x-2 x^2+e^4 (-146+6 x)+e^2 (-302+160 x-6 x^2))}{e^{4+3 x}+e^{4+2 x} (75-3 x)+e^{4+x} (1875-150 x+3 x^2)+e^4 (15625-1875 x+75 x^2-x^3)} \, dx\)

Optimal. Leaf size=23 \[ \left (3+\frac {1-\frac {x}{e^2}}{25+e^x-x}\right )^2 \]

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Rubi [F]  time = 1.27, 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^4 (152-6 x)+50 x+e^2 (-3800+148 x)+e^{2 x} \left (-6 e^4+e^2 (-6+6 x)\right )+e^x \left (2 x-2 x^2+e^4 (-146+6 x)+e^2 \left (-302+160 x-6 x^2\right )\right )}{e^{4+3 x}+e^{4+2 x} (75-3 x)+e^{4+x} \left (1875-150 x+3 x^2\right )+e^4 \left (15625-1875 x+75 x^2-x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^4*(152 - 6*x) + 50*x + E^2*(-3800 + 148*x) + E^(2*x)*(-6*E^4 + E^2*(-6 + 6*x)) + E^x*(2*x - 2*x^2 + E^4
*(-146 + 6*x) + E^2*(-302 + 160*x - 6*x^2)))/(E^(4 + 3*x) + E^(4 + 2*x)*(75 - 3*x) + E^(4 + x)*(1875 - 150*x +
3*x^2) + E^4*(15625 - 1875*x + 75*x^2 - x^3)),x]

[Out]

52*Defer[Int][(25 + E^x - x)^(-3), x] + 2*(77 - E^(-2))*Defer[Int][(25 + E^x - x)^(-2), x] - 6*(1 + E^(-2))*De
fer[Int][(25 + E^x - x)^(-1), x] - (2*(52 + E^2)*Defer[Int][x/(25 + E^x - x)^3, x])/E^2 + (2*(1 - 76*E^2 - 3*E
^4)*Defer[Int][x/(25 + E^x - x)^2, x])/E^4 + (6*Defer[Int][x/(25 + E^x - x), x])/E^2 + (4*(13 + E^2)*Defer[Int
][x^2/(25 + E^x - x)^3, x])/E^4 - (2*(1 - 3*E^2)*Defer[Int][x^2/(25 + E^x - x)^2, x])/E^4 - (2*Defer[Int][x^3/
(25 + E^x - x)^3, x])/E^4

