3.74.46 \(\int \frac {e^{2 x+\frac {16 e^{2 x} x^2}{729+1458 x+729 x^2}} (32 x+32 x^2+32 x^3)}{729+2187 x+2187 x^2+729 x^3} \, dx\)

Optimal. Leaf size=19 \[ e^{\frac {16 e^{2 x} x^2}{729 (1+x)^2}} \]

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Rubi [F]  time = 1.67, 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 {e^{2 x+\frac {16 e^{2 x} x^2}{729+1458 x+729 x^2}} \left (32 x+32 x^2+32 x^3\right )}{729+2187 x+2187 x^2+729 x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(2*x + (16*E^(2*x)*x^2)/(729 + 1458*x + 729*x^2))*(32*x + 32*x^2 + 32*x^3))/(729 + 2187*x + 2187*x^2 +
729*x^3),x]

[Out]

(32*Defer[Int][E^(2*x*(1 + (8*E^(2*x)*x)/(729*(1 + x)^2))), x])/729 - (32*Defer[Int][E^(2*x*(1 + (8*E^(2*x)*x)
/(729*(1 + x)^2)))/(1 + x)^3, x])/729 + (64*Defer[Int][E^(2*x*(1 + (8*E^(2*x)*x)/(729*(1 + x)^2)))/(1 + x)^2,
x])/729 - (64*Defer[Int][E^(2*x*(1 + (8*E^(2*x)*x)/(729*(1 + x)^2)))/(1 + x), x])/729

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{2 x+\frac {16 e^{2 x} x^2}{729+1458 x+729 x^2}} x \left (32+32 x+32 x^2\right )}{729+2187 x+2187 x^2+729 x^3} \, dx\\ &=\int \frac {32 e^{2 x \left (1+\frac {8 e^{2 x} x}{729 (1+x)^2}\right )} x \left (1+x+x^2\right )}{729 (1+x)^3} \, dx\\ &=\frac {32}{729} \int \frac {e^{2 x \left (1+\frac {8 e^{2 x} x}{729 (1+x)^2}\right )} x \left (1+x+x^2\right )}{(1+x)^3} \, dx\\ &=\frac {32}{729} \int \left (e^{2 x \left (1+\frac {8 e^{2 x} x}{729 (1+x)^2}\right )}-\frac {e^{2 x \left (1+\frac {8 e^{2 x} x}{729 (1+x)^2}\right )}}{(1+x)^3}+\frac {2 e^{2 x \left (1+\frac {8 e^{2 x} x}{729 (1+x)^2}\right )}}{(1+x)^2}-\frac {2 e^{2 x \left (1+\frac {8 e^{2 x} x}{729 (1+x)^2}\right )}}{1+x}\right ) \, dx\\ &=\frac {32}{729} \int e^{2 x \left (1+\frac {8 e^{2 x} x}{729 (1+x)^2}\right )} \, dx-\frac {32}{729} \int \frac {e^{2 x \left (1+\frac {8 e^{2 x} x}{729 (1+x)^2}\right )}}{(1+x)^3} \, dx+\frac {64}{729} \int \frac {e^{2 x \left (1+\frac {8 e^{2 x} x}{729 (1+x)^2}\right )}}{(1+x)^2} \, dx-\frac {64}{729} \int \frac {e^{2 x \left (1+\frac {8 e^{2 x} x}{729 (1+x)^2}\right )}}{1+x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.75, size = 19, normalized size = 1.00 \begin {gather*} e^{\frac {16 e^{2 x} x^2}{729 (1+x)^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(2*x + (16*E^(2*x)*x^2)/(729 + 1458*x + 729*x^2))*(32*x + 32*x^2 + 32*x^3))/(729 + 2187*x + 2187*
x^2 + 729*x^3),x]

[Out]

E^((16*E^(2*x)*x^2)/(729*(1 + x)^2))

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fricas [B]  time = 0.57, size = 40, normalized size = 2.11 \begin {gather*} e^{\left (-2 \, x + \frac {2 \, {\left (729 \, x^{3} + 8 \, x^{2} e^{\left (2 \, x\right )} + 1458 \, x^{2} + 729 \, x\right )}}{729 \, {\left (x^{2} + 2 \, x + 1\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((32*x^3+32*x^2+32*x)*exp(x)^2*exp(16*x^2*exp(x)^2/(729*x^2+1458*x+729))/(729*x^3+2187*x^2+2187*x+729
),x, algorithm="fricas")

