3.83.10 \(\int \frac {e^{\frac {3-x}{2}+2 x} (16 x^3-6 x^4)+e^{\frac {3-x}{2}+x} (-32 x^6+8 x^7)}{e^{3 x}+12 e^{2 x} x^3+48 e^x x^6+64 x^9} \, dx\)

Optimal. Leaf size=31 \[ \frac {e^{3+\frac {1}{2} (-3-x)+x}}{\left (\frac {e^x}{2 x^2}+2 x\right )^2} \]

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Rubi [F]  time = 2.25, 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^{\frac {3-x}{2}+2 x} \left (16 x^3-6 x^4\right )+e^{\frac {3-x}{2}+x} \left (-32 x^6+8 x^7\right )}{e^{3 x}+12 e^{2 x} x^3+48 e^x x^6+64 x^9} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((3 - x)/2 + 2*x)*(16*x^3 - 6*x^4) + E^((3 - x)/2 + x)*(-32*x^6 + 8*x^7))/(E^(3*x) + 12*E^(2*x)*x^3 + 4
8*E^x*x^6 + 64*x^9),x]

[Out]

-96*Defer[Int][(E^(3/2 + x/2)*x^6)/(E^x + 4*x^3)^3, x] + 32*Defer[Int][(E^(3/2 + x/2)*x^7)/(E^x + 4*x^3)^3, x]
+ 16*Defer[Int][(E^(3/2 + x/2)*x^3)/(E^x + 4*x^3)^2, x] - 6*Defer[Int][(E^(3/2 + x/2)*x^4)/(E^x + 4*x^3)^2, x
]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 e^{\frac {3}{2}+\frac {x}{2}} x^3 \left (4 (-4+x) x^3-e^x (-8+3 x)\right )}{\left (e^x+4 x^3\right )^3} \, dx\\ &=2 \int \frac {e^{\frac {3}{2}+\frac {x}{2}} x^3 \left (4 (-4+x) x^3-e^x (-8+3 x)\right )}{\left (e^x+4 x^3\right )^3} \, dx\\ &=2 \int \left (\frac {16 e^{\frac {3}{2}+\frac {x}{2}} (-3+x) x^6}{\left (e^x+4 x^3\right )^3}-\frac {e^{\frac {3}{2}+\frac {x}{2}} x^3 (-8+3 x)}{\left (e^x+4 x^3\right )^2}\right ) \, dx\\ &=-\left (2 \int \frac {e^{\frac {3}{2}+\frac {x}{2}} x^3 (-8+3 x)}{\left (e^x+4 x^3\right )^2} \, dx\right )+32 \int \frac {e^{\frac {3}{2}+\frac {x}{2}} (-3+x) x^6}{\left (e^x+4 x^3\right )^3} \, dx\\ &=-\left (2 \int \left (-\frac {8 e^{\frac {3}{2}+\frac {x}{2}} x^3}{\left (e^x+4 x^3\right )^2}+\frac {3 e^{\frac {3}{2}+\frac {x}{2}} x^4}{\left (e^x+4 x^3\right )^2}\right ) \, dx\right )+32 \int \left (-\frac {3 e^{\frac {3}{2}+\frac {x}{2}} x^6}{\left (e^x+4 x^3\right )^3}+\frac {e^{\frac {3}{2}+\frac {x}{2}} x^7}{\left (e^x+4 x^3\right )^3}\right ) \, dx\\ &=-\left (6 \int \frac {e^{\frac {3}{2}+\frac {x}{2}} x^4}{\left (e^x+4 x^3\right )^2} \, dx\right )+16 \int \frac {e^{\frac {3}{2}+\frac {x}{2}} x^3}{\left (e^x+4 x^3\right )^2} \, dx+32 \int \frac {e^{\frac {3}{2}+\frac {x}{2}} x^7}{\left (e^x+4 x^3\right )^3} \, dx-96 \int \frac {e^{\frac {3}{2}+\frac {x}{2}} x^6}{\left (e^x+4 x^3\right )^3} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.33, size = 27, normalized size = 0.87 \begin {gather*} \frac {4 e^{\frac {3}{2}+\frac {x}{2}} x^4}{\left (e^x+4 x^3\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((3 - x)/2 + 2*x)*(16*x^3 - 6*x^4) + E^((3 - x)/2 + x)*(-32*x^6 + 8*x^7))/(E^(3*x) + 12*E^(2*x)*x
^3 + 48*E^x*x^6 + 64*x^9),x]

