Optimal. Leaf size=21 \[ 25 x \left (5-x-e^{\frac {x}{\log (3)}} x\right )^2 \]
________________________________________________________________________________________
Rubi [B] time = 0.88, antiderivative size = 278, normalized size of antiderivative = 13.24, number of steps used = 25, number of rules used = 6, integrand size = 69, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {6688, 12, 6742, 2196, 2176, 2194} \begin {gather*} 25 x^3+50 x^3 e^{\frac {x}{\log (3)}}+25 x^3 e^{\frac {2 x}{\log (3)}}-250 x^2+\frac {25}{2} x^2 \log (27) e^{\frac {2 x}{\log (3)}}-50 x^2 (5-\log (27)) e^{\frac {x}{\log (3)}}-150 x^2 \log (3) e^{\frac {x}{\log (3)}}-\frac {75}{2} x^2 \log (3) e^{\frac {2 x}{\log (3)}}+625 x-300 \log ^3(3) e^{\frac {x}{\log (3)}}-\frac {75}{4} \log ^3(3) e^{\frac {2 x}{\log (3)}}+300 x \log ^2(3) e^{\frac {x}{\log (3)}}+\frac {75}{2} x \log ^2(3) e^{\frac {2 x}{\log (3)}}+\frac {25}{4} \log ^2(3) \log (27) e^{\frac {2 x}{\log (3)}}-100 \log ^2(3) (5-\log (27)) e^{\frac {x}{\log (3)}}-125 x \log (81) e^{\frac {x}{\log (3)}}-\frac {25}{2} x \log (3) \log (27) e^{\frac {2 x}{\log (3)}}+100 x \log (3) (5-\log (27)) e^{\frac {x}{\log (3)}}+125 \log (3) \log (81) e^{\frac {x}{\log (3)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 2176
Rule 2194
Rule 2196
Rule 6688
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {25 \left (5-x-e^{\frac {x}{\log (3)}} x\right ) \left (-((-5+3 x) \log (3))-e^{\frac {x}{\log (3)}} x (2 x+\log (27))\right )}{\log (3)} \, dx\\ &=\frac {25 \int \left (5-x-e^{\frac {x}{\log (3)}} x\right ) \left (-((-5+3 x) \log (3))-e^{\frac {x}{\log (3)}} x (2 x+\log (27))\right ) \, dx}{\log (3)}\\ &=\frac {25 \int \left (\left (25-20 x+3 x^2\right ) \log (3)+e^{\frac {2 x}{\log (3)}} x^2 (2 x+\log (27))+e^{\frac {x}{\log (3)}} x \left (2 x^2-2 x (5-\log (27))-5 \log (81)\right )\right ) \, dx}{\log (3)}\\ &=25 \int \left (25-20 x+3 x^2\right ) \, dx+\frac {25 \int e^{\frac {2 x}{\log (3)}} x^2 (2 x+\log (27)) \, dx}{\log (3)}+\frac {25 \int e^{\frac {x}{\log (3)}} x \left (2 x^2-2 x (5-\log (27))-5 \log (81)\right ) \, dx}{\log (3)}\\ &=625 x-250 x^2+25 x^3+\frac {25 \int \left (2 e^{\frac {2 x}{\log (3)}} x^3+e^{\frac {2 x}{\log (3)}} x^2 \log (27)\right ) \, dx}{\log (3)}+\frac {25 \int \left (2 e^{\frac {x}{\log (3)}} x^3+2 e^{\frac {x}{\log (3)}} x^2 (-5+\log (27))-5 e^{\frac {x}{\log (3)}} x \log (81)\right ) \, dx}{\log (3)}\\ &=625 x-250 x^2+25 x^3+\frac {50 \int e^{\frac {x}{\log (3)}} x^3 \, dx}{\log (3)}+\frac {50 \int e^{\frac {2 x}{\log (3)}} x^3 \, dx}{\log (3)}-\frac {(50 (5-\log (27))) \int e^{\frac {x}{\log (3)}} x^2 \, dx}{\log (3)}+\frac {(25 \log (27)) \int e^{\frac {2 x}{\log (3)}} x^2 \, dx}{\log (3)}-\frac {(125 \log (81)) \int e^{\frac {x}{\log (3)}} x \, dx}{\log (3)}\\ &=625 x-250 x^2+25 x^3+50 e^{\frac {x}{\log (3)}} x^3+25 e^{\frac {2 x}{\log (3)}} x^3-50 e^{\frac {x}{\log (3)}} x^2 (5-\log (27))+\frac {25}{2} e^{\frac {2 x}{\log (3)}} x^2 \log (27)-125 e^{\frac {x}{\log (3)}} x \log (81)-75 \int e^{\frac {2 x}{\log (3)}} x^2 \, dx-150 \int e^{\frac {x}{\log (3)}} x^2 \, dx+(100 (5-\log (27))) \int e^{\frac {x}{\log (3)}} x \, dx-(25 \log (27)) \int e^{\frac {2 x}{\log (3)}} x \, dx+(125 \log (81)) \int e^{\frac {x}{\log (3)}} \, dx\\ &=625 x-250 x^2+25 x^3+50 e^{\frac {x}{\log (3)}} x^3+25 e^{\frac {2 x}{\log (3)}} x^3-150 e^{\frac {x}{\log (3)}} x^2 \log (3)-\frac {75}{2} e^{\frac {2 x}{\log (3)}} x^2 \log (3)-50 e^{\frac {x}{\log (3)}} x^2 (5-\log (27))+100 e^{\frac {x}{\log (3)}} x \log (3) (5-\log (27))+\frac {25}{2} e^{\frac {2 x}{\log (3)}} x^2 \log (27)-\frac {25}{2} e^{\frac {2 x}{\log (3)}} x \log (3) \log (27)-125 e^{\frac {x}{\log (3)}} x \log (81)+125 e^{\frac {x}{\log (3)}} \log (3) \log (81)+(75 \log (3)) \int e^{\frac {2 x}{\log (3)}} x \, dx+(300 \log (3)) \int e^{\frac {x}{\log (3)}} x \, dx-(100 \log (3) (5-\log (27))) \int e^{\frac {x}{\log (3)}} \, dx+\frac {1}{2} (25 \log (3) \log (27)) \int e^{\frac {2 x}{\log (3)}} \, dx\\ &=625 x-250 x^2+25 x^3+50 e^{\frac {x}{\log (3)}} x^3+25 e^{\frac {2 x}{\log (3)}} x^3-150 e^{\frac {x}{\log (3)}} x^2 \log (3)-\frac {75}{2} e^{\frac {2 x}{\log (3)}} x^2 \log (3)+300 e^{\frac {x}{\log (3)}} x \log ^2(3)+\frac {75}{2} e^{\frac {2 x}{\log (3)}} x \log ^2(3)-50 e^{\frac {x}{\log (3)}} x^2 (5-\log (27))+100 e^{\frac {x}{\log (3)}} x \log (3) (5-\log (27))-100 e^{\frac {x}{\log (3)}} \log ^2(3) (5-\log (27))+\frac {25}{2} e^{\frac {2 x}{\log (3)}} x^2 \log (27)-\frac {25}{2} e^{\frac {2 x}{\log (3)}} x \log (3) \log (27)+\frac {25}{4} e^{\frac {2 x}{\log (3)}} \log ^2(3) \log (27)-125 e^{\frac {x}{\log (3)}} x \log (81)+125 e^{\frac {x}{\log (3)}} \log (3) \log (81)-\frac {1}{2} \left (75 \log ^2(3)\right ) \int e^{\frac {2 x}{\log (3)}} \, dx-\left (300 \log ^2(3)\right ) \int e^{\frac {x}{\log (3)}} \, dx\\ &=625 x-250 x^2+25 x^3+50 e^{\frac {x}{\log (3)}} x^3+25 e^{\frac {2 x}{\log (3)}} x^3-150 e^{\frac {x}{\log (3)}} x^2 \log (3)-\frac {75}{2} e^{\frac {2 x}{\log (3)}} x^2 \log (3)+300 e^{\frac {x}{\log (3)}} x \log ^2(3)+\frac {75}{2} e^{\frac {2 x}{\log (3)}} x \log ^2(3)-300 e^{\frac {x}{\log (3)}} \log ^3(3)-\frac {75}{4} e^{\frac {2 x}{\log (3)}} \log ^3(3)-50 e^{\frac {x}{\log (3)}} x^2 (5-\log (27))+100 e^{\frac {x}{\log (3)}} x \log (3) (5-\log (27))-100 e^{\frac {x}{\log (3)}} \log ^2(3) (5-\log (27))+\frac {25}{2} e^{\frac {2 x}{\log (3)}} x^2 \log (27)-\frac {25}{2} e^{\frac {2 x}{\log (3)}} x \log (3) \log (27)+\frac {25}{4} e^{\frac {2 x}{\log (3)}} \log ^2(3) \log (27)-125 e^{\frac {x}{\log (3)}} x \log (81)+125 e^{\frac {x}{\log (3)}} \log (3) \log (81)\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [B] time = 0.20, size = 82, normalized size = 3.90 \begin {gather*} 25 \left (25 x-10 x^2+x^3+e^{\frac {2 x}{\log (3)}} x^3+e^{\frac {x}{\log (3)}} \left (-10 x^2+2 x^3-6 \log ^3(3)+x \left (6 \log ^2(3)-\log (9) \log (27)\right )+\log (27) \log (243)+\log ^2(3) (-15+\log (729))\right )\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.68, size = 45, normalized size = 2.14 \begin {gather*} 25 \, x^{3} e^{\left (\frac {2 \, x}{\log \relax (3)}\right )} + 25 \, x^{3} - 250 \, x^{2} + 50 \, {\left (x^{3} - 5 \, x^{2}\right )} e^{\frac {x}{\log \relax (3)}} + 625 \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.14, size = 51, normalized size = 2.43 \begin {gather*} 25 \, x^{3} e^{\left (\frac {2 \, x}{\log \relax (3)}\right )} + 50 \, x^{3} e^{\frac {x}{\log \relax (3)}} + 25 \, x^{3} - 250 \, x^{2} e^{\frac {x}{\log \relax (3)}} - 250 \, x^{2} + 625 \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.18, size = 47, normalized size = 2.24
method | result | size |
risch | \(25 \,{\mathrm e}^{\frac {2 x}{\ln \relax (3)}} x^{3}+25 x^{3}-250 x^{2}+625 x +\left (50 x^{3}-250 x^{2}\right ) {\mathrm e}^{\frac {x}{\ln \relax (3)}}\) | \(47\) |
norman | \(625 x -250 x^{2}+25 x^{3}-250 x^{2} {\mathrm e}^{\frac {x}{\ln \relax (3)}}+50 \,{\mathrm e}^{\frac {x}{\ln \relax (3)}} x^{3}+25 \,{\mathrm e}^{\frac {2 x}{\ln \relax (3)}} x^{3}\) | \(53\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.51, size = 45, normalized size = 2.14 \begin {gather*} 25 \, x^{3} e^{\left (\frac {2 \, x}{\log \relax (3)}\right )} + 25 \, x^{3} - 250 \, x^{2} + 50 \, {\left (x^{3} - 5 \, x^{2}\right )} e^{\frac {x}{\log \relax (3)}} + 625 \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.56, size = 17, normalized size = 0.81 \begin {gather*} 25\,x\,{\left (x+x\,{\mathrm {e}}^{\frac {x}{\ln \relax (3)}}-5\right )}^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.22, size = 42, normalized size = 2.00 \begin {gather*} 25 x^{3} e^{\frac {2 x}{\log {\relax (3 )}}} + 25 x^{3} - 250 x^{2} + 625 x + \left (50 x^{3} - 250 x^{2}\right ) e^{\frac {x}{\log {\relax (3 )}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________