3.75.25 \(\int \frac {-48-24 x^2-3 x^4+e^{x+\frac {3}{4+x^2}} (-32 x-80 x^2-74 x^3-44 x^4-28 x^5-13 x^6-4 x^7-x^8)}{16+32 x+24 x^2+16 x^3+9 x^4+2 x^5+x^6} \, dx\)

Optimal. Leaf size=28 \[ -e^{x+\frac {3}{4+x^2}} x^2+\frac {4+x}{1+x} \]

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Rubi [F]  time = 1.26, 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 {-48-24 x^2-3 x^4+e^{x+\frac {3}{4+x^2}} \left (-32 x-80 x^2-74 x^3-44 x^4-28 x^5-13 x^6-4 x^7-x^8\right )}{16+32 x+24 x^2+16 x^3+9 x^4+2 x^5+x^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-48 - 24*x^2 - 3*x^4 + E^(x + 3/(4 + x^2))*(-32*x - 80*x^2 - 74*x^3 - 44*x^4 - 28*x^5 - 13*x^6 - 4*x^7 -
x^8))/(16 + 32*x + 24*x^2 + 16*x^3 + 9*x^4 + 2*x^5 + x^6),x]

[Out]

3/(1 + x) - 3*Defer[Int][E^(x + 3/(4 + x^2))/(2*I - x), x] - 2*Defer[Int][E^(x + 3/(4 + x^2))*x, x] - Defer[In
t][E^(x + 3/(4 + x^2))*x^2, x] + 3*Defer[Int][E^(x + 3/(4 + x^2))/(2*I + x), x] - 24*Defer[Int][(E^(x + 3/(4 +
x^2))*x)/(4 + x^2)^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-3 \left (4+x^2\right )^2-e^{x+\frac {3}{4+x^2}} x (1+x)^2 \left (32+16 x+10 x^2+8 x^3+2 x^4+x^5\right )}{\left (4+4 x+x^2+x^3\right )^2} \, dx\\ &=\int \left (-\frac {3}{(1+x)^2}-\frac {e^{x+\frac {3}{4+x^2}} x \left (32+16 x+10 x^2+8 x^3+2 x^4+x^5\right )}{\left (4+x^2\right )^2}\right ) \, dx\\ &=\frac {3}{1+x}-\int \frac {e^{x+\frac {3}{4+x^2}} x \left (32+16 x+10 x^2+8 x^3+2 x^4+x^5\right )}{\left (4+x^2\right )^2} \, dx\\ &=\frac {3}{1+x}-\int \left (2 e^{x+\frac {3}{4+x^2}} x+e^{x+\frac {3}{4+x^2}} x^2+\frac {24 e^{x+\frac {3}{4+x^2}} x}{\left (4+x^2\right )^2}-\frac {6 e^{x+\frac {3}{4+x^2}} x}{4+x^2}\right ) \, dx\\ &=\frac {3}{1+x}-2 \int e^{x+\frac {3}{4+x^2}} x \, dx+6 \int \frac {e^{x+\frac {3}{4+x^2}} x}{4+x^2} \, dx-24 \int \frac {e^{x+\frac {3}{4+x^2}} x}{\left (4+x^2\right )^2} \, dx-\int e^{x+\frac {3}{4+x^2}} x^2 \, dx\\ &=\frac {3}{1+x}-2 \int e^{x+\frac {3}{4+x^2}} x \, dx+6 \int \left (-\frac {e^{x+\frac {3}{4+x^2}}}{2 (2 i-x)}+\frac {e^{x+\frac {3}{4+x^2}}}{2 (2 i+x)}\right ) \, dx-24 \int \frac {e^{x+\frac {3}{4+x^2}} x}{\left (4+x^2\right )^2} \, dx-\int e^{x+\frac {3}{4+x^2}} x^2 \, dx\\ &=\frac {3}{1+x}-2 \int e^{x+\frac {3}{4+x^2}} x \, dx-3 \int \frac {e^{x+\frac {3}{4+x^2}}}{2 i-x} \, dx+3 \int \frac {e^{x+\frac {3}{4+x^2}}}{2 i+x} \, dx-24 \int \frac {e^{x+\frac {3}{4+x^2}} x}{\left (4+x^2\right )^2} \, dx-\int e^{x+\frac {3}{4+x^2}} x^2 \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.36, size = 26, normalized size = 0.93 \begin {gather*} -e^{x+\frac {3}{4+x^2}} x^2+\frac {3}{1+x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-48 - 24*x^2 - 3*x^4 + E^(x + 3/(4 + x^2))*(-32*x - 80*x^2 - 74*x^3 - 44*x^4 - 28*x^5 - 13*x^6 - 4*
x^7 - x^8))/(16 + 32*x + 24*x^2 + 16*x^3 + 9*x^4 + 2*x^5 + x^6),x]

[Out]

-(E^(x + 3/(4 + x^2))*x^2) + 3/(1 + x)

