\(\int \frac {-560+840 x+e^x (112-56 x-84 x^2)}{400 x^2-600 x^3+225 x^4+e^x (-160 x^2+240 x^3-90 x^4)+e^{2 x} (16 x^2-24 x^3+9 x^4)} \, dx\) [10294]

Optimal result

Integrand size = 80, antiderivative size = 25 \[ \int \frac {-560+840 x+e^x \left (112-56 x-84 x^2\right )}{400 x^2-600 x^3+225 x^4+e^x \left (-160 x^2+240 x^3-90 x^4\right )+e^{2 x} \left (16 x^2-24 x^3+9 x^4\right )} \, dx=\frac {28}{3 \left (5-e^x\right ) \left (\frac {4}{3}-x\right ) x} \]

[Out]

84/(45-9*exp(x))/(4/3-x)/x

Rubi [F]

\[ \int \frac {-560+840 x+e^x \left (112-56 x-84 x^2\right )}{400 x^2-600 x^3+225 x^4+e^x \left (-160 x^2+240 x^3-90 x^4\right )+e^{2 x} \left (16 x^2-24 x^3+9 x^4\right )} \, dx=\int \frac {-560+840 x+e^x \left (112-56 x-84 x^2\right )}{400 x^2-600 x^3+225 x^4+e^x \left (-160 x^2+240 x^3-90 x^4\right )+e^{2 x} \left (16 x^2-24 x^3+9 x^4\right )} \, dx \]

[In]

Int[(-560 + 840*x + E^x*(112 - 56*x - 84*x^2))/(400*x^2 - 600*x^3 + 225*x^4 + E^x*(-160*x^2 + 240*x^3 - 90*x^4
) + E^(2*x)*(16*x^2 - 24*x^3 + 9*x^4)),x]

[Out]

7*Defer[Int][1/((-5 + E^x)*x^2), x] + 35*Defer[Int][1/((-5 + E^x)^2*x), x] + 7*Defer[Int][1/((-5 + E^x)*x), x]
- 63*Defer[Int][1/((-5 + E^x)*(-4 + 3*x)^2), x] - 105*Defer[Int][1/((-5 + E^x)^2*(-4 + 3*x)), x] - 21*Defer[I
nt][1/((-5 + E^x)*(-4 + 3*x)), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {28 \left (-20+30 x-e^x \left (-4+2 x+3 x^2\right )\right )}{\left (5-e^x\right )^2 (4-3 x)^2 x^2} \, dx \\ & = 28 \int \frac {-20+30 x-e^x \left (-4+2 x+3 x^2\right )}{\left (5-e^x\right )^2 (4-3 x)^2 x^2} \, dx \\ & = 28 \int \left (-\frac {5}{\left (-5+e^x\right )^2 x (-4+3 x)}-\frac {-4+2 x+3 x^2}{\left (-5+e^x\right ) x^2 (-4+3 x)^2}\right ) \, dx \\ & = -\left (28 \int \frac {-4+2 x+3 x^2}{\left (-5+e^x\right ) x^2 (-4+3 x)^2} \, dx\right )-140 \int \frac {1}{\left (-5+e^x\right )^2 x (-4+3 x)} \, dx \\ & = -\left (28 \int \left (-\frac {1}{4 \left (-5+e^x\right ) x^2}-\frac {1}{4 \left (-5+e^x\right ) x}+\frac {9}{4 \left (-5+e^x\right ) (-4+3 x)^2}+\frac {3}{4 \left (-5+e^x\right ) (-4+3 x)}\right ) \, dx\right )-140 \int \left (-\frac {1}{4 \left (-5+e^x\right )^2 x}+\frac {3}{4 \left (-5+e^x\right )^2 (-4+3 x)}\right ) \, dx \\ & = 7 \int \frac {1}{\left (-5+e^x\right ) x^2} \, dx+7 \int \frac {1}{\left (-5+e^x\right ) x} \, dx-21 \int \frac {1}{\left (-5+e^x\right ) (-4+3 x)} \, dx+35 \int \frac {1}{\left (-5+e^x\right )^2 x} \, dx-63 \int \frac {1}{\left (-5+e^x\right ) (-4+3 x)^2} \, dx-105 \int \frac {1}{\left (-5+e^x\right )^2 (-4+3 x)} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 1.16 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \frac {-560+840 x+e^x \left (112-56 x-84 x^2\right )}{400 x^2-600 x^3+225 x^4+e^x \left (-160 x^2+240 x^3-90 x^4\right )+e^{2 x} \left (16 x^2-24 x^3+9 x^4\right )} \, dx=\frac {28}{\left (-5+e^x\right ) x (-4+3 x)} \]

[In]

Integrate[(-560 + 840*x + E^x*(112 - 56*x - 84*x^2))/(400*x^2 - 600*x^3 + 225*x^4 + E^x*(-160*x^2 + 240*x^3 -
90*x^4) + E^(2*x)*(16*x^2 - 24*x^3 + 9*x^4)),x]

[Out]

28/((-5 + E^x)*x*(-4 + 3*x))

Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76

method	result	size
norman	\(\frac {28}{x \left ({\mathrm e}^{x}-5\right ) \left (-4+3 x \right )}\)	\(19\)
risch	\(\frac {28}{x \left ({\mathrm e}^{x}-5\right ) \left (-4+3 x \right )}\)	\(19\)
parallelrisch	\(\frac {28}{x \left ({\mathrm e}^{x}-5\right ) \left (-4+3 x \right )}\)	\(19\)

[In]

int(((-84*x^2-56*x+112)*exp(x)+840*x-560)/((9*x^4-24*x^3+16*x^2)*exp(x)^2+(-90*x^4+240*x^3-160*x^2)*exp(x)+225
*x^4-600*x^3+400*x^2),x,method=_RETURNVERBOSE)

[Out]

28/x/(exp(x)-5)/(-4+3*x)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04 \[ \int \frac {-560+840 x+e^x \left (112-56 x-84 x^2\right )}{400 x^2-600 x^3+225 x^4+e^x \left (-160 x^2+240 x^3-90 x^4\right )+e^{2 x} \left (16 x^2-24 x^3+9 x^4\right )} \, dx=-\frac {28}{15 \, x^{2} - {\left (3 \, x^{2} - 4 \, x\right )} e^{x} - 20 \, x} \]

[In]

integrate(((-84*x^2-56*x+112)*exp(x)+840*x-560)/((9*x^4-24*x^3+16*x^2)*exp(x)^2+(-90*x^4+240*x^3-160*x^2)*exp(
x)+225*x^4-600*x^3+400*x^2),x, algorithm="fricas")

[Out]

-28/(15*x^2 - (3*x^2 - 4*x)*e^x - 20*x)

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \frac {-560+840 x+e^x \left (112-56 x-84 x^2\right )}{400 x^2-600 x^3+225 x^4+e^x \left (-160 x^2+240 x^3-90 x^4\right )+e^{2 x} \left (16 x^2-24 x^3+9 x^4\right )} \, dx=\frac {252}{- 135 x^{2} + 180 x + \left (27 x^{2} - 36 x\right ) e^{x}} \]

[In]

integrate(((-84*x**2-56*x+112)*exp(x)+840*x-560)/((9*x**4-24*x**3+16*x**2)*exp(x)**2+(-90*x**4+240*x**3-160*x*
*2)*exp(x)+225*x**4-600*x**3+400*x**2),x)

[Out]

252/(-135*x**2 + 180*x + (27*x**2 - 36*x)*exp(x))

Maxima [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04 \[ \int \frac {-560+840 x+e^x \left (112-56 x-84 x^2\right )}{400 x^2-600 x^3+225 x^4+e^x \left (-160 x^2+240 x^3-90 x^4\right )+e^{2 x} \left (16 x^2-24 x^3+9 x^4\right )} \, dx=-\frac {28}{15 \, x^{2} - {\left (3 \, x^{2} - 4 \, x\right )} e^{x} - 20 \, x} \]

[In]

integrate(((-84*x^2-56*x+112)*exp(x)+840*x-560)/((9*x^4-24*x^3+16*x^2)*exp(x)^2+(-90*x^4+240*x^3-160*x^2)*exp(
x)+225*x^4-600*x^3+400*x^2),x, algorithm="maxima")

[Out]

-28/(15*x^2 - (3*x^2 - 4*x)*e^x - 20*x)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {-560+840 x+e^x \left (112-56 x-84 x^2\right )}{400 x^2-600 x^3+225 x^4+e^x \left (-160 x^2+240 x^3-90 x^4\right )+e^{2 x} \left (16 x^2-24 x^3+9 x^4\right )} \, dx=\frac {28}{3 \, x^{2} e^{x} - 15 \, x^{2} - 4 \, x e^{x} + 20 \, x} \]

[In]

integrate(((-84*x^2-56*x+112)*exp(x)+840*x-560)/((9*x^4-24*x^3+16*x^2)*exp(x)^2+(-90*x^4+240*x^3-160*x^2)*exp(
x)+225*x^4-600*x^3+400*x^2),x, algorithm="giac")

[Out]

28/(3*x^2*e^x - 15*x^2 - 4*x*e^x + 20*x)

Mupad [B] (verification not implemented)

Time = 16.71 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {-560+840 x+e^x \left (112-56 x-84 x^2\right )}{400 x^2-600 x^3+225 x^4+e^x \left (-160 x^2+240 x^3-90 x^4\right )+e^{2 x} \left (16 x^2-24 x^3+9 x^4\right )} \, dx=-\frac {28\,\left (4\,x-3\,x^2\right )}{x^2\,{\left (3\,x-4\right )}^2\,\left ({\mathrm {e}}^x-5\right )} \]

[In]

int(-(exp(x)*(56*x + 84*x^2 - 112) - 840*x + 560)/(exp(2*x)*(16*x^2 - 24*x^3 + 9*x^4) - exp(x)*(160*x^2 - 240*
x^3 + 90*x^4) + 400*x^2 - 600*x^3 + 225*x^4),x)

[Out]

-(28*(4*x - 3*x^2))/(x^2*(3*x - 4)^2*(exp(x) - 5))