3.75.46 \(\int \frac {-x^6+e^6 (32 x^3-32 x^4)+e^3 (8 x^4-12 x^5)+e^{3+x} (-4 x^3+3 x^4-x^5+e^6 (-32 x+32 x^2-16 x^3)+e^3 (-24 x^2+20 x^3-8 x^4))}{e^{15+3 x} (-16+24 x-12 x^2+2 x^3)+e^{12+2 x} (-48 x^2+72 x^3-36 x^4+6 x^5)+e^{9+x} (-48 x^4+72 x^5-36 x^6+6 x^7)+e^6 (-16 x^6+24 x^7-12 x^8+2 x^9)} \, dx\)

Optimal. Leaf size=32 \[ \frac {x^2 \left (-4-\frac {x}{e^3}\right )^2}{(4-2 x)^2 \left (e^{3+x}+x^2\right )^2} \]

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Rubi [F]  time = 3.73, 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 {-x^6+e^6 \left (32 x^3-32 x^4\right )+e^3 \left (8 x^4-12 x^5\right )+e^{3+x} \left (-4 x^3+3 x^4-x^5+e^6 \left (-32 x+32 x^2-16 x^3\right )+e^3 \left (-24 x^2+20 x^3-8 x^4\right )\right )}{e^{15+3 x} \left (-16+24 x-12 x^2+2 x^3\right )+e^{12+2 x} \left (-48 x^2+72 x^3-36 x^4+6 x^5\right )+e^{9+x} \left (-48 x^4+72 x^5-36 x^6+6 x^7\right )+e^6 \left (-16 x^6+24 x^7-12 x^8+2 x^9\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-x^6 + E^6*(32*x^3 - 32*x^4) + E^3*(8*x^4 - 12*x^5) + E^(3 + x)*(-4*x^3 + 3*x^4 - x^5 + E^6*(-32*x + 32*x
^2 - 16*x^3) + E^3*(-24*x^2 + 20*x^3 - 8*x^4)))/(E^(15 + 3*x)*(-16 + 24*x - 12*x^2 + 2*x^3) + E^(12 + 2*x)*(-4
8*x^2 + 72*x^3 - 36*x^4 + 6*x^5) + E^(9 + x)*(-48*x^4 + 72*x^5 - 36*x^6 + 6*x^7) + E^6*(-16*x^6 + 24*x^7 - 12*
x^8 + 2*x^9)),x]

[Out]

(8*(1 + 2*E^3)^2*Defer[Int][(E^(3 + x) + x^2)^(-3), x])/E^6 + (16*(1 + 2*E^3)^2*Defer[Int][1/((-2 + x)*(E^(3 +
x) + x^2)^3), x])/E^6 + (4*(1 + 2*E^3)^2*Defer[Int][x/(E^(3 + x) + x^2)^3, x])/E^6 + (2*(1 + 2*E^3)^2*Defer[I
nt][x^2/(E^(3 + x) + x^2)^3, x])/E^6 + ((1 + 4*E^3)*Defer[Int][x^3/(E^(3 + x) + x^2)^3, x])/E^6 + Defer[Int][x
^4/(E^(3 + x) + x^2)^3, x]/(2*E^6) - ((5 + 14*E^3 + 8*E^6)*Defer[Int][(E^(3 + x) + x^2)^(-2), x])/E^6 - (8*(1
+ 2*E^3)^2*Defer[Int][1/((-2 + x)^3*(E^(3 + x) + x^2)^2), x])/E^6 - (8*(2 + 7*E^3 + 6*E^6)*Defer[Int][1/((-2 +
x)^2*(E^(3 + x) + x^2)^2), x])/E^6 - (16*(1 + 3*E^3 + 2*E^6)*Defer[Int][1/((-2 + x)*(E^(3 + x) + x^2)^2), x])
/E^6 - ((3 + 8*E^3)*Defer[Int][x/(E^(3 + x) + x^2)^2, x])/(2*E^6) - Defer[Int][x^2/(E^(3 + x) + x^2)^2, x]/(2*
E^6)

