3.37.41 \(\int \frac {25 e^2-40 x^3-12 x^4+400 x^6+160 x^7+16 x^8+e (5+200 x^3+40 x^4)+e^{2 x} (400 x^4+160 x^5+16 x^6)+e^x (-20 x^2-28 x^3-4 x^4+800 x^5+320 x^6+32 x^7+e (200 x^2+40 x^3))}{50 e^2+800 x^6+320 x^7+32 x^8+e (400 x^3+80 x^4)+e^{2 x} (800 x^4+320 x^5+32 x^6)+e^x (1600 x^5+640 x^6+64 x^7+e (400 x^2+80 x^3))} \, dx\)

Optimal. Leaf size=31 \[ x+\frac {1}{2} \left (-x+\frac {x}{5 e+4 x^2 (5+x) \left (e^x+x\right )}\right ) \]

________________________________________________________________________________________

Rubi [F]  time = 2.32, 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 {25 e^2-40 x^3-12 x^4+400 x^6+160 x^7+16 x^8+e \left (5+200 x^3+40 x^4\right )+e^{2 x} \left (400 x^4+160 x^5+16 x^6\right )+e^x \left (-20 x^2-28 x^3-4 x^4+800 x^5+320 x^6+32 x^7+e \left (200 x^2+40 x^3\right )\right )}{50 e^2+800 x^6+320 x^7+32 x^8+e \left (400 x^3+80 x^4\right )+e^{2 x} \left (800 x^4+320 x^5+32 x^6\right )+e^x \left (1600 x^5+640 x^6+64 x^7+e \left (400 x^2+80 x^3\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(25*E^2 - 40*x^3 - 12*x^4 + 400*x^6 + 160*x^7 + 16*x^8 + E*(5 + 200*x^3 + 40*x^4) + E^(2*x)*(400*x^4 + 160
*x^5 + 16*x^6) + E^x*(-20*x^2 - 28*x^3 - 4*x^4 + 800*x^5 + 320*x^6 + 32*x^7 + E*(200*x^2 + 40*x^3)))/(50*E^2 +
800*x^6 + 320*x^7 + 32*x^8 + E*(400*x^3 + 80*x^4) + E^(2*x)*(800*x^4 + 320*x^5 + 32*x^6) + E^x*(1600*x^5 + 64
0*x^6 + 64*x^7 + E*(400*x^2 + 80*x^3))),x]

[Out]

x/2 + (15*E*Defer[Int][(5*E + 20*E^x*x^2 + 20*x^3 + 4*E^x*x^3 + 4*x^4)^(-2), x])/2 + (5*E*Defer[Int][x/(5*E +
20*E^x*x^2 + 20*x^3 + 4*E^x*x^3 + 4*x^4)^2, x])/2 - 10*Defer[Int][x^3/(5*E + 20*E^x*x^2 + 20*x^3 + 4*E^x*x^3 +
4*x^4)^2, x] + 8*Defer[Int][x^4/(5*E + 20*E^x*x^2 + 20*x^3 + 4*E^x*x^3 + 4*x^4)^2, x] + 2*Defer[Int][x^5/(5*E
+ 20*E^x*x^2 + 20*x^3 + 4*E^x*x^3 + 4*x^4)^2, x] - (25*E*Defer[Int][1/((5 + x)*(5*E + 20*E^x*x^2 + 20*x^3 + 4
*E^x*x^3 + 4*x^4)^2), x])/2 - Defer[Int][(5*E + 20*E^x*x^2 + 20*x^3 + 4*E^x*x^3 + 4*x^4)^(-1), x] - Defer[Int]
[x/(5*E + 20*E^x*x^2 + 20*x^3 + 4*E^x*x^3 + 4*x^4), x]/2 + (5*Defer[Int][1/((5 + x)*(5*E + 20*E^x*x^2 + 20*x^3
+ 4*E^x*x^3 + 4*x^4)), x])/2

