3.29.25 \(\int \frac {-80 x+\frac {e^{3+2 x} (-128+128 x)}{x^2}}{\frac {16 e^{6+4 x}}{x^3}+10816 x-1040 x^2+25 x^3+\frac {e^{3+2 x} (832 x-40 x^2)}{x^2}} \, dx\)

Optimal. Leaf size=23 \[ \frac {4}{-26-\frac {e^{3+2 x}}{x^2}+\frac {5 x}{4}} \]

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Rubi [F]  time = 1.32, 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 {-80 x+\frac {e^{3+2 x} (-128+128 x)}{x^2}}{\frac {16 e^{6+4 x}}{x^3}+10816 x-1040 x^2+25 x^3+\frac {e^{3+2 x} \left (832 x-40 x^2\right )}{x^2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-80*x + (E^(3 + 2*x)*(-128 + 128*x))/x^2)/((16*E^(6 + 4*x))/x^3 + 10816*x - 1040*x^2 + 25*x^3 + (E^(3 + 2
*x)*(832*x - 40*x^2))/x^2),x]

[Out]

3328*Defer[Int][x^3/(-4*E^(3 + 2*x) - 104*x^2 + 5*x^3)^2, x] - 3568*Defer[Int][x^4/(-4*E^(3 + 2*x) - 104*x^2 +
5*x^3)^2, x] + 160*Defer[Int][x^5/(-4*E^(3 + 2*x) - 104*x^2 + 5*x^3)^2, x] + 32*Defer[Int][x/(-4*E^(3 + 2*x)
- 104*x^2 + 5*x^3), x] - 32*Defer[Int][x^2/(-4*E^(3 + 2*x) - 104*x^2 + 5*x^3), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {128 e^{3+2 x} (-1+x) x-80 x^4}{\left (4 e^{3+2 x}+(104-5 x) x^2\right )^2} \, dx\\ &=\int \left (\frac {16 x^3 \left (208-223 x+10 x^2\right )}{\left (-4 e^{3+2 x}-104 x^2+5 x^3\right )^2}-\frac {32 (-1+x) x}{-4 e^{3+2 x}-104 x^2+5 x^3}\right ) \, dx\\ &=16 \int \frac {x^3 \left (208-223 x+10 x^2\right )}{\left (-4 e^{3+2 x}-104 x^2+5 x^3\right )^2} \, dx-32 \int \frac {(-1+x) x}{-4 e^{3+2 x}-104 x^2+5 x^3} \, dx\\ &=16 \int \left (\frac {208 x^3}{\left (-4 e^{3+2 x}-104 x^2+5 x^3\right )^2}-\frac {223 x^4}{\left (-4 e^{3+2 x}-104 x^2+5 x^3\right )^2}+\frac {10 x^5}{\left (-4 e^{3+2 x}-104 x^2+5 x^3\right )^2}\right ) \, dx-32 \int \left (-\frac {x}{-4 e^{3+2 x}-104 x^2+5 x^3}+\frac {x^2}{-4 e^{3+2 x}-104 x^2+5 x^3}\right ) \, dx\\ &=32 \int \frac {x}{-4 e^{3+2 x}-104 x^2+5 x^3} \, dx-32 \int \frac {x^2}{-4 e^{3+2 x}-104 x^2+5 x^3} \, dx+160 \int \frac {x^5}{\left (-4 e^{3+2 x}-104 x^2+5 x^3\right )^2} \, dx+3328 \int \frac {x^3}{\left (-4 e^{3+2 x}-104 x^2+5 x^3\right )^2} \, dx-3568 \int \frac {x^4}{\left (-4 e^{3+2 x}-104 x^2+5 x^3\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.36, size = 26, normalized size = 1.13 \begin {gather*} \frac {16 x^2}{-4 e^{3+2 x}+x^2 (-104+5 x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-80*x + (E^(3 + 2*x)*(-128 + 128*x))/x^2)/((16*E^(6 + 4*x))/x^3 + 10816*x - 1040*x^2 + 25*x^3 + (E^
(3 + 2*x)*(832*x - 40*x^2))/x^2),x]

