3.64.97 \(\int \frac {-1476225+25 e^{15}+4920714 x-6561024 x^2+4374000 x^3-1458000 x^4+194400 x^5+e^{12} (-1125+750 x)+e^9 (20250-27000 x+9000 x^2)+e^6 (-182250+364500 x-243000 x^2+54000 x^3)+e^3 (820125-2186996 x+2187000 x^2-972000 x^3+162000 x^4)}{-1476225+25 e^{15}+4920750 x-6561000 x^2+4374000 x^3-1458000 x^4+194400 x^5+e^{12} (-1125+750 x)+e^9 (20250-27000 x+9000 x^2)+e^6 (-182250+364500 x-243000 x^2+54000 x^3)+e^3 (820125-2187000 x+2187000 x^2-972000 x^3+162000 x^4)} \, dx\)

Optimal. Leaf size=22 \[ x+\frac {2 x^2}{25 \left (e^3-3 (3-2 x)\right )^4} \]

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Rubi [B]  time = 0.27, antiderivative size = 66, normalized size of antiderivative = 3.00, number of steps used = 2, number of rules used = 1, integrand size = 195, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.005, Rules used = {2074} \begin {gather*} x+\frac {1}{450 \left (-6 x-e^3+9\right )^2}-\frac {9-e^3}{225 \left (-6 x-e^3+9\right )^3}+\frac {\left (9-e^3\right )^2}{450 \left (-6 x-e^3+9\right )^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1476225 + 25*E^15 + 4920714*x - 6561024*x^2 + 4374000*x^3 - 1458000*x^4 + 194400*x^5 + E^12*(-1125 + 750
*x) + E^9*(20250 - 27000*x + 9000*x^2) + E^6*(-182250 + 364500*x - 243000*x^2 + 54000*x^3) + E^3*(820125 - 218
6996*x + 2187000*x^2 - 972000*x^3 + 162000*x^4))/(-1476225 + 25*E^15 + 4920750*x - 6561000*x^2 + 4374000*x^3 -
1458000*x^4 + 194400*x^5 + E^12*(-1125 + 750*x) + E^9*(20250 - 27000*x + 9000*x^2) + E^6*(-182250 + 364500*x
- 243000*x^2 + 54000*x^3) + E^3*(820125 - 2187000*x + 2187000*x^2 - 972000*x^3 + 162000*x^4)),x]

[Out]

(9 - E^3)^2/(450*(9 - E^3 - 6*x)^4) - (9 - E^3)/(225*(9 - E^3 - 6*x)^3) + 1/(450*(9 - E^3 - 6*x)^2) + x

Rule 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (1-\frac {4 \left (-9+e^3\right )^2}{75 \left (-9+e^3+6 x\right )^5}+\frac {2 \left (-9+e^3\right )}{25 \left (-9+e^3+6 x\right )^4}-\frac {2}{75 \left (-9+e^3+6 x\right )^3}\right ) \, dx\\ &=\frac {\left (9-e^3\right )^2}{450 \left (9-e^3-6 x\right )^4}-\frac {9-e^3}{225 \left (9-e^3-6 x\right )^3}+\frac {1}{450 \left (9-e^3-6 x\right )^2}+x\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.05, size = 60, normalized size = 2.73 \begin {gather*} \frac {81-e^6+6 e^3 (3-2 x)+54 (-3+2 x)+\left (-9+e^3+6 x\right )^2+75 \left (-9+e^3+6 x\right )^5}{450 \left (-9+e^3+6 x\right )^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1476225 + 25*E^15 + 4920714*x - 6561024*x^2 + 4374000*x^3 - 1458000*x^4 + 194400*x^5 + E^12*(-1125
+ 750*x) + E^9*(20250 - 27000*x + 9000*x^2) + E^6*(-182250 + 364500*x - 243000*x^2 + 54000*x^3) + E^3*(820125
- 2186996*x + 2187000*x^2 - 972000*x^3 + 162000*x^4))/(-1476225 + 25*E^15 + 4920750*x - 6561000*x^2 + 4374000
*x^3 - 1458000*x^4 + 194400*x^5 + E^12*(-1125 + 750*x) + E^9*(20250 - 27000*x + 9000*x^2) + E^6*(-182250 + 364
500*x - 243000*x^2 + 54000*x^3) + E^3*(820125 - 2187000*x + 2187000*x^2 - 972000*x^3 + 162000*x^4)),x]