[Out]

(4*E^(3/2 + x/2)*x^4)/(E^x + 4*x^3)^2

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fricas [A]  time = 0.80, size = 36, normalized size = 1.16 \begin {gather*} \frac {4 \, x^{4} e^{\left (\frac {1}{2} \, x + \frac {15}{2}\right )}}{16 \, x^{6} e^{6} + 8 \, x^{3} e^{\left (x + 6\right )} + e^{\left (2 \, x + 6\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x^4+16*x^3)*exp(3/2-1/2*x)*exp(x)^2+(8*x^7-32*x^6)*exp(3/2-1/2*x)*exp(x))/(exp(x)^3+12*exp(x)^2
*x^3+48*x^6*exp(x)+64*x^9),x, algorithm="fricas")

[Out]

4*x^4*e^(1/2*x + 15/2)/(16*x^6*e^6 + 8*x^3*e^(x + 6) + e^(2*x + 6))

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giac [A]  time = 0.20, size = 30, normalized size = 0.97 \begin {gather*} \frac {8 \, x^{4} e^{\left (\frac {1}{2} \, x + \frac {3}{2}\right )}}{16 \, x^{6} + 8 \, x^{3} e^{x} + e^{\left (2 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x^4+16*x^3)*exp(3/2-1/2*x)*exp(x)^2+(8*x^7-32*x^6)*exp(3/2-1/2*x)*exp(x))/(exp(x)^3+12*exp(x)^2
*x^3+48*x^6*exp(x)+64*x^9),x, algorithm="giac")

[Out]

8*x^4*e^(1/2*x + 3/2)/(16*x^6 + 8*x^3*e^x + e^(2*x))

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maple [A]  time = 0.06, size = 22, normalized size = 0.71

method	result	size
risch	\(\frac {4 x^{4} {\mathrm e}^{\frac {3}{2}+\frac {x}{2}}}{\left (4 x^{3}+{\mathrm e}^{x}\right )^{2}}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-6*x^4+16*x^3)*exp(3/2-1/2*x)*exp(x)^2+(8*x^7-32*x^6)*exp(3/2-1/2*x)*exp(x))/(exp(x)^3+12*exp(x)^2*x^3+4
8*x^6*exp(x)+64*x^9),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.43, size = 30, normalized size = 0.97 \begin {gather*} \frac {4 \, x^{4} e^{\left (\frac {1}{2} \, x + \frac {3}{2}\right )}}{16 \, x^{6} + 8 \, x^{3} e^{x} + e^{\left (2 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x^4+16*x^3)*exp(3/2-1/2*x)*exp(x)^2+(8*x^7-32*x^6)*exp(3/2-1/2*x)*exp(x))/(exp(x)^3+12*exp(x)^2
*x^3+48*x^6*exp(x)+64*x^9),x, algorithm="maxima")

[Out]

4*x^4*e^(1/2*x + 3/2)/(16*x^6 + 8*x^3*e^x + e^(2*x))

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mupad [B]  time = 5.47, size = 30, normalized size = 0.97 \begin {gather*} \frac {4\,x^4\,{\mathrm {e}}^{x/2}\,{\mathrm {e}}^{3/2}}{{\mathrm {e}}^{2\,x}+8\,x^3\,{\mathrm {e}}^x+16\,x^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(2*x)*exp(3/2 - x/2)*(16*x^3 - 6*x^4) - exp(3/2 - x/2)*exp(x)*(32*x^6 - 8*x^7))/(exp(3*x) + 48*x^6*exp
(x) + 12*x^3*exp(2*x) + 64*x^9),x)

[Out]

(4*x^4*exp(x/2)*exp(3/2))/(exp(2*x) + 8*x^3*exp(x) + 16*x^6)

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sympy [A]  time = 0.16, size = 34, normalized size = 1.10 \begin {gather*} \frac {4 x^{4} e^{\frac {3}{2}} \sqrt {e^{x}}}{16 x^{6} + 8 x^{3} e^{x} + e^{2 x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x**4+16*x**3)*exp(3/2-1/2*x)*exp(x)**2+(8*x**7-32*x**6)*exp(3/2-1/2*x)*exp(x))/(exp(x)**3+12*ex
p(x)**2*x**3+48*x**6*exp(x)+64*x**9),x)

[Out]

4*x**4*exp(3/2)*sqrt(exp(x))/(16*x**6 + 8*x**3*exp(x) + exp(2*x))

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