3.54.91 \(\int \frac {9+e^{\frac {1}{2} (3-2 x)} (-3-3 x)+e^x (-48-48 x+e^{\frac {1}{2} (3-2 x)} (16+32 x))}{27 x^2-288 e^x x^2+768 e^{2 x} x^2} \, dx\)

Optimal. Leaf size=29 \[ \frac {3-e^{\frac {3}{2}-x}}{3 \left (-3+16 e^x\right ) x} \]

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Rubi [F]  time = 1.06, 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 {9+e^{\frac {1}{2} (3-2 x)} (-3-3 x)+e^x \left (-48-48 x+e^{\frac {1}{2} (3-2 x)} (16+32 x)\right )}{27 x^2-288 e^x x^2+768 e^{2 x} x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(9 + E^((3 - 2*x)/2)*(-3 - 3*x) + E^x*(-48 - 48*x + E^((3 - 2*x)/2)*(16 + 32*x)))/(27*x^2 - 288*E^x*x^2 +
768*E^(2*x)*x^2),x]

[Out]

E^(3/2 - x)/(9*x) - ((9 - 16*E^(3/2))*Defer[Int][1/((-3 + 16*E^x)*x^2), x])/9 - ((9 - 16*E^(3/2))*Defer[Int][1
/((-3 + 16*E^x)^2*x), x])/3 - ((9 - 16*E^(3/2))*Defer[Int][1/((-3 + 16*E^x)*x), x])/9

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {9+e^{\frac {1}{2} (3-2 x)} (-3-3 x)+e^x \left (-48-48 x+e^{\frac {1}{2} (3-2 x)} (16+32 x)\right )}{3 \left (3-16 e^x\right )^2 x^2} \, dx\\ &=\frac {1}{3} \int \frac {9+e^{\frac {1}{2} (3-2 x)} (-3-3 x)+e^x \left (-48-48 x+e^{\frac {1}{2} (3-2 x)} (16+32 x)\right )}{\left (3-16 e^x\right )^2 x^2} \, dx\\ &=\frac {1}{3} \int \left (\frac {-9+16 e^{3/2}}{\left (-3+16 e^x\right )^2 x}-\frac {e^{\frac {3}{2}-x} (1+x)}{3 x^2}+\frac {\left (-9+16 e^{3/2}\right ) (1+x)}{3 \left (-3+16 e^x\right ) x^2}\right ) \, dx\\ &=-\left (\frac {1}{9} \int \frac {e^{\frac {3}{2}-x} (1+x)}{x^2} \, dx\right )+\frac {1}{9} \left (-9+16 e^{3/2}\right ) \int \frac {1+x}{\left (-3+16 e^x\right ) x^2} \, dx+\frac {1}{3} \left (-9+16 e^{3/2}\right ) \int \frac {1}{\left (-3+16 e^x\right )^2 x} \, dx\\ &=\frac {e^{\frac {3}{2}-x}}{9 x}+\frac {1}{9} \left (-9+16 e^{3/2}\right ) \int \left (\frac {1}{\left (-3+16 e^x\right ) x^2}+\frac {1}{\left (-3+16 e^x\right ) x}\right ) \, dx+\frac {1}{3} \left (-9+16 e^{3/2}\right ) \int \frac {1}{\left (-3+16 e^x\right )^2 x} \, dx\\ &=\frac {e^{\frac {3}{2}-x}}{9 x}+\frac {1}{9} \left (-9+16 e^{3/2}\right ) \int \frac {1}{\left (-3+16 e^x\right ) x^2} \, dx+\frac {1}{9} \left (-9+16 e^{3/2}\right ) \int \frac {1}{\left (-3+16 e^x\right ) x} \, dx+\frac {1}{3} \left (-9+16 e^{3/2}\right ) \int \frac {1}{\left (-3+16 e^x\right )^2 x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.74, size = 29, normalized size = 1.00 \begin {gather*} -\frac {3-e^{\frac {3}{2}-x}}{3 \left (3 x-16 e^x x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(9 + E^((3 - 2*x)/2)*(-3 - 3*x) + E^x*(-48 - 48*x + E^((3 - 2*x)/2)*(16 + 32*x)))/(27*x^2 - 288*E^x*
x^2 + 768*E^(2*x)*x^2),x]

