3.72.14 \(\int \frac {e^{6 x} (2 x+x^2)+e^{5 x} (-10 x^2+5 x^3+5 x^4)+e^{4 x} (8 x-4 x^2+20 x^3-30 x^4+10 x^6)+e^{3 x} (-16 x^2+12 x^3-24 x^4+50 x^5-30 x^6-10 x^7+10 x^8)+e^{2 x} (8 x-12 x^2+8 x^3+4 x^4-6 x^5-31 x^6+40 x^7-10 x^8-10 x^9+5 x^{10})+e^x (8 x^2-28 x^3+4 x^4-12 x^5+26 x^6-11 x^7-11 x^8+10 x^9-3 x^{11}+x^{12})+e^{6 x} (7 e^{6 x}+e^{5 x} (-35 x+35 x^2)+e^{4 x} (70 x^2-140 x^3+70 x^4)+e^{3 x} (-70 x^3+210 x^4-210 x^5+70 x^6)+e^{2 x} (35 x^4-140 x^5+210 x^6-140 x^7+35 x^8)+e^x (-7 x^5+35 x^6-70 x^7+70 x^8-35 x^9+7 x^{10}))+e^{3 x} (e^{6 x} (-2-8 x)+e^{5 x} (10 x+30 x^2-40 x^3)+e^{4 x} (-4-8 x-20 x^2-40 x^3+140 x^4-80 x^5)+e^{3 x} (4 x+36 x^2-12 x^3+20 x^4-180 x^5+220 x^6-80 x^7)+e^{2 x} (4 x^2-64 x^3+90 x^4-40 x^5+100 x^6-200 x^7+150 x^8-40 x^9)+e^x (-4 x^3+36 x^4-74 x^5+58 x^6-36 x^7+60 x^8-70 x^9+38 x^{10}-8 x^{11}))}{e^{5 x}-x^5+5 x^6-10 x^7+10 x^8-5 x^9+x^{10}+e^{4 x} (-5 x+5 x^2)+e^{3 x} (10 x^2-20 x^3+10 x^4)+e^{2 x} (-10 x^3+30 x^4-30 x^5+10 x^6)+e^x (5 x^4-20 x^5+30 x^6-20 x^7+5 x^8)} \, dx\)

Optimal. Leaf size=30 \[ e^x \left (-e^{3 x}+x+\frac {2 x}{\left (e^x-x+x^2\right )^2}\right )^2 \]

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Rubi [F]  time = 145.59, 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 {e^{6 x} \left (2 x+x^2\right )+e^{5 x} \left (-10 x^2+5 x^3+5 x^4\right )+e^{4 x} \left (8 x-4 x^2+20 x^3-30 x^4+10 x^6\right )+e^{3 x} \left (-16 x^2+12 x^3-24 x^4+50 x^5-30 x^6-10 x^7+10 x^8\right )+e^{2 x} \left (8 x-12 x^2+8 x^3+4 x^4-6 x^5-31 x^6+40 x^7-10 x^8-10 x^9+5 x^{10}\right )+e^x \left (8 x^2-28 x^3+4 x^4-12 x^5+26 x^6-11 x^7-11 x^8+10 x^9-3 x^{11}+x^{12}\right )+e^{6 x} \left (7 e^{6 x}+e^{5 x} \left (-35 x+35 x^2\right )+e^{4 x} \left (70 x^2-140 x^3+70 x^4\right )+e^{3 x} \left (-70 x^3+210 x^4-210 x^5+70 x^6\right )+e^{2 x} \left (35 x^4-140 x^5+210 x^6-140 x^7+35 x^8\right )+e^x \left (-7 x^5+35 x^6-70 x^7+70 x^8-35 x^9+7 x^{10}\right )\right )+e^{3 x} \left (e^{6 x} (-2-8 x)+e^{5 x} \left (10 x+30 x^2-40 x^3\right )+e^{4 x} \left (-4-8 x-20 x^2-40 x^3+140 x^4-80 x^5\right )+e^{3 x} \left (4 x+36 x^2-12 x^3+20 x^4-180 x^5+220 x^6-80 x^7\right )+e^{2 x} \left (4 x^2-64 x^3+90 x^4-40 x^5+100 x^6-200 x^7+150 x^8-40 x^9\right )+e^x \left (-4 x^3+36 x^4-74 x^5+58 x^6-36 x^7+60 x^8-70 x^9+38 x^{10}-8 x^{11}\right )\right )}{e^{5 x}-x^5+5 x^6-10 x^7+10 x^8-5 x^9+x^{10}+e^{4 x} \left (-5 x+5 x^2\right )+e^{3 x} \left (10 x^2-20 x^3+10 x^4\right )+e^{2 x} \left (-10 x^3+30 x^4-30 x^5+10 x^6\right )+e^x \left (5 x^4-20 x^5+30 x^6-20 x^7+5 x^8\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(6*x)*(2*x + x^2) + E^(5*x)*(-10*x^2 + 5*x^3 + 5*x^4) + E^(4*x)*(8*x - 4*x^2 + 20*x^3 - 30*x^4 + 10*x^6
) + E^(3*x)*(-16*x^2 + 12*x^3 - 24*x^4 + 50*x^5 - 30*x^6 - 10*x^7 + 10*x^8) + E^(2*x)*(8*x - 12*x^2 + 8*x^3 +
4*x^4 - 6*x^5 - 31*x^6 + 40*x^7 - 10*x^8 - 10*x^9 + 5*x^10) + E^x*(8*x^2 - 28*x^3 + 4*x^4 - 12*x^5 + 26*x^6 -
11*x^7 - 11*x^8 + 10*x^9 - 3*x^11 + x^12) + E^(6*x)*(7*E^(6*x) + E^(5*x)*(-35*x + 35*x^2) + E^(4*x)*(70*x^2 -
140*x^3 + 70*x^4) + E^(3*x)*(-70*x^3 + 210*x^4 - 210*x^5 + 70*x^6) + E^(2*x)*(35*x^4 - 140*x^5 + 210*x^6 - 140
*x^7 + 35*x^8) + E^x*(-7*x^5 + 35*x^6 - 70*x^7 + 70*x^8 - 35*x^9 + 7*x^10)) + E^(3*x)*(E^(6*x)*(-2 - 8*x) + E^
(5*x)*(10*x + 30*x^2 - 40*x^3) + E^(4*x)*(-4 - 8*x - 20*x^2 - 40*x^3 + 140*x^4 - 80*x^5) + E^(3*x)*(4*x + 36*x
^2 - 12*x^3 + 20*x^4 - 180*x^5 + 220*x^6 - 80*x^7) + E^(2*x)*(4*x^2 - 64*x^3 + 90*x^4 - 40*x^5 + 100*x^6 - 200
*x^7 + 150*x^8 - 40*x^9) + E^x*(-4*x^3 + 36*x^4 - 74*x^5 + 58*x^6 - 36*x^7 + 60*x^8 - 70*x^9 + 38*x^10 - 8*x^1
1)))/(E^(5*x) - x^5 + 5*x^6 - 10*x^7 + 10*x^8 - 5*x^9 + x^10 + E^(4*x)*(-5*x + 5*x^2) + E^(3*x)*(10*x^2 - 20*x
^3 + 10*x^4) + E^(2*x)*(-10*x^3 + 30*x^4 - 30*x^5 + 10*x^6) + E^x*(5*x^4 - 20*x^5 + 30*x^6 - 20*x^7 + 5*x^8)),
x]

[Out]

