3.32.93 \(\int \frac {2 e^{5 x}+e^{4 x} (14-8 x)+8 x+24 x^2+14 x^3-2 x^4+e^{3 x} (-14-40 x+12 x^2)+e^{2 x} (8+58 x+36 x^2-8 x^3)+e^x (-30 x-60 x^2-8 x^3+2 x^4)}{64+e^{3 x}+e^{2 x} (12-3 x)-48 x+12 x^2-x^3+e^x (48-24 x+3 x^2)} \, dx\)

Optimal. Leaf size=24 \[ \frac {\left (\left (e^x-x\right )^2+x\right )^2}{\left (4+e^x-x\right )^2} \]

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Rubi [F]  time = 1.12, 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 {2 e^{5 x}+e^{4 x} (14-8 x)+8 x+24 x^2+14 x^3-2 x^4+e^{3 x} \left (-14-40 x+12 x^2\right )+e^{2 x} \left (8+58 x+36 x^2-8 x^3\right )+e^x \left (-30 x-60 x^2-8 x^3+2 x^4\right )}{64+e^{3 x}+e^{2 x} (12-3 x)-48 x+12 x^2-x^3+e^x \left (48-24 x+3 x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(2*E^(5*x) + E^(4*x)*(14 - 8*x) + 8*x + 24*x^2 + 14*x^3 - 2*x^4 + E^(3*x)*(-14 - 40*x + 12*x^2) + E^(2*x)*
(8 + 58*x + 36*x^2 - 8*x^3) + E^x*(-30*x - 60*x^2 - 8*x^3 + 2*x^4))/(64 + E^(3*x) + E^(2*x)*(12 - 3*x) - 48*x
+ 12*x^2 - x^3 + E^x*(48 - 24*x + 3*x^2)),x]

[Out]

2*E^x + E^(2*x) + 10*x + x^2 - 2*E^x*(5 + x) + 2560*Defer[Int][(4 + E^x - x)^(-3), x] - 1760*Defer[Int][(4 + E
^x - x)^(-2), x] + 240*Defer[Int][(4 + E^x - x)^(-1), x] - 192*Defer[Int][x/(4 + E^x - x)^3, x] + 114*Defer[In
t][x/(4 + E^x - x)^2, x] + 16*Defer[Int][x/(4 + E^x - x), x] - 54*Defer[Int][x^2/(4 + E^x - x)^3, x] + 14*Defe
r[Int][x^2/(4 + E^x - x)^2, x] - 2*Defer[Int][x^3/(4 + E^x - x)^3, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (e^{5 x}+e^{4 x} (7-4 x)+e^{3 x} \left (-7-20 x+6 x^2\right )+e^{2 x} \left (4+29 x+18 x^2-4 x^3\right )+x \left (4+12 x+7 x^2-x^3\right )+e^x x \left (-15-30 x-4 x^2+x^3\right )\right )}{\left (4+e^x-x\right )^3} \, dx\\ &=2 \int \frac {e^{5 x}+e^{4 x} (7-4 x)+e^{3 x} \left (-7-20 x+6 x^2\right )+e^{2 x} \left (4+29 x+18 x^2-4 x^3\right )+x \left (4+12 x+7 x^2-x^3\right )+e^x x \left (-15-30 x-4 x^2+x^3\right )}{\left (4+e^x-x\right )^3} \, dx\\ &=2 \int \left (5+e^{2 x}+x-e^x (5+x)+\frac {8 (15+x)}{4+e^x-x}-\frac {(-5+x) (16+x)^2}{\left (4+e^x-x\right )^3}+\frac {-880+57 x+7 x^2}{\left (4+e^x-x\right )^2}\right ) \, dx\\ &=10 x+x^2+2 \int e^{2 x} \, dx-2 \int e^x (5+x) \, dx-2 \int \frac {(-5+x) (16+x)^2}{\left (4+e^x-x\right )^3} \, dx+2 \int \frac {-880+57 x+7 x^2}{\left (4+e^x-x\right )^2} \, dx+16 \int \frac {15+x}{4+e^x-x} \, dx\\ &=e^{2 x}+10 x+x^2-2 e^x (5+x)+2 \int e^x \, dx+2 \int \left (-\frac {880}{\left (4+e^x-x\right )^2}+\frac {57 x}{\left (4+e^x-x\right )^2}+\frac {7 x^2}{\left (4+e^x-x\right )^2}\right ) \, dx-2 \int \left (-\frac {1280}{\left (4+e^x-x\right )^3}+\frac {96 x}{\left (4+e^x-x\right )^3}+\frac {27 x^2}{\left (4+e^x-x\right )^3}+\frac {x^3}{\left (4+e^x-x\right )^3}\right ) \, dx+16 \int \left (\frac {15}{4+e^x-x}+\frac {x}{4+e^x-x}\right ) \, dx\\ &=2 e^x+e^{2 x}+10 x+x^2-2 e^x (5+x)-2 \int \frac {x^3}{\left (4+e^x-x\right )^3} \, dx+14 \int \frac {x^2}{\left (4+e^x-x\right )^2} \, dx+16 \int \frac {x}{4+e^x-x} \, dx-54 \int \frac {x^2}{\left (4+e^x-x\right )^3} \, dx+114 \int \frac {x}{\left (4+e^x-x\right )^2} \, dx-192 \int \frac {x}{\left (4+e^x-x\right )^3} \, dx+240 \int \frac {1}{4+e^x-x} \, dx-1760 \int \frac {1}{\left (4+e^x-x\right )^2} \, dx+2560 \int \frac {1}{\left (4+e^x-x\right )^3} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.08, size = 51, normalized size = 2.12 \begin {gather*} e^{2 x}+10 x+x^2-2 e^x (4+x)-\frac {16 (16+x)}{4+e^x-x}+\frac {(16+x)^2}{\left (4+e^x-x\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2*E^(5*x) + E^(4*x)*(14 - 8*x) + 8*x + 24*x^2 + 14*x^3 - 2*x^4 + E^(3*x)*(-14 - 40*x + 12*x^2) + E^
(2*x)*(8 + 58*x + 36*x^2 - 8*x^3) + E^x*(-30*x - 60*x^2 - 8*x^3 + 2*x^4))/(64 + E^(3*x) + E^(2*x)*(12 - 3*x) -
48*x + 12*x^2 - x^3 + E^x*(48 - 24*x + 3*x^2)),x]