[Out]

(81 - E^6 + 6*E^3*(3 - 2*x) + 54*(-3 + 2*x) + (-9 + E^3 + 6*x)^2 + 75*(-9 + E^3 + 6*x)^5)/(450*(-9 + E^3 + 6*x
)^4)

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fricas [B]  time = 0.76, size = 151, normalized size = 6.86 \begin {gather*} \frac {32400 \, x^{5} - 194400 \, x^{4} + 437400 \, x^{3} - 437398 \, x^{2} + 25 \, x e^{12} + 300 \, {\left (2 \, x^{2} - 3 \, x\right )} e^{9} + 1350 \, {\left (4 \, x^{3} - 12 \, x^{2} + 9 \, x\right )} e^{6} + 2700 \, {\left (8 \, x^{4} - 36 \, x^{3} + 54 \, x^{2} - 27 \, x\right )} e^{3} + 164025 \, x}{25 \, {\left (1296 \, x^{4} - 7776 \, x^{3} + 17496 \, x^{2} + 12 \, {\left (2 \, x - 3\right )} e^{9} + 54 \, {\left (4 \, x^{2} - 12 \, x + 9\right )} e^{6} + 108 \, {\left (8 \, x^{3} - 36 \, x^{2} + 54 \, x - 27\right )} e^{3} - 17496 \, x + e^{12} + 6561\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((25*exp(3)^5+(750*x-1125)*exp(3)^4+(9000*x^2-27000*x+20250)*exp(3)^3+(54000*x^3-243000*x^2+364500*x-
182250)*exp(3)^2+(162000*x^4-972000*x^3+2187000*x^2-2186996*x+820125)*exp(3)+194400*x^5-1458000*x^4+4374000*x^
3-6561024*x^2+4920714*x-1476225)/(25*exp(3)^5+(750*x-1125)*exp(3)^4+(9000*x^2-27000*x+20250)*exp(3)^3+(54000*x
^3-243000*x^2+364500*x-182250)*exp(3)^2+(162000*x^4-972000*x^3+2187000*x^2-2187000*x+820125)*exp(3)+194400*x^5
-1458000*x^4+4374000*x^3-6561000*x^2+4920750*x-1476225),x, algorithm="fricas")

[Out]

1/25*(32400*x^5 - 194400*x^4 + 437400*x^3 - 437398*x^2 + 25*x*e^12 + 300*(2*x^2 - 3*x)*e^9 + 1350*(4*x^3 - 12*
x^2 + 9*x)*e^6 + 2700*(8*x^4 - 36*x^3 + 54*x^2 - 27*x)*e^3 + 164025*x)/(1296*x^4 - 7776*x^3 + 17496*x^2 + 12*(
2*x - 3)*e^9 + 54*(4*x^2 - 12*x + 9)*e^6 + 108*(8*x^3 - 36*x^2 + 54*x - 27)*e^3 - 17496*x + e^12 + 6561)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((25*exp(3)^5+(750*x-1125)*exp(3)^4+(9000*x^2-27000*x+20250)*exp(3)^3+(54000*x^3-243000*x^2+364500*x-
182250)*exp(3)^2+(162000*x^4-972000*x^3+2187000*x^2-2186996*x+820125)*exp(3)+194400*x^5-1458000*x^4+4374000*x^
3-6561024*x^2+4920714*x-1476225)/(25*exp(3)^5+(750*x-1125)*exp(3)^4+(9000*x^2-27000*x+20250)*exp(3)^3+(54000*x
^3-243000*x^2+364500*x-182250)*exp(3)^2+(162000*x^4-972000*x^3+2187000*x^2-2187000*x+820125)*exp(3)+194400*x^5
-1458000*x^4+4374000*x^3-6561000*x^2+4920750*x-1476225),x, algorithm="giac")