2*E^(2*x) + E^(4*x)/2 + E^(7*x) - 7*E^x*x^2 + 8*E^x*x^3 - 2*E^(2*x)*(1 + 2*x) - (E^(4*x)*(1 + 4*x))/2 + 16*Def
er[Int][(E^x*x^2)/(E^x - x + x^2)^5, x] - 48*Defer[Int][(E^x*x^3)/(E^x - x + x^2)^5, x] + 16*Defer[Int][(E^x*x
^4)/(E^x - x + x^2)^5, x] + 8*Defer[Int][(E^x*x)/(E^x - x + x^2)^4, x] - 12*Defer[Int][(E^x*x^2)/(E^x - x + x^
2)^4, x] + 8*Defer[Int][(E^x*x^2)/(E^x - x + x^2)^3, x] - 24*Defer[Int][(E^x*x^3)/(E^x - x + x^2)^3, x] + 48*D
efer[Int][(E^x*x^5)/(E^x - x + x^2)^3, x] - 104*Defer[Int][(E^x*x^6)/(E^x - x + x^2)^3, x] + 104*Defer[Int][(E
^x*x^7)/(E^x - x + x^2)^3, x] - 48*Defer[Int][(E^x*x^8)/(E^x - x + x^2)^3, x] + 8*Defer[Int][(E^x*x^9)/(E^x -
x + x^2)^3, x] + 8*Defer[Int][(E^x*x)/(E^x - x + x^2)^2, x] - 4*Defer[Int][(E^x*x^2)/(E^x - x + x^2)^2, x] - 2
8*Defer[Int][(E^x*x^3)/(E^x - x + x^2)^2, x] + 124*Defer[Int][(E^x*x^4)/(E^x - x + x^2)^2, x] - 180*Defer[Int]
[(E^x*x^5)/(E^x - x + x^2)^2, x] + 100*Defer[Int][(E^x*x^6)/(E^x - x + x^2)^2, x] - 16*Defer[Int][(E^x*x^7)/(E
^x - x + x^2)^2, x] - 36*Defer[Int][(E^x*x^2)/(E^x - x + x^2), x] + 96*Defer[Int][(E^x*x^3)/(E^x - x + x^2), x
] - 60*Defer[Int][(E^x*x^4)/(E^x - x + x^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^x \left (7 e^{11 x}+35 e^{10 x} (-1+x) x+70 e^{9 x} (-1+x)^2 x^2+2 e^{2 x} x^2 \left (-8+6 x-12 x^2+25 x^3-15 x^4-5 x^5+5 x^6\right )+e^{8 x} \left (-2-8 x-70 x^3+210 x^4-210 x^5+70 x^6\right )-e^{5 x} x \left (-6-37 x+12 x^2-20 x^3+180 x^4-220 x^5+80 x^6\right )+5 e^{7 x} x \left (2+6 x-8 x^2+7 x^3-28 x^4+42 x^5-28 x^6+7 x^7\right )-e^{4 x} x^2 \left (6+59 x-95 x^2+40 x^3-100 x^4+200 x^5-150 x^6+40 x^7\right )+e^x x \left (8-12 x+8 x^2+4 x^3-6 x^4-31 x^5+40 x^6-10 x^7-10 x^8+5 x^9\right )+x^2 \left (8-28 x+4 x^2-12 x^3+26 x^4-11 x^5-11 x^6+10 x^7-3 x^9+x^{10}\right )-2 e^{3 x} x \left (-4+2 x-8 x^2-3 x^3+37 x^4-34 x^5+18 x^6-30 x^7+35 x^8-19 x^9+4 x^{10}\right )+e^{6 x} \left (-4-8 x-20 x^2-40 x^3+140 x^4-87 x^5+35 x^6-70 x^7+70 x^8-35 x^9+7 x^{10}\right )\right )}{\left (e^x+(-1+x) x\right )^5} \, dx\\ &=\int \left (7 e^{7 x}-4 e^{2 x} (1+2 x)-2 e^{4 x} (1+4 x)+\frac {16 e^x x^2 \left (1-3 x+x^2\right )}{\left (e^x-x+x^2\right )^5}-\frac {4 e^x x (-2+3 x)}{\left (e^x-x+x^2\right )^4}-\frac {12 e^x x^2 \left (3-8 x+5 x^2\right )}{e^x-x+x^2}+e^x x \left (-14+17 x+8 x^2\right )-\frac {4 e^x x \left (-2+x+7 x^2-31 x^3+45 x^4-25 x^5+4 x^6\right )}{\left (e^x-x+x^2\right )^2}+\frac {8 e^x x^2 \left (1-3 x+6 x^3-13 x^4+13 x^5-6 x^6+x^7\right )}{\left (e^x-x+x^2\right )^3}\right ) \, dx\\ &=-\left (2 \int e^{4 x} (1+4 x) \, dx\right )-4 \int e^{2 x} (1+2 x) \, dx-4 \int \frac {e^x x (-2+3 x)}{\left (e^x-x+x^2\right )^4} \, dx-4 \int \frac {e^x x \left (-2+x+7 x^2-31 x^3+45 x^4-25 x^5+4 x^6\right )}{\left (e^x-x+x^2\right )^2} \, dx+7 \int e^{7 x} \, dx+8 \int \frac {e^x x^2 \left (1-3 x+6 x^3-13 x^4+13 x^5-6 x^6+x^7\right )}{\left (e^x-x+x^2\right )^3} \, dx-12 \int \frac {e^x x^2 \left (3-8 x+5 x^2\right )}{e^x-x+x^2} \, dx+16 \int \frac {e^x x^2 \left (1-3 x+x^2\right )}{\left (e^x-x+x^2\right )^5} \, dx+\int e^x x \left (-14+17 x+8 x^2\right ) \, dx\\ &=e^{7 x}-2 e^{2 x} (1+2 x)-\frac {1}{2} e^{4 x} (1+4 x)+2 \int e^{4 x} \, dx+4 \int e^{2 x} \, dx-4 \int \left (-\frac {2 e^x x}{\left (e^x-x+x^2\right )^4}+\frac {3 e^x x^2}{\left (e^x-x+x^2\right )^4}\right ) \, dx-4 \int \left (-\frac {2 e^x x}{\left (e^x-x+x^2\right )^2}+\frac {e^x x^2}{\left (e^x-x+x^2\right )^2}+\frac {7 e^x x^3}{\left (e^x-x+x^2\right )^2}-\frac {31 e^x x^4}{\left (e^x-x+x^2\right )^2}+\frac {45 e^x x^5}{\left (e^x-x+x^2\right )^2}-\frac {25 e^x x^6}{\left (e^x-x+x^2\right )^2}+\frac {4 e^x x^7}{\left (e^x-x+x^2\right )^2}\right ) \, dx+8 \int \left (\frac {e^x x^2}{\left (e^x-x+x^2\right )^3}-\frac {3 e^x x^3}{\left (e^x-x+x^2\right )^3}+\frac {6 e^x x^5}{\left (e^x-x+x^2\right )^3}-\frac {13 e^x x^6}{\left (e^x-x+x^2\right )^3}+\frac {13 e^x x^7}{\left (e^x-x+x^2\right )^3}-\frac {6 e^x x^8}{\left (e^x-x+x^2\right )^3}+\frac {e^x x^9}{\left (e^x-x+x^2\right )^3}\right ) \, dx-12 \int \left (\frac {3 e^x x^2}{e^x-x+x^2}-\frac {8 e^x x^3}{e^x-x+x^2}+\frac {5 e^x x^4}{e^x-x+x^2}\right ) \, dx+16 \int \left (\frac {e^x x^2}{\left (e^x-x+x^2\right )^5}-\frac {3 e^x x^3}{\left (e^x-x+x^2\right )^5}+\frac {e^x x^4}{\left (e^x-x+x^2\right )^5}\right ) \, dx+\int \left (-14 e^x x+17 e^x x^2+8 e^x x^3\right ) \, dx\\ &=2 e^{2 x}+\frac {e^{4 x}}{2}+e^{7 x}-2 e^{2 x} (1+2 x)-\frac {1}{2} e^{4 x} (1+4 x)-4 \int \frac {e^x x^2}{\left (e^x-x+x^2\right )^2} \, dx+8 \int e^x x^3 \, dx+8 \int \frac {e^x x}{\left (e^x-x+x^2\right )^4} \, dx+8 \int \frac {e^x x^2}{\left (e^x-x+x^2\right )^3} \, dx+8 \int \frac {e^x x^9}{\left (e^x-x+x^2\right )^3} \, dx+8 \int \frac {e^x x}{\left (e^x-x+x^2\right )^2} \, dx-12 \int \frac {e^x x^2}{\left (e^x-x+x^2\right )^4} \, dx-14 \int e^x x \, dx+16 \int \frac {e^x x^2}{\left (e^x-x+x^2\right )^5} \, dx+16 \int \frac {e^x x^4}{\left (e^x-x+x^2\right )^5} \, dx-16 \int \frac {e^x x^7}{\left (e^x-x+x^2\right )^2} \, dx+17 \int e^x x^2 \, dx-24 \int \frac {e^x x^3}{\left (e^x-x+x^2\right )^3} \, dx-28 \int \frac {e^x x^3}{\left (e^x-x+x^2\right )^2} \, dx-36 \int \frac {e^x x^2}{e^x-x+x^2} \, dx-48 \int \frac {e^x x^3}{\left (e^x-x+x^2\right )^5} \, dx+48 \int \frac {e^x x^5}{\left (e^x-x+x^2\right )^3} \, dx-48 \int \frac {e^x x^8}{\left (e^x-x+x^2\right )^3} \, dx-60 \int \frac {e^x x^4}{e^x-x+x^2} \, dx+96 \int \frac {e^x x^3}{e^x-x+x^2} \, dx+100 \int \frac {e^x x^6}{\left (e^x-x+x^2\right )^2} \, dx-104 \int \frac {e^x x^6}{\left (e^x-x+x^2\right )^3} \, dx+104 \int \frac {e^x x^7}{\left (e^x-x+x^2\right )^3} \, dx+124 \int \frac {e^x x^4}{\left (e^x-x+x^2\right )^2} \, dx-180 \int \frac {e^x x^5}{\left (e^x-x+x^2\right )^2} \, dx\\ &=2 e^{2 x}+\frac {e^{4 x}}{2}+e^{7 x}-14 e^x x+17 e^x x^2+8 e^x x^3-2 e^{2 x} (1+2 x)-\frac {1}{2} e^{4 x} (1+4 x)-4 \int \frac {e^x x^2}{\left (e^x-x+x^2\right )^2} \, dx+8 \int \frac {e^x x}{\left (e^x-x+x^2\right )^4} \, dx+8 \int \frac {e^x x^2}{\left (e^x-x+x^2\right )^3} \, dx+8 \int \frac {e^x x^9}{\left (e^x-x+x^2\right )^3} \, dx+8 \int \frac {e^x x}{\left (e^x-x+x^2\right )^2} \, dx-12 \int \frac {e^x x^2}{\left (e^x-x+x^2\right )^4} \, dx+14 \int e^x \, dx+16 \int \frac {e^x x^2}{\left (e^x-x+x^2\right )^5} \, dx+16 \int \frac {e^x x^4}{\left (e^x-x+x^2\right )^5} \, dx-16 \int \frac {e^x x^7}{\left (e^x-x+x^2\right )^2} \, dx-24 \int e^x x^2 \, dx-24 \int \frac {e^x x^3}{\left (e^x-x+x^2\right )^3} \, dx-28 \int \frac {e^x x^3}{\left (e^x-x+x^2\right )^2} \, dx-34 \int e^x x \, dx-36 \int \frac {e^x x^2}{e^x-x+x^2} \, dx-48 \int \frac {e^x x^3}{\left (e^x-x+x^2\right )^5} \, dx+48 \int \frac {e^x x^5}{\left (e^x-x+x^2\right )^3} \, dx-48 \int \frac {e^x x^8}{\left (e^x-x+x^2\right )^3} \, dx-60 \int \frac {e^x x^4}{e^x-x+x^2} \, dx+96 \int \frac {e^x x^3}{e^x-x+x^2} \, dx+100 \int \frac {e^x x^6}{\left (e^x-x+x^2\right )^2} \, dx-104 \int \frac {e^x x^6}{\left (e^x-x+x^2\right )^3} \, dx+104 \int \frac {e^x x^7}{\left (e^x-x+x^2\right )^3} \, dx+124 \int \frac {e^x x^4}{\left (e^x-x+x^2\right )^2} \, dx-180 \int \frac {e^x x^5}{\left (e^x-x+x^2\right )^2} \, dx\\ &=14 e^x+2 e^{2 x}+\frac {e^{4 x}}{2}+e^{7 x}-48 e^x x-7 e^x x^2+8 e^x x^3-2 e^{2 x} (1+2 x)-\frac {1}{2} e^{4 x} (1+4 x)-4 \int \frac {e^x x^2}{\left (e^x-x+x^2\right )^2} \, dx+8 \int \frac {e^x x}{\left (e^x-x+x^2\right )^4} \, dx+8 \int \frac {e^x x^2}{\left (e^x-x+x^2\right )^3} \, dx+8 \int \frac {e^x x^9}{\left (e^x-x+x^2\right )^3} \, dx+8 \int \frac {e^x x}{\left (e^x-x+x^2\right )^2} \, dx-12 \int \frac {e^x x^2}{\left (e^x-x+x^2\right )^4} \, dx+16 \int \frac {e^x x^2}{\left (e^x-x+x^2\right )^5} \, dx+16 \int \frac {e^x x^4}{\left (e^x-x+x^2\right )^5} \, dx-16 \int \frac {e^x x^7}{\left (e^x-x+x^2\right )^2} \, dx-24 \int \frac {e^x x^3}{\left (e^x-x+x^2\right )^3} \, dx-28 \int \frac {e^x x^3}{\left (e^x-x+x^2\right )^2} \, dx+34 \int e^x \, dx-36 \int \frac {e^x x^2}{e^x-x+x^2} \, dx+48 \int e^x x \, dx-48 \int \frac {e^x x^3}{\left (e^x-x+x^2\right )^5} \, dx+48 \int \frac {e^x x^5}{\left (e^x-x+x^2\right )^3} \, dx-48 \int \frac {e^x x^8}{\left (e^x-x+x^2\right )^3} \, dx-60 \int \frac {e^x x^4}{e^x-x+x^2} \, dx+96 \int \frac {e^x x^3}{e^x-x+x^2} \, dx+100 \int \frac {e^x x^6}{\left (e^x-x+x^2\right )^2} \, dx-104 \int \frac {e^x x^6}{\left (e^x-x+x^2\right )^3} \, dx+104 \int \frac {e^x x^7}{\left (e^x-x+x^2\right )^3} \, dx+124 \int \frac {e^x x^4}{\left (e^x-x+x^2\right )^2} \, dx-180 \int \frac {e^x x^5}{\left (e^x-x+x^2\right )^2} \, dx\\ &=48 e^x+2 e^{2 x}+\frac {e^{4 x}}{2}+e^{7 x}-7 e^x x^2+8 e^x x^3-2 e^{2 x} (1+2 x)-\frac {1}{2} e^{4 x} (1+4 x)-4 \int \frac {e^x x^2}{\left (e^x-x+x^2\right )^2} \, dx+8 \int \frac {e^x x}{\left (e^x-x+x^2\right )^4} \, dx+8 \int \frac {e^x x^2}{\left (e^x-x+x^2\right )^3} \, dx+8 \int \frac {e^x x^9}{\left (e^x-x+x^2\right )^3} \, dx+8 \int \frac {e^x x}{\left (e^x-x+x^2\right )^2} \, dx-12 \int \frac {e^x x^2}{\left (e^x-x+x^2\right )^4} \, dx+16 \int \frac {e^x x^2}{\left (e^x-x+x^2\right )^5} \, dx+16 \int \frac {e^x x^4}{\left (e^x-x+x^2\right )^5} \, dx-16 \int \frac {e^x x^7}{\left (e^x-x+x^2\right )^2} \, dx-24 \int \frac {e^x x^3}{\left (e^x-x+x^2\right )^3} \, dx-28 \int \frac {e^x x^3}{\left (e^x-x+x^2\right )^2} \, dx-36 \int \frac {e^x x^2}{e^x-x+x^2} \, dx-48 \int e^x \, dx-48 \int \frac {e^x x^3}{\left (e^x-x+x^2\right )^5} \, dx+48 \int \frac {e^x x^5}{\left (e^x-x+x^2\right )^3} \, dx-48 \int \frac {e^x x^8}{\left (e^x-x+x^2\right )^3} \, dx-60 \int \frac {e^x x^4}{e^x-x+x^2} \, dx+96 \int \frac {e^x x^3}{e^x-x+x^2} \, dx+100 \int \frac {e^x x^6}{\left (e^x-x+x^2\right )^2} \, dx-104 \int \frac {e^x x^6}{\left (e^x-x+x^2\right )^3} \, dx+104 \int \frac {e^x x^7}{\left (e^x-x+x^2\right )^3} \, dx+124 \int \frac {e^x x^4}{\left (e^x-x+x^2\right )^2} \, dx-180 \int \frac {e^x x^5}{\left (e^x-x+x^2\right )^2} \, dx\\ &=2 e^{2 x}+\frac {e^{4 x}}{2}+e^{7 x}-7 e^x x^2+8 e^x x^3-2 e^{2 x} (1+2 x)-\frac {1}{2} e^{4 x} (1+4 x)-4 \int \frac {e^x x^2}{\left (e^x-x+x^2\right )^2} \, dx+8 \int \frac {e^x x}{\left (e^x-x+x^2\right )^4} \, dx+8 \int \frac {e^x x^2}{\left (e^x-x+x^2\right )^3} \, dx+8 \int \frac {e^x x^9}{\left (e^x-x+x^2\right )^3} \, dx+8 \int \frac {e^x x}{\left (e^x-x+x^2\right )^2} \, dx-12 \int \frac {e^x x^2}{\left (e^x-x+x^2\right )^4} \, dx+16 \int \frac {e^x x^2}{\left (e^x-x+x^2\right )^5} \, dx+16 \int \frac {e^x x^4}{\left (e^x-x+x^2\right )^5} \, dx-16 \int \frac {e^x x^7}{\left (e^x-x+x^2\right )^2} \, dx-24 \int \frac {e^x x^3}{\left (e^x-x+x^2\right )^3} \, dx-28 \int \frac {e^x x^3}{\left (e^x-x+x^2\right )^2} \, dx-36 \int \frac {e^x x^2}{e^x-x+x^2} \, dx-48 \int \frac {e^x x^3}{\left (e^x-x+x^2\right )^5} \, dx+48 \int \frac {e^x x^5}{\left (e^x-x+x^2\right )^3} \, dx-48 \int \frac {e^x x^8}{\left (e^x-x+x^2\right )^3} \, dx-60 \int \frac {e^x x^4}{e^x-x+x^2} \, dx+96 \int \frac {e^x x^3}{e^x-x+x^2} \, dx+100 \int \frac {e^x x^6}{\left (e^x-x+x^2\right )^2} \, dx-104 \int \frac {e^x x^6}{\left (e^x-x+x^2\right )^3} \, dx+104 \int \frac {e^x x^7}{\left (e^x-x+x^2\right )^3} \, dx+124 \int \frac {e^x x^4}{\left (e^x-x+x^2\right )^2} \, dx-180 \int \frac {e^x x^5}{\left (e^x-x+x^2\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.24, size = 83, normalized size = 2.77 \begin {gather*} \frac {e^x \left (e^{5 x}-e^{2 x} x+2 e^{4 x} (-1+x) x-2 e^x (-1+x) x^2+e^{3 x} (-1+x)^2 x^2-x \left (2+x^2-2 x^3+x^4\right )\right )^2}{\left (e^x+(-1+x) x\right )^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(6*x)*(2*x + x^2) + E^(5*x)*(-10*x^2 + 5*x^3 + 5*x^4) + E^(4*x)*(8*x - 4*x^2 + 20*x^3 - 30*x^4 +
10*x^6) + E^(3*x)*(-16*x^2 + 12*x^3 - 24*x^4 + 50*x^5 - 30*x^6 - 10*x^7 + 10*x^8) + E^(2*x)*(8*x - 12*x^2 + 8*
x^3 + 4*x^4 - 6*x^5 - 31*x^6 + 40*x^7 - 10*x^8 - 10*x^9 + 5*x^10) + E^x*(8*x^2 - 28*x^3 + 4*x^4 - 12*x^5 + 26*
x^6 - 11*x^7 - 11*x^8 + 10*x^9 - 3*x^11 + x^12) + E^(6*x)*(7*E^(6*x) + E^(5*x)*(-35*x + 35*x^2) + E^(4*x)*(70*
x^2 - 140*x^3 + 70*x^4) + E^(3*x)*(-70*x^3 + 210*x^4 - 210*x^5 + 70*x^6) + E^(2*x)*(35*x^4 - 140*x^5 + 210*x^6
- 140*x^7 + 35*x^8) + E^x*(-7*x^5 + 35*x^6 - 70*x^7 + 70*x^8 - 35*x^9 + 7*x^10)) + E^(3*x)*(E^(6*x)*(-2 - 8*x
) + E^(5*x)*(10*x + 30*x^2 - 40*x^3) + E^(4*x)*(-4 - 8*x - 20*x^2 - 40*x^3 + 140*x^4 - 80*x^5) + E^(3*x)*(4*x
+ 36*x^2 - 12*x^3 + 20*x^4 - 180*x^5 + 220*x^6 - 80*x^7) + E^(2*x)*(4*x^2 - 64*x^3 + 90*x^4 - 40*x^5 + 100*x^6
- 200*x^7 + 150*x^8 - 40*x^9) + E^x*(-4*x^3 + 36*x^4 - 74*x^5 + 58*x^6 - 36*x^7 + 60*x^8 - 70*x^9 + 38*x^10 -
8*x^11)))/(E^(5*x) - x^5 + 5*x^6 - 10*x^7 + 10*x^8 - 5*x^9 + x^10 + E^(4*x)*(-5*x + 5*x^2) + E^(3*x)*(10*x^2
- 20*x^3 + 10*x^4) + E^(2*x)*(-10*x^3 + 30*x^4 - 30*x^5 + 10*x^6) + E^x*(5*x^4 - 20*x^5 + 30*x^6 - 20*x^7 + 5*
x^8)),x]