[Out]

E^(2*x) + 10*x + x^2 - 2*E^x*(4 + x) - (16*(16 + x))/(4 + E^x - x) + (16 + x)^2/(4 + E^x - x)^2

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fricas [B]  time = 0.74, size = 80, normalized size = 3.33 \begin {gather*} \frac {x^{4} + 2 \, x^{3} - 47 \, x^{2} - 4 \, x e^{\left (3 \, x\right )} + 2 \, {\left (3 \, x^{2} + x - 24\right )} e^{\left (2 \, x\right )} - 4 \, {\left (x^{3} + x^{2} - 24 \, x + 96\right )} e^{x} + 384 \, x + e^{\left (4 \, x\right )} - 768}{x^{2} - 2 \, {\left (x - 4\right )} e^{x} - 8 \, x + e^{\left (2 \, x\right )} + 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(x)^5+(-8*x+14)*exp(x)^4+(12*x^2-40*x-14)*exp(x)^3+(-8*x^3+36*x^2+58*x+8)*exp(x)^2+(2*x^4-8*x^
3-60*x^2-30*x)*exp(x)-2*x^4+14*x^3+24*x^2+8*x)/(exp(x)^3+(-3*x+12)*exp(x)^2+(3*x^2-24*x+48)*exp(x)-x^3+12*x^2-
48*x+64),x, algorithm="fricas")

[Out]

(x^4 + 2*x^3 - 47*x^2 - 4*x*e^(3*x) + 2*(3*x^2 + x - 24)*e^(2*x) - 4*(x^3 + x^2 - 24*x + 96)*e^x + 384*x + e^(
4*x) - 768)/(x^2 - 2*(x - 4)*e^x - 8*x + e^(2*x) + 16)

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giac [B]  time = 0.41, size = 98, normalized size = 4.08 \begin {gather*} \frac {x^{4} - 4 \, x^{3} e^{x} + 2 \, x^{3} + 6 \, x^{2} e^{\left (2 \, x\right )} - 4 \, x^{2} e^{x} - 47 \, x^{2} - 4 \, x e^{\left (3 \, x\right )} + 2 \, x e^{\left (2 \, x\right )} + 96 \, x e^{x} + 384 \, x + e^{\left (4 \, x\right )} - 48 \, e^{\left (2 \, x\right )} - 384 \, e^{x} - 768}{x^{2} - 2 \, x e^{x} - 8 \, x + e^{\left (2 \, x\right )} + 8 \, e^{x} + 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(x)^5+(-8*x+14)*exp(x)^4+(12*x^2-40*x-14)*exp(x)^3+(-8*x^3+36*x^2+58*x+8)*exp(x)^2+(2*x^4-8*x^
3-60*x^2-30*x)*exp(x)-2*x^4+14*x^3+24*x^2+8*x)/(exp(x)^3+(-3*x+12)*exp(x)^2+(3*x^2-24*x+48)*exp(x)-x^3+12*x^2-
48*x+64),x, algorithm="giac")

[Out]

(x^4 - 4*x^3*e^x + 2*x^3 + 6*x^2*e^(2*x) - 4*x^2*e^x - 47*x^2 - 4*x*e^(3*x) + 2*x*e^(2*x) + 96*x*e^x + 384*x +
e^(4*x) - 48*e^(2*x) - 384*e^x - 768)/(x^2 - 2*x*e^x - 8*x + e^(2*x) + 8*e^x + 16)