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {25 e^2+40 e^{1+x} x^2 (5+x)+16 e^{2 x} x^4 (5+x)^2+5 e \left (1+40 x^3+8 x^4\right )+4 x^3 \left (-10-3 x+100 x^3+40 x^4+4 x^5\right )+4 e^x x^2 \left (-5-7 x-x^2+200 x^3+80 x^4+8 x^5\right )}{2 \left (5 e+4 e^x x^2 (5+x)+4 x^3 (5+x)\right )^2} \, dx\\ &=\frac {1}{2} \int \frac {25 e^2+40 e^{1+x} x^2 (5+x)+16 e^{2 x} x^4 (5+x)^2+5 e \left (1+40 x^3+8 x^4\right )+4 x^3 \left (-10-3 x+100 x^3+40 x^4+4 x^5\right )+4 e^x x^2 \left (-5-7 x-x^2+200 x^3+80 x^4+8 x^5\right )}{\left (5 e+4 e^x x^2 (5+x)+4 x^3 (5+x)\right )^2} \, dx\\ &=\frac {1}{2} \int \left (1-\frac {5+7 x+x^2}{(5+x) \left (5 e+20 e^x x^2+20 x^3+4 e^x x^3+4 x^4\right )}+\frac {50 e+40 e x+5 e x^2-100 x^3+60 x^4+36 x^5+4 x^6}{(5+x) \left (5 e+20 e^x x^2+20 x^3+4 e^x x^3+4 x^4\right )^2}\right ) \, dx\\ &=\frac {x}{2}-\frac {1}{2} \int \frac {5+7 x+x^2}{(5+x) \left (5 e+20 e^x x^2+20 x^3+4 e^x x^3+4 x^4\right )} \, dx+\frac {1}{2} \int \frac {50 e+40 e x+5 e x^2-100 x^3+60 x^4+36 x^5+4 x^6}{(5+x) \left (5 e+20 e^x x^2+20 x^3+4 e^x x^3+4 x^4\right )^2} \, dx\\ &=\frac {x}{2}+\frac {1}{2} \int \left (\frac {15 e}{\left (5 e+20 e^x x^2+20 x^3+4 e^x x^3+4 x^4\right )^2}+\frac {5 e x}{\left (5 e+20 e^x x^2+20 x^3+4 e^x x^3+4 x^4\right )^2}-\frac {20 x^3}{\left (5 e+20 e^x x^2+20 x^3+4 e^x x^3+4 x^4\right )^2}+\frac {16 x^4}{\left (5 e+20 e^x x^2+20 x^3+4 e^x x^3+4 x^4\right )^2}+\frac {4 x^5}{\left (5 e+20 e^x x^2+20 x^3+4 e^x x^3+4 x^4\right )^2}-\frac {25 e}{(5+x) \left (5 e+20 e^x x^2+20 x^3+4 e^x x^3+4 x^4\right )^2}\right ) \, dx-\frac {1}{2} \int \left (\frac {2}{5 e+20 e^x x^2+20 x^3+4 e^x x^3+4 x^4}+\frac {x}{5 e+20 e^x x^2+20 x^3+4 e^x x^3+4 x^4}-\frac {5}{(5+x) \left (5 e+20 e^x x^2+20 x^3+4 e^x x^3+4 x^4\right )}\right ) \, dx\\ &=\frac {x}{2}-\frac {1}{2} \int \frac {x}{5 e+20 e^x x^2+20 x^3+4 e^x x^3+4 x^4} \, dx+2 \int \frac {x^5}{\left (5 e+20 e^x x^2+20 x^3+4 e^x x^3+4 x^4\right )^2} \, dx+\frac {5}{2} \int \frac {1}{(5+x) \left (5 e+20 e^x x^2+20 x^3+4 e^x x^3+4 x^4\right )} \, dx+8 \int \frac {x^4}{\left (5 e+20 e^x x^2+20 x^3+4 e^x x^3+4 x^4\right )^2} \, dx-10 \int \frac {x^3}{\left (5 e+20 e^x x^2+20 x^3+4 e^x x^3+4 x^4\right )^2} \, dx+\frac {1}{2} (5 e) \int \frac {x}{\left (5 e+20 e^x x^2+20 x^3+4 e^x x^3+4 x^4\right )^2} \, dx+\frac {1}{2} (15 e) \int \frac {1}{\left (5 e+20 e^x x^2+20 x^3+4 e^x x^3+4 x^4\right )^2} \, dx-\frac {1}{2} (25 e) \int \frac {1}{(5+x) \left (5 e+20 e^x x^2+20 x^3+4 e^x x^3+4 x^4\right )^2} \, dx-\int \frac {1}{5 e+20 e^x x^2+20 x^3+4 e^x x^3+4 x^4} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.09, size = 32, normalized size = 1.03 \begin {gather*} \frac {1}{2} x \left (1+\frac {1}{5 e+4 e^x x^2 (5+x)+4 x^3 (5+x)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(25*E^2 - 40*x^3 - 12*x^4 + 400*x^6 + 160*x^7 + 16*x^8 + E*(5 + 200*x^3 + 40*x^4) + E^(2*x)*(400*x^4
+ 160*x^5 + 16*x^6) + E^x*(-20*x^2 - 28*x^3 - 4*x^4 + 800*x^5 + 320*x^6 + 32*x^7 + E*(200*x^2 + 40*x^3)))/(50
*E^2 + 800*x^6 + 320*x^7 + 32*x^8 + E*(400*x^3 + 80*x^4) + E^(2*x)*(800*x^4 + 320*x^5 + 32*x^6) + E^x*(1600*x^
5 + 640*x^6 + 64*x^7 + E*(400*x^2 + 80*x^3))),x]

