Optimal. Leaf size=14 \[ x+\left (5+e^x-x^2\right )^4 \]
________________________________________________________________________________________
Rubi [B] time = 0.42, antiderivative size = 98, normalized size of antiderivative = 7.00, number of steps used = 57, number of rules used = 3, integrand size = 103, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {2194, 2196, 2176} \begin {gather*} x^8-4 e^x x^6-20 x^6+60 e^x x^4+6 e^{2 x} x^4+150 x^4-300 e^x x^2-60 e^{2 x} x^2-4 e^{3 x} x^2-500 x^2+x+500 e^x+150 e^{2 x}+20 e^{3 x}+e^{4 x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2176
Rule 2194
Rule 2196
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=x-500 x^2+150 x^4-20 x^6+x^8+4 \int e^{4 x} \, dx+\int e^{3 x} \left (60-8 x-12 x^2\right ) \, dx+\int e^{2 x} \left (300-120 x-120 x^2+24 x^3+12 x^4\right ) \, dx+\int e^x \left (500-600 x-300 x^2+240 x^3+60 x^4-24 x^5-4 x^6\right ) \, dx\\ &=e^{4 x}+x-500 x^2+150 x^4-20 x^6+x^8+\int \left (60 e^{3 x}-8 e^{3 x} x-12 e^{3 x} x^2\right ) \, dx+\int \left (300 e^{2 x}-120 e^{2 x} x-120 e^{2 x} x^2+24 e^{2 x} x^3+12 e^{2 x} x^4\right ) \, dx+\int \left (500 e^x-600 e^x x-300 e^x x^2+240 e^x x^3+60 e^x x^4-24 e^x x^5-4 e^x x^6\right ) \, dx\\ &=e^{4 x}+x-500 x^2+150 x^4-20 x^6+x^8-4 \int e^x x^6 \, dx-8 \int e^{3 x} x \, dx-12 \int e^{3 x} x^2 \, dx+12 \int e^{2 x} x^4 \, dx+24 \int e^{2 x} x^3 \, dx-24 \int e^x x^5 \, dx+60 \int e^{3 x} \, dx+60 \int e^x x^4 \, dx-120 \int e^{2 x} x \, dx-120 \int e^{2 x} x^2 \, dx+240 \int e^x x^3 \, dx+300 \int e^{2 x} \, dx-300 \int e^x x^2 \, dx+500 \int e^x \, dx-600 \int e^x x \, dx\\ &=500 e^x+150 e^{2 x}+20 e^{3 x}+e^{4 x}+x-600 e^x x-60 e^{2 x} x-\frac {8}{3} e^{3 x} x-500 x^2-300 e^x x^2-60 e^{2 x} x^2-4 e^{3 x} x^2+240 e^x x^3+12 e^{2 x} x^3+150 x^4+60 e^x x^4+6 e^{2 x} x^4-24 e^x x^5-20 x^6-4 e^x x^6+x^8+\frac {8}{3} \int e^{3 x} \, dx+8 \int e^{3 x} x \, dx-24 \int e^{2 x} x^3 \, dx+24 \int e^x x^5 \, dx-36 \int e^{2 x} x^2 \, dx+60 \int e^{2 x} \, dx+120 \int e^{2 x} x \, dx+120 \int e^x x^4 \, dx-240 \int e^x x^3 \, dx+600 \int e^x \, dx+600 \int e^x x \, dx-720 \int e^x x^2 \, dx\\ &=1100 e^x+180 e^{2 x}+\frac {188 e^{3 x}}{9}+e^{4 x}+x-500 x^2-1020 e^x x^2-78 e^{2 x} x^2-4 e^{3 x} x^2+150 x^4+180 e^x x^4+6 e^{2 x} x^4-20 x^6-4 e^x x^6+x^8-\frac {8}{3} \int e^{3 x} \, dx+36 \int e^{2 x} x \, dx+36 \int e^{2 x} x^2 \, dx-60 \int e^{2 x} \, dx-120 \int e^x x^4 \, dx-480 \int e^x x^3 \, dx-600 \int e^x \, dx+720 \int e^x x^2 \, dx+1440 \int e^x x \, dx\\ &=500 e^x+150 e^{2 x}+20 e^{3 x}+e^{4 x}+x+1440 e^x x+18 e^{2 x} x-500 x^2-300 e^x x^2-60 e^{2 x} x^2-4 e^{3 x} x^2-480 e^x x^3+150 x^4+60 e^x x^4+6 e^{2 x} x^4-20 x^6-4 e^x x^6+x^8-18 \int e^{2 x} \, dx-36 \int e^{2 x} x \, dx+480 \int e^x x^3 \, dx-1440 \int e^x \, dx-1440 \int e^x x \, dx+1440 \int e^x x^2 \, dx\\ &=-940 e^x+141 e^{2 x}+20 e^{3 x}+e^{4 x}+x-500 x^2+1140 e^x x^2-60 e^{2 x} x^2-4 e^{3 x} x^2+150 x^4+60 e^x x^4+6 e^{2 x} x^4-20 x^6-4 e^x x^6+x^8+18 \int e^{2 x} \, dx+1440 \int e^x \, dx-1440 \int e^x x^2 \, dx-2880 \int e^x x \, dx\\ &=500 e^x+150 e^{2 x}+20 e^{3 x}+e^{4 x}+x-2880 e^x x-500 x^2-300 e^x x^2-60 e^{2 x} x^2-4 e^{3 x} x^2+150 x^4+60 e^x x^4+6 e^{2 x} x^4-20 x^6-4 e^x