3.70.45 \(\int \frac {-2125 x^2+11750 x^3-13125 x^4-9000 x^5+e^{\frac {x}{-1+3 x}} (75-375 x+575 x^2)}{5625 x^2-46500 x^3+131350 x^4-136700 x^5+18025 x^6+28200 x^7+3600 x^8+e^{\frac {2 x}{-1+3 x}} (9-78 x+241 x^2-312 x^3+144 x^4)+e^{\frac {x}{-1+3 x}} (-450 x+3810 x^2-11270 x^3+13190 x^4-4080 x^5-1440 x^6)} \, dx\)

Optimal. Leaf size=32 \[ \frac {5}{\left (5-\frac {e^{\frac {x}{-1+3 x}}}{5 x}+x\right ) (-3+4 x)} \]

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Rubi [F]  time = 4.24, 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 {-2125 x^2+11750 x^3-13125 x^4-9000 x^5+e^{\frac {x}{-1+3 x}} \left (75-375 x+575 x^2\right )}{5625 x^2-46500 x^3+131350 x^4-136700 x^5+18025 x^6+28200 x^7+3600 x^8+e^{\frac {2 x}{-1+3 x}} \left (9-78 x+241 x^2-312 x^3+144 x^4\right )+e^{\frac {x}{-1+3 x}} \left (-450 x+3810 x^2-11270 x^3+13190 x^4-4080 x^5-1440 x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-2125*x^2 + 11750*x^3 - 13125*x^4 - 9000*x^5 + E^(x/(-1 + 3*x))*(75 - 375*x + 575*x^2))/(5625*x^2 - 46500
*x^3 + 131350*x^4 - 136700*x^5 + 18025*x^6 + 28200*x^7 + 3600*x^8 + E^((2*x)/(-1 + 3*x))*(9 - 78*x + 241*x^2 -
312*x^3 + 144*x^4) + E^(x/(-1 + 3*x))*(-450*x + 3810*x^2 - 11270*x^3 + 13190*x^4 - 4080*x^5 - 1440*x^6)),x]

[Out]

