3.60.32 \(\int \frac {3 x^3+e^{4+2 x} (x-4 x^2)+e^x (-x^2-2 x^3)}{2 e^{20+10 x}+2 e^{5 x} x^5-10 e^{4 x} x^6+20 e^{3 x} x^7-20 e^{2 x} x^8+10 e^x x^9-2 x^{10}+e^{16+8 x} (10 e^x x-10 x^2)+e^{12+6 x} (20 e^{2 x} x^2-40 e^x x^3+20 x^4)+e^{8+4 x} (20 e^{3 x} x^3-60 e^{2 x} x^4+60 e^x x^5-20 x^6)+e^{4+2 x} (10 e^{4 x} x^4-40 e^{3 x} x^5+60 e^{2 x} x^6-40 e^x x^7+10 x^8)} \, dx\)

Optimal. Leaf size=29 \[ 2+\frac {1}{4 \left (e^x+\frac {e^{4+2 x}}{x}-x\right )^4 x^2} \]

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Rubi [F]  time = 3.37, 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 {3 x^3+e^{4+2 x} \left (x-4 x^2\right )+e^x \left (-x^2-2 x^3\right )}{2 e^{20+10 x}+2 e^{5 x} x^5-10 e^{4 x} x^6+20 e^{3 x} x^7-20 e^{2 x} x^8+10 e^x x^9-2 x^{10}+e^{16+8 x} \left (10 e^x x-10 x^2\right )+e^{12+6 x} \left (20 e^{2 x} x^2-40 e^x x^3+20 x^4\right )+e^{8+4 x} \left (20 e^{3 x} x^3-60 e^{2 x} x^4+60 e^x x^5-20 x^6\right )+e^{4+2 x} \left (10 e^{4 x} x^4-40 e^{3 x} x^5+60 e^{2 x} x^6-40 e^x x^7+10 x^8\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(3*x^3 + E^(4 + 2*x)*(x - 4*x^2) + E^x*(-x^2 - 2*x^3))/(2*E^(20 + 10*x) + 2*E^(5*x)*x^5 - 10*E^(4*x)*x^6 +
20*E^(3*x)*x^7 - 20*E^(2*x)*x^8 + 10*E^x*x^9 - 2*x^10 + E^(16 + 8*x)*(10*E^x*x - 10*x^2) + E^(12 + 6*x)*(20*E
^(2*x)*x^2 - 40*E^x*x^3 + 20*x^4) + E^(8 + 4*x)*(20*E^(3*x)*x^3 - 60*E^(2*x)*x^4 + 60*E^x*x^5 - 20*x^6) + E^(4
+ 2*x)*(10*E^(4*x)*x^4 - 40*E^(3*x)*x^5 + 60*E^(2*x)*x^6 - 40*E^x*x^7 + 10*x^8)),x]

[Out]