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x \left (4 e^3+x\right ) \left (8 e^3 (-1+x) x^2+x^4+e^{3+x} x \left (4-3 x+x^2\right )+4 e^{6+x} \left (2-2 x+x^2\right )\right )}{2 e^6 (2-x)^3 \left (e^{3+x}+x^2\right )^3} \, dx\\ &=\frac {\int \frac {x \left (4 e^3+x\right ) \left (8 e^3 (-1+x) x^2+x^4+e^{3+x} x \left (4-3 x+x^2\right )+4 e^{6+x} \left (2-2 x+x^2\right )\right )}{(2-x)^3 \left (e^{3+x}+x^2\right )^3} \, dx}{2 e^6}\\ &=\frac {\int \left (\frac {x^3 \left (4 e^3+x\right )^2}{(-2+x) \left (e^{3+x}+x^2\right )^3}+\frac {x \left (32 e^6+8 e^3 \left (3-4 e^3\right ) x+4 \left (1-5 e^3+4 e^6\right ) x^2-\left (3-8 e^3\right ) x^3+x^4\right )}{(2-x)^3 \left (e^{3+x}+x^2\right )^2}\right ) \, dx}{2 e^6}\\ &=\frac {\int \frac {x^3 \left (4 e^3+x\right )^2}{(-2+x) \left (e^{3+x}+x^2\right )^3} \, dx}{2 e^6}+\frac {\int \frac {x \left (32 e^6+8 e^3 \left (3-4 e^3\right ) x+4 \left (1-5 e^3+4 e^6\right ) x^2-\left (3-8 e^3\right ) x^3+x^4\right )}{(2-x)^3 \left (e^{3+x}+x^2\right )^2} \, dx}{2 e^6}\\ &=\frac {\int \left (\frac {16 \left (1+2 e^3\right )^2}{\left (e^{3+x}+x^2\right )^3}+\frac {32 \left (1+2 e^3\right )^2}{(-2+x) \left (e^{3+x}+x^2\right )^3}+\frac {8 \left (1+2 e^3\right )^2 x}{\left (e^{3+x}+x^2\right )^3}+\frac {4 \left (1+2 e^3\right )^2 x^2}{\left (e^{3+x}+x^2\right )^3}+\frac {2 \left (1+4 e^3\right ) x^3}{\left (e^{3+x}+x^2\right )^3}+\frac {x^4}{\left (e^{3+x}+x^2\right )^3}\right ) \, dx}{2 e^6}+\frac {\int \left (-\frac {2 \left (5+14 e^3+8 e^6\right )}{\left (e^{3+x}+x^2\right )^2}-\frac {16 \left (1+2 e^3\right )^2}{(-2+x)^3 \left (e^{3+x}+x^2\right )^2}-\frac {16 \left (2+7 e^3+6 e^6\right )}{(-2+x)^2 \left (e^{3+x}+x^2\right )^2}-\frac {32 \left (1+3 e^3+2 e^6\right )}{(-2+x) \left (e^{3+x}+x^2\right )^2}-\frac {\left (3+8 e^3\right ) x}{\left (e^{3+x}+x^2\right )^2}-\frac {x^2}{\left (e^{3+x}+x^2\right )^2}\right ) \, dx}{2 e^6}\\ &=\frac {\int \frac {x^4}{\left (e^{3+x}+x^2\right )^3} \, dx}{2 e^6}-\frac {\int \frac {x^2}{\left (e^{3+x}+x^2\right )^2} \, dx}{2 e^6}+\frac {\left (2 \left (1+2 e^3\right )^2\right ) \int \frac {x^2}{\left (e^{3+x}+x^2\right )^3} \, dx}{e^6}+\frac {\left (4 \left (1+2 e^3\right )^2\right ) \int \frac {x}{\left (e^{3+x}+x^2\right )^3} \, dx}{e^6}+\frac {\left (8 \left (1+2 e^3\right )^2\right ) \int \frac {1}{\left (e^{3+x}+x^2\right )^3} \, dx}{e^6}-\frac {\left (8 \left (1+2 e^3\right )^2\right ) \int \frac {1}{(-2+x)^3 \left (e^{3+x}+x^2\right )^2} \, dx}{e^6}+\frac {\left (16 \left (1+2 e^3\right )^2\right ) \int \frac {1}{(-2+x) \left (e^{3+x}+x^2\right )^3} \, dx}{e^6}+\frac {\left (1+4 e^3\right ) \int \frac {x^3}{\left (e^{3+x}+x^2\right )^3} \, dx}{e^6}-\frac {\left (3+8 e^3\right ) \int \frac {x}{\left (e^{3+x}+x^2\right )^2} \, dx}{2 e^6}-\frac {\left (16 \left (1+3 e^3+2 e^6\right )\right ) \int \frac {1}{(-2+x) \left (e^{3+x}+x^2\right )^2} \, dx}{e^6}-\frac {\left (8 \left (2+7 e^3+6 e^6\right )\right ) \int \frac {1}{(-2+x)^2 \left (e^{3+x}+x^2\right )^2} \, dx}{e^6}-\frac {\left (5+14 e^3+8 e^6\right ) \int \frac {1}{\left (e^{3+x}+x^2\right )^2} \, dx}{e^6}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.53, size = 35, normalized size = 1.09 \begin {gather*} \frac {x^2 \left (4 e^3+x\right )^2}{4 e^6 (-2+x)^2 \left (e^{3+x}+x^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-x^6 + E^6*(32*x^3 - 32*x^4) + E^3*(8*x^4 - 12*x^5) + E^(3 + x)*(-4*x^3 + 3*x^4 - x^5 + E^6*(-32*x
+ 32*x^2 - 16*x^3) + E^3*(-24*x^2 + 20*x^3 - 8*x^4)))/(E^(15 + 3*x)*(-16 + 24*x - 12*x^2 + 2*x^3) + E^(12 + 2*
x)*(-48*x^2 + 72*x^3 - 36*x^4 + 6*x^5) + E^(9 + x)*(-48*x^4 + 72*x^5 - 36*x^6 + 6*x^7) + E^6*(-16*x^6 + 24*x^7
- 12*x^8 + 2*x^9)),x]