[Out]

e^(-2*x + 2/729*(729*x^3 + 8*x^2*e^(2*x) + 1458*x^2 + 729*x)/(x^2 + 2*x + 1))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {32 \, {\left (x^{3} + x^{2} + x\right )} e^{\left (\frac {16 \, x^{2} e^{\left (2 \, x\right )}}{729 \, {\left (x^{2} + 2 \, x + 1\right )}} + 2 \, x\right )}}{729 \, {\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((32*x^3+32*x^2+32*x)*exp(x)^2*exp(16*x^2*exp(x)^2/(729*x^2+1458*x+729))/(729*x^3+2187*x^2+2187*x+729
),x, algorithm="giac")

[Out]

integrate(32/729*(x^3 + x^2 + x)*e^(16/729*x^2*e^(2*x)/(x^2 + 2*x + 1) + 2*x)/(x^3 + 3*x^2 + 3*x + 1), x)

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maple [A]  time = 0.07, size = 16, normalized size = 0.84

method	result	size
risch	\({\mathrm e}^{\frac {16 x^{2} {\mathrm e}^{2 x}}{729 \left (x +1\right )^{2}}}\)	\(16\)
norman	\(\frac {x^{2} {\mathrm e}^{\frac {16 x^{2} {\mathrm e}^{2 x}}{729 x^{2}+1458 x +729}}+2 x \,{\mathrm e}^{\frac {16 x^{2} {\mathrm e}^{2 x}}{729 x^{2}+1458 x +729}}+{\mathrm e}^{\frac {16 x^{2} {\mathrm e}^{2 x}}{729 x^{2}+1458 x +729}}}{\left (x +1\right )^{2}}\)	\(81\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((32*x^3+32*x^2+32*x)*exp(x)^2*exp(16*x^2*exp(x)^2/(729*x^2+1458*x+729))/(729*x^3+2187*x^2+2187*x+729),x,me
thod=_RETURNVERBOSE)

[Out]

exp(16/729*x^2*exp(2*x)/(x+1)^2)

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maxima [B]  time = 0.50, size = 35, normalized size = 1.84 \begin {gather*} e^{\left (\frac {16 \, e^{\left (2 \, x\right )}}{729 \, {\left (x^{2} + 2 \, x + 1\right )}} - \frac {32 \, e^{\left (2 \, x\right )}}{729 \, {\left (x + 1\right )}} + \frac {16}{729} \, e^{\left (2 \, x\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((32*x^3+32*x^2+32*x)*exp(x)^2*exp(16*x^2*exp(x)^2/(729*x^2+1458*x+729))/(729*x^3+2187*x^2+2187*x+729
),x, algorithm="maxima")

[Out]

e^(16/729*e^(2*x)/(x^2 + 2*x + 1) - 32/729*e^(2*x)/(x + 1) + 16/729*e^(2*x))

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mupad [B]  time = 4.76, size = 22, normalized size = 1.16 \begin {gather*} {\mathrm {e}}^{\frac {16\,x^2\,{\mathrm {e}}^{2\,x}}{729\,x^2+1458\,x+729}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(2*x)*exp((16*x^2*exp(2*x))/(1458*x + 729*x^2 + 729))*(32*x + 32*x^2 + 32*x^3))/(2187*x + 2187*x^2 + 7
29*x^3 + 729),x)

[Out]

exp((16*x^2*exp(2*x))/(1458*x + 729*x^2 + 729))

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sympy [A]  time = 0.27, size = 20, normalized size = 1.05 \begin {gather*} e^{\frac {16 x^{2} e^{2 x}}{729 x^{2} + 1458 x + 729}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((32*x**3+32*x**2+32*x)*exp(x)**2*exp(16*x**2*exp(x)**2/(729*x**2+1458*x+729))/(729*x**3+2187*x**2+21
87*x+729),x)

[Out]

exp(16*x**2*exp(2*x)/(729*x**2 + 1458*x + 729))

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