3.60.82 \(\int \frac {4 e^{8 x} x+32 e^{7 x} x^2+16 x^5-32 x^6+32 x^7-16 x^8+4 x^9+e^{6 x} (-16 x^2+112 x^3)+e^{5 x} (-10 x+15 x^2-96 x^3+224 x^4)+e^{4 x} (-20 x^2+72 x^3-240 x^4+280 x^5)+e^{3 x} (10 x^2-10 x^3+158 x^4-320 x^5+224 x^6)+e^{2 x} (-12 x^4+192 x^5-240 x^6+112 x^7)+e^x (-20 x^4-44 x^5+123 x^6-96 x^7+32 x^8)}{2 e^{8 x}+16 e^{7 x} x+8 x^4-16 x^5+16 x^6-8 x^7+2 x^8+e^{6 x} (-8 x+56 x^2)+e^{5 x} (-48 x^2+112 x^3)+e^{4 x} (16 x^2-120 x^3+140 x^4)+e^{3 x} (64 x^3-160 x^4+112 x^5)+e^{2 x} (-16 x^3+96 x^4-120 x^5+56 x^6)+e^x (-32 x^4+64 x^5-48 x^6+16 x^7)} \, dx\)

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

________________________________________________________________________________________

Rubi [F]  time = 15.39, 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 {4 e^{8 x} x+32 e^{7 x} x^2+16 x^5-32 x^6+32 x^7-16 x^8+4 x^9+e^{6 x} \left (-16 x^2+112 x^3\right )+e^{5 x} \left (-10 x+15 x^2-96 x^3+224 x^4\right )+e^{4 x} \left (-20 x^2+72 x^3-240 x^4+280 x^5\right )+e^{3 x} \left (10 x^2-10 x^3+158 x^4-320 x^5+224 x^6\right )+e^{2 x} \left (-12 x^4+192 x^5-240 x^6+112 x^7\right )+e^x \left (-20 x^4-44 x^5+123 x^6-96 x^7+32 x^8\right )}{2 e^{8 x}+16 e^{7 x} x+8 x^4-16 x^5+16 x^6-8 x^7+2 x^8+e^{6 x} \left (-8 x+56 x^2\right )+e^{5 x} \left (-48 x^2+112 x^3\right )+e^{4 x} \left (16 x^2-120 x^3+140 x^4\right )+e^{3 x} \left (64 x^3-160 x^4+112 x^5\right )+e^{2 x} \left (-16 x^3+96 x^4-120 x^5+56 x^6\right )+e^x \left (-32 x^4+64 x^5-48 x^6+16 x^7\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(4*E^(8*x)*x + 32*E^(7*x)*x^2 + 16*x^5 - 32*x^6 + 32*x^7 - 16*x^8 + 4*x^9 + E^(6*x)*(-16*x^2 + 112*x^3) +
E^(5*x)*(-10*x + 15*x^2 - 96*x^3 + 224*x^4) + E^(4*x)*(-20*x^2 + 72*x^3 - 240*x^4 + 280*x^5) + E^(3*x)*(10*x^2
- 10*x^3 + 158*x^4 - 320*x^5 + 224*x^6) + E^(2*x)*(-12*x^4 + 192*x^5 - 240*x^6 + 112*x^7) + E^x*(-20*x^4 - 44
*x^5 + 123*x^6 - 96*x^7 + 32*x^8))/(2*E^(8*x) + 16*E^(7*x)*x + 8*x^4 - 16*x^5 + 16*x^6 - 8*x^7 + 2*x^8 + E^(6*
x)*(-8*x + 56*x^2) + E^(5*x)*(-48*x^2 + 112*x^3) + E^(4*x)*(16*x^2 - 120*x^3 + 140*x^4) + E^(3*x)*(64*x^3 - 16
0*x^4 + 112*x^5) + E^(2*x)*(-16*x^3 + 96*x^4 - 120*x^5 + 56*x^6) + E^x*(-32*x^4 + 64*x^5 - 48*x^6 + 16*x^7)),x
]

[Out]