[Out]

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

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fricas [B]  time = 0.89, size = 390, normalized size = 13.00 \begin {gather*} \frac {4 \, {\left (x^{2} - x\right )} e^{\left (10 \, x\right )} + 6 \, {\left (x^{4} - 2 \, x^{3} + x^{2}\right )} e^{\left (9 \, x\right )} + 2 \, {\left (2 \, x^{6} - 6 \, x^{5} + 6 \, x^{4} - 2 \, x^{3} - x\right )} e^{\left (8 \, x\right )} + {\left (x^{8} - 4 \, x^{7} + 6 \, x^{6} - 4 \, x^{5} + x^{4} - 8 \, x^{3} + 8 \, x^{2}\right )} e^{\left (7 \, x\right )} - 4 \, {\left (3 \, x^{5} - 6 \, x^{4} + 3 \, x^{3} + x\right )} e^{\left (6 \, x\right )} - {\left (8 \, x^{7} - 24 \, x^{6} + 24 \, x^{5} - 8 \, x^{4} + 8 \, x^{3} - 9 \, x^{2}\right )} e^{\left (5 \, x\right )} - 2 \, {\left (x^{9} - 4 \, x^{8} + 6 \, x^{7} - 4 \, x^{6} + 3 \, x^{5} - 6 \, x^{4} + 4 \, x^{3}\right )} e^{\left (4 \, x\right )} + 2 \, {\left (3 \, x^{6} - 6 \, x^{5} + 3 \, x^{4} + 2 \, x^{2}\right )} e^{\left (3 \, x\right )} + 4 \, {\left (x^{8} - 3 \, x^{7} + 3 \, x^{6} - x^{5} + 2 \, x^{4} - 2 \, x^{3}\right )} e^{\left (2 \, x\right )} + {\left (x^{10} - 4 \, x^{9} + 6 \, x^{8} - 4 \, x^{7} + 5 \, x^{6} - 8 \, x^{5} + 4 \, x^{4} + 4 \, x^{2}\right )} e^{x} + e^{\left (11 \, x\right )}}{x^{8} - 4 \, x^{7} + 6 \, x^{6} - 4 \, x^{5} + x^{4} + 4 \, {\left (x^{2} - x\right )} e^{\left (3 \, x\right )} + 6 \, {\left (x^{4} - 2 \, x^{3} + x^{2}\right )} e^{\left (2 \, x\right )} + 4 \, {\left (x^{6} - 3 \, x^{5} + 3 \, x^{4} - x^{3}\right )} e^{x} + e^{\left (4 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((7*exp(x)^6+(35*x^2-35*x)*exp(x)^5+(70*x^4-140*x^3+70*x^2)*exp(x)^4+(70*x^6-210*x^5+210*x^4-70*x^3)
*exp(x)^3+(35*x^8-140*x^7+210*x^6-140*x^5+35*x^4)*exp(x)^2+(7*x^10-35*x^9+70*x^8-70*x^7+35*x^6-7*x^5)*exp(x))*
exp(3*x)^2+((-8*x-2)*exp(x)^6+(-40*x^3+30*x^2+10*x)*exp(x)^5+(-80*x^5+140*x^4-40*x^3-20*x^2-8*x-4)*exp(x)^4+(-
80*x^7+220*x^6-180*x^5+20*x^4-12*x^3+36*x^2+4*x)*exp(x)^3+(-40*x^9+150*x^8-200*x^7+100*x^6-40*x^5+90*x^4-64*x^
3+4*x^2)*exp(x)^2+(-8*x^11+38*x^10-70*x^9+60*x^8-36*x^7+58*x^6-74*x^5+36*x^4-4*x^3)*exp(x))*exp(3*x)+(x^2+2*x)
*exp(x)^6+(5*x^4+5*x^3-10*x^2)*exp(x)^5+(10*x^6-30*x^4+20*x^3-4*x^2+8*x)*exp(x)^4+(10*x^8-10*x^7-30*x^6+50*x^5
-24*x^4+12*x^3-16*x^2)*exp(x)^3+(5*x^10-10*x^9-10*x^8+40*x^7-31*x^6-6*x^5+4*x^4+8*x^3-12*x^2+8*x)*exp(x)^2+(x^
12-3*x^11+10*x^9-11*x^8-11*x^7+26*x^6-12*x^5+4*x^4-28*x^3+8*x^2)*exp(x))/(exp(x)^5+(5*x^2-5*x)*exp(x)^4+(10*x^
4-20*x^3+10*x^2)*exp(x)^3+(10*x^6-30*x^5+30*x^4-10*x^3)*exp(x)^2+(5*x^8-20*x^7+30*x^6-20*x^5+5*x^4)*exp(x)+x^1
0-5*x^9+10*x^8-10*x^7+5*x^6-x^5),x, algorithm="fricas")

