3.42.88 \(\int \frac {1200 x+240 e^x x+12 e^{2 x} x+e^{18 x} (e^{2 x} (-4+18 x)+e^x (-40+190 x))}{100 x^5+20 e^x x^5+e^{2 x} x^5} \, dx\)

Optimal. Leaf size=24 \[ \frac {-4+\frac {e^{18 x}}{x+10 e^{-x} x}}{x^3} \]

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Rubi [F]  time = 1.71, 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 {1200 x+240 e^x x+12 e^{2 x} x+e^{18 x} \left (e^{2 x} (-4+18 x)+e^x (-40+190 x)\right )}{100 x^5+20 e^x x^5+e^{2 x} x^5} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(1200*x + 240*E^x*x + 12*E^(2*x)*x + E^(18*x)*(E^(2*x)*(-4 + 18*x) + E^x*(-40 + 190*x)))/(100*x^5 + 20*E^x
*x^5 + E^(2*x)*x^5),x]

[Out]

1000000000000000000/x^4 - (100000000000000000*E^x)/x^4 + (10000000000000000*E^(2*x))/x^4 - (1000000000000000*E
^(3*x))/x^4 + (100000000000000*E^(4*x))/x^4 - (10000000000000*E^(5*x))/x^4 + (1000000000000*E^(6*x))/x^4 - (10
0000000000*E^(7*x))/x^4 + (10000000000*E^(8*x))/x^4 - (1000000000*E^(9*x))/x^4 + (100000000*E^(10*x))/x^4 - (1
0000000*E^(11*x))/x^4 + (1000000*E^(12*x))/x^4 - (100000*E^(13*x))/x^4 + (10000*E^(14*x))/x^4 - (1000*E^(15*x)
)/x^4 + (100*E^(16*x))/x^4 - (10*E^(17*x))/x^4 + E^(18*x)/x^4 - 4/x^3 + 40000000000000000000*Defer[Int][1/((10
+ E^x)*x^5), x] - 100000000000000000000*Defer[Int][1/((10 + E^x)^2*x^4), x] + 10000000000000000000*Defer[Int]
[1/((10 + E^x)*x^4), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (600 x+120 e^x x+6 e^{2 x} x+e^{20 x} (-2+9 x)+5 e^{19 x} (-4+19 x)\right )}{\left (10+e^x\right )^2 x^5} \, dx\\ &=2 \int \frac {600 x+120 e^x x+6 e^{2 x} x+e^{20 x} (-2+9 x)+5 e^{19 x} (-4+19 x)}{\left (10+e^x\right )^2 x^5} \, dx\\ &=2 \int \left (-\frac {50000000000000000 e^x (-4+x)}{x^5}+\frac {10000000000000000 e^{2 x} (-2+x)}{x^5}+\frac {200000000000000 e^{4 x} (-1+x)}{x^5}-\frac {50000000000000000000}{\left (10+e^x\right )^2 x^4}+\frac {5000000000000000000 (4+x)}{\left (10+e^x\right ) x^5}+\frac {20000000000 e^{8 x} (-1+2 x)}{x^5}+\frac {2 (-1000000000000000000+3 x)}{x^5}-\frac {500000000000000 e^{3 x} (-4+3 x)}{x^5}+\frac {1000000000000 e^{6 x} (-2+3 x)}{x^5}+\frac {2000000 e^{12 x} (-1+3 x)}{x^5}+\frac {200 e^{16 x} (-1+4 x)}{x^5}-\frac {5000000000000 e^{5 x} (-4+5 x)}{x^5}+\frac {100000000 e^{10 x} (-2+5 x)}{x^5}-\frac {50000000000 e^{7 x} (-4+7 x)}{x^5}+\frac {10000 e^{14 x} (-2+7 x)}{x^5}-\frac {500000000 e^{9 x} (-4+9 x)}{x^5}+\frac {e^{18 x} (-2+9 x)}{x^5}-\frac {5000000 e^{11 x} (-4+11 x)}{x^5}-\frac {50000 e^{13 x} (-4+13 x)}{x^5}-\frac {500 e^{15 x} (-4+15 x)}{x^5}-\frac {5 e^{17 x} (-4+17 x)}{x^5}\right ) \, dx\\ &=2 \int \frac {e^{18 x} (-2+9 x)}{x^5} \, dx+4 \int \frac {-1000000000000000000+3 x}{x^5} \, dx-10 \int \frac {e^{17 x} (-4+17 x)}{x^5} \, dx+400 \int \frac {e^{16 x} (-1+4 x)}{x^5} \, dx-1000 \int \frac {e^{15 x} (-4+15 x)}{x^5} \, dx+20000 \int \frac {e^{14 x} (-2+7 x)}{x^5} \, dx-100000 \int \frac {e^{13 x} (-4+13 x)}{x^5} \, dx+4000000 \int \frac {e^{12 x} (-1+3 x)}{x^5} \, dx-10000000 \int \frac {e^{11 x} (-4+11 x)}{x^5} \, dx+200000000 \int \frac {e^{10 x} (-2+5 x)}{x^5} \, dx-1000000000 \int \frac {e^{9 x} (-4+9 x)}{x^5} \, dx+40000000000 \int \frac {e^{8 x} (-1+2 x)}{x^5} \, dx-100000000000 \int \frac {e^{7 x} (-4+7 x)}{x^5} \, dx+2000000000000 \int \frac {e^{6 x} (-2+3 x)}{x^5} \, dx-10000000000000 \int \frac {e^{5 x} (-4+5 x)}{x^5} \, dx+400000000000000 \int \frac {e^{4 x} (-1+x)}{x^5} \, dx-1000000000000000 \int \frac {e^{3 x} (-4+3 x)}{x^5} \, dx+20000000000000000 \int \frac {e^{2 x} (-2+x)}{x^5} \, dx-100000000000000000 \int \frac {e^x (-4+x)}{x^5} \, dx+10000000000000000000 \int \frac {4+x}{\left (10+e^x\right ) x^5} \, dx-100000000000000000000 \int \frac {1}{\left (10+e^x\right )^2 x^4} \, dx\\ &=-\frac {100000000000000000 e^x}{x^4}+\frac {10000000000000000 e^{2 x}}{x^4}-\frac {1000000000000000 e^{3 x}}{x^4}+\frac {100000000000000 e^{4 x}}{x^4}-\frac {10000000000000 e^{5 x}}{x^4}+\frac {1000000000000 e^{6 x}}{x^4}-\frac {100000000000 e^{7 x}}{x^4}+\frac {10000000000 e^{8 x}}{x^4}-\frac {1000000000 e^{9 x}}{x^4}+\frac {100000000 e^{10 x}}{x^4}-\frac {10000000 e^{11 x}}{x^4}+\frac {1000000 e^{12 x}}{x^4}-\frac {100000 e^{13 x}}{x^4}+\frac {10000 e^{14 x}}{x^4}-\frac {1000 e^{15 x}}{x^4}+\frac {100 e^{16 x}}{x^4}-\frac {10 e^{17 x}}{x^4}+\frac {e^{18 x}}{x^4}+4 \int \left (-\frac {1000000000000000000}{x^5}+\frac {3}{x^4}\right ) \, dx+10000000000000000000 \int \left (\frac {4}{\left (10+e^x\right ) x^5}+\frac {1}{\left (10+e^x\right ) x^4}\right ) \, dx-100000000000000000000 \int \frac {1}{\left (10+e^x\right )^2 x^4} \, dx\\ &=\frac {1000000000000000000}{x^4}-\frac {100000000000000000 e^x}{x^4}+\frac {10000000000000000 e^{2 x}}{x^4}-\frac {1000000000000000 e^{3 x}}{x^4}+\frac {100000000000000 e^{4 x}}{x^4}-\frac {10000000000000 e^{5 x}}{x^4}+\frac {1000000000000 e^{6 x}}{x^4}-\frac {100000000000 e^{7 x}}{x^4}+\frac {10000000000 e^{8 x}}{x^4}-\frac {1000000000 e^{9 x}}{x^4}+\frac {100000000 e^{10 x}}{x^4}-\frac {10000000 e^{11 x}}{x^4}+\frac {1000000 e^{12 x}}{x^4}-\frac {100000 e^{13 x}}{x^4}+\frac {10000 e^{14 x}}{x^4}-\frac {1000 e^{15 x}}{x^4}+\frac {100 e^{16 x}}{x^4}-\frac {10 e^{17 x}}{x^4}+\frac {e^{18 x}}{x^4}-\frac {4}{x^3}+10000000000000000000 \int \frac {1}{\left (10+e^x\right ) x^4} \, dx+40000000000000000000 \int \frac {1}{\left (10+e^x\right ) x^5} \, dx-100000000000000000000 \int \frac {1}{\left (10+e^x\right )^2 x^4} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.68, size = 26, normalized size = 1.08 \begin {gather*} \frac {e^{19 x}-40 x-4 e^x x}{\left (10+e^x\right ) x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1200*x + 240*E^x*x + 12*E^(2*x)*x + E^(18*x)*(E^(2*x)*(-4 + 18*x) + E^x*(-40 + 190*x)))/(100*x^5 +
20*E^x*x^5 + E^(2*x)*x^5),x]

