3.99.57 \(\int \frac {24 x+72 x^2+48 x^3+32 x^5+80 x^6+64 x^7+16 x^8+e^x (64 x^4+192 x^5+256 x^6+192 x^7+64 x^8)}{9-24 x^4-24 x^5+16 x^8+32 x^9+16 x^{10}+e^{2 x} (256 x^6+512 x^7+256 x^8)+e^x (-96 x^3-96 x^4+128 x^7+256 x^8+128 x^9)} \, dx\)

Optimal. Leaf size=29 \[ \frac {1+x}{-x \left (4 e^x+x\right )+\frac {3}{x^2 (4+4 x)}} \]

________________________________________________________________________________________

Rubi [F]  time = 3.08, 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 {24 x+72 x^2+48 x^3+32 x^5+80 x^6+64 x^7+16 x^8+e^x \left (64 x^4+192 x^5+256 x^6+192 x^7+64 x^8\right )}{9-24 x^4-24 x^5+16 x^8+32 x^9+16 x^{10}+e^{2 x} \left (256 x^6+512 x^7+256 x^8\right )+e^x \left (-96 x^3-96 x^4+128 x^7+256 x^8+128 x^9\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(24*x + 72*x^2 + 48*x^3 + 32*x^5 + 80*x^6 + 64*x^7 + 16*x^8 + E^x*(64*x^4 + 192*x^5 + 256*x^6 + 192*x^7 +
64*x^8))/(9 - 24*x^4 - 24*x^5 + 16*x^8 + 32*x^9 + 16*x^10 + E^(2*x)*(256*x^6 + 512*x^7 + 256*x^8) + E^x*(-96*x
^3 - 96*x^4 + 128*x^7 + 256*x^8 + 128*x^9)),x]

[Out]