x^2 - 5*Defer[Int][(E^(3*x)*x^2)/(E^(4*x) - 2*E^(2*x)*x + 4*E^(3*x)*x + 2*x^2 - 4*E^x*x^2 + 6*E^(2*x)*x^2 - 2*
x^3 + 4*E^x*x^3 + x^4)^2, x] + 10*Defer[Int][(E^x*x^3)/(E^(4*x) - 2*E^(2*x)*x + 4*E^(3*x)*x + 2*x^2 - 4*E^x*x^
2 + 6*E^(2*x)*x^2 - 2*x^3 + 4*E^x*x^3 + x^4)^2, x] - 20*Defer[Int][x^4/(E^(4*x) - 2*E^(2*x)*x + 4*E^(3*x)*x +
2*x^2 - 4*E^x*x^2 + 6*E^(2*x)*x^2 - 2*x^3 + 4*E^x*x^3 + x^4)^2, x] + 5*Defer[Int][(E^x*x^4)/(E^(4*x) - 2*E^(2*
x)*x + 4*E^(3*x)*x + 2*x^2 - 4*E^x*x^2 + 6*E^(2*x)*x^2 - 2*x^3 + 4*E^x*x^3 + x^4)^2, x] - 20*Defer[Int][(E^(2*
x)*x^4)/(E^(4*x) - 2*E^(2*x)*x + 4*E^(3*x)*x + 2*x^2 - 4*E^x*x^2 + 6*E^(2*x)*x^2 - 2*x^3 + 4*E^x*x^3 + x^4)^2,
x] + 10*Defer[Int][(E^(3*x)*x^4)/(E^(4*x) - 2*E^(2*x)*x + 4*E^(3*x)*x + 2*x^2 - 4*E^x*x^2 + 6*E^(2*x)*x^2 - 2
*x^3 + 4*E^x*x^3 + x^4)^2, x] + 40*Defer[Int][x^5/(E^(4*x) - 2*E^(2*x)*x + 4*E^(3*x)*x + 2*x^2 - 4*E^x*x^2 + 6
*E^(2*x)*x^2 - 2*x^3 + 4*E^x*x^3 + x^4)^2, x] - 50*Defer[Int][(E^x*x^5)/(E^(4*x) - 2*E^(2*x)*x + 4*E^(3*x)*x +
2*x^2 - 4*E^x*x^2 + 6*E^(2*x)*x^2 - 2*x^3 + 4*E^x*x^3 + x^4)^2, x] + 30*Defer[Int][(E^(2*x)*x^5)/(E^(4*x) - 2
*E^(2*x)*x + 4*E^(3*x)*x + 2*x^2 - 4*E^x*x^2 + 6*E^(2*x)*x^2 - 2*x^3 + 4*E^x*x^3 + x^4)^2, x] - 30*Defer[Int][
x^6/(E^(4*x) - 2*E^(2*x)*x + 4*E^(3*x)*x + 2*x^2 - 4*E^x*x^2 + 6*E^(2*x)*x^2 - 2*x^3 + 4*E^x*x^3 + x^4)^2, x]
+ 30*Defer[Int][(E^x*x^6)/(E^(4*x) - 2*E^(2*x)*x + 4*E^(3*x)*x + 2*x^2 - 4*E^x*x^2 + 6*E^(2*x)*x^2 - 2*x^3 + 4
*E^x*x^3 + x^4)^2, x] + 10*Defer[Int][x^7/(E^(4*x) - 2*E^(2*x)*x + 4*E^(3*x)*x + 2*x^2 - 4*E^x*x^2 + 6*E^(2*x)
*x^2 - 2*x^3 + 4*E^x*x^3 + x^4)^2, x] - 5*Defer[Int][(E^x*x)/(E^(4*x) - 2*E^(2*x)*x + 4*E^(3*x)*x + 2*x^2 - 4*
E^x*x^2 + 6*E^(2*x)*x^2 - 2*x^3 + 4*E^x*x^3 + x^4), x] + 10*Defer[Int][x^2/(E^(4*x) - 2*E^(2*x)*x + 4*E^(3*x)*
x + 2*x^2 - 4*E^x*x^2 + 6*E^(2*x)*x^2 - 2*x^3 + 4*E^x*x^3 + x^4), x] + (15*Defer[Int][(E^x*x^2)/(E^(4*x) - 2*E
^(2*x)*x + 4*E^(3*x)*x + 2*x^2 - 4*E^x*x^2 + 6*E^(2*x)*x^2 - 2*x^3 + 4*E^x*x^3 + x^4), x])/2 - 10*Defer[Int][x
^3/(E^(4*x) - 2*E^(2*x)*x + 4*E^(3*x)*x + 2*x^2 - 4*E^x*x^2 + 6*E^(2*x)*x^2 - 2*x^3 + 4*E^x*x^3 + x^4), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x \left (4 e^{8 x}+32 e^{7 x} x+16 e^{6 x} x (-1+7 x)+4 x^4 \left (2-2 x+x^2\right )^2+4 e^{2 x} x^3 \left (-3+48 x-60 x^2+28 x^3\right )+4 e^{4 x} x \left (-5+18 x-60 x^2+70 x^3\right )+e^{5 x} \left (-10+15 x-96 x^2+224 x^3\right )+e^x x^3 \left (-20-44 x+123 x^2-96 x^3+32 x^4\right )+2 e^{3 x} x \left (5-5 x+79 x^2-160 x^3+112 x^4\right )\right )}{2 \left (e^{4 x}+4 e^{3 x} x+4 e^x (-1+x) x^2+2 e^{2 x} x (-1+3 x)+x^2 \left (2-2 x+x^2\right )\right )^2} \, dx\\ &=\frac {1}{2} \int \frac {x \left (4 e^{8 x}+32 e^{7 x} x+16 e^{6 x} x (-1+7 x)+4 x^4 \left (2-2 x+x^2\right )^2+4 e^{2 x} x^3 \left (-3+48 x-60 x^2+28 x^3\right )+4 e^{4 x} x \left (-5+18 x-60 x^2+70 x^3\right )+e^{5 x} \left (-10+15 x-96 x^2+224 x^3\right )+e^x x^3 \left (-20-44 x+123 x^2-96 x^3+32 x^4\right )+2 