[Out]

(x^2*(4*E^3 + x)^2)/(4*E^6*(-2 + x)^2*(E^(3 + x) + x^2)^2)

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fricas [B]  time = 0.92, size = 78, normalized size = 2.44 \begin {gather*} \frac {x^{4} e^{6} + 8 \, x^{3} e^{9} + 16 \, x^{2} e^{12}}{4 \, {\left ({\left (x^{6} - 4 \, x^{5} + 4 \, x^{4}\right )} e^{12} + {\left (x^{2} - 4 \, x + 4\right )} e^{\left (2 \, x + 18\right )} + 2 \, {\left (x^{4} - 4 \, x^{3} + 4 \, x^{2}\right )} e^{\left (x + 15\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-16*x^3+32*x^2-32*x)*exp(3)^2+(-8*x^4+20*x^3-24*x^2)*exp(3)-x^5+3*x^4-4*x^3)*exp(3+x)+(-32*x^4+32
*x^3)*exp(3)^2+(-12*x^5+8*x^4)*exp(3)-x^6)/((2*x^3-12*x^2+24*x-16)*exp(3)^2*exp(3+x)^3+(6*x^5-36*x^4+72*x^3-48
*x^2)*exp(3)^2*exp(3+x)^2+(6*x^7-36*x^6+72*x^5-48*x^4)*exp(3)^2*exp(3+x)+(2*x^9-12*x^8+24*x^7-16*x^6)*exp(3)^2
),x, algorithm="fricas")

[Out]

1/4*(x^4*e^6 + 8*x^3*e^9 + 16*x^2*e^12)/((x^6 - 4*x^5 + 4*x^4)*e^12 + (x^2 - 4*x + 4)*e^(2*x + 18) + 2*(x^4 -
4*x^3 + 4*x^2)*e^(x + 15))