-Defer[Int][(E^x*x^2)/(E^(4 + 2*x) + E^x*x - x^2)^5, x] + Defer[Int][(E^x*x^3)/(E^(4 + 2*x) + E^x*x - x^2)^5,
x] - 2*Defer[Int][x^3/(-E^(4 + 2*x) - E^x*x + x^2)^5, x] + 2*Defer[Int][x^4/(-E^(4 + 2*x) - E^x*x + x^2)^5, x]
+ Defer[Int][x/(-E^(4 + 2*x) - E^x*x + x^2)^4, x]/2 - 2*Defer[Int][x^2/(-E^(4 + 2*x) - E^x*x + x^2)^4, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x \left (3 x^2-e^x x (1+2 x)-e^{4+2 x} (-1+4 x)\right )}{2 \left (e^{4+2 x}+e^x x-x^2\right )^5} \, dx\\ &=\frac {1}{2} \int \frac {x \left (3 x^2-e^x x (1+2 x)-e^{4+2 x} (-1+4 x)\right )}{\left (e^{4+2 x}+e^x x-x^2\right )^5} \, dx\\ &=\frac {1}{2} \int \left (\frac {2 \left (e^x-2 x\right ) (-1+x) x^2}{\left (e^{4+2 x}+e^x x-x^2\right )^5}-\frac {x (-1+4 x)}{\left (-e^{4+2 x}-e^x x+x^2\right )^4}\right ) \, dx\\ &=-\left (\frac {1}{2} \int \frac {x (-1+4 x)}{\left (-e^{4+2 x}-e^x x+x^2\right )^4} \, dx\right )+\int \frac {\left (e^x-2 x\right ) (-1+x) x^2}{\left (e^{4+2 x}+e^x x-x^2\right )^5} \, dx\\ &=-\left (\frac {1}{2} \int \left (-\frac {x}{\left (-e^{4+2 x}-e^x x+x^2\right )^4}+\frac {4 x^2}{\left (-e^{4+2 x}-e^x x+x^2\right )^4}\right ) \, dx\right )+\int \left (-\frac {\left (e^x-2 x\right ) x^2}{\left (e^{4+2 x}+e^x x-x^2\right )^5}+\frac {\left (e^x-2 x\right ) x^3}{\left (e^{4+2 x}+e^x x-x^2\right )^5}\right ) \, dx\\ &=\frac {1}{2} \int \frac {x}{\left (-e^{4+2 x}-e^x x+x^2\right )^4} \, dx-2 \int \frac {x^2}{\left (-e^{4+2 x}-e^x x+x^2\right )^4} \, dx-\int \frac {\left (e^x-2 x\right ) x^2}{\left (e^{4+2 x}+e^x x-x^2\right )^5} \, dx+\int \frac {\left (e^x-2 x\right ) x^3}{\left (e^{4+2 x}+e^x x-x^2\right )^5} \, dx\\ &=\frac {1}{2} \int \frac {x}{\left (-e^{4+2 x}-e^x x+x^2\right )^4} \, dx-2 \int \frac {x^2}{\left (-e^{4+2 x}-e^x x+x^2\right )^4} \, dx-\int \left (\frac {e^x x^2}{\left (e^{4+2 x}+e^x x-x^2\right )^5}+\frac {2 x^3}{\left (-e^{4+2 x}-e^x x+x^2\right )^5}\right ) \, dx+\int \left (\frac {e^x x^3}{\left (e^{4+2 x}+e^x x-x^2\right )^5}+\frac {2 x^4}{\left (-e^{4+2 x}-e^x x+x^2\right )^5}\right ) \, dx\\ &=\frac {1}{2} \int \frac {x}{\left (-e^{4+2 x}-e^x x+x^2\right )^4} \, dx-2 \int \frac {x^3}{\left (-e^{4+2 x}-e^x x+x^2\right )^5} \, dx+2 \int \frac {x^4}{\left (-e^{4+2 x}-e^x x+x^2\right )^5} \, dx-2 \int \frac {x^2}{\left (-e^{4+2 x}-e^x x+x^2\right )^4} \, dx-\int \frac {e^x x^2}{\left (e^{4+2 x}+e^x x-x^2\right )^5} \, dx+\int \frac {e^x x^3}{\left (e^{4+2 x}+e^x x-x^2\right )^5} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.52, size = 27, normalized size = 0.93 \begin {gather*} \frac {x^2}{4 \left (e^{4+2 x}+e^x x-x^2\right )^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3*x^3 + E^(4 + 2*x)*(x - 4*x^2) + E^x*(-x^2 - 2*x^3))/(2*E^(20 + 10*x) + 2*E^(5*x)*x^5 - 10*E^(4*x)
*x^6 + 20*E^(3*x)*x^7 - 20*E^(2*x)*x^8 + 10*E^x*x^9 - 2*x^10 + E^(16 + 8*x)*(10*E^x*x - 10*x^2) + E^(12 + 6*x)
*(20*E^(2*x)*x^2 - 40*E^x*x^3 + 20*x^4) + E^(8 + 4*x)*(20*E^(3*x)*x^3 - 60*E^(2*x)*x^4 + 60*E^x*x^5 - 20*x^6)
+ E^(4 + 2*x)*(10*E^(4*x)*x^4 - 40*E^(3*x)*x^5 + 60*E^(2*x)*x^6 - 40*E^x*x^7 + 10*x^8)),x]

[Out]

x^2/(4*(E^(4 + 2*x) + E^x*x - x^2)^4)