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 0.80, size = 62, normalized size = 2.00 \begin {gather*} \frac {4 \, x^{5} + 20 \, x^{4} + 5 \, x e + 4 \, {\left (x^{4} + 5 \, x^{3}\right )} e^{x} + x}{2 \, {\left (4 \, x^{4} + 20 \, x^{3} + 4 \, {\left (x^{3} + 5 \, x^{2}\right )} e^{x} + 5 \, e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*x^6+160*x^5+400*x^4)*exp(x)^2+((40*x^3+200*x^2)*exp(1)+32*x^7+320*x^6+800*x^5-4*x^4-28*x^3-20*x
^2)*exp(x)+25*exp(1)^2+(40*x^4+200*x^3+5)*exp(1)+16*x^8+160*x^7+400*x^6-12*x^4-40*x^3)/((32*x^6+320*x^5+800*x^
4)*exp(x)^2+((80*x^3+400*x^2)*exp(1)+64*x^7+640*x^6+1600*x^5)*exp(x)+50*exp(1)^2+(80*x^4+400*x^3)*exp(1)+32*x^
8+320*x^7+800*x^6),x, algorithm="fricas")

[Out]

1/2*(4*x^5 + 20*x^4 + 5*x*e + 4*(x^4 + 5*x^3)*e^x + x)/(4*x^4 + 20*x^3 + 4*(x^3 + 5*x^2)*e^x + 5*e)

________________________________________________________________________________________

giac [B]  time = 2.25, size = 64, normalized size = 2.06 \begin {gather*} \frac {4 \, x^{5} + 4 \, x^{4} e^{x} + 20 \, x^{4} + 20 \, x^{3} e^{x} + 5 \, x e + x}{2 \, {\left (4 \, x^{4} + 4 \, x^{3} e^{x} + 20 \, x^{3} + 20 \, x^{2} e^{x} + 5 \, e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*x^6+160*x^5+400*x^4)*exp(x)^2+((40*x^3+200*x^2)*exp(1)+32*x^7+320*x^6+800*x^5-4*x^4-28*x^3-20*x
^2)*exp(x)+25*exp(1)^2+(40*x^4+200*x^3+5)*exp(1)+16*x^8+160*x^7+400*x^6-12*x^4-40*x^3)/((32*x^6+320*x^5+800*x^
4)*exp(x)^2+((80*x^3+400*x^2)*exp(1)+64*x^7+640*x^6+1600*x^5)*exp(x)+50*exp(1)^2+(80*x^4+400*x^3)*exp(1)+32*x^
8+320*x^7+800*x^6),x, algorithm="giac")

