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3.91.99 \(\int \frac {7198-28800 x+e^x (3150-10800 x-7200 x^2)}{91125000-2186392500 x+21860281350 x^2-116581690799 x^3+349764501600 x^4-559716480000 x^5+373248000000 x^6+e^{3 x} (11390625-273375000 x+2733750000 x^2-14580000000 x^3+43740000000 x^4-69984000000 x^5+46656000000 x^6)+e^{2 x} (68343750-1640098125 x+16400070000 x^2-87465420000 x^3+262401120000 x^4-419865120000 x^5+279936000000 x^6)+e^x (136687500-3279892500 x+32795280675 x^2-174901685400 x^3+524724490800 x^4-839652480000 x^5+559872000000 x^6)} \, dx\)

Optimal. Leaf size=25 \[ \frac {1}{\left (x+\frac {225 \left (2+e^x\right ) \left (-x+4 x^2\right )^2}{x^2}\right )^2} \]

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Rubi [F]  time = 1.74, 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 {7198-28800 x+e^x \left (3150-10800 x-7200 x^2\right )}{91125000-2186392500 x+21860281350 x^2-116581690799 x^3+349764501600 x^4-559716480000 x^5+373248000000 x^6+e^{3 x} \left (11390625-273375000 x+2733750000 x^2-14580000000 x^3+43740000000 x^4-69984000000 x^5+46656000000 x^6\right )+e^{2 x} \left (68343750-1640098125 x+16400070000 x^2-87465420000 x^3+262401120000 x^4-419865120000 x^5+279936000000 x^6\right )+e^x \left (136687500-3279892500 x+32795280675 x^2-174901685400 x^3+524724490800 x^4-839652480000 x^5+559872000000 x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(7198 - 28800*x + E^x*(3150 - 10800*x - 7200*x^2))/(91125000 - 2186392500*x + 21860281350*x^2 - 1165816907
99*x^3 + 349764501600*x^4 - 559716480000*x^5 + 373248000000*x^6 + E^(3*x)*(11390625 - 273375000*x + 2733750000
*x^2 - 14580000000*x^3 + 43740000000*x^4 - 69984000000*x^5 + 46656000000*x^6) + E^(2*x)*(68343750 - 1640098125
*x + 16400070000*x^2 - 87465420000*x^3 + 262401120000*x^4 - 419865120000*x^5 + 279936000000*x^6) + E^x*(136687
500 - 3279892500*x + 32795280675*x^2 - 174901685400*x^3 + 524724490800*x^4 - 839652480000*x^5 + 559872000000*x
^6)),x]

[Out]