e^{3 x} x \left (5-5 x+79 x^2-160 x^3+112 x^4\right )\right )}{\left (e^{4 x}+4 e^{3 x} x+4 e^x (-1+x) x^2+2 e^{2 x} x (-1+3 x)+x^2 \left (2-2 x+x^2\right )\right )^2} \, dx\\ &=\frac {1}{2} \int \left (4 x-\frac {5 x \left (2 e^x-4 x-3 e^x x+4 x^2\right )}{e^{4 x}-2 e^{2 x} x+4 e^{3 x} x+2 x^2-4 e^x x^2+6 e^{2 x} x^2-2 x^3+4 e^x x^3+x^4}+\frac {10 x^2 \left (-e^{3 x}+2 e^x x-4 x^2+e^x x^2-4 e^{2 x} x^2+2 e^{3 x} x^2+8 x^3-10 e^x x^3+6 e^{2 x} x^3-6 x^4+6 e^x x^4+2 x^5\right )}{\left (e^{4 x}-2 e^{2 x} x+4 e^{3 x} x+2 x^2-4 e^x x^2+6 e^{2 x} x^2-2 x^3+4 e^x x^3+x^4\right )^2}\right ) \, dx\\ &=x^2-\frac {5}{2} \int \frac {x \left (2 e^x-4 x-3 e^x x+4 x^2\right )}{e^{4 x}-2 e^{2 x} x+4 e^{3 x} x+2 x^2-4 e^x x^2+6 e^{2 x} x^2-2 x^3+4 e^x x^3+x^4} \, dx+5 \int \frac {x^2 \left (-e^{3 x}+2 e^x x-4 x^2+e^x x^2-4 e^{2 x} x^2+2 e^{3 x} x^2+8 x^3-10 e^x x^3+6 e^{2 x} x^3-6 x^4+6 e^x x^4+2 x^5\right )}{\left (e^{4 x}-2 e^{2 x} x+4 e^{3 x} x+2 x^2-4 e^x x^2+6 e^{2 x} x^2-2 x^3+4 e^x x^3+x^4\right )^2} \, dx\\ &=x^2-\frac {5}{2} \int \left (\frac {2 e^x x}{e^{4 x}-2 e^{2 x} x+4 e^{3 x} x+2 x^2-4 e^x x^2+6 e^{2 x} x^2-2 x^3+4 e^x x^3+x^4}-\frac {4 x^2}{e^{4 x}-2 e^{2 x} x+4 e^{3 x} x+2 x^2-4 e^x x^2+6 e^{2 x} x^2-2 x^3+4 e^x x^3+x^4}-\frac {3 e^x x^2}{e^{4 x}-2 e^{2 x} x+4 e^{3 x} x+2 x^2-4 e^x x^2+6 e^{2 x} x^2-2 x^3+4 e^x x^3+x^4}+\frac {4 x^3}{e^{4 x}-2 e^{2 x} x+4 e^{3 x} x+2 x^2-4 e^x x^2+6 e^{2 x} x^2-2 x^3+4 e^x x^3+x^4}\right ) \, dx+5 \int \left (-\frac {e^{3 x} x^2}{\left (e^{4 x}-2 e^{2 x} x+4 e^{3 x} x+2 x^2-4 e^x x^2+6 e^{2 x} x^2-2 x^3+4 e^x x^3+x^4\right )^2}+\frac {2 e^x x^3}{\left (e^{4 x}-2 e^{2 x} x+4 e^{3 x} x+2 x^2-4 e^x x^2+6 e^{2 x} x^2-2 x^3+4 e^x x^3+x^4\right )^2}-\frac {4 x^4}{\left (e^{4 x}-2 e^{2 x} x+4 e^{3 x} x+2 x^2-4 e^x x^2+6 e^{2 x} x^2-2 x^3+4 e^x x^3+x^4\right )^2}+\frac {e^x x^4}{\left (e^{4 x}-2 e^{2 x} x+4 e^{3 x} x+2 x^2-4 e^x x^2+6 e^{2 x} x^2-2 x^3+4 e^x x^3+x^4\right )^2}-\frac {4 e^{2 x} x^4}{\left (e^{4 x}-2 e^{2 x} x+4 e^{3 x} x+2 x^2-4 e^x x^2+6 e^{2 x} x^2-2 x^3+4 e^x x^3+x^4\right )^2}+\frac {2 e^{3 x} x^4}{\left (e^{4 x}-2 e^{2 x} x+4 e^{3 x} x+2 x^2-4 e^x x^2+6 e^{2 x} x^2-2 x^3+4 e^x x^3+x^4\right )^2}+\frac {8 x^5}{\left (e^{4 x}-2 e^{2 x} x+4 e^{3 x} x+2 x^2-4 e^x x^2+6 e^{2 x} x^2-2 x^3+4 e^x x^3+x^4\right )^2}-\frac {10 e^x x^5}{\left (e^{4 x}-2 e^{2 x} x+4 e^{3 x} x+2 x^2-4 e^x x^2+6 e^{2 x} x^2-2 x^3+4 e^x x^3+x^4\right )^2}+\frac {6 e^{2 x} x^5}{\left (e^{4 x}-2 e^{2 x} x+4 e^{3 x} x+2 x^2-4 e^x x^2+6 e^{2 x} x^2-2 x^3+4 e^x x^3+x^4\right )^2}-\frac {6 x^6}{\left (e^{4 x}-2 e^{2 x} x+4 e^{3 x} x+2 x^2-4 e^x x^2+6 e^{2 x} x^2-2 x^3+4 e^x x^3+x^4\right )^2}+\frac {6 e^x x^6}{\left (e^{4 x}-2 e^{2 x} x+4 e^{3 x} x+2 x^2-4 e^x x^2+6 e^{2 x} x^2-2 x^3+4 e^x x^3+x^4\right )^2}+\frac {2 x^7}{\left (e^{4 x}-2 e^{2 x} x+4 e^{3 x} x+2 x^2-4 e^x x^2+6 e^{2 x} x^2-2 x^3+4 e^x x^3+x^4\right )^2}\right ) \, dx\\ &=x^2-5 \int \frac {e^{3 x} x^2}{\left (e^{4 x}-2 e^{2 x} x+4 e^{3 x} x+2 x^2-4 e^x x^2+6 e^{2 x} x^2-2 x^3+4 e^x x^3+x^4\right )^2} \, dx+5 \int \frac {e^x x^4}{\left (e^{4 x}-2 e^{2 x} x+4 e^{3 x} x+2 x^2-4 e^x x^2+6 e^{2 x} x^2-2 x^3+4 e^x x^3+x^4\right )^2} \, dx-5 \int \frac {e^x x}{e^{4 x}-2 e^{2 x} x+4 e^{3 x} x+2 x^2-4 e^x x^2+6 e^{2 x} x^2-2 x^3+4 e^x x^3+x^4} \, dx+\frac {15}{2} \int \frac {e^x x^2}{e^{4 x}-2 e^{2 x} x+4 e^{3 x} x+2 x^2-4 e^x x^2+6 e^{2 x} x^2-2 x^3+4 e^x x^3+x^4} \, dx+10 \int \frac {e^x x^3}{\left (e^{4 x}-2 e^{2 x} x+4 e^{3 x} x+2 x^2-4 e^x x^2+6 e^{2 x} x^2-2 x^3+4 e^x x^3+x^4\right )^2} \, dx+10 \int \frac {e^{3 x} x^4}{\left (e^{4 x}-2 e^{2 x} x+4 e^{3 x} x+2 x^2-4 e^x x^2+6 e^{2 x} x^2-2 x^3+4 e^x x^3+x^4\right )^2} \, dx+10 \int \frac {x^7}{\left (e^{4 x}-2 e^{2 x} x+4 e^{3 x} x+2 x^2-4 e^x x^2+6 e^{2 x} x^2-2 x^3+4 e^x x^3+x^4\right )^2} \, dx+10 \int \frac {x^2}{e^{4 x}-2 e^{2 x} x+4 e^{3 x} x+2 x^2-4 e^x x^2+6 e^{2 x} x^2-2 x^3+4 e^x x^3+x^4} \, dx-10 \int \frac {x^3}{e^{4 x}-2 e^{2 x} x+4 e^{3 x} x+2 x^2-4 e^x x^2+6 e^{2 x} x^2-2 x^3+4 e^x x^3+x^4} \, dx-20 \int \frac {x^4}{\left (e^{4 x}-2 e^{2 x} x+4 e^{3 x} x+2 x^2-4 e^x x^2+6 e^{2 x} x^2-2 x^3+4 e^x x^3+x^4\right )^2} \, dx-20 \int \frac {e^{2 x} x^4}{\left (e^{4 x}-2 e^{2 x} x+4 e^{3 x} x+2 x^2-4 e^x x^2+6 e^{2 x} x^2-2 x^3+4 e^x x^3+x^4\right )^2} \, dx+30 \int \frac {e^{2 x} x^5}{\left (e^{4 x}-2 e^{2 x} x+4 e^{3 x} x+2 x^2-4 e^x x^2+6 e^{2 x} x^2-2 x^3+4 e^x x^3+x^4\right )^2} \, dx-30 \int \frac {x^6}{\left (e^{4 x}-2 e^{2 x} x+4 e^{3 x} x+2 x^2-4 e^x x^2+6 e^{2 x} x^2-2 x^3+4 e^x x^3+x^4\right )^2} \, dx+30 \int \frac {e^x x^6}{\left (e^{4 x}-2 e^{2 x} x+4 e^{3 x} x+2 x^2-4 e^x x^2+6 e^{2 x} x^2-2 x^3+4 e^x x^3+x^4\right )^2} \, dx+40 \int \frac {x^5}{\left (e^{4 x}-2 e^{2 x} x+4 e^{3 x} x+2 x^2-4 e^x x^2+6 e^{2 x} x^2-2 x^3+4 e^x x^3+x^4\right )^2} \, dx-50 \int \frac {e^x x^5}{\left (e^{4 x}-2 e^{2 x} x+4 e^{3 x} x+2 x^2-4 e^x x^2+6 e^{2 x} x^2-2 x^3+4 e^x x^3+x^4\right )^2} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [B]  time = 0.21, size = 66, normalized size = 2.13 \begin {gather*} \frac {1}{2} x^2 \left (2-\frac {5 e^x}{e^{4 x}+4 e^{3 x} x+4 e^x (-1+x) x^2+2 e^{2 x} x (-1+3 x)+x^2 \left (2-2 x+x^2\right )}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4*E^(8*x)*x + 32*E^(7*x)*x^2 + 16*x^5 - 32*x^6 + 32*x^7 - 16*x^8 + 4*x^9 + E^(6*x)*(-16*x^2 + 112*x
^3) + E^(5*x)*(-10*x + 15*x^2 - 96*x^3 + 224*x^4) + E^(4*x)*(-20*x^2 + 72*x^3 - 240*x^4 + 280*x^5) + E^(3*x)*(
10*x^2 - 10*x^3 + 158*x^4 - 320*x^5 + 224*x^6) + E^(2*x)*(-12*x^4 + 192*x^5 - 240*x^6 + 112*x^7) + E^x*(-20*x^
4 - 44*x^5 + 123*x^6 - 96*x^7 + 32*x^8))/(2*E^(8*x) + 16*E^(7*x)*x + 8*x^4 - 16*x^5 + 16*x^6 - 8*x^7 + 2*x^8 +
E^(6*x)*(-8*x + 56*x^2) + E^(5*x)*(-48*x^2 + 112*x^3) + E^(4*x)*(16*x^2 - 120*x^3 + 140*x^4) + E^(3*x)*(64*x^
3 - 160*x^4 + 112*x^5) + E^(2*x)*(-16*x^3 + 96*x^4 - 120*x^5 + 56*x^6) + E^x*(-32*x^4 + 64*x^5 - 48*x^6 + 16*x
^7)),x]