[Out]

(E^(19*x) - 40*x - 4*E^x*x)/((10 + E^x)*x^4)

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fricas [A]  time = 1.84, size = 31, normalized size = 1.29 \begin {gather*} -\frac {4 \, x e^{x} + 40 \, x - e^{\left (19 \, x\right )}}{x^{4} e^{x} + 10 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((18*x-4)*exp(x)^2+(190*x-40)*exp(x))*exp(9*x)^2+12*x*exp(x)^2+240*exp(x)*x+1200*x)/(x^5*exp(x)^2+2
0*x^5*exp(x)+100*x^5),x, algorithm="fricas")

[Out]

-(4*x*e^x + 40*x - e^(19*x))/(x^4*e^x + 10*x^4)

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giac [A]  time = 0.15, size = 31, normalized size = 1.29 \begin {gather*} -\frac {4 \, x e^{x} + 40 \, x - e^{\left (19 \, x\right )}}{x^{4} e^{x} + 10 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((18*x-4)*exp(x)^2+(190*x-40)*exp(x))*exp(9*x)^2+12*x*exp(x)^2+240*exp(x)*x+1200*x)/(x^5*exp(x)^2+2
0*x^5*exp(x)+100*x^5),x, algorithm="giac")

[Out]

-(4*x*e^x + 40*x - e^(19*x))/(x^4*e^x + 10*x^4)

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maple [B]  time = 0.11, size = 181, normalized size = 7.54