36*Defer[Int][x/(-3 + 16*E^x*x^3 + 4*x^4 + 16*E^x*x^4 + 4*x^5)^2, x] + 96*Defer[Int][x^2/(-3 + 16*E^x*x^3 + 4*
x^4 + 16*E^x*x^4 + 4*x^5)^2, x] + 72*Defer[Int][x^3/(-3 + 16*E^x*x^3 + 4*x^4 + 16*E^x*x^4 + 4*x^5)^2, x] + 12*
Defer[Int][x^4/(-3 + 16*E^x*x^3 + 4*x^4 + 16*E^x*x^4 + 4*x^5)^2, x] + 16*Defer[Int][x^5/(-3 + 16*E^x*x^3 + 4*x
^4 + 16*E^x*x^4 + 4*x^5)^2, x] + 32*Defer[Int][x^6/(-3 + 16*E^x*x^3 + 4*x^4 + 16*E^x*x^4 + 4*x^5)^2, x] - 32*D
efer[Int][x^8/(-3 + 16*E^x*x^3 + 4*x^4 + 16*E^x*x^4 + 4*x^5)^2, x] - 16*Defer[Int][x^9/(-3 + 16*E^x*x^3 + 4*x^
4 + 16*E^x*x^4 + 4*x^5)^2, x] + 4*Defer[Int][x/(-3 + 16*E^x*x^3 + 4*x^4 + 16*E^x*x^4 + 4*x^5), x] + 8*Defer[In
t][x^2/(-3 + 16*E^x*x^3 + 4*x^4 + 16*E^x*x^4 + 4*x^5), x] + 8*Defer[Int][x^3/(-3 + 16*E^x*x^3 + 4*x^4 + 16*E^x
*x^4 + 4*x^5), x] + 4*Defer[Int][x^4/(-3 + 16*E^x*x^3 + 4*x^4 + 16*E^x*x^4 + 4*x^5), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {8 x (1+x) \left (3+6 x+8 e^x x^3+4 \left (1+4 e^x\right ) x^4+2 \left (3+8 e^x\right ) x^5+\left (2+8 e^x\right ) x^6\right )}{\left (3-4 x^4-4 x^5-16 e^x x^3 (1+x)\right )^2} \, dx\\ &=8 \int \frac {x (1+x) \left (3+6 x+8 e^x x^3+4 \left (1+4 e^x\right ) x^4+2 \left (3+8 e^x\right ) x^5+\left (2+8 e^x\right ) x^6\right )}{\left (3-4 x^4-4 x^5-16 e^x x^3 (1+x)\right )^2} \, dx\\ &=8 \int \left (\frac {x \left (1+2 x+2 x^2+x^3\right )}{2 \left (-3+16 e^x x^3+4 x^4+16 e^x x^4+4 x^5\right )}-\frac {x \left (-9-24 x-18 x^2-3 x^3-4 x^4-8 x^5+8 x^7+4 x^8\right )}{2 \left (-3+16 e^x x^3+4 x^4+16 e^x x^4+4 x^5\right )^2}\right ) \, dx\\ &=4 \int \frac {x \left (1+2 x+2 x^2+x^3\right )}{-3+16 e^x x^3+4 x^4+16 e^x x^4+4 x^5} \, dx-4 \int \frac {x \left (-9-24 x-18 x^2-3 x^3-4 x^4-8 x^5+8 x^7+4 x^8\right )}{\left (-3+16 e^x x^3+4 x^4+16 e^x x^4+4 x^5\right )^2} \, dx\\ &=-\left (4 \int \left (-\frac {9 x}{\left (-3+16 e^x x^3+4 x^4+16 e^x x^4+4 x^5\right )^2}-\frac {24 x^2}{\left (-3+16 e^x x^3+4 x^4+16 e^x x^4+4 x^5\right )^2}-\frac {18 x^3}{\left (-3+16 e^x x^3+4 x^4+16 e^x x^4+4 x^5\right )^2}-\frac {3 x^4}{\left (-3+16 e^x x^3+4 x^4+16 e^x x^4+4 x^5\right )^2}-\frac {4 x^5}{\left (-3+16 e^x x^3+4 x^4+16 e^x x^4+4 x^5\right )^2}-\frac {8 x^6}{\left (-3+16 e^x x^3+4 x^4+16 e^x x^4+4 x^5\right )^2}+\frac {8 x^8}{\left (-3+16 e^x x^3+4 x^4+16 e^x x^4+4 x^5\right )^2}+\frac {4 x^9}{\left (-3+16 e^x x^3+4 x^4+16 e^x x^4+4 x^5\right )^2}\right ) \, dx\right )+4 \int \left (\frac {x}{-3+16 e^x x^3+4 x^4+16 e^x x^4+4 x^5}+\frac {2 x^2}{-3+16 e^x x^3+4 x^4+16 e^x x^4+4 x^5}+\frac {2 x^3}{-3+16 e^x x^3+4 x^4+16 e^x x^4+4 x^5}+\frac {x^4}{-3+16 e^x x^3+4 x^4+16 e^x x^4+4 x^5}\right ) \, dx\\ &=4 \int \frac {x}{-3+16 e^x x^3+4 x^4+16 e^x x^4+4 x^5} \, dx+4 \int \frac {x^4}{-3+16 e^x x^3+4 x^4+16 e^x x^4+4 x^5} \, dx+8 \int \frac {x^2}{-3+16 e^x x^3+4 x^4+16 e^x x^4+4 x^5} \, dx+8 \int \frac {x^3}{-3+16 e^x x^3+4 x^4+16 e^x x^4+4 x^5} \, dx+12 \int \frac {x^4}{\left (-3+16 e^x x^3+4 x^4+16 e^x x^4+4 x^5\right )^2} \, dx+16 \int \frac {x^5}{\left (-3+16 e^x x^3+4 x^4+16 e^x x^4+4 x^5\right )^2} \, dx-16 \int \frac {x^9}{\left (-3+16 e^x x^3+4 x^4+16 e^x x^4+4 x^5\right )^2} \, dx+32 \int \frac {x^6}{\left (-3+16 e^x x^3+4 x^4+16 e^x x^4+4 x^5\right )^2} \, dx-32 \int \frac {x^8}{\left (-3+16 e^x x^3+4 x^4+16 e^x x^4+4 x^5\right )^2} \, dx+36 \int \frac {x}{\left (-3+16 e^x x^3+4 x^4+16 e^x x^4+4 x^5\right )^2} \, dx+72 \int \frac {x^3}{\left (-3+16 e^x x^3+4 x^4+16 e^x x^4+4 x^5\right )^2} \, dx+96 \int \frac {x^2}{\left (-3+16 e^x x^3+4 x^4+16 e^x x^4+4 x^5\right )^2} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.75, size = 35, normalized size = 1.21 \begin {gather*} -\frac {8 x^2 (1+x)^2}{-6+8 x^4+8 x^5+32 e^x x^3 (1+x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(24*x + 72*x^2 + 48*x^3 + 32*x^5 + 80*x^6 + 64*x^7 + 16*x^8 + E^x*(64*x^4 + 192*x^5 + 256*x^6 + 192*
x^7 + 64*x^8))/(9 - 24*x^4 - 24*x^5 + 16*x^8 + 32*x^9 + 16*x^10 + E^(2*x)*(256*x^6 + 512*x^7 + 256*x^8) + E^x*
(-96*x^3 - 96*x^4 + 128*x^7 + 256*x^8 + 128*x^9)),x]