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 0.75, size = 127, normalized size = 4.10 \begin {gather*} \frac {2 \, x^{6} - 4 \, x^{5} + 4 \, x^{4} + 8 \, x^{3} e^{\left (3 \, x\right )} + 2 \, x^{2} e^{\left (4 \, x\right )} + 4 \, {\left (3 \, x^{4} - x^{3}\right )} e^{\left (2 \, x\right )} + {\left (8 \, x^{5} - 8 \, x^{4} - 5 \, x^{2}\right )} e^{x}}{2 \, {\left (x^{4} - 2 \, x^{3} + 2 \, x^{2} + 4 \, x e^{\left (3 \, x\right )} + 2 \, {\left (3 \, x^{2} - x\right )} e^{\left (2 \, x\right )} + 4 \, {\left (x^{3} - x^{2}\right )} e^{x} + e^{\left (4 \, x\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x*exp(x)^8+32*x^2*exp(x)^7+(112*x^3-16*x^2)*exp(x)^6+(224*x^4-96*x^3+15*x^2-10*x)*exp(x)^5+(280*x
^5-240*x^4+72*x^3-20*x^2)*exp(x)^4+(224*x^6-320*x^5+158*x^4-10*x^3+10*x^2)*exp(x)^3+(112*x^7-240*x^6+192*x^5-1
2*x^4)*exp(x)^2+(32*x^8-96*x^7+123*x^6-44*x^5-20*x^4)*exp(x)+4*x^9-16*x^8+32*x^7-32*x^6+16*x^5)/(2*exp(x)^8+16
*x*exp(x)^7+(56*x^2-8*x)*exp(x)^6+(112*x^3-48*x^2)*exp(x)^5+(140*x^4-120*x^3+16*x^2)*exp(x)^4+(112*x^5-160*x^4
+64*x^3)*exp(x)^3+(56*x^6-120*x^5+96*x^4-16*x^3)*exp(x)^2+(16*x^7-48*x^6+64*x^5-32*x^4)*exp(x)+2*x^8-8*x^7+16*
x^6-16*x^5+8*x^4),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [B]  time = 6.29, size = 128, normalized size = 4.13 \begin {gather*} \frac {x^{6} + 4 \, x^{5} e^{x} - 2 \, x^{5} + 6 \, x^{4} e^{\left (2 \, x\right )} - 4 \, x^{4} e^{x} + 2 \, x^{4} + 4 \, x^{3} e^{\left (3 \, x\right )} - 2 \, x^{3} e^{\left (2 \, x\right )} + x^{2} e^{\left (4 \, x\right )} - 5 \, x^{2} e^{x}}{x^{4} + 4 \, x^{3} e^{x} - 2 \, x^{3} + 6 \, x^{2} e^{\left (2 \, x\right )} - 4 \, x^{2} e^{x} + 2 \, x^{2} + 4 \, x e^{\left (3 \, x\right )} - 2 \, x e^{\left (2 \, x\right )} + e^{\left (4 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x*exp(x)^8+32*x^2*exp(x)^7+(112*x^3-16*x^2)*exp(x)^6+(224*x^4-96*x^3+15*x^2-10*x)*exp(x)^5+(280*x
^5-240*x^4+72*x^3-20*x^2)*exp(x)^4+(224*x^6-320*x^5+158*x^4-10*x^3+10*x^2)*exp(x)^3+(112*x^7-240*x^6+192*x^5-1
2*x^4)*exp(x)^2+(32*x^8-96*x^7+123*x^6-44*x^5-20*x^4)*exp(x)+4*x^9-16*x^8+32*x^7-32*x^6+16*x^5)/(2*exp(x)^8+16
*x*exp(x)^7+(56*x^2-8*x)*exp(x)^6+(112*x^3-48*x^2)*exp(x)^5+(140*x^4-120*x^3+16*x^2)*exp(x)^4+(112*x^5-160*x^4
+64*x^3)*exp(x)^3+(56*x^6-120*x^5+96*x^4-16*x^3)*exp(x)^2+(16*x^7-48*x^6+64*x^5-32*x^4)*exp(x)+2*x^8-8*x^7+16*
x^6-16*x^5+8*x^4),x, algorithm="giac")