[Out]

-1/3*(3 - E^(3/2 - x))/(3*x - 16*E^x*x)

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fricas [A]  time = 0.67, size = 24, normalized size = 0.83 \begin {gather*} -\frac {e^{\frac {3}{2}} - 3 \, e^{x}}{3 \, {\left (16 \, x e^{\left (2 \, x\right )} - 3 \, x e^{x}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((32*x+16)*exp(3/2-x)-48*x-48)*exp(x)+(-3*x-3)*exp(3/2-x)+9)/(768*exp(x)^2*x^2-288*exp(x)*x^2+27*x^
2),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.19, size = 24, normalized size = 0.83 \begin {gather*} -\frac {e^{\frac {3}{2}} - 3 \, e^{x}}{3 \, {\left (16 \, x e^{\left (2 \, x\right )} - 3 \, x e^{x}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((32*x+16)*exp(3/2-x)-48*x-48)*exp(x)+(-3*x-3)*exp(3/2-x)+9)/(768*exp(x)^2*x^2-288*exp(x)*x^2+27*x^
2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.12, size = 24, normalized size = 0.83

method	result	size
norman	\(\frac {\left (-\frac {{\mathrm e}^{\frac {3}{2}}}{3}+{\mathrm e}^{x}\right ) {\mathrm e}^{-x}}{x \left (16 \,{\mathrm e}^{x}-3\right )}\)	\(24\)
risch	\(\frac {{\mathrm e}^{\frac {3}{2}-x}}{9 x}-\frac {16 \,{\mathrm e}^{\frac {3}{2}}-9}{9 x \left (16 \,{\mathrm e}^{x}-3\right )}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((32*x+16)*exp(3/2-x)-48*x-48)*exp(x)+(-3*x-3)*exp(3/2-x)+9)/(768*exp(x)^2*x^2-288*exp(x)*x^2+27*x^2),x,m
ethod=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.40, size = 11, normalized size = 0.38 \begin {gather*} \frac {1}{16 \, x e^{x} - 3 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((32*x+16)*exp(3/2-x)-48*x-48)*exp(x)+(-3*x-3)*exp(3/2-x)+9)/(768*exp(x)^2*x^2-288*exp(x)*x^2+27*x^
2),x, algorithm="maxima")

[Out]

1/(16*x*e^x - 3*x)

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mupad [B]  time = 0.14, size = 24, normalized size = 0.83 \begin {gather*} -\frac {{\mathrm {e}}^{-x}\,\left ({\mathrm {e}}^{3/2}-3\,{\mathrm {e}}^x\right )}{3\,x\,\left (16\,{\mathrm {e}}^x-3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(3/2 - x)*(3*x + 3) + exp(x)*(48*x - exp(3/2 - x)*(32*x + 16) + 48) - 9)/(768*x^2*exp(2*x) - 288*x^2*
exp(x) + 27*x^2),x)

[Out]

-(exp(-x)*(exp(3/2) - 3*exp(x)))/(3*x*(16*exp(x) - 3))

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sympy [A]  time = 0.19, size = 29, normalized size = 1.00 \begin {gather*} \frac {9 - 16 e^{\frac {3}{2}}}{144 x e^{x} - 27 x} + \frac {e^{\frac {3}{2}} e^{- x}}{9 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((32*x+16)*exp(3/2-x)-48*x-48)*exp(x)+(-3*x-3)*exp(3/2-x)+9)/(768*exp(x)**2*x**2-288*exp(x)*x**2+27
*x**2),x)

[Out]

(9 - 16*exp(3/2))/(144*x*exp(x) - 27*x) + exp(3/2)*exp(-x)/(9*x)

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