[Out]

(-8*x^2*(1 + x)^2)/(-6 + 8*x^4 + 8*x^5 + 32*E^x*x^3*(1 + x))

________________________________________________________________________________________

fricas [A]  time = 0.60, size = 39, normalized size = 1.34 \begin {gather*} -\frac {4 \, {\left (x^{4} + 2 \, x^{3} + x^{2}\right )}}{4 \, x^{5} + 4 \, x^{4} + 16 \, {\left (x^{4} + x^{3}\right )} e^{x} - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((64*x^8+192*x^7+256*x^6+192*x^5+64*x^4)*exp(x)+16*x^8+64*x^7+80*x^6+32*x^5+48*x^3+72*x^2+24*x)/((25
6*x^8+512*x^7+256*x^6)*exp(x)^2+(128*x^9+256*x^8+128*x^7-96*x^4-96*x^3)*exp(x)+16*x^10+32*x^9+16*x^8-24*x^5-24
*x^4+9),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.16, size = 42, normalized size = 1.45 \begin {gather*} -\frac {4 \, {\left (x^{4} + 2 \, x^{3} + x^{2}\right )}}{4 \, x^{5} + 16 \, x^{4} e^{x} + 4 \, x^{4} + 16 \, x^{3} e^{x} - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((64*x^8+192*x^7+256*x^6+192*x^5+64*x^4)*exp(x)+16*x^8+64*x^7+80*x^6+32*x^5+48*x^3+72*x^2+24*x)/((25
6*x^8+512*x^7+256*x^6)*exp(x)^2+(128*x^9+256*x^8+128*x^7-96*x^4-96*x^3)*exp(x)+16*x^10+32*x^9+16*x^8-24*x^5-24
*x^4+9),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.13, size = 39, normalized size = 1.34

method	result	size
risch	\(-\frac {4 x^{2} \left (x +1\right )^{2}}{16 \,{\mathrm e}^{x} x^{4}+4 x^{5}+16 \,{\mathrm e}^{x} x^{3}+4 x^{4}-3}\)	\(39\)
norman	\(\frac {-4 x^{4}-8 x^{3}-4 x^{2}}{16 \,{\mathrm e}^{x} x^{4}+4 x^{5}+16 \,{\mathrm e}^{x} x^{3}+4 x^{4}-3}\)	\(46\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((64*x^8+192*x^7+256*x^6+192*x^5+64*x^4)*exp(x)+16*x^8+64*x^7+80*x^6+32*x^5+48*x^3+72*x^2+24*x)/((256*x^8+
512*x^7+256*x^6)*exp(x)^2+(128*x^9+256*x^8+128*x^7-96*x^4-96*x^3)*exp(x)+16*x^10+32*x^9+16*x^8-24*x^5-24*x^4+9
),x,method=_RETURNVERBOSE)