[Out]

(x^6 + 4*x^5*e^x - 2*x^5 + 6*x^4*e^(2*x) - 4*x^4*e^x + 2*x^4 + 4*x^3*e^(3*x) - 2*x^3*e^(2*x) + x^2*e^(4*x) - 5
*x^2*e^x)/(x^4 + 4*x^3*e^x - 2*x^3 + 6*x^2*e^(2*x) - 4*x^2*e^x + 2*x^2 + 4*x*e^(3*x) - 2*x*e^(2*x) + e^(4*x))

________________________________________________________________________________________

maple [B]  time = 0.09, size = 69, normalized size = 2.23

method	result	size
risch	\(x^{2}-\frac {5 x^{2} {\mathrm e}^{x}}{2 \left ({\mathrm e}^{4 x}+4 x \,{\mathrm e}^{3 x}+6 \,{\mathrm e}^{2 x} x^{2}+4 \,{\mathrm e}^{x} x^{3}+x^{4}-2 x \,{\mathrm e}^{2 x}-4 \,{\mathrm e}^{x} x^{2}-2 x^{3}+2 x^{2}\right )}\)	\(69\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x*exp(x)^8+32*x^2*exp(x)^7+(112*x^3-16*x^2)*exp(x)^6+(224*x^4-96*x^3+15*x^2-10*x)*exp(x)^5+(280*x^5-240
*x^4+72*x^3-20*x^2)*exp(x)^4+(224*x^6-320*x^5+158*x^4-10*x^3+10*x^2)*exp(x)^3+(112*x^7-240*x^6+192*x^5-12*x^4)
*exp(x)^2+(32*x^8-96*x^7+123*x^6-44*x^5-20*x^4)*exp(x)+4*x^9-16*x^8+32*x^7-32*x^6+16*x^5)/(2*exp(x)^8+16*x*exp
(x)^7+(56*x^2-8*x)*exp(x)^6+(112*x^3-48*x^2)*exp(x)^5+(140*x^4-120*x^3+16*x^2)*exp(x)^4+(112*x^5-160*x^4+64*x^
3)*exp(x)^3+(56*x^6-120*x^5+96*x^4-16*x^3)*exp(x)^2+(16*x^7-48*x^6+64*x^5-32*x^4)*exp(x)+2*x^8-8*x^7+16*x^6-16
*x^5+8*x^4),x,method=_RETURNVERBOSE)