902*Defer[Int][(450 + 225*E^x - 3599*x - 1800*E^x*x + 7200*x^2 + 3600*E^x*x^2)^(-3), x] - 7198*Defer[Int][x/(4
50 + 225*E^x - 3599*x - 1800*E^x*x + 7200*x^2 + 3600*E^x*x^2)^3, x] + 14400*Defer[Int][x^2/(450 + 225*E^x - 35
99*x - 1800*E^x*x + 7200*x^2 + 3600*E^x*x^2)^3, x] + 4*Defer[Int][1/((-1 + 4*x)*(450 + 225*E^x - 3599*x - 1800
*E^x*x + 7200*x^2 + 3600*E^x*x^2)^3), x] - 2*Defer[Int][(450 + 225*E^x - 3599*x - 1800*E^x*x + 7200*x^2 + 3600
*E^x*x^2)^(-2), x] - 16*Defer[Int][1/((-1 + 4*x)*(450 + 225*E^x - 3599*x - 1800*E^x*x + 7200*x^2 + 3600*E^x*x^
2)^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {7198-28800 x-450 e^x \left (-7+24 x+16 x^2\right )}{\left (450+225 e^x (1-4 x)^2-3599 x+7200 x^2\right )^3} \, dx\\ &=\int \left (-\frac {2 (7+4 x)}{(-1+4 x) \left (450+225 e^x-3599 x-1800 e^x x+7200 x^2+3600 e^x x^2\right )^2}+\frac {2 \left (-449+5403 x-21596 x^2+28800 x^3\right )}{(-1+4 x) \left (450+225 e^x-3599 x-1800 e^x x+7200 x^2+3600 e^x x^2\right )^3}\right ) \, dx\\ &=-\left (2 \int \frac {7+4 x}{(-1+4 x) \left (450+225 e^x-3599 x-1800 e^x x+7200 x^2+3600 e^x x^2\right )^2} \, dx\right )+2 \int \frac {-449+5403 x-21596 x^2+28800 x^3}{(-1+4 x) \left (450+225 e^x-3599 x-1800 e^x x+7200 x^2+3600 e^x x^2\right )^3} \, dx\\ &=2 \int \left (\frac {451}{\left (450+225 e^x-3599 x-1800 e^x x+7200 x^2+3600 e^x x^2\right )^3}-\frac {3599 x}{\left (450+225 e^x-3599 x-1800 e^x x+7200 x^2+3600 e^x x^2\right )^3}+\frac {7200 x^2}{\left (450+225 e^x-3599 x-1800 e^x x+7200 x^2+3600 e^x x^2\right )^3}+\frac {2}{(-1+4 x) \left (450+225 e^x-3599 x-1800 e^x x+7200 x^2+3600 e^x x^2\right )^3}\right ) \, dx-2 \int \left (\frac {1}{\left (450+225 e^x-3599 x-1800 e^x x+7200 x^2+3600 e^x x^2\right )^2}+\frac {8}{(-1+4 x) \left (450+225 e^x-3599 x-1800 e^x x+7200 x^2+3600 e^x x^2\right )^2}\right ) \, dx\\ &=-\left (2 \int \frac {1}{\left (450+225 e^x-3599 x-1800 e^x x+7200 x^2+3600 e^x x^2\right )^2} \, dx\right )+4 \int \frac {1}{(-1+4 x) \left (450+225 e^x-3599 x-1800 e^x x+7200 x^2+3600 e^x x^2\right )^3} \, dx-16 \int \frac {1}{(-1+4 x) \left (450+225 e^x-3599 x-1800 e^x x+7200 x^2+3600 e^x x^2\right )^2} \, dx+902 \int \frac {1}{\left (450+225 e^x-3599 x-1800 e^x x+7200 x^2+3600 e^x x^2\right )^3} \, dx-7198 \int \frac {x}{\left (450+225 e^x-3599 x-1800 e^x x+7200 x^2+3600 e^x x^2\right )^3} \, dx+14400 \int \frac {x^2}{\left (450+225 e^x-3599 x-1800 e^x x+7200 x^2+3600 e^x x^2\right )^3} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.44, size = 24, normalized size = 0.96 \begin {gather*} \frac {1}{\left (450+225 e^x (1-4 x)^2-3599 x+7200 x^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(7198 - 28800*x + E^x*(3150 - 10800*x - 7200*x^2))/(91125000 - 2186392500*x + 21860281350*x^2 - 1165
81690799*x^3 + 349764501600*x^4 - 559716480000*x^5 + 373248000000*x^6 + E^(3*x)*(11390625 - 273375000*x + 2733
750000*x^2 - 14580000000*x^3 + 43740000000*x^4 - 69984000000*x^5 + 46656000000*x^6) + E^(2*x)*(68343750 - 1640
098125*x + 16400070000*x^2 - 87465420000*x^3 + 262401120000*x^4 - 419865120000*x^5 + 279936000000*x^6) + E^x*(
136687500 - 3279892500*x + 32795280675*x^2 - 174901685400*x^3 + 524724490800*x^4 - 839652480000*x^5 + 55987200
0000*x^6)),x]

[Out]

(450 + 225*E^x*(1 - 4*x)^2 - 3599*x + 7200*x^2)^(-2)

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fricas [B]  time = 0.90, size = 72, normalized size = 2.88 \begin {gather*} \frac {1}{51840000 \, x^{4} - 51825600 \, x^{3} + 19432801 \, x^{2} + 50625 \, {\left (256 \, x^{4} - 256 \, x^{3} + 96 \, x^{2} - 16 \, x + 1\right )} e^{\left (2 \, x\right )} + 450 \, {\left (115200 \, x^{4} - 115184 \, x^{3} + 43192 \, x^{2} - 7199 \, x + 450\right )} e^{x} - 3239100 \, x + 202500} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-7200*x^2-10800*x+3150)*exp(x)-28800*x+7198)/((46656000000*x^6-69984000000*x^5+43740000000*x^4-145
80000000*x^3+2733750000*x^2-273375000*x+11390625)*exp(x)^3+(279936000000*x^6-419865120000*x^5+262401120000*x^4
-87465420000*x^3+16400070000*x^2-1640098125*x+68343750)*exp(x)^2+(559872000000*x^6-839652480000*x^5+5247244908
00*x^4-174901685400*x^3+32795280675*x^2-3279892500*x+136687500)*exp(x)+373248000000*x^6-559716480000*x^5+34976
4501600*x^4-116581690799*x^3+21860281350*x^2-2186392500*x+91125000),x, algorithm="fricas")

[Out]

1/(51840000*x^4 - 51825600*x^3 + 19432801*x^2 + 50625*(256*x^4 - 256*x^3 + 96*x^2 - 16*x + 1)*e^(2*x) + 450*(1
15200*x^4 - 115184*x^3 + 43192*x^2 - 7199*x + 450)*e^x - 3239100*x + 202500)