[Out]

Timed out

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maple [B]  time = 0.43, size = 80, normalized size = 3.64

method	result	size
risch	\(x +\frac {2 x^{2}}{25 \left ({\mathrm e}^{12}+24 x \,{\mathrm e}^{9}+216 x^{2} {\mathrm e}^{6}+864 x^{3} {\mathrm e}^{3}+1296 x^{4}-36 \,{\mathrm e}^{9}-648 x \,{\mathrm e}^{6}-3888 x^{2} {\mathrm e}^{3}-7776 x^{3}+486 \,{\mathrm e}^{6}+5832 x \,{\mathrm e}^{3}+17496 x^{2}-2916 \,{\mathrm e}^{3}-17496 x +6561\right )}\)	\(80\)
norman	\(\frac {\left (-360 \,{\mathrm e}^{6}+6480 \,{\mathrm e}^{3}-29160\right ) x^{3}+\left (-120 \,{\mathrm e}^{9}+3240 \,{\mathrm e}^{6}-29160 \,{\mathrm e}^{3}+\frac {2187002}{25}\right ) x^{2}+\left (-15 \,{\mathrm e}^{12}+540 \,{\mathrm e}^{9}-7290 \,{\mathrm e}^{6}+43740 \,{\mathrm e}^{3}-98415\right ) x +1296 x^{5}-\frac {2 \,{\mathrm e}^{15}}{3}+30 \,{\mathrm e}^{12}-540 \,{\mathrm e}^{9}+4860 \,{\mathrm e}^{6}-21870 \,{\mathrm e}^{3}+39366}{\left ({\mathrm e}^{3}+6 x -9\right )^{4}}\)	\(110\)
default	\(x +\frac {2 \left (\munderset {\textit {\_R} =\RootOf \left (7776 \textit {\_Z}^{5}+\left (6480 \,{\mathrm e}^{3}-58320\right ) \textit {\_Z}^{4}+\left (-38880 \,{\mathrm e}^{3}+2160 \,{\mathrm e}^{6}+174960\right ) \textit {\_Z}^{3}+\left (87480 \,{\mathrm e}^{3}+360 \,{\mathrm e}^{9}-9720 \,{\mathrm e}^{6}-262440\right ) \textit {\_Z}^{2}+\left (-87480 \,{\mathrm e}^{3}+30 \,{\mathrm e}^{12}-1080 \,{\mathrm e}^{9}+14580 \,{\mathrm e}^{6}+196830\right ) \textit {\_Z} +32805 \,{\mathrm e}^{3}+{\mathrm e}^{15}-45 \,{\mathrm e}^{12}+810 \,{\mathrm e}^{9}-7290 \,{\mathrm e}^{6}-59049\right )}{\sum }\frac {\left (-6 \textit {\_R}^{2}+\left ({\mathrm e}^{3}-9\right ) \textit {\_R} \right ) \ln \left (x -\textit {\_R} \right )}{6561+{\mathrm e}^{12}+24 \textit {\_R} \,{\mathrm e}^{9}+216 \textit {\_R}^{2} {\mathrm e}^{6}+864 \textit {\_R}^{3} {\mathrm e}^{3}+1296 \textit {\_R}^{4}-36 \,{\mathrm e}^{9}-648 \textit {\_R} \,{\mathrm e}^{6}-3888 \textit {\_R}^{2} {\mathrm e}^{3}-7776 \textit {\_R}^{3}+486 \,{\mathrm e}^{6}+5832 \textit {\_R} \,{\mathrm e}^{3}+17496 \textit {\_R}^{2}-2916 \,{\mathrm e}^{3}-17496 \textit {\_R}}\right )}{375}\)	\(187\)
gosper	\(-\frac {50 \,{\mathrm e}^{15}+1125 x \,{\mathrm e}^{12}+9000 x^{2} {\mathrm e}^{9}+27000 x^{3} {\mathrm e}^{6}-97200 x^{5}-2250 \,{\mathrm e}^{12}-40500 x \,{\mathrm e}^{9}-243000 x^{2} {\mathrm e}^{6}-486000 x^{3} {\mathrm e}^{3}+40500 \,{\mathrm e}^{9}+546750 x \,{\mathrm e}^{6}+2187000 x^{2} {\mathrm e}^{3}+2187000 x^{3}-364500 \,{\mathrm e}^{6}-3280500 x \,{\mathrm e}^{3}-6561006 x^{2}+1640250 \,{\mathrm e}^{3}+7381125 x -2952450}{75 \left ({\mathrm e}^{12}+24 x \,{\mathrm e}^{9}+216 x^{2} {\mathrm e}^{6}+864 x^{3} {\mathrm e}^{3}+1296 x^{4}-36 \,{\mathrm e}^{9}-648 x \,{\mathrm e}^{6}-3888 x^{2} {\mathrm e}^{3}-7776 x^{3}+486 \,{\mathrm e}^{6}+5832 x \,{\mathrm e}^{3}+17496 x^{2}-2916 \,{\mathrm e}^{3}-17496 x +6561\right )}\)	\(202\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((25*exp(3)^5+(750*x-1125)*exp(3)^4+(9000*x^2-27000*x+20250)*exp(3)^3+(54000*x^3-243000*x^2+364500*x-182250
)*exp(3)^2+(162000*x^4-972000*x^3+2187000*x^2-2186996*x+820125)*exp(3)+194400*x^5-1458000*x^4+4374000*x^3-6561
024*x^2+4920714*x-1476225)/(25*exp(3)^5+(750*x-1125)*exp(3)^4+(9000*x^2-27000*x+20250)*exp(3)^3+(54000*x^3-243
000*x^2+364500*x-182250)*exp(3)^2+(162000*x^4-972000*x^3+2187000*x^2-2187000*x+820125)*exp(3)+194400*x^5-14580
00*x^4+4374000*x^3-6561000*x^2+4920750*x-1476225),x,method=_RETURNVERBOSE)