[Out]

1/2*(4*x^5 + 4*x^4*e^x + 20*x^4 + 20*x^3*e^x + 5*x*e + x)/(4*x^4 + 4*x^3*e^x + 20*x^3 + 20*x^2*e^x + 5*e)

________________________________________________________________________________________

maple [A]  time = 0.41, size = 39, normalized size = 1.26

method	result	size
risch	\(\frac {x}{2}+\frac {x}{8 \,{\mathrm e}^{x} x^{3}+8 x^{4}+40 \,{\mathrm e}^{x} x^{2}+40 x^{3}+10 \,{\mathrm e}}\)	\(39\)
norman	\(\frac {\left (\frac {1}{2}+\frac {5 \,{\mathrm e}}{2}\right ) x -50 x^{3}-50 \,{\mathrm e}^{x} x^{2}+2 x^{5}+2 \,{\mathrm e}^{x} x^{4}-\frac {25 \,{\mathrm e}}{2}}{4 \,{\mathrm e}^{x} x^{3}+4 x^{4}+20 \,{\mathrm e}^{x} x^{2}+20 x^{3}+5 \,{\mathrm e}}\)	\(70\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((16*x^6+160*x^5+400*x^4)*exp(x)^2+((40*x^3+200*x^2)*exp(1)+32*x^7+320*x^6+800*x^5-4*x^4-28*x^3-20*x^2)*ex
p(x)+25*exp(1)^2+(40*x^4+200*x^3+5)*exp(1)+16*x^8+160*x^7+400*x^6-12*x^4-40*x^3)/((32*x^6+320*x^5+800*x^4)*exp
(x)^2+((80*x^3+400*x^2)*exp(1)+64*x^7+640*x^6+1600*x^5)*exp(x)+50*exp(1)^2+(80*x^4+400*x^3)*exp(1)+32*x^8+320*
x^7+800*x^6),x,method=_RETURNVERBOSE)

[Out]

1/2*x+1/2*x/(4*exp(x)*x^3+4*x^4+20*exp(x)*x^2+20*x^3+5*exp(1))

________________________________________________________________________________________

maxima [B]  time = 0.73, size = 64, normalized size = 2.06 \begin {gather*} \frac {4 \, x^{5} + 20 \, x^{4} + x {\left (5 \, e + 1\right )} + 4 \, {\left (x^{4} + 5 \, x^{3}\right )} e^{x}}{2 \, {\left (4 \, x^{4} + 20 \, x^{3} + 4 \, {\left (x^{3} + 5 \, x^{2}\right )} e^{x} + 5 \, e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*x^6+160*x^5+400*x^4)*exp(x)^2+((40*x^3+200*x^2)*exp(1)+32*x^7+320*x^6+800*x^5-4*x^4-28*x^3-20*x
^2)*exp(x)+25*exp(1)^2+(40*x^4+200*x^3+5)*exp(1)+16*x^8+160*x^7+400*x^6-12*x^4-40*x^3)/((32*x^6+320*x^5+800*x^
4)*exp(x)^2+((80*x^3+400*x^2)*exp(1)+64*x^7+640*x^6+1600*x^5)*exp(x)+50*exp(1)^2+(80*x^4+400*x^3)*exp(1)+32*x^
8+320*x^7+800*x^6),x, algorithm="maxima")

[Out]

