3.85.14 \(\int \frac {3072+2432 x+268 x^2+8 x^3+e^{e^{x/4}} (-64+e^{x/4} (-16 x-x^2))}{1024+128 x+4 x^2} \, dx\)

Optimal. Leaf size=29 \[ -e^4+3 x+x^2-\frac {e^{e^{x/4}} x}{16+x} \]

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Rubi [F]  time = 0.50, 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 {3072+2432 x+268 x^2+8 x^3+e^{e^{x/4}} \left (-64+e^{x/4} \left (-16 x-x^2\right )\right )}{1024+128 x+4 x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(3072 + 2432*x + 268*x^2 + 8*x^3 + E^E^(x/4)*(-64 + E^(x/4)*(-16*x - x^2)))/(1024 + 128*x + 4*x^2),x]

[Out]

-E^E^(x/4) + 3*x + x^2 - 16*Defer[Int][E^E^(x/4)/(16 + x)^2, x] + 4*Defer[Int][E^((4*E^(x/4) + x)/4)/(16 + x),
x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3072+2432 x+268 x^2+8 x^3+e^{e^{x/4}} \left (-64+e^{x/4} \left (-16 x-x^2\right )\right )}{4 (16+x)^2} \, dx\\ &=\frac {1}{4} \int \frac {3072+2432 x+268 x^2+8 x^3+e^{e^{x/4}} \left (-64+e^{x/4} \left (-16 x-x^2\right )\right )}{(16+x)^2} \, dx\\ &=\frac {1}{4} \int \left (\frac {e^{\frac {1}{4} \left (4 e^{x/4}+x\right )} x}{-16-x}+\frac {3072}{(16+x)^2}-\frac {64 e^{e^{x/4}}}{(16+x)^2}+\frac {2432 x}{(16+x)^2}+\frac {268 x^2}{(16+x)^2}+\frac {8 x^3}{(16+x)^2}\right ) \, dx\\ &=-\frac {768}{16+x}+\frac {1}{4} \int \frac {e^{\frac {1}{4} \left (4 e^{x/4}+x\right )} x}{-16-x} \, dx+2 \int \frac {x^3}{(16+x)^2} \, dx-16 \int \frac {e^{e^{x/4}}}{(16+x)^2} \, dx+67 \int \frac {x^2}{(16+x)^2} \, dx+608 \int \frac {x}{(16+x)^2} \, dx\\ &=-\frac {768}{16+x}+\frac {1}{4} \int \left (-e^{\frac {1}{4} \left (4 e^{x/4}+x\right )}+\frac {16 e^{\frac {1}{4} \left (4 e^{x/4}+x\right )}}{16+x}\right ) \, dx+2 \int \left (-32+x-\frac {4096}{(16+x)^2}+\frac {768}{16+x}\right ) \, dx-16 \int \frac {e^{e^{x/4}}}{(16+x)^2} \, dx+67 \int \left (1+\frac {256}{(16+x)^2}-\frac {32}{16+x}\right ) \, dx+608 \int \left (-\frac {16}{(16+x)^2}+\frac {1}{16+x}\right ) \, dx\\ &=3 x+x^2-\frac {1}{4} \int e^{\frac {1}{4} \left (4 e^{x/4}+x\right )} \, dx+4 \int \frac {e^{\frac {1}{4} \left (4 e^{x/4}+x\right )}}{16+x} \, dx-16 \int \frac {e^{e^{x/4}}}{(16+x)^2} \, dx\\ &=3 x+x^2+4 \int \frac {e^{\frac {1}{4} \left (4 e^{x/4}+x\right )}}{16+x} \, dx-16 \int \frac {e^{e^{x/4}}}{(16+x)^2} \, dx-\operatorname {Subst}\left (\int e^{e^x+x} \, dx,x,\frac {x}{4}\right )\\ &=3 x+x^2+4 \int \frac {e^{\frac {1}{4} \left (4 e^{x/4}+x\right )}}{16+x} \, dx-16 \int \frac {e^{e^{x/4}}}{(16+x)^2} \, dx-\operatorname {Subst}\left (\int e^x \, dx,x,e^{x/4}\right )\\ &=-e^{e^{x/4}}+3 x+x^2+4 \int \frac {e^{\frac {1}{4} \left (4 e^{x/4}+x\right )}}{16+x} \, dx-16 \int \frac {e^{e^{x/4}}}{(16+x)^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.25, size = 26, normalized size = 0.90 \begin {gather*} \frac {x \left (48-e^{e^{x/4}}+19 x+x^2\right )}{16+x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3072 + 2432*x + 268*x^2 + 8*x^3 + E^E^(x/4)*(-64 + E^(x/4)*(-16*x - x^2)))/(1024 + 128*x + 4*x^2),x
]