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (-3 e^{4+2 x}-e^2 (1900-74 x)-e^{4+x} (73-3 x)+3 e^{2+2 x} (-1+x)+25 x-e^x (-1+x) x-e^4 (-76+3 x)-e^{2+x} \left (151-80 x+3 x^2\right )\right )}{e^4 \left (25+e^x-x\right )^3} \, dx\\ &=\frac {2 \int \frac {-3 e^{4+2 x}-e^2 (1900-74 x)-e^{4+x} (73-3 x)+3 e^{2+2 x} (-1+x)+25 x-e^x (-1+x) x-e^4 (-76+3 x)-e^{2+x} \left (151-80 x+3 x^2\right )}{\left (25+e^x-x\right )^3} \, dx}{e^4}\\ &=\frac {2 \int \left (-\frac {3 e^2 \left (1+e^2-x\right )}{25+e^x-x}-\frac {(-26+x) \left (-e^2+x\right )^2}{\left (25+e^x-x\right )^3}+\frac {-e^2 \left (1-77 e^2\right )+\left (1-76 e^2-3 e^4\right ) x-\left (1-3 e^2\right ) x^2}{\left (25+e^x-x\right )^2}\right ) \, dx}{e^4}\\ &=-\frac {2 \int \frac {(-26+x) \left (-e^2+x\right )^2}{\left (25+e^x-x\right )^3} \, dx}{e^4}+\frac {2 \int \frac {-e^2 \left (1-77 e^2\right )+\left (1-76 e^2-3 e^4\right ) x-\left (1-3 e^2\right ) x^2}{\left (25+e^x-x\right )^2} \, dx}{e^4}-\frac {6 \int \frac {1+e^2-x}{25+e^x-x} \, dx}{e^2}\\ &=\frac {2 \int \left (\frac {e^2 \left (-1+77 e^2\right )}{\left (25+e^x-x\right )^2}-\frac {\left (-1+76 e^2+3 e^4\right ) x}{\left (25+e^x-x\right )^2}+\frac {\left (-1+3 e^2\right ) x^2}{\left (25+e^x-x\right )^2}\right ) \, dx}{e^4}-\frac {2 \int \left (-\frac {26 e^4}{\left (25+e^x-x\right )^3}+\frac {e^2 \left (52+e^2\right ) x}{\left (25+e^x-x\right )^3}-\frac {2 \left (13+e^2\right ) x^2}{\left (25+e^x-x\right )^3}+\frac {x^3}{\left (25+e^x-x\right )^3}\right ) \, dx}{e^4}-\frac {6 \int \left (\frac {1+e^2}{25+e^x-x}-\frac {x}{25+e^x-x}\right ) \, dx}{e^2}\\ &=52 \int \frac {1}{\left (25+e^x-x\right )^3} \, dx+\left (2 \left (77-\frac {1}{e^2}\right )\right ) \int \frac {1}{\left (25+e^x-x\right )^2} \, dx-\left (6 \left (1+\frac {1}{e^2}\right )\right ) \int \frac {1}{25+e^x-x} \, dx-\frac {2 \int \frac {x^3}{\left (25+e^x-x\right )^3} \, dx}{e^4}+\frac {6 \int \frac {x}{25+e^x-x} \, dx}{e^2}-\frac {\left (2 \left (1-3 e^2\right )\right ) \int \frac {x^2}{\left (25+e^x-x\right )^2} \, dx}{e^4}+\frac {\left (4 \left (13+e^2\right )\right ) \int \frac {x^2}{\left (25+e^x-x\right )^3} \, dx}{e^4}-\frac {\left (2 \left (52+e^2\right )\right ) \int \frac {x}{\left (25+e^x-x\right )^3} \, dx}{e^2}+\frac {\left (2 \left (1-76 e^2-3 e^4\right )\right ) \int \frac {x}{\left (25+e^x-x\right )^2} \, dx}{e^4}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 40, normalized size = 1.74 \begin {gather*} -\frac {\left (e^2-x\right ) \left (-6 e^{2+x}+x+e^2 (-151+6 x)\right )}{e^4 \left (25+e^x-x\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^4*(152 - 6*x) + 50*x + E^2*(-3800 + 148*x) + E^(2*x)*(-6*E^4 + E^2*(-6 + 6*x)) + E^x*(2*x - 2*x^2
+ E^4*(-146 + 6*x) + E^2*(-302 + 160*x - 6*x^2)))/(E^(4 + 3*x) + E^(4 + 2*x)*(75 - 3*x) + E^(4 + x)*(1875 - 1
50*x + 3*x^2) + E^4*(15625 - 1875*x + 75*x^2 - x^3)),x]

[Out]

-(((E^2 - x)*(-6*E^(2 + x) + x + E^2*(-151 + 6*x)))/(E^4*(25 + E^x - x)^2))

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fricas [B]  time = 0.70, size = 74, normalized size = 3.22 \begin {gather*} \frac {x^{2} e^{4} - {\left (6 \, x - 151\right )} e^{8} + 2 \, {\left (3 \, x^{2} - 76 \, x\right )} e^{6} - 6 \, {\left (x e^{2} - e^{4}\right )} e^{\left (x + 4\right )}}{{\left (x^{2} - 50 \, x + 625\right )} e^{8} - 2 \, {\left (x - 25\right )} e^{\left (x + 8\right )} + e^{\left (2 \, x + 8\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*exp(2)^2+(6*x-6)*exp(2))*exp(x)^2+((6*x-146)*exp(2)^2+(-6*x^2+160*x-302)*exp(2)-2*x^2+2*x)*exp(
x)+(-6*x+152)*exp(2)^2+(148*x-3800)*exp(2)+50*x)/(exp(2)^2*exp(x)^3+(-3*x+75)*exp(2)^2*exp(x)^2+(3*x^2-150*x+1
875)*exp(2)^2*exp(x)+(-x^3+75*x^2-1875*x+15625)*exp(2)^2),x, algorithm="fricas")

[Out]

(x^2*e^4 - (6*x - 151)*e^8 + 2*(3*x^2 - 76*x)*e^6 - 6*(x*e^2 - e^4)*e^(x + 4))/((x^2 - 50*x + 625)*e^8 - 2*(x
- 25)*e^(x + 8) + e^(2*x + 8))