1/2*(4*x^5 + 20*x^4 + x*(5*e + 1) + 4*(x^4 + 5*x^3)*e^x)/(4*x^4 + 20*x^3 + 4*(x^3 + 5*x^2)*e^x + 5*e)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {25\,{\mathrm {e}}^2+{\mathrm {e}}^x\,\left (\mathrm {e}\,\left (40\,x^3+200\,x^2\right )-20\,x^2-28\,x^3-4\,x^4+800\,x^5+320\,x^6+32\,x^7\right )+\mathrm {e}\,\left (40\,x^4+200\,x^3+5\right )+{\mathrm {e}}^{2\,x}\,\left (16\,x^6+160\,x^5+400\,x^4\right )-40\,x^3-12\,x^4+400\,x^6+160\,x^7+16\,x^8}{50\,{\mathrm {e}}^2+\mathrm {e}\,\left (80\,x^4+400\,x^3\right )+{\mathrm {e}}^{2\,x}\,\left (32\,x^6+320\,x^5+800\,x^4\right )+{\mathrm {e}}^x\,\left (\mathrm {e}\,\left (80\,x^3+400\,x^2\right )+1600\,x^5+640\,x^6+64\,x^7\right )+800\,x^6+320\,x^7+32\,x^8} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((25*exp(2) + exp(x)*(exp(1)*(200*x^2 + 40*x^3) - 20*x^2 - 28*x^3 - 4*x^4 + 800*x^5 + 320*x^6 + 32*x^7) + e
xp(1)*(200*x^3 + 40*x^4 + 5) + exp(2*x)*(400*x^4 + 160*x^5 + 16*x^6) - 40*x^3 - 12*x^4 + 400*x^6 + 160*x^7 + 1
6*x^8)/(50*exp(2) + exp(1)*(400*x^3 + 80*x^4) + exp(2*x)*(800*x^4 + 320*x^5 + 32*x^6) + exp(x)*(exp(1)*(400*x^
2 + 80*x^3) + 1600*x^5 + 640*x^6 + 64*x^7) + 800*x^6 + 320*x^7 + 32*x^8),x)

[Out]

int((25*exp(2) + exp(x)*(exp(1)*(200*x^2 + 40*x^3) - 20*x^2 - 28*x^3 - 4*x^4 + 800*x^5 + 320*x^6 + 32*x^7) + e
xp(1)*(200*x^3 + 40*x^4 + 5) + exp(2*x)*(400*x^4 + 160*x^5 + 16*x^6) - 40*x^3 - 12*x^4 + 400*x^6 + 160*x^7 + 1
6*x^8)/(50*exp(2) + exp(1)*(400*x^3 + 80*x^4) + exp(2*x)*(800*x^4 + 320*x^5 + 32*x^6) + exp(x)*(exp(1)*(400*x^
2 + 80*x^3) + 1600*x^5 + 640*x^6 + 64*x^7) + 800*x^6 + 320*x^7 + 32*x^8), x)

________________________________________________________________________________________

sympy [A]  time = 0.42, size = 32, normalized size = 1.03 \begin {gather*} \frac {x}{2} + \frac {x}{8 x^{4} + 40 x^{3} + \left (8 x^{3} + 40 x^{2}\right ) e^{x} + 10 e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*x**6+160*x**5+400*x**4)*exp(x)**2+((40*x**3+200*x**2)*exp(1)+32*x**7+320*x**6+800*x**5-4*x**4-2
8*x**3-20*x**2)*exp(x)+25*exp(1)**2+(40*x**4+200*x**3+5)*exp(1)+16*x**8+160*x**7+400*x**6-12*x**4-40*x**3)/((3
2*x**6+320*x**5+800*x**4)*exp(x)**2+((80*x**3+400*x**2)*exp(1)+64*x**7+640*x**6+1600*x**5)*exp(x)+50*exp(1)**2
+(80*x**4+400*x**3)*exp(1)+32*x**8+320*x**7+800*x**6),x)

[Out]

x/2 + x/(8*x**4 + 40*x**3 + (8*x**3 + 40*x**2)*exp(x) + 10*E)

________________________________________________________________________________________