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giac [B]  time = 0.20, size = 92, normalized size = 3.68 \begin {gather*} \frac {1}{12960000 \, x^{4} e^{\left (2 \, x\right )} + 51840000 \, x^{4} e^{x} + 51840000 \, x^{4} - 12960000 \, x^{3} e^{\left (2 \, x\right )} - 51832800 \, x^{3} e^{x} - 51825600 \, x^{3} + 4860000 \, x^{2} e^{\left (2 \, x\right )} + 19436400 \, x^{2} e^{x} + 19432801 \, x^{2} - 810000 \, x e^{\left (2 \, x\right )} - 3239550 \, x e^{x} - 3239100 \, x + 50625 \, e^{\left (2 \, x\right )} + 202500 \, e^{x} + 202500} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-7200*x^2-10800*x+3150)*exp(x)-28800*x+7198)/((46656000000*x^6-69984000000*x^5+43740000000*x^4-145
80000000*x^3+2733750000*x^2-273375000*x+11390625)*exp(x)^3+(279936000000*x^6-419865120000*x^5+262401120000*x^4
-87465420000*x^3+16400070000*x^2-1640098125*x+68343750)*exp(x)^2+(559872000000*x^6-839652480000*x^5+5247244908
00*x^4-174901685400*x^3+32795280675*x^2-3279892500*x+136687500)*exp(x)+373248000000*x^6-559716480000*x^5+34976
4501600*x^4-116581690799*x^3+21860281350*x^2-2186392500*x+91125000),x, algorithm="giac")

[Out]

1/(12960000*x^4*e^(2*x) + 51840000*x^4*e^x + 51840000*x^4 - 12960000*x^3*e^(2*x) - 51832800*x^3*e^x - 51825600
*x^3 + 4860000*x^2*e^(2*x) + 19436400*x^2*e^x + 19432801*x^2 - 810000*x*e^(2*x) - 3239550*x*e^x - 3239100*x +
50625*e^(2*x) + 202500*e^x + 202500)

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maple [A]  time = 0.27, size = 29, normalized size = 1.16

method result size
norman \(\frac {1}{\left (3600 \,{\mathrm e}^{x} x^{2}-1800 \,{\mathrm e}^{x} x +7200 x^{2}+225 \,{\mathrm e}^{x}-3599 x +450\right )^{2}}\) \(29\)
risch \(\frac {1}{\left (3600 \,{\mathrm e}^{x} x^{2}-1800 \,{\mathrm e}^{x} x +7200 x^{2}+225 \,{\mathrm e}^{x}-3599 x +450\right )^{2}}\) \(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-7200*x^2-10800*x+3150)*exp(x)-28800*x+7198)/((46656000000*x^6-69984000000*x^5+43740000000*x^4-145800000
00*x^3+2733750000*x^2-273375000*x+11390625)*exp(x)^3+(279936000000*x^6-419865120000*x^5+262401120000*x^4-87465
420000*x^3+16400070000*x^2-1640098125*x+68343750)*exp(x)^2+(559872000000*x^6-839652480000*x^5+524724490800*x^4
-174901685400*x^3+32795280675*x^2-3279892500*x+136687500)*exp(x)+373248000000*x^6-559716480000*x^5+34976450160
0*x^4-116581690799*x^3+21860281350*x^2-2186392500*x+91125000),x,method=_RETURNVERBOSE)

[Out]

1/(3600*exp(x)*x^2-1800*exp(x)*x+7200*x^2+225*exp(x)-3599*x+450)^2

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maxima [B]  time = 0.55, size = 72, normalized size = 2.88 \begin {gather*} \frac {1}{51840000 \, x^{4} - 51825600 \, x^{3} + 19432801 \, x^{2} + 50625 \, {\left (256 \, x^{4} - 256 \, x^{3} + 96 \, x^{2} - 16 \, x + 1\right )} e^{\left (2 \, x\right )} + 450 \, {\left (115200 \, x^{4} - 115184 \, x^{3} + 43192 \, x^{2} - 7199 \, x + 450\right )} e^{x} - 3239100 \, x + 202500} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-7200*x^2-10800*x+3150)*exp(x)-28800*x+7198)/((46656000000*x^6-69984000000*x^5+43740000000*x^4-145
80000000*x^3+2733750000*x^2-273375000*x+11390625)*exp(x)^3+(279936000000*x^6-419865120000*x^5+262401120000*x^4
-87465420000*x^3+16400070000*x^2-1640098125*x+68343750)*exp(x)^2+(559872000000*x^6-839652480000*x^5+5247244908
00*x^4-174901685400*x^3+32795280675*x^2-3279892500*x+136687500)*exp(x)+373248000000*x^6-559716480000*x^5+34976
4501600*x^4-116581690799*x^3+21860281350*x^2-2186392500*x+91125000),x, algorithm="maxima")