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fricas [B]  time = 0.65, size = 136, normalized size = 4.69 \begin {gather*} \frac {x^{2}}{4 \, {\left (x^{8} - 4 \, x^{7} e^{x} - 2 \, {\left (2 \, x^{2} e^{12} - 3 \, x^{2} e^{8}\right )} e^{\left (6 \, x\right )} - 4 \, {\left (3 \, x^{3} e^{8} - x^{3} e^{4}\right )} e^{\left (5 \, x\right )} + {\left (6 \, x^{4} e^{8} - 12 \, x^{4} e^{4} + x^{4}\right )} e^{\left (4 \, x\right )} + 4 \, {\left (3 \, x^{5} e^{4} - x^{5}\right )} e^{\left (3 \, x\right )} - 2 \, {\left (2 \, x^{6} e^{4} - 3 \, x^{6}\right )} e^{\left (2 \, x\right )} + 4 \, x e^{\left (7 \, x + 12\right )} + e^{\left (8 \, x + 16\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^2+x)*exp(2*x+4)+(-2*x^3-x^2)*exp(x)+3*x^3)/(2*exp(2*x+4)^5+(10*exp(x)*x-10*x^2)*exp(2*x+4)^4+
(20*exp(x)^2*x^2-40*exp(x)*x^3+20*x^4)*exp(2*x+4)^3+(20*x^3*exp(x)^3-60*exp(x)^2*x^4+60*x^5*exp(x)-20*x^6)*exp
(2*x+4)^2+(10*x^4*exp(x)^4-40*x^5*exp(x)^3+60*x^6*exp(x)^2-40*x^7*exp(x)+10*x^8)*exp(2*x+4)+2*x^5*exp(x)^5-10*
x^6*exp(x)^4+20*x^7*exp(x)^3-20*x^8*exp(x)^2+10*x^9*exp(x)-2*x^10),x, algorithm="fricas")

[Out]

1/4*x^2/(x^8 - 4*x^7*e^x - 2*(2*x^2*e^12 - 3*x^2*e^8)*e^(6*x) - 4*(3*x^3*e^8 - x^3*e^4)*e^(5*x) + (6*x^4*e^8 -
12*x^4*e^4 + x^4)*e^(4*x) + 4*(3*x^5*e^4 - x^5)*e^(3*x) - 2*(2*x^6*e^4 - 3*x^6)*e^(2*x) + 4*x*e^(7*x + 12) +
e^(8*x + 16))

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giac [B]  time = 4.15, size = 147, normalized size = 5.07 \begin {gather*} \frac {x^{2}}{2 \, {\left (x^{8} - 4 \, x^{7} e^{x} + 6 \, x^{6} e^{\left (2 \, x\right )} - 4 \, x^{6} e^{\left (2 \, x + 4\right )} - 4 \, x^{5} e^{\left (3 \, x\right )} + 12 \, x^{5} e^{\left (3 \, x + 4\right )} + x^{4} e^{\left (4 \, x\right )} + 6 \, x^{4} e^{\left (4 \, x + 8\right )} - 12 \, x^{4} e^{\left (4 \, x + 4\right )} - 12 \, x^{3} e^{\left (5 \, x + 8\right )} + 4 \, x^{3} e^{\left (5 \, x + 4\right )} - 4 \, x^{2} e^{\left (6 \, x + 12\right )} + 6 \, x^{2} e^{\left (6 \, x + 8\right )} + 4 \, x e^{\left (7 \, x + 12\right )} + e^{\left (8 \, x + 16\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^2+x)*exp(2*x+4)+(-2*x^3-x^2)*exp(x)+3*x^3)/(2*exp(2*x+4)^5+(10*exp(x)*x-10*x^2)*exp(2*x+4)^4+
(20*exp(x)^2*x^2-40*exp(x)*x^3+20*x^4)*exp(2*x+4)^3+(20*x^3*exp(x)^3-60*exp(x)^2*x^4+60*x^5*exp(x)-20*x^6)*exp
(2*x+4)^2+(10*x^4*exp(x)^4-40*x^5*exp(x)^3+60*x^6*exp(x)^2-40*x^7*exp(x)+10*x^8)*exp(2*x+4)+2*x^5*exp(x)^5-10*
x^6*exp(x)^4+20*x^7*exp(x)^3-20*x^8*exp(x)^2+10*x^9*exp(x)-2*x^10),x, algorithm="giac")