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giac [B]  time = 0.33, size = 97, normalized size = 3.03 \begin {gather*} \frac {x^{4} + 8 \, x^{3} e^{3} + 16 \, x^{2} e^{6}}{4 \, {\left (x^{6} e^{6} - 4 \, x^{5} e^{6} + 4 \, x^{4} e^{6} + 2 \, x^{4} e^{\left (x + 9\right )} - 8 \, x^{3} e^{\left (x + 9\right )} + x^{2} e^{\left (2 \, x + 12\right )} + 8 \, x^{2} e^{\left (x + 9\right )} - 4 \, x e^{\left (2 \, x + 12\right )} + 4 \, e^{\left (2 \, x + 12\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-16*x^3+32*x^2-32*x)*exp(3)^2+(-8*x^4+20*x^3-24*x^2)*exp(3)-x^5+3*x^4-4*x^3)*exp(3+x)+(-32*x^4+32
*x^3)*exp(3)^2+(-12*x^5+8*x^4)*exp(3)-x^6)/((2*x^3-12*x^2+24*x-16)*exp(3)^2*exp(3+x)^3+(6*x^5-36*x^4+72*x^3-48
*x^2)*exp(3)^2*exp(3+x)^2+(6*x^7-36*x^6+72*x^5-48*x^4)*exp(3)^2*exp(3+x)+(2*x^9-12*x^8+24*x^7-16*x^6)*exp(3)^2
),x, algorithm="giac")

[Out]

1/4*(x^4 + 8*x^3*e^3 + 16*x^2*e^6)/(x^6*e^6 - 4*x^5*e^6 + 4*x^4*e^6 + 2*x^4*e^(x + 9) - 8*x^3*e^(x + 9) + x^2*
e^(2*x + 12) + 8*x^2*e^(x + 9) - 4*x*e^(2*x + 12) + 4*e^(2*x + 12))

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maple [A]  time = 0.13, size = 41, normalized size = 1.28

method	result	size
risch	\(\frac {x^{2} \left (16 \,{\mathrm e}^{6}+8 x \,{\mathrm e}^{3}+x^{2}\right ) {\mathrm e}^{-6}}{4 \left (x^{2}-4 x +4\right ) \left (x^{2}+{\mathrm e}^{3+x}\right )^{2}}\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-16*x^3+32*x^2-32*x)*exp(3)^2+(-8*x^4+20*x^3-24*x^2)*exp(3)-x^5+3*x^4-4*x^3)*exp(3+x)+(-32*x^4+32*x^3)*
exp(3)^2+(-12*x^5+8*x^4)*exp(3)-x^6)/((2*x^3-12*x^2+24*x-16)*exp(3)^2*exp(3+x)^3+(6*x^5-36*x^4+72*x^3-48*x^2)*
exp(3)^2*exp(3+x)^2+(6*x^7-36*x^6+72*x^5-48*x^4)*exp(3)^2*exp(3+x)+(2*x^9-12*x^8+24*x^7-16*x^6)*exp(3)^2),x,me
thod=_RETURNVERBOSE)

[Out]

1/4*x^2*(16*exp(6)+8*x*exp(3)+x^2)/(x^2-4*x+4)/(x^2+exp(3+x))^2*exp(-6)

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maxima [B]  time = 0.51, size = 89, normalized size = 2.78 \begin {gather*} \frac {x^{4} + 8 \, x^{3} e^{3} + 16 \, x^{2} e^{6}}{4 \, {\left (x^{6} e^{6} - 4 \, x^{5} e^{6} + 4 \, x^{4} e^{6} + {\left (x^{2} e^{12} - 4 \, x e^{12} + 4 \, e^{12}\right )} e^{\left (2 \, x\right )} + 2 \, {\left (x^{4} e^{9} - 4 \, x^{3} e^{9} + 4 \, x^{2} e^{9}\right )} e^{x}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-16*x^3+32*x^2-32*x)*exp(3)^2+(-8*x^4+20*x^3-24*x^2)*exp(3)-x^5+3*x^4-4*x^3)*exp(3+x)+(-32*x^4+32
*x^3)*exp(3)^2+(-12*x^5+8*x^4)*exp(3)-x^6)/((2*x^3-12*x^2+24*x-16)*exp(3)^2*exp(3+x)^3+(6*x^5-36*x^4+72*x^3-48
*x^2)*exp(3)^2*exp(3+x)^2+(6*x^7-36*x^6+72*x^5-48*x^4)*exp(3)^2*exp(3+x)+(2*x^9-12*x^8+24*x^7-16*x^6)*exp(3)^2
),x, algorithm="maxima")

[Out]