[Out]

1/(51840000*x^4 - 51825600*x^3 + 19432801*x^2 + 50625*(256*x^4 - 256*x^3 + 96*x^2 - 16*x + 1)*e^(2*x) + 450*(1
15200*x^4 - 115184*x^3 + 43192*x^2 - 7199*x + 450)*e^x - 3239100*x + 202500)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {28800\,x+{\mathrm {e}}^x\,\left (7200\,x^2+10800\,x-3150\right )-7198}{{\mathrm {e}}^{3\,x}\,\left (46656000000\,x^6-69984000000\,x^5+43740000000\,x^4-14580000000\,x^3+2733750000\,x^2-273375000\,x+11390625\right )-2186392500\,x+{\mathrm {e}}^{2\,x}\,\left (279936000000\,x^6-419865120000\,x^5+262401120000\,x^4-87465420000\,x^3+16400070000\,x^2-1640098125\,x+68343750\right )+{\mathrm {e}}^x\,\left (559872000000\,x^6-839652480000\,x^5+524724490800\,x^4-174901685400\,x^3+32795280675\,x^2-3279892500\,x+136687500\right )+21860281350\,x^2-116581690799\,x^3+349764501600\,x^4-559716480000\,x^5+373248000000\,x^6+91125000} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(28800*x + exp(x)*(10800*x + 7200*x^2 - 3150) - 7198)/(exp(3*x)*(2733750000*x^2 - 273375000*x - 145800000
00*x^3 + 43740000000*x^4 - 69984000000*x^5 + 46656000000*x^6 + 11390625) - 2186392500*x + exp(2*x)*(1640007000
0*x^2 - 1640098125*x - 87465420000*x^3 + 262401120000*x^4 - 419865120000*x^5 + 279936000000*x^6 + 68343750) +
exp(x)*(32795280675*x^2 - 3279892500*x - 174901685400*x^3 + 524724490800*x^4 - 839652480000*x^5 + 559872000000
*x^6 + 136687500) + 21860281350*x^2 - 116581690799*x^3 + 349764501600*x^4 - 559716480000*x^5 + 373248000000*x^
6 + 91125000),x)

[Out]

int(-(28800*x + exp(x)*(10800*x + 7200*x^2 - 3150) - 7198)/(exp(3*x)*(2733750000*x^2 - 273375000*x - 145800000
00*x^3 + 43740000000*x^4 - 69984000000*x^5 + 46656000000*x^6 + 11390625) - 2186392500*x + exp(2*x)*(1640007000
0*x^2 - 1640098125*x - 87465420000*x^3 + 262401120000*x^4 - 419865120000*x^5 + 279936000000*x^6 + 68343750) +
exp(x)*(32795280675*x^2 - 3279892500*x - 174901685400*x^3 + 524724490800*x^4 - 839652480000*x^5 + 559872000000
*x^6 + 136687500) + 21860281350*x^2 - 116581690799*x^3 + 349764501600*x^4 - 559716480000*x^5 + 373248000000*x^
6 + 91125000), x)

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sympy [B]  time = 0.67, size = 70, normalized size = 2.80 \begin {gather*} \frac {1}{51840000 x^{4} - 51825600 x^{3} + 19432801 x^{2} - 3239100 x + \left (12960000 x^{4} - 12960000 x^{3} + 4860000 x^{2} - 810000 x + 50625\right ) e^{2 x} + \left (51840000 x^{4} - 51832800 x^{3} + 19436400 x^{2} - 3239550 x + 202500\right ) e^{x} + 202500} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-7200*x**2-10800*x+3150)*exp(x)-28800*x+7198)/((46656000000*x**6-69984000000*x**5+43740000000*x**4
-14580000000*x**3+2733750000*x**2-273375000*x+11390625)*exp(x)**3+(279936000000*x**6-419865120000*x**5+2624011
20000*x**4-87465420000*x**3+16400070000*x**2-1640098125*x+68343750)*exp(x)**2+(559872000000*x**6-839652480000*
x**5+524724490800*x**4-174901685400*x**3+32795280675*x**2-3279892500*x+136687500)*exp(x)+373248000000*x**6-559
716480000*x**5+349764501600*x**4-116581690799*x**3+21860281350*x**2-2186392500*x+91125000),x)

[Out]

1/(51840000*x**4 - 51825600*x**3 + 19432801*x**2 - 3239100*x + (12960000*x**4 - 12960000*x**3 + 4860000*x**2 -
 810000*x + 50625)*exp(2*x) + (51840000*x**4 - 51832800*x**3 + 19436400*x**2 - 3239550*x + 202500)*exp(x) + 20
2500)

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