[Out]

(4*(x^2 - x)*e^(10*x) + 6*(x^4 - 2*x^3 + x^2)*e^(9*x) + 2*(2*x^6 - 6*x^5 + 6*x^4 - 2*x^3 - x)*e^(8*x) + (x^8 -
4*x^7 + 6*x^6 - 4*x^5 + x^4 - 8*x^3 + 8*x^2)*e^(7*x) - 4*(3*x^5 - 6*x^4 + 3*x^3 + x)*e^(6*x) - (8*x^7 - 24*x^
6 + 24*x^5 - 8*x^4 + 8*x^3 - 9*x^2)*e^(5*x) - 2*(x^9 - 4*x^8 + 6*x^7 - 4*x^6 + 3*x^5 - 6*x^4 + 4*x^3)*e^(4*x)
+ 2*(3*x^6 - 6*x^5 + 3*x^4 + 2*x^2)*e^(3*x) + 4*(x^8 - 3*x^7 + 3*x^6 - x^5 + 2*x^4 - 2*x^3)*e^(2*x) + (x^10 -
4*x^9 + 6*x^8 - 4*x^7 + 5*x^6 - 8*x^5 + 4*x^4 + 4*x^2)*e^x + e^(11*x))/(x^8 - 4*x^7 + 6*x^6 - 4*x^5 + x^4 + 4*
(x^2 - x)*e^(3*x) + 6*(x^4 - 2*x^3 + x^2)*e^(2*x) + 4*(x^6 - 3*x^5 + 3*x^4 - x^3)*e^x + e^(4*x))

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giac [B]  time = 0.79, size = 548, normalized size = 18.27 \begin {gather*} \frac {x^{10} e^{x} - 2 \, x^{9} e^{\left (4 \, x\right )} - 4 \, x^{9} e^{x} + x^{8} e^{\left (7 \, x\right )} + 8 \, x^{8} e^{\left (4 \, x\right )} + 4 \, x^{8} e^{\left (2 \, x\right )} + 6 \, x^{8} e^{x} - 4 \, x^{7} e^{\left (7 \, x\right )} - 8 \, x^{7} e^{\left (5 \, x\right )} - 12 \, x^{7} e^{\left (4 \, x\right )} - 12 \, x^{7} e^{\left (2 \, x\right )} - 4 \, x^{7} e^{x} + 4 \, x^{6} e^{\left (8 \, x\right )} + 6 \, x^{6} e^{\left (7 \, x\right )} + 24 \, x^{6} e^{\left (5 \, x\right )} + 8 \, x^{6} e^{\left (4 \, x\right )} + 6 \, x^{6} e^{\left (3 \, x\right )} + 12 \, x^{6} e^{\left (2 \, x\right )} + 5 \, x^{6} e^{x} - 12 \, x^{5} e^{\left (8 \, x\right )} - 4 \, x^{5} e^{\left (7 \, x\right )} - 12 \, x^{5} e^{\left (6 \, x\right )} - 24 \, x^{5} e^{\left (5 \, x\right )} - 6 \, x^{5} e^{\left (4 \, x\right )} - 12 \, x^{5} e^{\left (3 \, x\right )} - 4 \, x^{5} e^{\left (2 \, x\right )} - 8 \, x^{5} e^{x} + 6 \, x^{4} e^{\left (9 \, x\right )} + 12 \, x^{4} e^{\left (8 \, x\right )} + x^{4} e^{\left (7 \, x\right )} + 24 \, x^{4} e^{\left (6 \, x\right )} + 8 \, x^{4} e^{\left (5 \, x\right )} + 12 \, x^{4} e^{\left (4 \, x\right )} + 6 \, x^{4} e^{\left (3 \, x\right )} + 8 \, x^{4} e^{\left (2 \, x\right )} + 4 \, x^{4} e^{x} - 12 \, x^{3} e^{\left (9 \, x\right )} - 4 \, x^{3} e^{\left (8 \, x\right )} - 8 \, x^{3} e^{\left (7 \, x\right )} - 12 \, x^{3} e^{\left (6 \, x\right )} - 8 \, x^{3} e^{\left (5 \, x\right )} - 8 \, x^{3} e^{\left (4 \, x\right )} - 8 \, x^{3} e^{\left (2 \, x\right )} + 4 \, x^{2} e^{\left (10 \, x\right )} + 6 \, x^{2} e^{\left (9 \, x\right )} + 8 \, x^{2} e^{\left (7 \, x\right )} + 9 \, x^{2} e^{\left (5 \, x\right )} + 4 \, x^{2} e^{\left (3 \, x\right )} + 4 \, x^{2} e^{x} - 4 \, x e^{\left (10 \, x\right )} - 2 \, x e^{\left (8 \, x\right )} - 4 \, x e^{\left (6 \, x\right )} + e^{\left (11 \, x\right )}}{x^{8} - 4 \, x^{7} + 4 \, x^{6} e^{x} + 6 \, x^{6} - 12 \, x^{5} e^{x} - 4 \, x^{5} + 6 \, x^{4} e^{\left (2 \, x\right )} + 12 \, x^{4} e^{x} + x^{4} - 12 \, x^{3} e^{\left (2 \, x\right )} - 4 \, x^{3} e^{x} + 4 \, x^{2} e^{\left (3 \, x\right )} + 6 \, x^{2} e^{\left (2 \, x\right )} - 4 \, x e^{\left (3 \, x\right )} + e^{\left (4 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((7*exp(x)^6+(35*x^2-35*x)*exp(x)^5+(70*x^4-140*x^3+70*x^2)*exp(x)^4+(70*x^6-210*x^5+210*x^4-70*x^3)
*exp(x)^3+(35*x^8-140*x^7+210*x^6-140*x^5+35*x^4)*exp(x)^2+(7*x^10-35*x^9+70*x^8-70*x^7+35*x^6-7*x^5)*exp(x))*
exp(3*x)^2+((-8*x-2)*exp(x)^6+(-40*x^3+30*x^2+10*x)*exp(x)^5+(-80*x^5+140*x^4-40*x^3-20*x^2-8*x-4)*exp(x)^4+(-
80*x^7+220*x^6-180*x^5+20*x^4-12*x^3+36*x^2+4*x)*exp(x)^3+(-40*x^9+150*x^8-200*x^7+100*x^6-40*x^5+90*x^4-64*x^
3+4*x^2)*exp(x)^2+(-8*x^11+38*x^10-70*x^9+60*x^8-36*x^7+58*x^6-74*x^5+36*x^4-4*x^3)*exp(x))*exp(3*x)+(x^2+2*x)
*exp(x)^6+(5*x^4+5*x^3-10*x^2)*exp(x)^5+(10*x^6-30*x^4+20*x^3-4*x^2+8*x)*exp(x)^4+(10*x^8-10*x^7-30*x^6+50*x^5
-24*x^4+12*x^3-16*x^2)*exp(x)^3+(5*x^10-10*x^9-10*x^8+40*x^7-31*x^6-6*x^5+4*x^4+8*x^3-12*x^2+8*x)*exp(x)^2+(x^
12-3*x^11+10*x^9-11*x^8-11*x^7+26*x^6-12*x^5+4*x^4-28*x^3+8*x^2)*exp(x))/(exp(x)^5+(5*x^2-5*x)*exp(x)^4+(10*x^
4-20*x^3+10*x^2)*exp(x)^3+(10*x^6-30*x^5+30*x^4-10*x^3)*exp(x)^2+(5*x^8-20*x^7+30*x^6-20*x^5+5*x^4)*exp(x)+x^1
0-5*x^9+10*x^8-10*x^7+5*x^6-x^5),x, algorithm="giac")

[Out]