1/4*(x^4 + 8*x^3*e^3 + 16*x^2*e^6)/(x^6*e^6 - 4*x^5*e^6 + 4*x^4*e^6 + (x^2*e^12 - 4*x*e^12 + 4*e^12)*e^(2*x) +
2*(x^4*e^9 - 4*x^3*e^9 + 4*x^2*e^9)*e^x)

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mupad [B]  time = 5.45, size = 98, normalized size = 3.06 \begin {gather*} -\frac {\frac {{\mathrm {e}}^{-6}\,x^7}{4}+\frac {{\mathrm {e}}^{-6}\,\left (8\,{\mathrm {e}}^3-4\right )\,x^6}{4}+\frac {{\mathrm {e}}^{-6}\,\left (16\,{\mathrm {e}}^6-32\,{\mathrm {e}}^3+4\right )\,x^5}{4}+\frac {{\mathrm {e}}^{-6}\,\left (32\,{\mathrm {e}}^3-64\,{\mathrm {e}}^6\right )\,x^4}{4}+16\,x^3}{\left (2\,x-x^2\right )\,{\left (x-2\right )}^3\,\left ({\mathrm {e}}^{2\,x+6}+2\,x^2\,{\mathrm {e}}^{x+3}+x^4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x + 3)*(exp(6)*(32*x - 32*x^2 + 16*x^3) + exp(3)*(24*x^2 - 20*x^3 + 8*x^4) + 4*x^3 - 3*x^4 + x^5) - e
xp(3)*(8*x^4 - 12*x^5) - exp(6)*(32*x^3 - 32*x^4) + x^6)/(exp(6)*(16*x^6 - 24*x^7 + 12*x^8 - 2*x^9) - exp(6)*e
xp(3*x + 9)*(24*x - 12*x^2 + 2*x^3 - 16) + exp(x + 3)*exp(6)*(48*x^4 - 72*x^5 + 36*x^6 - 6*x^7) + exp(6)*exp(2
*x + 6)*(48*x^2 - 72*x^3 + 36*x^4 - 6*x^5)),x)

[Out]

-((x^7*exp(-6))/4 + 16*x^3 + (x^5*exp(-6)*(16*exp(6) - 32*exp(3) + 4))/4 + (x^4*exp(-6)*(32*exp(3) - 64*exp(6)
))/4 + (x^6*exp(-6)*(8*exp(3) - 4))/4)/((2*x - x^2)*(x - 2)^3*(exp(2*x + 6) + 2*x^2*exp(x + 3) + x^4))

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sympy [B]  time = 0.43, size = 102, normalized size = 3.19 \begin {gather*} \frac {x^{4} + 8 x^{3} e^{3} + 16 x^{2} e^{6}}{4 x^{6} e^{6} - 16 x^{5} e^{6} + 16 x^{4} e^{6} + \left (4 x^{2} e^{6} - 16 x e^{6} + 16 e^{6}\right ) e^{2 x + 6} + \left (8 x^{4} e^{6} - 32 x^{3} e^{6} + 32 x^{2} e^{6}\right ) e^{x + 3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-16*x**3+32*x**2-32*x)*exp(3)**2+(-8*x**4+20*x**3-24*x**2)*exp(3)-x**5+3*x**4-4*x**3)*exp(3+x)+(-
32*x**4+32*x**3)*exp(3)**2+(-12*x**5+8*x**4)*exp(3)-x**6)/((2*x**3-12*x**2+24*x-16)*exp(3)**2*exp(3+x)**3+(6*x
**5-36*x**4+72*x**3-48*x**2)*exp(3)**2*exp(3+x)**2+(6*x**7-36*x**6+72*x**5-48*x**4)*exp(3)**2*exp(3+x)+(2*x**9
-12*x**8+24*x**7-16*x**6)*exp(3)**2),x)

[Out]

(x**4 + 8*x**3*exp(3) + 16*x**2*exp(6))/(4*x**6*exp(6) - 16*x**5*exp(6) + 16*x**4*exp(6) + (4*x**2*exp(6) - 16
*x*exp(6) + 16*exp(6))*exp(2*x + 6) + (8*x**4*exp(6) - 32*x**3*exp(6) + 32*x**2*exp(6))*exp(x + 3))

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