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fricas [A]  time = 1.02, size = 34, normalized size = 1.21 \begin {gather*} -\frac {{\left (x^{3} + x^{2}\right )} e^{\left (\frac {x^{3} + 4 \, x + 3}{x^{2} + 4}\right )} - 3}{x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^8-4*x^7-13*x^6-28*x^5-44*x^4-74*x^3-80*x^2-32*x)*exp(3/(x^2+4))*exp(x)-3*x^4-24*x^2-48)/(x^6+2*
x^5+9*x^4+16*x^3+24*x^2+32*x+16),x, algorithm="fricas")

[Out]

-((x^3 + x^2)*e^((x^3 + 4*x + 3)/(x^2 + 4)) - 3)/(x + 1)

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giac [B]  time = 0.34, size = 69, normalized size = 2.46 \begin {gather*} -\frac {x^{3} e^{\left (\frac {4 \, x^{3} - 3 \, x^{2} + 16 \, x}{4 \, {\left (x^{2} + 4\right )}} + \frac {3}{4}\right )} + x^{2} e^{\left (\frac {4 \, x^{3} - 3 \, x^{2} + 16 \, x}{4 \, {\left (x^{2} + 4\right )}} + \frac {3}{4}\right )} - 3}{x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^8-4*x^7-13*x^6-28*x^5-44*x^4-74*x^3-80*x^2-32*x)*exp(3/(x^2+4))*exp(x)-3*x^4-24*x^2-48)/(x^6+2*
x^5+9*x^4+16*x^3+24*x^2+32*x+16),x, algorithm="giac")

[Out]

-(x^3*e^(1/4*(4*x^3 - 3*x^2 + 16*x)/(x^2 + 4) + 3/4) + x^2*e^(1/4*(4*x^3 - 3*x^2 + 16*x)/(x^2 + 4) + 3/4) - 3)
/(x + 1)

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maple [A]  time = 0.08, size = 31, normalized size = 1.11

method	result	size
risch	\(\frac {3}{x +1}-x^{2} {\mathrm e}^{\frac {x^{3}+4 x +3}{x^{2}+4}}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x^8-4*x^7-13*x^6-28*x^5-44*x^4-74*x^3-80*x^2-32*x)*exp(3/(x^2+4))*exp(x)-3*x^4-24*x^2-48)/(x^6+2*x^5+9*
x^4+16*x^3+24*x^2+32*x+16),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [B]  time = 0.53, size = 91, normalized size = 3.25 \begin {gather*} -x^{2} e^{\left (x + \frac {3}{x^{2} + 4}\right )} + \frac {6 \, {\left (11 \, x^{2} - 5 \, x + 24\right )}}{25 \, {\left (x^{3} + x^{2} + 4 \, x + 4\right )}} + \frac {3 \, {\left (7 \, x^{2} - 10 \, x - 12\right )}}{25 \, {\left (x^{3} + x^{2} + 4 \, x + 4\right )}} - \frac {12 \, {\left (x^{2} - 5 \, x - 16\right )}}{25 \, {\left (x^{3} + x^{2} + 4 \, x + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^8-4*x^7-13*x^6-28*x^5-44*x^4-74*x^3-80*x^2-32*x)*exp(3/(x^2+4))*exp(x)-3*x^4-24*x^2-48)/(x^6+2*
x^5+9*x^4+16*x^3+24*x^2+32*x+16),x, algorithm="maxima")

[Out]

-x^2*e^(x + 3/(x^2 + 4)) + 6/25*(11*x^2 - 5*x + 24)/(x^3 + x^2 + 4*x + 4) + 3/25*(7*x^2 - 10*x - 12)/(x^3 + x^
2 + 4*x + 4) - 12/25*(x^2 - 5*x - 16)/(x^3 + x^2 + 4*x + 4)

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mupad [B]  time = 5.09, size = 25, normalized size = 0.89 \begin {gather*} \frac {3}{x+1}-x^2\,{\mathrm {e}}^{\frac {3}{x^2+4}}\,{\mathrm {e}}^x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(24*x^2 + 3*x^4 + exp(3/(x^2 + 4))*exp(x)*(32*x + 80*x^2 + 74*x^3 + 44*x^4 + 28*x^5 + 13*x^6 + 4*x^7 + x^
8) + 48)/(32*x + 24*x^2 + 16*x^3 + 9*x^4 + 2*x^5 + x^6 + 16),x)

[Out]

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

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sympy [A]  time = 36.77, size = 19, normalized size = 0.68 \begin {gather*} - x^{2} e^{x} e^{\frac {3}{x^{2} + 4}} + \frac {3}{x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x**8-4*x**7-13*x**6-28*x**5-44*x**4-74*x**3-80*x**2-32*x)*exp(3/(x**2+4))*exp(x)-3*x**4-24*x**2-4
8)/(x**6+2*x**5+9*x**4+16*x**3+24*x**2+32*x+16),x)

[Out]

-x**2*exp(x)*exp(3/(x**2 + 4)) + 3/(x + 1)

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