3.50.1 \(\int \frac {e^{e^{16 x}} (6-12 x^2+e^{16 x} (96 x+704 x^2+192 x^3))}{9+132 x+520 x^2+264 x^3+36 x^4} \, dx\)

Optimal. Leaf size=24 \[ \frac {e^{e^{16 x}}}{5+\frac {3}{2 x}+3 (2+x)} \]

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Rubi [A]  time = 0.10, antiderivative size = 45, normalized size of antiderivative = 1.88, number of steps used = 1, number of rules used = 1, integrand size = 57, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.018, Rules used = {2288} \begin {gather*} \frac {2 e^{e^{16 x}} \left (6 x^3+22 x^2+3 x\right )}{36 x^4+264 x^3+520 x^2+132 x+9} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^E^(16*x)*(6 - 12*x^2 + E^(16*x)*(96*x + 704*x^2 + 192*x^3)))/(9 + 132*x + 520*x^2 + 264*x^3 + 36*x^4),x
]

[Out]

(2*E^E^(16*x)*(3*x + 22*x^2 + 6*x^3))/(9 + 132*x + 520*x^2 + 264*x^3 + 36*x^4)

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {2 e^{e^{16 x}} \left (3 x+22 x^2+6 x^3\right )}{9+132 x+520 x^2+264 x^3+36 x^4}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 22, normalized size = 0.92 \begin {gather*} \frac {2 e^{e^{16 x}} x}{3+22 x+6 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^E^(16*x)*(6 - 12*x^2 + E^(16*x)*(96*x + 704*x^2 + 192*x^3)))/(9 + 132*x + 520*x^2 + 264*x^3 + 36*
x^4),x]

[Out]

(2*E^E^(16*x)*x)/(3 + 22*x + 6*x^2)

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fricas [A]  time = 0.84, size = 20, normalized size = 0.83 \begin {gather*} \frac {2 \, x e^{\left (e^{\left (16 \, x\right )}\right )}}{6 \, x^{2} + 22 \, x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((192*x^3+704*x^2+96*x)*exp(2*x)^8-12*x^2+6)*exp(exp(2*x)^8)/(36*x^4+264*x^3+520*x^2+132*x+9),x, alg
orithm="fricas")

[Out]

2*x*e^(e^(16*x))/(6*x^2 + 22*x + 3)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {2 \, {\left (6 \, x^{2} - 16 \, {\left (6 \, x^{3} + 22 \, x^{2} + 3 \, x\right )} e^{\left (16 \, x\right )} - 3\right )} e^{\left (e^{\left (16 \, x\right )}\right )}}{36 \, x^{4} + 264 \, x^{3} + 520 \, x^{2} + 132 \, x + 9}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((192*x^3+704*x^2+96*x)*exp(2*x)^8-12*x^2+6)*exp(exp(2*x)^8)/(36*x^4+264*x^3+520*x^2+132*x+9),x, alg
orithm="giac")

[Out]

integrate(-2*(6*x^2 - 16*(6*x^3 + 22*x^2 + 3*x)*e^(16*x) - 3)*e^(e^(16*x))/(36*x^4 + 264*x^3 + 520*x^2 + 132*x
+ 9), x)

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maple [A]  time = 0.05, size = 21, normalized size = 0.88

method	result	size
risch	\(\frac {2 x \,{\mathrm e}^{{\mathrm e}^{16 x}}}{6 x^{2}+22 x +3}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((192*x^3+704*x^2+96*x)*exp(2*x)^8-12*x^2+6)*exp(exp(2*x)^8)/(36*x^4+264*x^3+520*x^2+132*x+9),x,method=_RE
TURNVERBOSE)

[Out]

2*x/(6*x^2+22*x+3)*exp(exp(16*x))

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maxima [A]  time = 0.40, size = 20, normalized size = 0.83 \begin {gather*} \frac {2 \, x e^{\left (e^{\left (16 \, x\right )}\right )}}{6 \, x^{2} + 22 \, x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((192*x^3+704*x^2+96*x)*exp(2*x)^8-12*x^2+6)*exp(exp(2*x)^8)/(36*x^4+264*x^3+520*x^2+132*x+9),x, alg
orithm="maxima")

[Out]

2*x*e^(e^(16*x))/(6*x^2 + 22*x + 3)

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mupad [B]  time = 3.36, size = 19, normalized size = 0.79 \begin {gather*} \frac {x\,{\mathrm {e}}^{{\mathrm {e}}^{16\,x}}}{3\,\left (x^2+\frac {11\,x}{3}+\frac {1}{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(exp(16*x))*(exp(16*x)*(96*x + 704*x^2 + 192*x^3) - 12*x^2 + 6))/(132*x + 520*x^2 + 264*x^3 + 36*x^4 +
9),x)

[Out]

(x*exp(exp(16*x)))/(3*((11*x)/3 + x^2 + 1/2))

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sympy [A]  time = 0.20, size = 19, normalized size = 0.79 \begin {gather*} \frac {2 x e^{e^{16 x}}}{6 x^{2} + 22 x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((192*x**3+704*x**2+96*x)*exp(2*x)**8-12*x**2+6)*exp(exp(2*x)**8)/(36*x**4+264*x**3+520*x**2+132*x+9
),x)

[Out]

2*x*exp(exp(16*x))/(6*x**2 + 22*x + 3)

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