3.104.5 \(\int \frac {400 x+800 x^2-432 x^3+48 x^4+e^x (-400+160 x-16 x^2)}{25 x^4+70 x^5+29 x^6-28 x^7+4 x^8+e^{2 x} (100-40 x+4 x^2)+e^x (-100 x^2-120 x^3+68 x^4-8 x^5)} \, dx\)

Optimal. Leaf size=32 \[ \frac {4}{e^x-\frac {x^2}{2}+x^2 \left (-x+\frac {x}{5-x}\right )} \]

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Rubi [F]  time = 1.44, 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 {400 x+800 x^2-432 x^3+48 x^4+e^x \left (-400+160 x-16 x^2\right )}{25 x^4+70 x^5+29 x^6-28 x^7+4 x^8+e^{2 x} \left (100-40 x+4 x^2\right )+e^x \left (-100 x^2-120 x^3+68 x^4-8 x^5\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(400*x + 800*x^2 - 432*x^3 + 48*x^4 + E^x*(-400 + 160*x - 16*x^2))/(25*x^4 + 70*x^5 + 29*x^6 - 28*x^7 + 4*
x^8 + E^(2*x)*(100 - 40*x + 4*x^2) + E^x*(-100*x^2 - 120*x^3 + 68*x^4 - 8*x^5)),x]

[Out]

40*Defer[Int][(-10*E^x + 2*E^x*x + 5*x^2 + 7*x^3 - 2*x^4)^(-1), x] + 400*Defer[Int][x/(10*E^x - 2*E^x*x - 5*x^
2 - 7*x^3 + 2*x^4)^2, x] + 600*Defer[Int][x^2/(10*E^x - 2*E^x*x - 5*x^2 - 7*x^3 + 2*x^4)^2, x] - 672*Defer[Int
][x^3/(10*E^x - 2*E^x*x - 5*x^2 - 7*x^3 + 2*x^4)^2, x] + 184*Defer[Int][x^4/(10*E^x - 2*E^x*x - 5*x^2 - 7*x^3
+ 2*x^4)^2, x] - 16*Defer[Int][x^5/(10*E^x - 2*E^x*x - 5*x^2 - 7*x^3 + 2*x^4)^2, x] + 8*Defer[Int][x/(10*E^x -
2*E^x*x - 5*x^2 - 7*x^3 + 2*x^4), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-16 e^x (-5+x)^2+16 x \left (25+50 x-27 x^2+3 x^3\right )}{\left (2 e^x (-5+x)+x^2 \left (5+7 x-2 x^2\right )\right )^2} \, dx\\ &=\int \left (-\frac {8 x \left (-50-75 x+84 x^2-23 x^3+2 x^4\right )}{\left (10 e^x-2 e^x x-5 x^2-7 x^3+2 x^4\right )^2}+\frac {8 (-5+x)}{10 e^x-2 e^x x-5 x^2-7 x^3+2 x^4}\right ) \, dx\\ &=-\left (8 \int \frac {x \left (-50-75 x+84 x^2-23 x^3+2 x^4\right )}{\left (10 e^x-2 e^x x-5 x^2-7 x^3+2 x^4\right )^2} \, dx\right )+8 \int \frac {-5+x}{10 e^x-2 e^x x-5 x^2-7 x^3+2 x^4} \, dx\\ &=-\left (8 \int \left (-\frac {50 x}{\left (10 e^x-2 e^x x-5 x^2-7 x^3+2 x^4\right )^2}-\frac {75 x^2}{\left (10 e^x-2 e^x x-5 x^2-7 x^3+2 x^4\right )^2}+\frac {84 x^3}{\left (10 e^x-2 e^x x-5 x^2-7 x^3+2 x^4\right )^2}-\frac {23 x^4}{\left (10 e^x-2 e^x x-5 x^2-7 x^3+2 x^4\right )^2}+\frac {2 x^5}{\left (10 e^x-2 e^x x-5 x^2-7 x^3+2 x^4\right )^2}\right ) \, dx\right )+8 \int \left (\frac {5}{-10 e^x+2 e^x x+5 x^2+7 x^3-2 x^4}+\frac {x}{10 e^x-2 e^x x-5 x^2-7 x^3+2 x^4}\right ) \, dx\\ &=8 \int \frac {x}{10 e^x-2 e^x x-5 x^2-7 x^3+2 x^4} \, dx-16 \int \frac {x^5}{\left (10 e^x-2 e^x x-5 x^2-7 x^3+2 x^4\right )^2} \, dx+40 \int \frac {1}{-10 e^x+2 e^x x+5 x^2+7 x^3-2 x^4} \, dx+184 \int \frac {x^4}{\left (10 e^x-2 e^x x-5 x^2-7 x^3+2 x^4\right )^2} \, dx+400 \int \frac {x}{\left (10 e^x-2 e^x x-5 x^2-7 x^3+2 x^4\right )^2} \, dx+600 \int \frac {x^2}{\left (10 e^x-2 e^x x-5 x^2-7 x^3+2 x^4\right )^2} \, dx-672 \int \frac {x^3}{\left (10 e^x-2 e^x x-5 x^2-7 x^3+2 x^4\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.43, size = 30, normalized size = 0.94 \begin {gather*} -\frac {8 (-5+x)}{-2 e^x (-5+x)+x^2 \left (-5-7 x+2 x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(400*x + 800*x^2 - 432*x^3 + 48*x^4 + E^x*(-400 + 160*x - 16*x^2))/(25*x^4 + 70*x^5 + 29*x^6 - 28*x^
7 + 4*x^8 + E^(2*x)*(100 - 40*x + 4*x^2) + E^x*(-100*x^2 - 120*x^3 + 68*x^4 - 8*x^5)),x]