(x^10*e^x - 2*x^9*e^(4*x) - 4*x^9*e^x + x^8*e^(7*x) + 8*x^8*e^(4*x) + 4*x^8*e^(2*x) + 6*x^8*e^x - 4*x^7*e^(7*x
) - 8*x^7*e^(5*x) - 12*x^7*e^(4*x) - 12*x^7*e^(2*x) - 4*x^7*e^x + 4*x^6*e^(8*x) + 6*x^6*e^(7*x) + 24*x^6*e^(5*
x) + 8*x^6*e^(4*x) + 6*x^6*e^(3*x) + 12*x^6*e^(2*x) + 5*x^6*e^x - 12*x^5*e^(8*x) - 4*x^5*e^(7*x) - 12*x^5*e^(6
*x) - 24*x^5*e^(5*x) - 6*x^5*e^(4*x) - 12*x^5*e^(3*x) - 4*x^5*e^(2*x) - 8*x^5*e^x + 6*x^4*e^(9*x) + 12*x^4*e^(
8*x) + x^4*e^(7*x) + 24*x^4*e^(6*x) + 8*x^4*e^(5*x) + 12*x^4*e^(4*x) + 6*x^4*e^(3*x) + 8*x^4*e^(2*x) + 4*x^4*e
^x - 12*x^3*e^(9*x) - 4*x^3*e^(8*x) - 8*x^3*e^(7*x) - 12*x^3*e^(6*x) - 8*x^3*e^(5*x) - 8*x^3*e^(4*x) - 8*x^3*e
^(2*x) + 4*x^2*e^(10*x) + 6*x^2*e^(9*x) + 8*x^2*e^(7*x) + 9*x^2*e^(5*x) + 4*x^2*e^(3*x) + 4*x^2*e^x - 4*x*e^(1
0*x) - 2*x*e^(8*x) - 4*x*e^(6*x) + e^(11*x))/(x^8 - 4*x^7 + 4*x^6*e^x + 6*x^6 - 12*x^5*e^x - 4*x^5 + 6*x^4*e^(
2*x) + 12*x^4*e^x + x^4 - 12*x^3*e^(2*x) - 4*x^3*e^x + 4*x^2*e^(3*x) + 6*x^2*e^(2*x) - 4*x*e^(3*x) + e^(4*x))

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maple [B]  time = 0.12, size = 259, normalized size = 8.63

method	result	size
risch	\({\mathrm e}^{7 x}-12 x^{5}-2 x \,{\mathrm e}^{4 x}+24 x^{4}-12 x^{3}-4 x \,{\mathrm e}^{2 x}+\left (8 x^{3}-7 x^{2}\right ) {\mathrm e}^{x}+\frac {4 \left (-50 x^{8} {\mathrm e}^{x}+12 x^{3} {\mathrm e}^{3 x}+100 x^{7} {\mathrm e}^{x}+10 x^{9} {\mathrm e}^{x}+50 x^{5} {\mathrm e}^{x}+3 x^{11}+{\mathrm e}^{3 x}+45 x^{7}-60 x^{8}-18 x^{10}+45 x^{9}-18 x^{6}+3 x^{5}+{\mathrm e}^{x}-4 x^{2} {\mathrm e}^{3 x}+11 \,{\mathrm e}^{2 x} x^{3}+66 x^{5} {\mathrm e}^{2 x}+2 \,{\mathrm e}^{2 x} x^{2}-2 x \,{\mathrm e}^{2 x}-100 x^{6} {\mathrm e}^{x}-9 \,{\mathrm e}^{x} x^{4}+{\mathrm e}^{x} x^{2}-2 \,{\mathrm e}^{x} x^{3}-44 x^{6} {\mathrm e}^{2 x}-44 \,{\mathrm e}^{2 x} x^{4}+11 \,{\mathrm e}^{2 x} x^{7}+4 \,{\mathrm e}^{3 x} x^{5}-12 \,{\mathrm e}^{3 x} x^{4}\right ) x^{2}}{\left (x^{2}+{\mathrm e}^{x}-x \right )^{4}}\)	\(259\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((7*exp(x)^6+(35*x^2-35*x)*exp(x)^5+(70*x^4-140*x^3+70*x^2)*exp(x)^4+(70*x^6-210*x^5+210*x^4-70*x^3)*exp(x
)^3+(35*x^8-140*x^7+210*x^6-140*x^5+35*x^4)*exp(x)^2+(7*x^10-35*x^9+70*x^8-70*x^7+35*x^6-7*x^5)*exp(x))*exp(3*
x)^2+((-8*x-2)*exp(x)^6+(-40*x^3+30*x^2+10*x)*exp(x)^5+(-80*x^5+140*x^4-40*x^3-20*x^2-8*x-4)*exp(x)^4+(-80*x^7
+220*x^6-180*x^5+20*x^4-12*x^3+36*x^2+4*x)*exp(x)^3+(-40*x^9+150*x^8-200*x^7+100*x^6-40*x^5+90*x^4-64*x^3+4*x^
2)*exp(x)^2+(-8*x^11+38*x^10-70*x^9+60*x^8-36*x^7+58*x^6-74*x^5+36*x^4-4*x^3)*exp(x))*exp(3*x)+(x^2+2*x)*exp(x
)^6+(5*x^4+5*x^3-10*x^2)*exp(x)^5+(10*x^6-30*x^4+20*x^3-4*x^2+8*x)*exp(x)^4+(10*x^8-10*x^7-30*x^6+50*x^5-24*x^
4+12*x^3-16*x^2)*exp(x)^3+(5*x^10-10*x^9-10*x^8+40*x^7-31*x^6-6*x^5+4*x^4+8*x^3-12*x^2+8*x)*exp(x)^2+(x^12-3*x
^11+10*x^9-11*x^8-11*x^7+26*x^6-12*x^5+4*x^4-28*x^3+8*x^2)*exp(x))/(exp(x)^5+(5*x^2-5*x)*exp(x)^4+(10*x^4-20*x
^3+10*x^2)*exp(x)^3+(10*x^6-30*x^5+30*x^4-10*x^3)*exp(x)^2+(5*x^8-20*x^7+30*x^6-20*x^5+5*x^4)*exp(x)+x^10-5*x^
9+10*x^8-10*x^7+5*x^6-x^5),x,method=_RETURNVERBOSE)

[Out]

exp(7*x)-12*x^5-2*x*exp(4*x)+24*x^4-12*x^3-4*x*exp(2*x)+(8*x^3-7*x^2)*exp(x)+4*(-50*x^8*exp(x)+12*x^3*exp(3*x)
+100*x^7*exp(x)+10*x^9*exp(x)+50*x^5*exp(x)+3*x^11+exp(3*x)+45*x^7-60*x^8-18*x^10+45*x^9-18*x^6+3*x^5+exp(x)-4
*x^2*exp(3*x)+11*exp(2*x)*x^3+66*x^5*exp(2*x)+2*exp(2*x)*x^2-2*x*exp(2*x)-100*x^6*exp(x)-9*exp(x)*x^4+exp(x)*x
^2-2*exp(x)*x^3-44*x^6*exp(2*x)-44*exp(2*x)*x^4+11*exp(2*x)*x^7+4*exp(3*x)*x^5-12*exp(3*x)*x^4)*x^2/(x^2+exp(x
)-x)^4