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maple [B]  time = 0.08, size = 49, normalized size = 2.04

method	result	size
risch	\(x^{2}+{\mathrm e}^{2 x}+10 x +\left (-2 x -8\right ) {\mathrm e}^{x}+\frac {17 x^{2}-16 \,{\mathrm e}^{x} x +224 x -256 \,{\mathrm e}^{x}-768}{\left (x -{\mathrm e}^{x}-4\right )^{2}}\)	\(49\)
norman	\(\frac {x^{4}+{\mathrm e}^{4 x}+x^{2}+2 x^{3}+2 x \,{\mathrm e}^{2 x}-4 x \,{\mathrm e}^{3 x}-4 \,{\mathrm e}^{x} x^{2}-4 \,{\mathrm e}^{x} x^{3}+6 \,{\mathrm e}^{2 x} x^{2}}{\left (x -{\mathrm e}^{x}-4\right )^{2}}\)	\(64\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*exp(x)^5+(-8*x+14)*exp(x)^4+(12*x^2-40*x-14)*exp(x)^3+(-8*x^3+36*x^2+58*x+8)*exp(x)^2+(2*x^4-8*x^3-60*x
^2-30*x)*exp(x)-2*x^4+14*x^3+24*x^2+8*x)/(exp(x)^3+(-3*x+12)*exp(x)^2+(3*x^2-24*x+48)*exp(x)-x^3+12*x^2-48*x+6
4),x,method=_RETURNVERBOSE)

[Out]

x^2+exp(2*x)+10*x+(-2*x-8)*exp(x)+(17*x^2-16*exp(x)*x+224*x-256*exp(x)-768)/(x-exp(x)-4)^2

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maxima [B]  time = 0.77, size = 80, normalized size = 3.33 \begin {gather*} \frac {x^{4} + 2 \, x^{3} - 47 \, x^{2} - 4 \, x e^{\left (3 \, x\right )} + 2 \, {\left (3 \, x^{2} + x - 24\right )} e^{\left (2 \, x\right )} - 4 \, {\left (x^{3} + x^{2} - 24 \, x + 96\right )} e^{x} + 384 \, x + e^{\left (4 \, x\right )} - 768}{x^{2} - 2 \, {\left (x - 4\right )} e^{x} - 8 \, x + e^{\left (2 \, x\right )} + 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(x)^5+(-8*x+14)*exp(x)^4+(12*x^2-40*x-14)*exp(x)^3+(-8*x^3+36*x^2+58*x+8)*exp(x)^2+(2*x^4-8*x^
3-60*x^2-30*x)*exp(x)-2*x^4+14*x^3+24*x^2+8*x)/(exp(x)^3+(-3*x+12)*exp(x)^2+(3*x^2-24*x+48)*exp(x)-x^3+12*x^2-
48*x+64),x, algorithm="maxima")

[Out]

(x^4 + 2*x^3 - 47*x^2 - 4*x*e^(3*x) + 2*(3*x^2 + x - 24)*e^(2*x) - 4*(x^3 + x^2 - 24*x + 96)*e^x + 384*x + e^(
4*x) - 768)/(x^2 - 2*(x - 4)*e^x - 8*x + e^(2*x) + 16)

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mupad [B]  time = 2.08, size = 26, normalized size = 1.08 \begin {gather*} \frac {{\left (x+{\mathrm {e}}^{2\,x}-2\,x\,{\mathrm {e}}^x+x^2\right )}^2}{{\left ({\mathrm {e}}^x-x+4\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((8*x + 2*exp(5*x) - exp(3*x)*(40*x - 12*x^2 + 14) - exp(x)*(30*x + 60*x^2 + 8*x^3 - 2*x^4) + exp(2*x)*(58*
x + 36*x^2 - 8*x^3 + 8) - exp(4*x)*(8*x - 14) + 24*x^2 + 14*x^3 - 2*x^4)/(exp(3*x) - 48*x + exp(x)*(3*x^2 - 24
*x + 48) - exp(2*x)*(3*x - 12) + 12*x^2 - x^3 + 64),x)

[Out]

(x + exp(2*x) - 2*x*exp(x) + x^2)^2/(exp(x) - x + 4)^2

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sympy [B]  time = 0.21, size = 63, normalized size = 2.62 \begin {gather*} x^{2} + 10 x + \left (- 2 x - 8\right ) e^{x} + \frac {17 x^{2} + 224 x + \left (- 16 x - 256\right ) e^{x} - 768}{x^{2} - 8 x + \left (8 - 2 x\right ) e^{x} + e^{2 x} + 16} + e^{2 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(x)**5+(-8*x+14)*exp(x)**4+(12*x**2-40*x-14)*exp(x)**3+(-8*x**3+36*x**2+58*x+8)*exp(x)**2+(2*x
**4-8*x**3-60*x**2-30*x)*exp(x)-2*x**4+14*x**3+24*x**2+8*x)/(exp(x)**3+(-3*x+12)*exp(x)**2+(3*x**2-24*x+48)*ex
p(x)-x**3+12*x**2-48*x+64),x)

[Out]

x**2 + 10*x + (-2*x - 8)*exp(x) + (17*x**2 + 224*x + (-16*x - 256)*exp(x) - 768)/(x**2 - 8*x + (8 - 2*x)*exp(x
) + exp(2*x) + 16) + exp(2*x)

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