method	result	size
risch	\(\frac {-4 x +1000000000000000000}{x^{4}}+\frac {{\mathrm e}^{18 x}}{x^{4}}-\frac {10 \,{\mathrm e}^{17 x}}{x^{4}}+\frac {100 \,{\mathrm e}^{16 x}}{x^{4}}-\frac {1000 \,{\mathrm e}^{15 x}}{x^{4}}+\frac {10000 \,{\mathrm e}^{14 x}}{x^{4}}-\frac {100000 \,{\mathrm e}^{13 x}}{x^{4}}+\frac {1000000 \,{\mathrm e}^{12 x}}{x^{4}}-\frac {10000000 \,{\mathrm e}^{11 x}}{x^{4}}+\frac {100000000 \,{\mathrm e}^{10 x}}{x^{4}}-\frac {1000000000 \,{\mathrm e}^{9 x}}{x^{4}}+\frac {10000000000 \,{\mathrm e}^{8 x}}{x^{4}}-\frac {100000000000 \,{\mathrm e}^{7 x}}{x^{4}}+\frac {1000000000000 \,{\mathrm e}^{6 x}}{x^{4}}-\frac {10000000000000 \,{\mathrm e}^{5 x}}{x^{4}}+\frac {100000000000000 \,{\mathrm e}^{4 x}}{x^{4}}-\frac {1000000000000000 \,{\mathrm e}^{3 x}}{x^{4}}+\frac {10000000000000000 \,{\mathrm e}^{2 x}}{x^{4}}-\frac {100000000000000000 \,{\mathrm e}^{x}}{x^{4}}-\frac {10000000000000000000}{x^{4} \left ({\mathrm e}^{x}+10\right )}\)	\(181\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((18*x-4)*exp(x)^2+(190*x-40)*exp(x))*exp(9*x)^2+12*x*exp(x)^2+240*exp(x)*x+1200*x)/(x^5*exp(x)^2+20*x^5*
exp(x)+100*x^5),x,method=_RETURNVERBOSE)

[Out]

(-4*x+1000000000000000000)/x^4+1/x^4*exp(18*x)-10/x^4*exp(17*x)+100/x^4*exp(16*x)-1000/x^4*exp(15*x)+10000/x^4
*exp(14*x)-100000/x^4*exp(13*x)+1000000/x^4*exp(12*x)-10000000/x^4*exp(11*x)[phone]/x^4*exp(10*x)-100000000
0/x^4*exp(9*x)[phone]/x^4*exp(8*x)-100000000000/x^4*exp(7*x)[phone]/x^4*exp(6*x)-10000000000000/x^
4*exp(5*x)[phone]*exp(4*x)/x^4-1000000000000000/x^4*exp(3*x)+10000000000000000*exp(2*x)/x^4-100000000
000000000*exp(x)/x^4-10000000000000000000/x^4/(exp(x)+10)

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maxima [A]  time = 0.40, size = 31, normalized size = 1.29 \begin {gather*} -\frac {4 \, x e^{x} + 40 \, x - e^{\left (19 \, x\right )}}{x^{4} e^{x} + 10 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((18*x-4)*exp(x)^2+(190*x-40)*exp(x))*exp(9*x)^2+12*x*exp(x)^2+240*exp(x)*x+1200*x)/(x^5*exp(x)^2+2
0*x^5*exp(x)+100*x^5),x, algorithm="maxima")

[Out]

-(4*x*e^x + 40*x - e^(19*x))/(x^4*e^x + 10*x^4)

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mupad [B]  time = 3.48, size = 181, normalized size = 7.54 \begin {gather*} \frac {10000000000000000\,{\mathrm {e}}^{2\,x}}{x^4}-\frac {4\,x-1000000000000000000}{x^4}-\frac {100000000000000000\,{\mathrm {e}}^x}{x^4}-\frac {1000000000000000\,{\mathrm {e}}^{3\,x}}{x^4}+\frac {100000000000000\,{\mathrm {e}}^{4\,x}}{x^4}-\frac {10000000000000\,{\mathrm {e}}^{5\,x}}{x^4}+\frac {1000000000000\,{\mathrm {e}}^{6\,x}}{x^4}-\frac {100000000000\,{\mathrm {e}}^{7\,x}}{x^4}+\frac {10000000000\,{\mathrm {e}}^{8\,x}}{x^4}-\frac {1000000000\,{\mathrm {e}}^{9\,x}}{x^4}+\frac {100000000\,{\mathrm {e}}^{10\,x}}{x^4}-\frac {10000000\,{\mathrm {e}}^{11\,x}}{x^4}+\frac {1000000\,{\mathrm {e}}^{12\,x}}{x^4}-\frac {100000\,{\mathrm {e}}^{13\,x}}{x^4}+\frac {10000\,{\mathrm {e}}^{14\,x}}{x^4}-\frac {1000\,{\mathrm {e}}^{15\,x}}{x^4}+\frac {100\,{\mathrm {e}}^{16\,x}}{x^4}-\frac {10\,{\mathrm {e}}^{17\,x}}{x^4}+\frac {{\mathrm {e}}^{18\,x}}{x^4}-\frac {10000000000000000000}{x^4\,\left ({\mathrm {e}}^x+10\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1200*x + 12*x*exp(2*x) + exp(18*x)*(exp(x)*(190*x - 40) + exp(2*x)*(18*x - 4)) + 240*x*exp(x))/(20*x^5*ex
p(x) + x^5*exp(2*x) + 100*x^5),x)