[Out]

(-8*(-5 + x))/(-2*E^x*(-5 + x) + x^2*(-5 - 7*x + 2*x^2))

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fricas [A]  time = 0.54, size = 30, normalized size = 0.94 \begin {gather*} -\frac {8 \, {\left (x - 5\right )}}{2 \, x^{4} - 7 \, x^{3} - 5 \, x^{2} - 2 \, {\left (x - 5\right )} e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-16*x^2+160*x-400)*exp(x)+48*x^4-432*x^3+800*x^2+400*x)/((4*x^2-40*x+100)*exp(x)^2+(-8*x^5+68*x^4-
120*x^3-100*x^2)*exp(x)+4*x^8-28*x^7+29*x^6+70*x^5+25*x^4),x, algorithm="fricas")

[Out]

-8*(x - 5)/(2*x^4 - 7*x^3 - 5*x^2 - 2*(x - 5)*e^x)

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giac [A]  time = 0.19, size = 32, normalized size = 1.00 \begin {gather*} -\frac {8 \, {\left (x - 5\right )}}{2 \, x^{4} - 7 \, x^{3} - 5 \, x^{2} - 2 \, x e^{x} + 10 \, e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-16*x^2+160*x-400)*exp(x)+48*x^4-432*x^3+800*x^2+400*x)/((4*x^2-40*x+100)*exp(x)^2+(-8*x^5+68*x^4-
120*x^3-100*x^2)*exp(x)+4*x^8-28*x^7+29*x^6+70*x^5+25*x^4),x, algorithm="giac")

[Out]

-8*(x - 5)/(2*x^4 - 7*x^3 - 5*x^2 - 2*x*e^x + 10*e^x)

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maple [A]  time = 0.09, size = 33, normalized size = 1.03

method	result	size
risch	\(-\frac {8 \left (x -5\right )}{2 x^{4}-7 x^{3}-5 x^{2}-2 \,{\mathrm e}^{x} x +10 \,{\mathrm e}^{x}}\)	\(33\)
norman	\(\frac {40-8 x}{2 x^{4}-7 x^{3}-5 x^{2}-2 \,{\mathrm e}^{x} x +10 \,{\mathrm e}^{x}}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-16*x^2+160*x-400)*exp(x)+48*x^4-432*x^3+800*x^2+400*x)/((4*x^2-40*x+100)*exp(x)^2+(-8*x^5+68*x^4-120*x^
3-100*x^2)*exp(x)+4*x^8-28*x^7+29*x^6+70*x^5+25*x^4),x,method=_RETURNVERBOSE)

[Out]

-8*(x-5)/(2*x^4-7*x^3-5*x^2-2*exp(x)*x+10*exp(x))

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maxima [A]  time = 0.39, size = 30, normalized size = 0.94 \begin {gather*} -\frac {8 \, {\left (x - 5\right )}}{2 \, x^{4} - 7 \, x^{3} - 5 \, x^{2} - 2 \, {\left (x - 5\right )} e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-16*x^2+160*x-400)*exp(x)+48*x^4-432*x^3+800*x^2+400*x)/((4*x^2-40*x+100)*exp(x)^2+(-8*x^5+68*x^4-
120*x^3-100*x^2)*exp(x)+4*x^8-28*x^7+29*x^6+70*x^5+25*x^4),x, algorithm="maxima")

[Out]

-8*(x - 5)/(2*x^4 - 7*x^3 - 5*x^2 - 2*(x - 5)*e^x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {400\,x-{\mathrm {e}}^x\,\left (16\,x^2-160\,x+400\right )+800\,x^2-432\,x^3+48\,x^4}{{\mathrm {e}}^{2\,x}\,\left (4\,x^2-40\,x+100\right )-{\mathrm {e}}^x\,\left (8\,x^5-68\,x^4+120\,x^3+100\,x^2\right )+25\,x^4+70\,x^5+29\,x^6-28\,x^7+4\,x^8} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((400*x - exp(x)*(16*x^2 - 160*x + 400) + 800*x^2 - 432*x^3 + 48*x^4)/(exp(2*x)*(4*x^2 - 40*x + 100) - exp(
x)*(100*x^2 + 120*x^3 - 68*x^4 + 8*x^5) + 25*x^4 + 70*x^5 + 29*x^6 - 28*x^7 + 4*x^8),x)

[Out]

int((400*x - exp(x)*(16*x^2 - 160*x + 400) + 800*x^2 - 432*x^3 + 48*x^4)/(exp(2*x)*(4*x^2 - 40*x + 100) - exp(
x)*(100*x^2 + 120*x^3 - 68*x^4 + 8*x^5) + 25*x^4 + 70*x^5 + 29*x^6 - 28*x^7 + 4*x^8), x)

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sympy [A]  time = 0.27, size = 27, normalized size = 0.84 \begin {gather*} \frac {8 x - 40}{- 2 x^{4} + 7 x^{3} + 5 x^{2} + \left (2 x - 10\right ) e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-16*x**2+160*x-400)*exp(x)+48*x**4-432*x**3+800*x**2+400*x)/((4*x**2-40*x+100)*exp(x)**2+(-8*x**5+
68*x**4-120*x**3-100*x**2)*exp(x)+4*x**8-28*x**7+29*x**6+70*x**5+25*x**4),x)

[Out]

(8*x - 40)/(-2*x**4 + 7*x**3 + 5*x**2 + (2*x - 10)*exp(x))

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