x^6+x^8+2880 \int e^x \, dx+2880 \int e^x x \, dx\\ &=3380 e^x+150 e^{2 x}+20 e^{3 x}+e^{4 x}+x-500 x^2-300 e^x x^2-60 e^{2 x} x^2-4 e^{3 x} x^2+150 x^4+60 e^x x^4+6 e^{2 x} x^4-20 x^6-4 e^x x^6+x^8-2880 \int e^x \, dx\\ &=500 e^x+150 e^{2 x}+20 e^{3 x}+e^{4 x}+x-500 x^2-300 e^x x^2-60 e^{2 x} x^2-4 e^{3 x} x^2+150 x^4+60 e^x x^4+6 e^{2 x} x^4-20 x^6-4 e^x x^6+x^8\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [B] time = 0.03, size = 70, normalized size = 5.00 \begin {gather*} e^{4 x}+x-500 x^2+150 x^4-20 x^6+x^8-4 e^x \left (-5+x^2\right )^3-\frac {4}{3} e^{3 x} \left (-15+3 x^2\right )+6 e^{2 x} \left (25-10 x^2+x^4\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.80, size = 70, normalized size = 5.00 \begin {gather*} x^{8} - 20 \, x^{6} + 150 \, x^{4} - 500 \, x^{2} - 4 \, {\left (x^{2} - 5\right )} e^{\left (3 \, x\right )} + 6 \, {\left (x^{4} - 10 \, x^{2} + 25\right )} e^{\left (2 \, x\right )} - 4 \, {\left (x^{6} - 15 \, x^{4} + 75 \, x^{2} - 125\right )} e^{x} + x + e^{\left (4 \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.14, size = 70, normalized size = 5.00 \begin {gather*} x^{8} - 20 \, x^{6} + 150 \, x^{4} - 500 \, x^{2} - 4 \, {\left (x^{2} - 5\right )} e^{\left (3 \, x\right )} + 6 \, {\left (x^{4} - 10 \, x^{2} + 25\right )} e^{\left (2 \, x\right )} - 4 \, {\left (x^{6} - 15 \, x^{4} + 75 \, x^{2} - 125\right )} e^{x} + x + e^{\left (4 \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.05, size = 74, normalized size = 5.29
method | result | size |
risch | \({\mathrm e}^{4 x}+\left (-4 x^{2}+20\right ) {\mathrm e}^{3 x}+\left (6 x^{4}-60 x^{2}+150\right ) {\mathrm e}^{2 x}+\left (-4 x^{6}+60 x^{4}-300 x^{2}+500\right ) {\mathrm e}^{x}+x^{8}-20 x^{6}+150 x^{4}-500 x^{2}+x\) | \(74\) |
default | \(x +20 \,{\mathrm e}^{3 x}-4 x^{2} {\mathrm e}^{3 x}+6 \,{\mathrm e}^{2 x} x^{4}-60 \,{\mathrm e}^{2 x} x^{2}+150 \,{\mathrm e}^{2 x}+60 \,{\mathrm e}^{x} x^{4}-300 \,{\mathrm e}^{x} x^{2}-4 x^{6} {\mathrm e}^{x}+500 \,{\mathrm e}^{x}-500 x^{2}+150 x^{4}-20 x^{6}+x^{8}+{\mathrm e}^{4 x}\) | \(89\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.35, size = 70, normalized size = 5.00 \begin {gather*} x^{8} - 20 \, x^{6} + 150 \, x^{4} - 500 \, x^{2} - 4 \, {\left (x^{2} - 5\right )} e^{\left (3 \, x\right )} + 6 \, {\left (x^{4} - 10 \, x^{2} + 25\right )} e^{\left (2 \, x\right )} - 4 \, {\left (x^{6} - 15 \, x^{4} + 75 \, x^{2} - 125\right )} e^{x} + x + e^{\left (4 \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 5.87, size = 88, normalized size = 6.29 \begin {gather*} x+150\,{\mathrm {e}}^{2\,x}+20\,{\mathrm {e}}^{3\,x}+{\mathrm {e}}^{4\,x}+500\,{\mathrm {e}}^x-300\,x^2\,{\mathrm {e}}^x+60\,x^4\,{\mathrm {e}}^x-4\,x^6\,{\mathrm {e}}^x-60\,x^2\,{\mathrm {e}}^{2\,x}-4\,x^2\,{\mathrm {e}}^{3\,x}+6\,x^4\,{\mathrm {e}}^{2\,x}-500\,x^2+150\,x^4-20\,x^6+x^8 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.18, size = 73, normalized size = 5.21 \begin {gather*} x^{8} - 20 x^{6} + 150 x^{4} - 500 x^{2} + x + \left (20 - 4 x^{2}\right ) e^{3 x} + \left (6 x^{4} - 60 x^{2} + 150\right ) e^{2 x} + \left (- 4 x^{6} + 60 x^{4} - 300 x^{2} + 500\right ) e^{x} + e^{4 x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________