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giac [B]  time = 0.23, size = 79, normalized size = 3.43 \begin {gather*} \frac {6 \, x^{2} e^{4} + x^{2} e^{2} - 6 \, x e^{6} - 152 \, x e^{4} - 6 \, x e^{\left (x + 4\right )} + 151 \, e^{6} + 6 \, e^{\left (x + 6\right )}}{x^{2} e^{6} - 50 \, x e^{6} - 2 \, x e^{\left (x + 6\right )} + 625 \, e^{6} + e^{\left (2 \, x + 6\right )} + 50 \, e^{\left (x + 6\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*exp(2)^2+(6*x-6)*exp(2))*exp(x)^2+((6*x-146)*exp(2)^2+(-6*x^2+160*x-302)*exp(2)-2*x^2+2*x)*exp(
x)+(-6*x+152)*exp(2)^2+(148*x-3800)*exp(2)+50*x)/(exp(2)^2*exp(x)^3+(-3*x+75)*exp(2)^2*exp(x)^2+(3*x^2-150*x+1
875)*exp(2)^2*exp(x)+(-x^3+75*x^2-1875*x+15625)*exp(2)^2),x, algorithm="giac")

[Out]

(6*x^2*e^4 + x^2*e^2 - 6*x*e^6 - 152*x*e^4 - 6*x*e^(x + 4) + 151*e^6 + 6*e^(x + 6))/(x^2*e^6 - 50*x*e^6 - 2*x*
e^(x + 6) + 625*e^6 + e^(2*x + 6) + 50*e^(x + 6))

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maple [B]  time = 0.31, size = 54, normalized size = 2.35

method	result	size
risch	\(-\frac {\left (6 x \,{\mathrm e}^{4}-6 \,{\mathrm e}^{4+x}-6 x^{2} {\mathrm e}^{2}+6 x \,{\mathrm e}^{2+x}-151 \,{\mathrm e}^{4}+152 \,{\mathrm e}^{2} x -x^{2}\right ) {\mathrm e}^{-4}}{\left (x -{\mathrm e}^{x}-25\right )^{2}}\)	\(54\)
norman	\(\frac {\left (-2 \left (3 \,{\mathrm e}^{4}-74 \,{\mathrm e}^{2}-25\right ) {\mathrm e}^{-2} x -\left (6 \,{\mathrm e}^{2}+1\right ) {\mathrm e}^{-2} {\mathrm e}^{2 x}+2 \left (3 \,{\mathrm e}^{4}-150 \,{\mathrm e}^{2}-25\right ) {\mathrm e}^{-2} {\mathrm e}^{x}+2 \left (3 \,{\mathrm e}^{2}+1\right ) {\mathrm e}^{-2} x \,{\mathrm e}^{x}+\left (151 \,{\mathrm e}^{4}-3750 \,{\mathrm e}^{2}-625\right ) {\mathrm e}^{-2}\right ) {\mathrm e}^{-2}}{\left (x -{\mathrm e}^{x}-25\right )^{2}}\)	\(103\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-6*exp(2)^2+(6*x-6)*exp(2))*exp(x)^2+((6*x-146)*exp(2)^2+(-6*x^2+160*x-302)*exp(2)-2*x^2+2*x)*exp(x)+(-6
*x+152)*exp(2)^2+(148*x-3800)*exp(2)+50*x)/(exp(2)^2*exp(x)^3+(-3*x+75)*exp(2)^2*exp(x)^2+(3*x^2-150*x+1875)*e
xp(2)^2*exp(x)+(-x^3+75*x^2-1875*x+15625)*exp(2)^2),x,method=_RETURNVERBOSE)

[Out]

-(6*x*exp(4)-6*exp(4+x)-6*x^2*exp(2)+6*x*exp(2+x)-151*exp(4)+152*exp(2)*x-x^2)/(x-exp(x)-25)^2*exp(-4)

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maxima [B]  time = 0.42, size = 78, normalized size = 3.39 \begin {gather*} \frac {x^{2} {\left (6 \, e^{2} + 1\right )} - 2 \, x {\left (3 \, e^{4} + 76 \, e^{2}\right )} - 6 \, {\left (x e^{2} - e^{4}\right )} e^{x} + 151 \, e^{4}}{x^{2} e^{4} - 50 \, x e^{4} - 2 \, {\left (x e^{4} - 25 \, e^{4}\right )} e^{x} + 625 \, e^{4} + e^{\left (2 \, x + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*exp(2)^2+(6*x-6)*exp(2))*exp(x)^2+((6*x-146)*exp(2)^2+(-6*x^2+160*x-302)*exp(2)-2*x^2+2*x)*exp(
x)+(-6*x+152)*exp(2)^2+(148*x-3800)*exp(2)+50*x)/(exp(2)^2*exp(x)^3+(-3*x+75)*exp(2)^2*exp(x)^2+(3*x^2-150*x+1
875)*exp(2)^2*exp(x)+(-x^3+75*x^2-1875*x+15625)*exp(2)^2),x, algorithm="maxima")