[Out]

x^2-5/2*x^2*exp(x)/(exp(4*x)+4*x*exp(3*x)+6*exp(2*x)*x^2+4*exp(x)*x^3+x^4-2*x*exp(2*x)-4*exp(x)*x^2-2*x^3+2*x^
2)

________________________________________________________________________________________

maxima [B]  time = 0.60, size = 127, normalized size = 4.10 \begin {gather*} \frac {2 \, x^{6} - 4 \, x^{5} + 4 \, x^{4} + 8 \, x^{3} e^{\left (3 \, x\right )} + 2 \, x^{2} e^{\left (4 \, x\right )} + 4 \, {\left (3 \, x^{4} - x^{3}\right )} e^{\left (2 \, x\right )} + {\left (8 \, x^{5} - 8 \, x^{4} - 5 \, x^{2}\right )} e^{x}}{2 \, {\left (x^{4} - 2 \, x^{3} + 2 \, x^{2} + 4 \, x e^{\left (3 \, x\right )} + 2 \, {\left (3 \, x^{2} - x\right )} e^{\left (2 \, x\right )} + 4 \, {\left (x^{3} - x^{2}\right )} e^{x} + e^{\left (4 \, x\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x*exp(x)^8+32*x^2*exp(x)^7+(112*x^3-16*x^2)*exp(x)^6+(224*x^4-96*x^3+15*x^2-10*x)*exp(x)^5+(280*x
^5-240*x^4+72*x^3-20*x^2)*exp(x)^4+(224*x^6-320*x^5+158*x^4-10*x^3+10*x^2)*exp(x)^3+(112*x^7-240*x^6+192*x^5-1
2*x^4)*exp(x)^2+(32*x^8-96*x^7+123*x^6-44*x^5-20*x^4)*exp(x)+4*x^9-16*x^8+32*x^7-32*x^6+16*x^5)/(2*exp(x)^8+16
*x*exp(x)^7+(56*x^2-8*x)*exp(x)^6+(112*x^3-48*x^2)*exp(x)^5+(140*x^4-120*x^3+16*x^2)*exp(x)^4+(112*x^5-160*x^4
+64*x^3)*exp(x)^3+(56*x^6-120*x^5+96*x^4-16*x^3)*exp(x)^2+(16*x^7-48*x^6+64*x^5-32*x^4)*exp(x)+2*x^8-8*x^7+16*
x^6-16*x^5+8*x^4),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} -\int \frac {{\mathrm {e}}^{6\,x}\,\left (16\,x^2-112\,x^3\right )-4\,x\,{\mathrm {e}}^{8\,x}-{\mathrm {e}}^{3\,x}\,\left (224\,x^6-320\,x^5+158\,x^4-10\,x^3+10\,x^2\right )-32\,x^2\,{\mathrm {e}}^{7\,x}+{\mathrm {e}}^{5\,x}\,\left (-224\,x^4+96\,x^3-15\,x^2+10\,x\right )+{\mathrm {e}}^x\,\left (-32\,x^8+96\,x^7-123\,x^6+44\,x^5+20\,x^4\right )+{\mathrm {e}}^{2\,x}\,\left (-112\,x^7+240\,x^6-192\,x^5+12\,x^4\right )+{\mathrm {e}}^{4\,x}\,\left (-280\,x^5+240\,x^4-72\,x^3+20\,x^2\right )-16\,x^5+32\,x^6-32\,x^7+16\,x^8-4\,x^9}{2\,{\mathrm {e}}^{8\,x}-{\mathrm {e}}^{6\,x}\,\left (8\,x-56\,x^2\right )+16\,x\,{\mathrm {e}}^{7\,x}-{\mathrm {e}}^{5\,x}\,\left (48\,x^2-112\,x^3\right )-{\mathrm {e}}^x\,\left (-16\,x^7+48\,x^6-64\,x^5+32\,x^4\right )+{\mathrm {e}}^{4\,x}\,\left (140\,x^4-120\,x^3+16\,x^2\right )+{\mathrm {e}}^{3\,x}\,\left (112\,x^5-160\,x^4+64\,x^3\right )-{\mathrm {e}}^{2\,x}\,\left (-56\,x^6+120\,x^5-96\,x^4+16\,x^3\right )+8\,x^4-16\,x^5+16\,x^6-8\,x^7+2\,x^8} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(6*x)*(16*x^2 - 112*x^3) - 4*x*exp(8*x) - exp(3*x)*(10*x^2 - 10*x^3 + 158*x^4 - 320*x^5 + 224*x^6) -
32*x^2*exp(7*x) + exp(5*x)*(10*x - 15*x^2 + 96*x^3 - 224*x^4) + exp(x)*(20*x^4 + 44*x^5 - 123*x^6 + 96*x^7 - 3
2*x^8) + exp(2*x)*(12*x^4 - 192*x^5 + 240*x^6 - 112*x^7) + exp(4*x)*(20*x^2 - 72*x^3 + 240*x^4 - 280*x^5) - 16
*x^5 + 32*x^6 - 32*x^7 + 16*x^8 - 4*x^9)/(2*exp(8*x) - exp(6*x)*(8*x - 56*x^2) + 16*x*exp(7*x) - exp(5*x)*(48*
x^2 - 112*x^3) - exp(x)*(32*x^4 - 64*x^5 + 48*x^6 - 16*x^7) + exp(4*x)*(16*x^2 - 120*x^3 + 140*x^4) + exp(3*x)
*(64*x^3 - 160*x^4 + 112*x^5) - exp(2*x)*(16*x^3 - 96*x^4 + 120*x^5 - 56*x^6) + 8*x^4 - 16*x^5 + 16*x^6 - 8*x^
7 + 2*x^8),x)