[Out]

1/2*x^2/(x^8 - 4*x^7*e^x + 6*x^6*e^(2*x) - 4*x^6*e^(2*x + 4) - 4*x^5*e^(3*x) + 12*x^5*e^(3*x + 4) + x^4*e^(4*x
) + 6*x^4*e^(4*x + 8) - 12*x^4*e^(4*x + 4) - 12*x^3*e^(5*x + 8) + 4*x^3*e^(5*x + 4) - 4*x^2*e^(6*x + 12) + 6*x
^2*e^(6*x + 8) + 4*x*e^(7*x + 12) + e^(8*x + 16))

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maple [A]  time = 0.12, size = 24, normalized size = 0.83

method	result	size
risch	\(\frac {x^{2}}{4 \left ({\mathrm e}^{x} x -x^{2}+{\mathrm e}^{2 x +4}\right )^{4}}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-4*x^2+x)*exp(2*x+4)+(-2*x^3-x^2)*exp(x)+3*x^3)/(2*exp(2*x+4)^5+(10*exp(x)*x-10*x^2)*exp(2*x+4)^4+(20*ex
p(x)^2*x^2-40*exp(x)*x^3+20*x^4)*exp(2*x+4)^3+(20*x^3*exp(x)^3-60*exp(x)^2*x^4+60*x^5*exp(x)-20*x^6)*exp(2*x+4
)^2+(10*x^4*exp(x)^4-40*x^5*exp(x)^3+60*x^6*exp(x)^2-40*x^7*exp(x)+10*x^8)*exp(2*x+4)+2*x^5*exp(x)^5-10*x^6*ex
p(x)^4+20*x^7*exp(x)^3-20*x^8*exp(x)^2+10*x^9*exp(x)-2*x^10),x,method=_RETURNVERBOSE)

[Out]

1/4*x^2/(exp(x)*x-x^2+exp(2*x+4))^4

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maxima [B]  time = 0.84, size = 117, normalized size = 4.03 \begin {gather*} \frac {x^{2}}{4 \, {\left (x^{8} - 2 \, x^{6} {\left (2 \, e^{4} - 3\right )} e^{\left (2 \, x\right )} - 4 \, x^{7} e^{x} + 4 \, x^{5} {\left (3 \, e^{4} - 1\right )} e^{\left (3 \, x\right )} + x^{4} {\left (6 \, e^{8} - 12 \, e^{4} + 1\right )} e^{\left (4 \, x\right )} - 4 \, x^{3} {\left (3 \, e^{8} - e^{4}\right )} e^{\left (5 \, x\right )} - 2 \, x^{2} {\left (2 \, e^{12} - 3 \, e^{8}\right )} e^{\left (6 \, x\right )} + 4 \, x e^{\left (7 \, x + 12\right )} + e^{\left (8 \, x + 16\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^2+x)*exp(2*x+4)+(-2*x^3-x^2)*exp(x)+3*x^3)/(2*exp(2*x+4)^5+(10*exp(x)*x-10*x^2)*exp(2*x+4)^4+
(20*exp(x)^2*x^2-40*exp(x)*x^3+20*x^4)*exp(2*x+4)^3+(20*x^3*exp(x)^3-60*exp(x)^2*x^4+60*x^5*exp(x)-20*x^6)*exp
(2*x+4)^2+(10*x^4*exp(x)^4-40*x^5*exp(x)^3+60*x^6*exp(x)^2-40*x^7*exp(x)+10*x^8)*exp(2*x+4)+2*x^5*exp(x)^5-10*
x^6*exp(x)^4+20*x^7*exp(x)^3-20*x^8*exp(x)^2+10*x^9*exp(x)-2*x^10),x, algorithm="maxima")

[Out]