[Out]

x+2/25*x^2/(exp(12)+24*x*exp(9)+216*x^2*exp(6)+864*x^3*exp(3)+1296*x^4-36*exp(9)-648*x*exp(6)-3888*x^2*exp(3)-
7776*x^3+486*exp(6)+5832*x*exp(3)+17496*x^2-2916*exp(3)-17496*x+6561)

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maxima [B]  time = 0.37, size = 67, normalized size = 3.05 \begin {gather*} x + \frac {2 \, x^{2}}{25 \, {\left (1296 \, x^{4} + 864 \, x^{3} {\left (e^{3} - 9\right )} + 216 \, x^{2} {\left (e^{6} - 18 \, e^{3} + 81\right )} + 24 \, x {\left (e^{9} - 27 \, e^{6} + 243 \, e^{3} - 729\right )} + e^{12} - 36 \, e^{9} + 486 \, e^{6} - 2916 \, e^{3} + 6561\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((25*exp(3)^5+(750*x-1125)*exp(3)^4+(9000*x^2-27000*x+20250)*exp(3)^3+(54000*x^3-243000*x^2+364500*x-
182250)*exp(3)^2+(162000*x^4-972000*x^3+2187000*x^2-2186996*x+820125)*exp(3)+194400*x^5-1458000*x^4+4374000*x^
3-6561024*x^2+4920714*x-1476225)/(25*exp(3)^5+(750*x-1125)*exp(3)^4+(9000*x^2-27000*x+20250)*exp(3)^3+(54000*x
^3-243000*x^2+364500*x-182250)*exp(3)^2+(162000*x^4-972000*x^3+2187000*x^2-2187000*x+820125)*exp(3)+194400*x^5
-1458000*x^4+4374000*x^3-6561000*x^2+4920750*x-1476225),x, algorithm="maxima")

[Out]

x + 2/25*x^2/(1296*x^4 + 864*x^3*(e^3 - 9) + 216*x^2*(e^6 - 18*e^3 + 81) + 24*x*(e^9 - 27*e^6 + 243*e^3 - 729)
+ e^12 - 36*e^9 + 486*e^6 - 2916*e^3 + 6561)