[Out]

-int((exp(6*x)*(16*x^2 - 112*x^3) - 4*x*exp(8*x) - exp(3*x)*(10*x^2 - 10*x^3 + 158*x^4 - 320*x^5 + 224*x^6) -
32*x^2*exp(7*x) + exp(5*x)*(10*x - 15*x^2 + 96*x^3 - 224*x^4) + exp(x)*(20*x^4 + 44*x^5 - 123*x^6 + 96*x^7 - 3
2*x^8) + exp(2*x)*(12*x^4 - 192*x^5 + 240*x^6 - 112*x^7) + exp(4*x)*(20*x^2 - 72*x^3 + 240*x^4 - 280*x^5) - 16
*x^5 + 32*x^6 - 32*x^7 + 16*x^8 - 4*x^9)/(2*exp(8*x) - exp(6*x)*(8*x - 56*x^2) + 16*x*exp(7*x) - exp(5*x)*(48*
x^2 - 112*x^3) - exp(x)*(32*x^4 - 64*x^5 + 48*x^6 - 16*x^7) + exp(4*x)*(16*x^2 - 120*x^3 + 140*x^4) + exp(3*x)
*(64*x^3 - 160*x^4 + 112*x^5) - exp(2*x)*(16*x^3 - 96*x^4 + 120*x^5 - 56*x^6) + 8*x^4 - 16*x^5 + 16*x^6 - 8*x^
7 + 2*x^8), x)

________________________________________________________________________________________

sympy [B]  time = 0.53, size = 68, normalized size = 2.19 \begin {gather*} x^{2} - \frac {5 x^{2} e^{x}}{2 x^{4} - 4 x^{3} + 4 x^{2} + 8 x e^{3 x} + \left (12 x^{2} - 4 x\right ) e^{2 x} + \left (8 x^{3} - 8 x^{2}\right ) e^{x} + 2 e^{4 x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x*exp(x)**8+32*x**2*exp(x)**7+(112*x**3-16*x**2)*exp(x)**6+(224*x**4-96*x**3+15*x**2-10*x)*exp(x)
**5+(280*x**5-240*x**4+72*x**3-20*x**2)*exp(x)**4+(224*x**6-320*x**5+158*x**4-10*x**3+10*x**2)*exp(x)**3+(112*
x**7-240*x**6+192*x**5-12*x**4)*exp(x)**2+(32*x**8-96*x**7+123*x**6-44*x**5-20*x**4)*exp(x)+4*x**9-16*x**8+32*
x**7-32*x**6+16*x**5)/(2*exp(x)**8+16*x*exp(x)**7+(56*x**2-8*x)*exp(x)**6+(112*x**3-48*x**2)*exp(x)**5+(140*x*
*4-120*x**3+16*x**2)*exp(x)**4+(112*x**5-160*x**4+64*x**3)*exp(x)**3+(56*x**6-120*x**5+96*x**4-16*x**3)*exp(x)
**2+(16*x**7-48*x**6+64*x**5-32*x**4)*exp(x)+2*x**8-8*x**7+16*x**6-16*x**5+8*x**4),x)

[Out]

x**2 - 5*x**2*exp(x)/(2*x**4 - 4*x**3 + 4*x**2 + 8*x*exp(3*x) + (12*x**2 - 4*x)*exp(2*x) + (8*x**3 - 8*x**2)*e
xp(x) + 2*exp(4*x))

________________________________________________________________________________________