1/4*x^2/(x^8 - 2*x^6*(2*e^4 - 3)*e^(2*x) - 4*x^7*e^x + 4*x^5*(3*e^4 - 1)*e^(3*x) + x^4*(6*e^8 - 12*e^4 + 1)*e^
(4*x) - 4*x^3*(3*e^8 - e^4)*e^(5*x) - 2*x^2*(2*e^12 - 3*e^8)*e^(6*x) + 4*x*e^(7*x + 12) + e^(8*x + 16))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {{\mathrm {e}}^{2\,x+4}\,\left (x-4\,x^2\right )-{\mathrm {e}}^x\,\left (2\,x^3+x^2\right )+3\,x^3}{2\,{\mathrm {e}}^{10\,x+20}+{\mathrm {e}}^{2\,x+4}\,\left (10\,x^4\,{\mathrm {e}}^{4\,x}-40\,x^7\,{\mathrm {e}}^x-40\,x^5\,{\mathrm {e}}^{3\,x}+60\,x^6\,{\mathrm {e}}^{2\,x}+10\,x^8\right )+10\,x^9\,{\mathrm {e}}^x+{\mathrm {e}}^{6\,x+12}\,\left (20\,x^2\,{\mathrm {e}}^{2\,x}-40\,x^3\,{\mathrm {e}}^x+20\,x^4\right )+{\mathrm {e}}^{8\,x+16}\,\left (10\,x\,{\mathrm {e}}^x-10\,x^2\right )+2\,x^5\,{\mathrm {e}}^{5\,x}-10\,x^6\,{\mathrm {e}}^{4\,x}+20\,x^7\,{\mathrm {e}}^{3\,x}-20\,x^8\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x+8}\,\left (60\,x^5\,{\mathrm {e}}^x+20\,x^3\,{\mathrm {e}}^{3\,x}-60\,x^4\,{\mathrm {e}}^{2\,x}-20\,x^6\right )-2\,x^{10}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(2*x + 4)*(x - 4*x^2) - exp(x)*(x^2 + 2*x^3) + 3*x^3)/(2*exp(10*x + 20) + exp(2*x + 4)*(10*x^4*exp(4*x
) - 40*x^7*exp(x) - 40*x^5*exp(3*x) + 60*x^6*exp(2*x) + 10*x^8) + 10*x^9*exp(x) + exp(6*x + 12)*(20*x^2*exp(2*
x) - 40*x^3*exp(x) + 20*x^4) + exp(8*x + 16)*(10*x*exp(x) - 10*x^2) + 2*x^5*exp(5*x) - 10*x^6*exp(4*x) + 20*x^
7*exp(3*x) - 20*x^8*exp(2*x) + exp(4*x + 8)*(60*x^5*exp(x) + 20*x^3*exp(3*x) - 60*x^4*exp(2*x) - 20*x^6) - 2*x
^10),x)

[Out]

int((exp(2*x + 4)*(x - 4*x^2) - exp(x)*(x^2 + 2*x^3) + 3*x^3)/(2*exp(10*x + 20) + exp(2*x + 4)*(10*x^4*exp(4*x
) - 40*x^7*exp(x) - 40*x^5*exp(3*x) + 60*x^6*exp(2*x) + 10*x^8) + 10*x^9*exp(x) + exp(6*x + 12)*(20*x^2*exp(2*
x) - 40*x^3*exp(x) + 20*x^4) + exp(8*x + 16)*(10*x*exp(x) - 10*x^2) + 2*x^5*exp(5*x) - 10*x^6*exp(4*x) + 20*x^
7*exp(3*x) - 20*x^8*exp(2*x) + exp(4*x + 8)*(60*x^5*exp(x) + 20*x^3*exp(3*x) - 60*x^4*exp(2*x) - 20*x^6) - 2*x
^10), x)

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sympy [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**2+x)*exp(2*x+4)+(-2*x**3-x**2)*exp(x)+3*x**3)/(2*exp(2*x+4)**5+(10*exp(x)*x-10*x**2)*exp(2*x
+4)**4+(20*exp(x)**2*x**2-40*exp(x)*x**3+20*x**4)*exp(2*x+4)**3+(20*x**3*exp(x)**3-60*exp(x)**2*x**4+60*x**5*e
xp(x)-20*x**6)*exp(2*x+4)**2+(10*x**4*exp(x)**4-40*x**5*exp(x)**3+60*x**6*exp(x)**2-40*x**7*exp(x)+10*x**8)*ex
p(2*x+4)+2*x**5*exp(x)**5-10*x**6*exp(x)**4+20*x**7*exp(x)**3-20*x**8*exp(x)**2+10*x**9*exp(x)-2*x**10),x)

[Out]

Timed out

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