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maxima [B]  time = 0.85, size = 390, normalized size = 13.00 \begin {gather*} \frac {4 \, {\left (x^{2} - x\right )} e^{\left (10 \, x\right )} + 6 \, {\left (x^{4} - 2 \, x^{3} + x^{2}\right )} e^{\left (9 \, x\right )} + 2 \, {\left (2 \, x^{6} - 6 \, x^{5} + 6 \, x^{4} - 2 \, x^{3} - x\right )} e^{\left (8 \, x\right )} + {\left (x^{8} - 4 \, x^{7} + 6 \, x^{6} - 4 \, x^{5} + x^{4} - 8 \, x^{3} + 8 \, x^{2}\right )} e^{\left (7 \, x\right )} - 4 \, {\left (3 \, x^{5} - 6 \, x^{4} + 3 \, x^{3} + x\right )} e^{\left (6 \, x\right )} - {\left (8 \, x^{7} - 24 \, x^{6} + 24 \, x^{5} - 8 \, x^{4} + 8 \, x^{3} - 9 \, x^{2}\right )} e^{\left (5 \, x\right )} - 2 \, {\left (x^{9} - 4 \, x^{8} + 6 \, x^{7} - 4 \, x^{6} + 3 \, x^{5} - 6 \, x^{4} + 4 \, x^{3}\right )} e^{\left (4 \, x\right )} + 2 \, {\left (3 \, x^{6} - 6 \, x^{5} + 3 \, x^{4} + 2 \, x^{2}\right )} e^{\left (3 \, x\right )} + 4 \, {\left (x^{8} - 3 \, x^{7} + 3 \, x^{6} - x^{5} + 2 \, x^{4} - 2 \, x^{3}\right )} e^{\left (2 \, x\right )} + {\left (x^{10} - 4 \, x^{9} + 6 \, x^{8} - 4 \, x^{7} + 5 \, x^{6} - 8 \, x^{5} + 4 \, x^{4} + 4 \, x^{2}\right )} e^{x} + e^{\left (11 \, x\right )}}{x^{8} - 4 \, x^{7} + 6 \, x^{6} - 4 \, x^{5} + x^{4} + 4 \, {\left (x^{2} - x\right )} e^{\left (3 \, x\right )} + 6 \, {\left (x^{4} - 2 \, x^{3} + x^{2}\right )} e^{\left (2 \, x\right )} + 4 \, {\left (x^{6} - 3 \, x^{5} + 3 \, x^{4} - x^{3}\right )} e^{x} + e^{\left (4 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((7*exp(x)^6+(35*x^2-35*x)*exp(x)^5+(70*x^4-140*x^3+70*x^2)*exp(x)^4+(70*x^6-210*x^5+210*x^4-70*x^3)
*exp(x)^3+(35*x^8-140*x^7+210*x^6-140*x^5+35*x^4)*exp(x)^2+(7*x^10-35*x^9+70*x^8-70*x^7+35*x^6-7*x^5)*exp(x))*
exp(3*x)^2+((-8*x-2)*exp(x)^6+(-40*x^3+30*x^2+10*x)*exp(x)^5+(-80*x^5+140*x^4-40*x^3-20*x^2-8*x-4)*exp(x)^4+(-
80*x^7+220*x^6-180*x^5+20*x^4-12*x^3+36*x^2+4*x)*exp(x)^3+(-40*x^9+150*x^8-200*x^7+100*x^6-40*x^5+90*x^4-64*x^
3+4*x^2)*exp(x)^2+(-8*x^11+38*x^10-70*x^9+60*x^8-36*x^7+58*x^6-74*x^5+36*x^4-4*x^3)*exp(x))*exp(3*x)+(x^2+2*x)
*exp(x)^6+(5*x^4+5*x^3-10*x^2)*exp(x)^5+(10*x^6-30*x^4+20*x^3-4*x^2+8*x)*exp(x)^4+(10*x^8-10*x^7-30*x^6+50*x^5
-24*x^4+12*x^3-16*x^2)*exp(x)^3+(5*x^10-10*x^9-10*x^8+40*x^7-31*x^6-6*x^5+4*x^4+8*x^3-12*x^2+8*x)*exp(x)^2+(x^
12-3*x^11+10*x^9-11*x^8-11*x^7+26*x^6-12*x^5+4*x^4-28*x^3+8*x^2)*exp(x))/(exp(x)^5+(5*x^2-5*x)*exp(x)^4+(10*x^
4-20*x^3+10*x^2)*exp(x)^3+(10*x^6-30*x^5+30*x^4-10*x^3)*exp(x)^2+(5*x^8-20*x^7+30*x^6-20*x^5+5*x^4)*exp(x)+x^1
0-5*x^9+10*x^8-10*x^7+5*x^6-x^5),x, algorithm="maxima")

[Out]

(4*(x^2 - x)*e^(10*x) + 6*(x^4 - 2*x^3 + x^2)*e^(9*x) + 2*(2*x^6 - 6*x^5 + 6*x^4 - 2*x^3 - x)*e^(8*x) + (x^8 -
4*x^7 + 6*x^6 - 4*x^5 + x^4 - 8*x^3 + 8*x^2)*e^(7*x) - 4*(3*x^5 - 6*x^4 + 3*x^3 + x)*e^(6*x) - (8*x^7 - 24*x^
6 + 24*x^5 - 8*x^4 + 8*x^3 - 9*x^2)*e^(5*x) - 2*(x^9 - 4*x^8 + 6*x^7 - 4*x^6 + 3*x^5 - 6*x^4 + 4*x^3)*e^(4*x)
+ 2*(3*x^6 - 6*x^5 + 3*x^4 + 2*x^2)*e^(3*x) + 4*(x^8 - 3*x^7 + 3*x^6 - x^5 + 2*x^4 - 2*x^3)*e^(2*x) + (x^10 -
4*x^9 + 6*x^8 - 4*x^7 + 5*x^6 - 8*x^5 + 4*x^4 + 4*x^2)*e^x + e^(11*x))/(x^8 - 4*x^7 + 6*x^6 - 4*x^5 + x^4 + 4*
(x^2 - x)*e^(3*x) + 6*(x^4 - 2*x^3 + x^2)*e^(2*x) + 4*(x^6 - 3*x^5 + 3*x^4 - x^3)*e^x + e^(4*x))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {{\mathrm {e}}^{6\,x}\,\left (7\,{\mathrm {e}}^{6\,x}-{\mathrm {e}}^{5\,x}\,\left (35\,x-35\,x^2\right )+{\mathrm {e}}^{2\,x}\,\left (35\,x^8-140\,x^7+210\,x^6-140\,x^5+35\,x^4\right )+{\mathrm {e}}^{4\,x}\,\left (70\,x^4-140\,x^3+70\,x^2\right )-{\mathrm {e}}^{3\,x}\,\left (-70\,x^6+210\,x^5-210\,x^4+70\,x^3\right )-{\mathrm {e}}^x\,\left (-7\,x^{10}+35\,x^9-70\,x^8+70\,x^7-35\,x^6+7\,x^5\right )\right )+{\mathrm {e}}^{3\,x}\,\left ({\mathrm {e}}^{3\,x}\,\left (-80\,x^7+220\,x^6-180\,x^5+20\,x^4-12\,x^3+36\,x^2+4\,x\right )+{\mathrm {e}}^{5\,x}\,\left (-40\,x^3+30\,x^2+10\,x\right )-{\mathrm {e}}^x\,\left (8\,x^{11}-38\,x^{10}+70\,x^9-60\,x^8+36\,x^7-58\,x^6+74\,x^5-36\,x^4+4\,x^3\right )-{\mathrm {e}}^{4\,x}\,\left (80\,x^5-140\,x^4+40\,x^3+20\,x^2+8\,x+4\right )+{\mathrm {e}}^{2\,x}\,\left (-40\,x^9+150\,x^8-200\,x^7+100\,x^6-40\,x^5+90\,x^4-64\,x^3+4\,x^2\right )-{\mathrm {e}}^{6\,x}\,\left (8\,x+2\right )\right )+{\mathrm {e}}^x\,\left (x^{12}-3\,x^{11}+10\,x^9-11\,x^8-11\,x^7+26\,x^6-12\,x^5+4\,x^4-28\,x^3+8\,x^2\right )-{\mathrm {e}}^{3\,x}\,\left (-10\,x^8+10\,x^7+30\,x^6-50\,x^5+24\,x^4-12\,x^3+16\,x^2\right )+{\mathrm {e}}^{5\,x}\,\left (5\,x^4+5\,x^3-10\,x^2\right )+{\mathrm {e}}^{6\,x}\,\left (x^2+2\,x\right )+{\mathrm {e}}^{4\,x}\,\left (10\,x^6-30\,x^4+20\,x^3-4\,x^2+8\,x\right )+{\mathrm {e}}^{2\,x}\,\left (5\,x^{10}-10\,x^9-10\,x^8+40\,x^7-31\,x^6-6\,x^5+4\,x^4+8\,x^3-12\,x^2+8\,x\right )}{{\mathrm {e}}^{5\,x}-{\mathrm {e}}^{4\,x}\,\left (5\,x-5\,x^2\right )+{\mathrm {e}}^{3\,x}\,\left (10\,x^4-20\,x^3+10\,x^2\right )+{\mathrm {e}}^x\,\left (5\,x^8-20\,x^7+30\,x^6-20\,x^5+5\,x^4\right )-{\mathrm {e}}^{2\,x}\,\left (-10\,x^6+30\,x^5-30\,x^4+10\,x^3\right )-x^5+5\,x^6-10\,x^7+10\,x^8-5\,x^9+x^{10}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(6*x)*(7*exp(6*x) - exp(5*x)*(35*x - 35*x^2) + exp(2*x)*(35*x^4 - 140*x^5 + 210*x^6 - 140*x^7 + 35*x^8
) + exp(4*x)*(70*x^2 - 140*x^3 + 70*x^4) - exp(3*x)*(70*x^3 - 210*x^4 + 210*x^5 - 70*x^6) - exp(x)*(7*x^5 - 35
*x^6 + 70*x^7 - 70*x^8 + 35*x^9 - 7*x^10)) + exp(3*x)*(exp(3*x)*(4*x + 36*x^2 - 12*x^3 + 20*x^4 - 180*x^5 + 22
0*x^6 - 80*x^7) + exp(5*x)*(10*x + 30*x^2 - 40*x^3) - exp(x)*(4*x^3 - 36*x^4 + 74*x^5 - 58*x^6 + 36*x^7 - 60*x
^8 + 70*x^9 - 38*x^10 + 8*x^11) - exp(4*x)*(8*x + 20*x^2 + 40*x^3 - 140*x^4 + 80*x^5 + 4) + exp(2*x)*(4*x^2 -
64*x^3 + 90*x^4 - 40*x^5 + 100*x^6 - 200*x^7 + 150*x^8 - 40*x^9) - exp(6*x)*(8*x + 2)) + exp(x)*(8*x^2 - 28*x^
3 + 4*x^4 - 12*x^5 + 26*x^6 - 11*x^7 - 11*x^8 + 10*x^9 - 3*x^11 + x^12) - exp(3*x)*(16*x^2 - 12*x^3 + 24*x^4 -
50*x^5 + 30*x^6 + 10*x^7 - 10*x^8) + exp(5*x)*(5*x^3 - 10*x^2 + 5*x^4) + exp(6*x)*(2*x + x^2) + exp(4*x)*(8*x
- 4*x^2 + 20*x^3 - 30*x^4 + 10*x^6) + exp(2*x)*(8*x - 12*x^2 + 8*x^3 + 4*x^4 - 6*x^5 - 31*x^6 + 40*x^7 - 10*x
^8 - 10*x^9 + 5*x^10))/(exp(5*x) - exp(4*x)*(5*x - 5*x^2) + exp(3*x)*(10*x^2 - 20*x^3 + 10*x^4) + exp(x)*(5*x^
4 - 20*x^5 + 30*x^6 - 20*x^7 + 5*x^8) - exp(2*x)*(10*x^3 - 30*x^4 + 30*x^5 - 10*x^6) - x^5 + 5*x^6 - 10*x^7 +
10*x^8 - 5*x^9 + x^10),x)