[Out]

(10000000000000000*exp(2*x))/x^4 - (4*x - 1000000000000000000)/x^4 - (100000000000000000*exp(x))/x^4 - (100000
0000000000*exp(3*x))/x^4 + (100000000000000*exp(4*x))/x^4 - (10000000000000*exp(5*x))/x^4 + (1000000000000*exp
(6*x))/x^4 - (100000000000*exp(7*x))/x^4 + (10000000000*exp(8*x))/x^4 - (1000000000*exp(9*x))/x^4 + (100000000
*exp(10*x))/x^4 - (10000000*exp(11*x))/x^4 + (1000000*exp(12*x))/x^4 - (100000*exp(13*x))/x^4 + (10000*exp(14*
x))/x^4 - (1000*exp(15*x))/x^4 + (100*exp(16*x))/x^4 - (10*exp(17*x))/x^4 + exp(18*x)/x^4 - 100000000000000000
00/(x^4*(exp(x) + 10))

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sympy [B]  time = 0.47, size = 204, normalized size = 8.50 \begin {gather*} - \frac {10000000000000000000}{x^{4} e^{x} + 10 x^{4}} + \frac {1000000000000000000 - 4 x}{x^{4}} + \frac {x^{68} e^{18 x} - 10 x^{68} e^{17 x} + 100 x^{68} e^{16 x} - 1000 x^{68} e^{15 x} + 10000 x^{68} e^{14 x} - 100000 x^{68} e^{13 x} + 1000000 x^{68} e^{12 x} - 10000000 x^{68} e^{11 x} + 100000000 x^{68} e^{10 x} - 1000000000 x^{68} e^{9 x} + 10000000000 x^{68} e^{8 x} - 100000000000 x^{68} e^{7 x} + 1000000000000 x^{68} e^{6 x} - 10000000000000 x^{68} e^{5 x} + 100000000000000 x^{68} e^{4 x} - 1000000000000000 x^{68} e^{3 x} + 10000000000000000 x^{68} e^{2 x} - 100000000000000000 x^{68} e^{x}}{x^{72}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((18*x-4)*exp(x)**2+(190*x-40)*exp(x))*exp(9*x)**2+12*x*exp(x)**2+240*exp(x)*x+1200*x)/(x**5*exp(x)
**2+20*x**5*exp(x)+100*x**5),x)

[Out]

-10000000000000000000/(x**4*exp(x) + 10*x**4) + (1000000000000000000 - 4*x)/x**4 + (x**68*exp(18*x) - 10*x**68
*exp(17*x) + 100*x**68*exp(16*x) - 1000*x**68*exp(15*x) + 10000*x**68*exp(14*x) - 100000*x**68*exp(13*x) + 100
0000*x**68*exp(12*x) - 10000000*x**68*exp(11*x) + 100000000*x**68*exp(10*x) - 1000000000*x**68*exp(9*x) + 1000
0000000*x**68*exp(8*x) - 100000000000*x**68*exp(7*x) + 1000000000000*x**68*exp(6*x) - 10000000000000*x**68*exp
(5*x) + 100000000000000*x**68*exp(4*x) - 1000000000000000*x**68*exp(3*x) + 10000000000000000*x**68*exp(2*x) -
100000000000000000*x**68*exp(x))/x**72

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