[Out]

(16*x^2)/(-4*E^(3 + 2*x) + x^2*(-104 + 5*x))

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fricas [A]  time = 0.62, size = 21, normalized size = 0.91 \begin {gather*} \frac {16}{5 \, x - 4 \, e^{\left (2 \, x - 2 \, \log \relax (x) + 3\right )} - 104} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((128*x-128)*exp(-2*log(x)+2*x+3)-80*x)/(16*x*exp(-2*log(x)+2*x+3)^2+(-40*x^2+832*x)*exp(-2*log(x)+2
*x+3)+25*x^3-1040*x^2+10816*x),x, algorithm="fricas")

[Out]

16/(5*x - 4*e^(2*x - 2*log(x) + 3) - 104)

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giac [A]  time = 0.21, size = 26, normalized size = 1.13 \begin {gather*} \frac {16 \, x^{2}}{5 \, x^{3} - 104 \, x^{2} - 4 \, e^{\left (2 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((128*x-128)*exp(-2*log(x)+2*x+3)-80*x)/(16*x*exp(-2*log(x)+2*x+3)^2+(-40*x^2+832*x)*exp(-2*log(x)+2
*x+3)+25*x^3-1040*x^2+10816*x),x, algorithm="giac")

[Out]

16*x^2/(5*x^3 - 104*x^2 - 4*e^(2*x + 3))

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maple [A]  time = 0.07, size = 21, normalized size = 0.91

method	result	size
risch	\(\frac {16}{5 x -\frac {4 \,{\mathrm e}^{2 x +3}}{x^{2}}-104}\)	\(21\)
norman	\(\frac {16}{5 x -4 \,{\mathrm e}^{-2 \ln \relax (x )+2 x +3}-104}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((128*x-128)*exp(-2*ln(x)+2*x+3)-80*x)/(16*x*exp(-2*ln(x)+2*x+3)^2+(-40*x^2+832*x)*exp(-2*ln(x)+2*x+3)+25*
x^3-1040*x^2+10816*x),x,method=_RETURNVERBOSE)

[Out]

16/(5*x-4/x^2*exp(2*x+3)-104)

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maxima [A]  time = 0.52, size = 26, normalized size = 1.13 \begin {gather*} \frac {16 \, x^{2}}{5 \, x^{3} - 104 \, x^{2} - 4 \, e^{\left (2 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((128*x-128)*exp(-2*log(x)+2*x+3)-80*x)/(16*x*exp(-2*log(x)+2*x+3)^2+(-40*x^2+832*x)*exp(-2*log(x)+2
*x+3)+25*x^3-1040*x^2+10816*x),x, algorithm="maxima")

[Out]

16*x^2/(5*x^3 - 104*x^2 - 4*e^(2*x + 3))

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mupad [B]  time = 1.81, size = 26, normalized size = 1.13 \begin {gather*} -\frac {16\,x^2}{4\,{\mathrm {e}}^{2\,x+3}+104\,x^2-5\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(80*x - exp(2*x - 2*log(x) + 3)*(128*x - 128))/(10816*x + exp(2*x - 2*log(x) + 3)*(832*x - 40*x^2) + 16*x
*exp(4*x - 4*log(x) + 6) - 1040*x^2 + 25*x^3),x)

[Out]

-(16*x^2)/(4*exp(2*x + 3) + 104*x^2 - 5*x^3)

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sympy [A]  time = 0.15, size = 24, normalized size = 1.04 \begin {gather*} - \frac {16 x^{2}}{- 5 x^{3} + 104 x^{2} + 4 e^{2 x + 3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((128*x-128)*exp(-2*ln(x)+2*x+3)-80*x)/(16*x*exp(-2*ln(x)+2*x+3)**2+(-40*x**2+832*x)*exp(-2*ln(x)+2*
x+3)+25*x**3-1040*x**2+10816*x),x)

[Out]

-16*x**2/(-5*x**3 + 104*x**2 + 4*exp(2*x + 3))

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