[Out]

int((exp(6*x)*(7*exp(6*x) - exp(5*x)*(35*x - 35*x^2) + exp(2*x)*(35*x^4 - 140*x^5 + 210*x^6 - 140*x^7 + 35*x^8
) + exp(4*x)*(70*x^2 - 140*x^3 + 70*x^4) - exp(3*x)*(70*x^3 - 210*x^4 + 210*x^5 - 70*x^6) - exp(x)*(7*x^5 - 35
*x^6 + 70*x^7 - 70*x^8 + 35*x^9 - 7*x^10)) + exp(3*x)*(exp(3*x)*(4*x + 36*x^2 - 12*x^3 + 20*x^4 - 180*x^5 + 22
0*x^6 - 80*x^7) + exp(5*x)*(10*x + 30*x^2 - 40*x^3) - exp(x)*(4*x^3 - 36*x^4 + 74*x^5 - 58*x^6 + 36*x^7 - 60*x
^8 + 70*x^9 - 38*x^10 + 8*x^11) - exp(4*x)*(8*x + 20*x^2 + 40*x^3 - 140*x^4 + 80*x^5 + 4) + exp(2*x)*(4*x^2 -
64*x^3 + 90*x^4 - 40*x^5 + 100*x^6 - 200*x^7 + 150*x^8 - 40*x^9) - exp(6*x)*(8*x + 2)) + exp(x)*(8*x^2 - 28*x^
3 + 4*x^4 - 12*x^5 + 26*x^6 - 11*x^7 - 11*x^8 + 10*x^9 - 3*x^11 + x^12) - exp(3*x)*(16*x^2 - 12*x^3 + 24*x^4 -
50*x^5 + 30*x^6 + 10*x^7 - 10*x^8) + exp(5*x)*(5*x^3 - 10*x^2 + 5*x^4) + exp(6*x)*(2*x + x^2) + exp(4*x)*(8*x
- 4*x^2 + 20*x^3 - 30*x^4 + 10*x^6) + exp(2*x)*(8*x - 12*x^2 + 8*x^3 + 4*x^4 - 6*x^5 - 31*x^6 + 40*x^7 - 10*x
^8 - 10*x^9 + 5*x^10))/(exp(5*x) - exp(4*x)*(5*x - 5*x^2) + exp(3*x)*(10*x^2 - 20*x^3 + 10*x^4) + exp(x)*(5*x^
4 - 20*x^5 + 30*x^6 - 20*x^7 + 5*x^8) - exp(2*x)*(10*x^3 - 30*x^4 + 30*x^5 - 10*x^6) - x^5 + 5*x^6 - 10*x^7 +
10*x^8 - 5*x^9 + x^10), x)

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sympy [B]  time = 0.95, size = 291, normalized size = 9.70 \begin {gather*} - 12 x^{5} + 24 x^{4} - 12 x^{3} - 2 x e^{4 x} - 4 x e^{2 x} + \left (8 x^{3} - 7 x^{2}\right ) e^{x} + e^{7 x} + \frac {12 x^{13} - 72 x^{12} + 180 x^{11} - 240 x^{10} + 180 x^{9} - 72 x^{8} + 12 x^{7} + \left (16 x^{7} - 48 x^{6} + 48 x^{5} - 16 x^{4} + 4 x^{2}\right ) e^{3 x} + \left (44 x^{9} - 176 x^{8} + 264 x^{7} - 176 x^{6} + 44 x^{5} + 8 x^{4} - 8 x^{3}\right ) e^{2 x} + \left (40 x^{11} - 200 x^{10} + 400 x^{9} - 400 x^{8} + 200 x^{7} - 36 x^{6} - 8 x^{5} + 4 x^{4} + 4 x^{2}\right ) e^{x}}{x^{8} - 4 x^{7} + 6 x^{6} - 4 x^{5} + x^{4} + \left (4 x^{2} - 4 x\right ) e^{3 x} + \left (6 x^{4} - 12 x^{3} + 6 x^{2}\right ) e^{2 x} + \left (4 x^{6} - 12 x^{5} + 12 x^{4} - 4 x^{3}\right ) e^{x} + e^{4 x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((7*exp(x)**6+(35*x**2-35*x)*exp(x)**5+(70*x**4-140*x**3+70*x**2)*exp(x)**4+(70*x**6-210*x**5+210*x*
*4-70*x**3)*exp(x)**3+(35*x**8-140*x**7+210*x**6-140*x**5+35*x**4)*exp(x)**2+(7*x**10-35*x**9+70*x**8-70*x**7+
35*x**6-7*x**5)*exp(x))*exp(3*x)**2+((-8*x-2)*exp(x)**6+(-40*x**3+30*x**2+10*x)*exp(x)**5+(-80*x**5+140*x**4-4
0*x**3-20*x**2-8*x-4)*exp(x)**4+(-80*x**7+220*x**6-180*x**5+20*x**4-12*x**3+36*x**2+4*x)*exp(x)**3+(-40*x**9+1
50*x**8-200*x**7+100*x**6-40*x**5+90*x**4-64*x**3+4*x**2)*exp(x)**2+(-8*x**11+38*x**10-70*x**9+60*x**8-36*x**7
+58*x**6-74*x**5+36*x**4-4*x**3)*exp(x))*exp(3*x)+(x**2+2*x)*exp(x)**6+(5*x**4+5*x**3-10*x**2)*exp(x)**5+(10*x
**6-30*x**4+20*x**3-4*x**2+8*x)*exp(x)**4+(10*x**8-10*x**7-30*x**6+50*x**5-24*x**4+12*x**3-16*x**2)*exp(x)**3+
(5*x**10-10*x**9-10*x**8+40*x**7-31*x**6-6*x**5+4*x**4+8*x**3-12*x**2+8*x)*exp(x)**2+(x**12-3*x**11+10*x**9-11
*x**8-11*x**7+26*x**6-12*x**5+4*x**4-28*x**3+8*x**2)*exp(x))/(exp(x)**5+(5*x**2-5*x)*exp(x)**4+(10*x**4-20*x**
3+10*x**2)*exp(x)**3+(10*x**6-30*x**5+30*x**4-10*x**3)*exp(x)**2+(5*x**8-20*x**7+30*x**6-20*x**5+5*x**4)*exp(x
)+x**10-5*x**9+10*x**8-10*x**7+5*x**6-x**5),x)

[Out]

-12*x**5 + 24*x**4 - 12*x**3 - 2*x*exp(4*x) - 4*x*exp(2*x) + (8*x**3 - 7*x**2)*exp(x) + exp(7*x) + (12*x**13 -
72*x**12 + 180*x**11 - 240*x**10 + 180*x**9 - 72*x**8 + 12*x**7 + (16*x**7 - 48*x**6 + 48*x**5 - 16*x**4 + 4*
x**2)*exp(3*x) + (44*x**9 - 176*x**8 + 264*x**7 - 176*x**6 + 44*x**5 + 8*x**4 - 8*x**3)*exp(2*x) + (40*x**11 -
200*x**10 + 400*x**9 - 400*x**8 + 200*x**7 - 36*x**6 - 8*x**5 + 4*x**4 + 4*x**2)*exp(x))/(x**8 - 4*x**7 + 6*x
**6 - 4*x**5 + x**4 + (4*x**2 - 4*x)*exp(3*x) + (6*x**4 - 12*x**3 + 6*x**2)*exp(2*x) + (4*x**6 - 12*x**5 + 12*
x**4 - 4*x**3)*exp(x) + exp(4*x))

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