[Out]

-4*x^2*(x+1)^2/(16*exp(x)*x^4+4*x^5+16*exp(x)*x^3+4*x^4-3)

________________________________________________________________________________________

maxima [A]  time = 0.40, size = 39, normalized size = 1.34 \begin {gather*} -\frac {4 \, {\left (x^{4} + 2 \, x^{3} + x^{2}\right )}}{4 \, x^{5} + 4 \, x^{4} + 16 \, {\left (x^{4} + x^{3}\right )} e^{x} - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((64*x^8+192*x^7+256*x^6+192*x^5+64*x^4)*exp(x)+16*x^8+64*x^7+80*x^6+32*x^5+48*x^3+72*x^2+24*x)/((25
6*x^8+512*x^7+256*x^6)*exp(x)^2+(128*x^9+256*x^8+128*x^7-96*x^4-96*x^3)*exp(x)+16*x^10+32*x^9+16*x^8-24*x^5-24
*x^4+9),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {24\,x+{\mathrm {e}}^x\,\left (64\,x^8+192\,x^7+256\,x^6+192\,x^5+64\,x^4\right )+72\,x^2+48\,x^3+32\,x^5+80\,x^6+64\,x^7+16\,x^8}{{\mathrm {e}}^{2\,x}\,\left (256\,x^8+512\,x^7+256\,x^6\right )+{\mathrm {e}}^x\,\left (128\,x^9+256\,x^8+128\,x^7-96\,x^4-96\,x^3\right )-24\,x^4-24\,x^5+16\,x^8+32\,x^9+16\,x^{10}+9} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((24*x + exp(x)*(64*x^4 + 192*x^5 + 256*x^6 + 192*x^7 + 64*x^8) + 72*x^2 + 48*x^3 + 32*x^5 + 80*x^6 + 64*x^
7 + 16*x^8)/(exp(2*x)*(256*x^6 + 512*x^7 + 256*x^8) + exp(x)*(128*x^7 - 96*x^4 - 96*x^3 + 256*x^8 + 128*x^9) -
24*x^4 - 24*x^5 + 16*x^8 + 32*x^9 + 16*x^10 + 9),x)

[Out]

int((24*x + exp(x)*(64*x^4 + 192*x^5 + 256*x^6 + 192*x^7 + 64*x^8) + 72*x^2 + 48*x^3 + 32*x^5 + 80*x^6 + 64*x^
7 + 16*x^8)/(exp(2*x)*(256*x^6 + 512*x^7 + 256*x^8) + exp(x)*(128*x^7 - 96*x^4 - 96*x^3 + 256*x^8 + 128*x^9) -
24*x^4 - 24*x^5 + 16*x^8 + 32*x^9 + 16*x^10 + 9), x)

________________________________________________________________________________________

sympy [B]  time = 0.36, size = 41, normalized size = 1.41 \begin {gather*} \frac {- 4 x^{4} - 8 x^{3} - 4 x^{2}}{4 x^{5} + 4 x^{4} + \left (16 x^{4} + 16 x^{3}\right ) e^{x} - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((64*x**8+192*x**7+256*x**6+192*x**5+64*x**4)*exp(x)+16*x**8+64*x**7+80*x**6+32*x**5+48*x**3+72*x**2
+24*x)/((256*x**8+512*x**7+256*x**6)*exp(x)**2+(128*x**9+256*x**8+128*x**7-96*x**4-96*x**3)*exp(x)+16*x**10+32
*x**9+16*x**8-24*x**5-24*x**4+9),x)

[Out]

(-4*x**4 - 8*x**3 - 4*x**2)/(4*x**5 + 4*x**4 + (16*x**4 + 16*x**3)*exp(x) - 3)

________________________________________________________________________________________