[Out]

(x*(48 - E^E^(x/4) + 19*x + x^2))/(16 + x)

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fricas [A]  time = 0.65, size = 26, normalized size = 0.90 \begin {gather*} \frac {x^{3} + 19 \, x^{2} - x e^{\left (e^{\left (\frac {1}{4} \, x\right )}\right )} + 48 \, x}{x + 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^2-16*x)*exp(1/4*x)-64)*exp(exp(1/4*x))+8*x^3+268*x^2+2432*x+3072)/(4*x^2+128*x+1024),x, algori
thm="fricas")

[Out]

(x^3 + 19*x^2 - x*e^(e^(1/4*x)) + 48*x)/(x + 16)

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giac [B]  time = 0.18, size = 53, normalized size = 1.83 \begin {gather*} \frac {x^{3} e^{\left (\frac {1}{4} \, x\right )} + 19 \, x^{2} e^{\left (\frac {1}{4} \, x\right )} + 48 \, x e^{\left (\frac {1}{4} \, x\right )} - x e^{\left (\frac {1}{4} \, x + e^{\left (\frac {1}{4} \, x\right )}\right )}}{x e^{\left (\frac {1}{4} \, x\right )} + 16 \, e^{\left (\frac {1}{4} \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^2-16*x)*exp(1/4*x)-64)*exp(exp(1/4*x))+8*x^3+268*x^2+2432*x+3072)/(4*x^2+128*x+1024),x, algori
thm="giac")

[Out]

(x^3*e^(1/4*x) + 19*x^2*e^(1/4*x) + 48*x*e^(1/4*x) - x*e^(1/4*x + e^(1/4*x)))/(x*e^(1/4*x) + 16*e^(1/4*x))

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maple [A]  time = 0.41, size = 21, normalized size = 0.72

method	result	size
risch	\(x^{2}+3 x -\frac {{\mathrm e}^{{\mathrm e}^{\frac {x}{4}}} x}{x +16}\)	\(21\)
norman	\(\frac {x^{3}+19 x^{2}-x \,{\mathrm e}^{{\mathrm e}^{\frac {x}{4}}}-768}{x +16}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-x^2-16*x)*exp(1/4*x)-64)*exp(exp(1/4*x))+8*x^3+268*x^2+2432*x+3072)/(4*x^2+128*x+1024),x,method=_RETUR
NVERBOSE)

[Out]

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

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maxima [A]  time = 0.41, size = 20, normalized size = 0.69 \begin {gather*} x^{2} + 3 \, x - \frac {x e^{\left (e^{\left (\frac {1}{4} \, x\right )}\right )}}{x + 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^2-16*x)*exp(1/4*x)-64)*exp(exp(1/4*x))+8*x^3+268*x^2+2432*x+3072)/(4*x^2+128*x+1024),x, algori
thm="maxima")

[Out]

x^2 + 3*x - x*e^(e^(1/4*x))/(x + 16)

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mupad [B]  time = 5.22, size = 22, normalized size = 0.76 \begin {gather*} \frac {x\,\left (19\,x-{\mathrm {e}}^{{\mathrm {e}}^{x/4}}+x^2+48\right )}{x+16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2432*x - exp(exp(x/4))*(exp(x/4)*(16*x + x^2) + 64) + 268*x^2 + 8*x^3 + 3072)/(128*x + 4*x^2 + 1024),x)

[Out]

(x*(19*x - exp(exp(x/4)) + x^2 + 48))/(x + 16)

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sympy [A]  time = 0.16, size = 17, normalized size = 0.59 \begin {gather*} x^{2} + 3 x - \frac {x e^{e^{\frac {x}{4}}}}{x + 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x**2-16*x)*exp(1/4*x)-64)*exp(exp(1/4*x))+8*x**3+268*x**2+2432*x+3072)/(4*x**2+128*x+1024),x)

[Out]

x**2 + 3*x - x*exp(exp(x/4))/(x + 16)

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