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mupad [B]  time = 4.33, size = 84, normalized size = 3.82 \begin {gather*} \frac {x\,\left (12150\,{\mathrm {e}}^6-72900\,{\mathrm {e}}^3-437398\,x-900\,{\mathrm {e}}^9+25\,{\mathrm {e}}^{12}+145800\,x\,{\mathrm {e}}^3-16200\,x\,{\mathrm {e}}^6+600\,x\,{\mathrm {e}}^9-97200\,x^2\,{\mathrm {e}}^3+21600\,x^3\,{\mathrm {e}}^3+5400\,x^2\,{\mathrm {e}}^6+437400\,x^2-194400\,x^3+32400\,x^4+164025\right )}{25\,{\left (6\,x+{\mathrm {e}}^3-9\right )}^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4920714*x + 25*exp(15) + exp(9)*(9000*x^2 - 27000*x + 20250) + exp(6)*(364500*x - 243000*x^2 + 54000*x^3
- 182250) + exp(3)*(2187000*x^2 - 2186996*x - 972000*x^3 + 162000*x^4 + 820125) - 6561024*x^2 + 4374000*x^3 -
1458000*x^4 + 194400*x^5 + exp(12)*(750*x - 1125) - 1476225)/(4920750*x + 25*exp(15) + exp(9)*(9000*x^2 - 2700
0*x + 20250) + exp(6)*(364500*x - 243000*x^2 + 54000*x^3 - 182250) + exp(3)*(2187000*x^2 - 2187000*x - 972000*
x^3 + 162000*x^4 + 820125) - 6561000*x^2 + 4374000*x^3 - 1458000*x^4 + 194400*x^5 + exp(12)*(750*x - 1125) - 1
476225),x)

[Out]

(x*(12150*exp(6) - 72900*exp(3) - 437398*x - 900*exp(9) + 25*exp(12) + 145800*x*exp(3) - 16200*x*exp(6) + 600*
x*exp(9) - 97200*x^2*exp(3) + 21600*x^3*exp(3) + 5400*x^2*exp(6) + 437400*x^2 - 194400*x^3 + 32400*x^4 + 16402
5))/(25*(6*x + exp(3) - 9)^4)

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sympy [B]  time = 0.83, size = 76, normalized size = 3.45 \begin {gather*} \frac {2 x^{2}}{32400 x^{4} + x^{3} \left (-194400 + 21600 e^{3}\right ) + x^{2} \left (- 97200 e^{3} + 437400 + 5400 e^{6}\right ) + x \left (- 16200 e^{6} - 437400 + 145800 e^{3} + 600 e^{9}\right ) - 900 e^{9} - 72900 e^{3} + 164025 + 25 e^{12} + 12150 e^{6}} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((25*exp(3)**5+(750*x-1125)*exp(3)**4+(9000*x**2-27000*x+20250)*exp(3)**3+(54000*x**3-243000*x**2+364
500*x-182250)*exp(3)**2+(162000*x**4-972000*x**3+2187000*x**2-2186996*x+820125)*exp(3)+194400*x**5-1458000*x**
4+4374000*x**3-6561024*x**2+4920714*x-1476225)/(25*exp(3)**5+(750*x-1125)*exp(3)**4+(9000*x**2-27000*x+20250)*
exp(3)**3+(54000*x**3-243000*x**2+364500*x-182250)*exp(3)**2+(162000*x**4-972000*x**3+2187000*x**2-2187000*x+8
20125)*exp(3)+194400*x**5-1458000*x**4+4374000*x**3-6561000*x**2+4920750*x-1476225),x)

[Out]

2*x**2/(32400*x**4 + x**3*(-194400 + 21600*exp(3)) + x**2*(-97200*exp(3) + 437400 + 5400*exp(6)) + x*(-16200*e
xp(6) - 437400 + 145800*exp(3) + 600*exp(9)) - 900*exp(9) - 72900*exp(3) + 164025 + 25*exp(12) + 12150*exp(6))
+ x

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