[Out]

(x^2*(6*e^2 + 1) - 2*x*(3*e^4 + 76*e^2) - 6*(x*e^2 - e^4)*e^x + 151*e^4)/(x^2*e^4 - 50*x*e^4 - 2*(x*e^4 - 25*e
^4)*e^x + 625*e^4 + e^(2*x + 4))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {50\,x-{\mathrm {e}}^{2\,x}\,\left (6\,{\mathrm {e}}^4-{\mathrm {e}}^2\,\left (6\,x-6\right )\right )+{\mathrm {e}}^x\,\left (2\,x-{\mathrm {e}}^2\,\left (6\,x^2-160\,x+302\right )-2\,x^2+{\mathrm {e}}^4\,\left (6\,x-146\right )\right )-{\mathrm {e}}^4\,\left (6\,x-152\right )+{\mathrm {e}}^2\,\left (148\,x-3800\right )}{{\mathrm {e}}^{3\,x}\,{\mathrm {e}}^4-{\mathrm {e}}^4\,\left (x^3-75\,x^2+1875\,x-15625\right )+{\mathrm {e}}^4\,{\mathrm {e}}^x\,\left (3\,x^2-150\,x+1875\right )-{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^4\,\left (3\,x-75\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((50*x - exp(2*x)*(6*exp(4) - exp(2)*(6*x - 6)) + exp(x)*(2*x - exp(2)*(6*x^2 - 160*x + 302) - 2*x^2 + exp(
4)*(6*x - 146)) - exp(4)*(6*x - 152) + exp(2)*(148*x - 3800))/(exp(3*x)*exp(4) - exp(4)*(1875*x - 75*x^2 + x^3
- 15625) + exp(4)*exp(x)*(3*x^2 - 150*x + 1875) - exp(2*x)*exp(4)*(3*x - 75)),x)

[Out]

int((50*x - exp(2*x)*(6*exp(4) - exp(2)*(6*x - 6)) + exp(x)*(2*x - exp(2)*(6*x^2 - 160*x + 302) - 2*x^2 + exp(
4)*(6*x - 146)) - exp(4)*(6*x - 152) + exp(2)*(148*x - 3800))/(exp(3*x)*exp(4) - exp(4)*(1875*x - 75*x^2 + x^3
- 15625) + exp(4)*exp(x)*(3*x^2 - 150*x + 1875) - exp(2*x)*exp(4)*(3*x - 75)), x)

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sympy [B]  time = 0.21, size = 87, normalized size = 3.78 \begin {gather*} \frac {x^{2} + 6 x^{2} e^{2} - 152 x e^{2} - 6 x e^{4} + \left (- 6 x e^{2} + 6 e^{4}\right ) e^{x} + 151 e^{4}}{x^{2} e^{4} - 50 x e^{4} + \left (- 2 x e^{4} + 50 e^{4}\right ) e^{x} + e^{4} e^{2 x} + 625 e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*exp(2)**2+(6*x-6)*exp(2))*exp(x)**2+((6*x-146)*exp(2)**2+(-6*x**2+160*x-302)*exp(2)-2*x**2+2*x)
*exp(x)+(-6*x+152)*exp(2)**2+(148*x-3800)*exp(2)+50*x)/(exp(2)**2*exp(x)**3+(-3*x+75)*exp(2)**2*exp(x)**2+(3*x
**2-150*x+1875)*exp(2)**2*exp(x)+(-x**3+75*x**2-1875*x+15625)*exp(2)**2),x)

[Out]

(x**2 + 6*x**2*exp(2) - 152*x*exp(2) - 6*x*exp(4) + (-6*x*exp(2) + 6*exp(4))*exp(x) + 151*exp(4))/(x**2*exp(4)
- 50*x*exp(4) + (-2*x*exp(4) + 50*exp(4))*exp(x) + exp(4)*exp(2*x) + 625*exp(4))

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