(-14875*Defer[Int][(E^(x/(-1 + 3*x)) - 25*x - 5*x^2)^(-2), x])/72 - (125*Defer[Int][x/(-E^(x/(-1 + 3*x)) + 25*
x + 5*x^2)^2, x])/2 + (400*Defer[Int][1/((-1 + 3*x)^2*(-E^(x/(-1 + 3*x)) + 25*x + 5*x^2)^2), x])/9 + (1145*Def
er[Int][1/((-1 + 3*x)*(-E^(x/(-1 + 3*x)) + 25*x + 5*x^2)^2), x])/9 - (6945*Defer[Int][1/((-3 + 4*x)*(-E^(x/(-1
+ 3*x)) + 25*x + 5*x^2)^2), x])/8 - 5*Defer[Int][1/((-1 + 3*x)^2*(-E^(x/(-1 + 3*x)) + 25*x + 5*x^2)), x] - 9*
Defer[Int][1/((-1 + 3*x)*(-E^(x/(-1 + 3*x)) + 25*x + 5*x^2)), x] - 75*Defer[Int][1/((-3 + 4*x)^2*(-E^(x/(-1 +
3*x)) + 25*x + 5*x^2)), x] + 12*Defer[Int][1/((-3 + 4*x)*(-E^(x/(-1 + 3*x)) + 25*x + 5*x^2)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {25 \left (-5 (1-3 x)^2 x^2 (17+8 x)+e^{\frac {x}{-1+3 x}} \left (3-15 x+23 x^2\right )\right )}{\left (3-13 x+12 x^2\right )^2 \left (e^{\frac {x}{-1+3 x}}-5 x (5+x)\right )^2} \, dx\\ &=25 \int \frac {-5 (1-3 x)^2 x^2 (17+8 x)+e^{\frac {x}{-1+3 x}} \left (3-15 x+23 x^2\right )}{\left (3-13 x+12 x^2\right )^2 \left (e^{\frac {x}{-1+3 x}}-5 x (5+x)\right )^2} \, dx\\ &=25 \int \left (-\frac {3-15 x+23 x^2}{(-1+3 x)^2 (-3+4 x)^2 \left (-e^{\frac {x}{-1+3 x}}+25 x+5 x^2\right )}-\frac {5 x \left (5-23 x+34 x^2+18 x^3\right )}{(-1+3 x)^2 (-3+4 x) \left (-e^{\frac {x}{-1+3 x}}+25 x+5 x^2\right )^2}\right ) \, dx\\ &=-\left (25 \int \frac {3-15 x+23 x^2}{(-1+3 x)^2 (-3+4 x)^2 \left (-e^{\frac {x}{-1+3 x}}+25 x+5 x^2\right )} \, dx\right )-125 \int \frac {x \left (5-23 x+34 x^2+18 x^3\right )}{(-1+3 x)^2 (-3+4 x) \left (-e^{\frac {x}{-1+3 x}}+25 x+5 x^2\right )^2} \, dx\\ &=-\left (25 \int \left (\frac {1}{5 (-1+3 x)^2 \left (-e^{\frac {x}{-1+3 x}}+25 x+5 x^2\right )}+\frac {9}{25 (-1+3 x) \left (-e^{\frac {x}{-1+3 x}}+25 x+5 x^2\right )}+\frac {3}{(-3+4 x)^2 \left (-e^{\frac {x}{-1+3 x}}+25 x+5 x^2\right )}-\frac {12}{25 (-3+4 x) \left (-e^{\frac {x}{-1+3 x}}+25 x+5 x^2\right )}\right ) \, dx\right )-125 \int \left (\frac {119}{72 \left (e^{\frac {x}{-1+3 x}}-25 x-5 x^2\right )^2}+\frac {x}{2 \left (-e^{\frac {x}{-1+3 x}}+25 x+5 x^2\right )^2}-\frac {16}{45 (-1+3 x)^2 \left (-e^{\frac {x}{-1+3 x}}+25 x+5 x^2\right )^2}-\frac {229}{225 (-1+3 x) \left (-e^{\frac {x}{-1+3 x}}+25 x+5 x^2\right )^2}+\frac {1389}{200 (-3+4 x) \left (-e^{\frac {x}{-1+3 x}}+25 x+5 x^2\right )^2}\right ) \, dx\\ &=-\left (5 \int \frac {1}{(-1+3 x)^2 \left (-e^{\frac {x}{-1+3 x}}+25 x+5 x^2\right )} \, dx\right )-9 \int \frac {1}{(-1+3 x) \left (-e^{\frac {x}{-1+3 x}}+25 x+5 x^2\right )} \, dx+12 \int \frac {1}{(-3+4 x) \left (-e^{\frac {x}{-1+3 x}}+25 x+5 x^2\right )} \, dx+\frac {400}{9} \int \frac {1}{(-1+3 x)^2 \left (-e^{\frac {x}{-1+3 x}}+25 x+5 x^2\right )^2} \, dx-\frac {125}{2} \int \frac {x}{\left (-e^{\frac {x}{-1+3 x}}+25 x+5 x^2\right )^2} \, dx-75 \int \frac {1}{(-3+4 x)^2 \left (-e^{\frac {x}{-1+3 x}}+25 x+5 x^2\right )} \, dx+\frac {1145}{9} \int \frac {1}{(-1+3 x) \left (-e^{\frac {x}{-1+3 x}}+25 x+5 x^2\right )^2} \, dx-\frac {14875}{72} \int \frac {1}{\left (e^{\frac {x}{-1+3 x}}-25 x-5 x^2\right )^2} \, dx-\frac {6945}{8} \int \frac {1}{(-3+4 x) \left (-e^{\frac {x}{-1+3 x}}+25 x+5 x^2\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.70, size = 32, normalized size = 1.00 \begin {gather*} \frac {25 x}{(-3+4 x) \left (-e^{\frac {x}{-1+3 x}}+5 x (5+x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2125*x^2 + 11750*x^3 - 13125*x^4 - 9000*x^5 + E^(x/(-1 + 3*x))*(75 - 375*x + 575*x^2))/(5625*x^2 -
46500*x^3 + 131350*x^4 - 136700*x^5 + 18025*x^6 + 28200*x^7 + 3600*x^8 + E^((2*x)/(-1 + 3*x))*(9 - 78*x + 241
*x^2 - 312*x^3 + 144*x^4) + E^(x/(-1 + 3*x))*(-450*x + 3810*x^2 - 11270*x^3 + 13190*x^4 - 4080*x^5 - 1440*x^6)
),x]

[Out]

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

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fricas [A]  time = 1.08, size = 36, normalized size = 1.12 \begin {gather*} \frac {25 \, x}{20 \, x^{3} + 85 \, x^{2} - {\left (4 \, x - 3\right )} e^{\left (\frac {x}{3 \, x - 1}\right )} - 75 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((575*x^2-375*x+75)*exp(x/(3*x-1))-9000*x^5-13125*x^4+11750*x^3-2125*x^2)/((144*x^4-312*x^3+241*x^2-
78*x+9)*exp(x/(3*x-1))^2+(-1440*x^6-4080*x^5+13190*x^4-11270*x^3+3810*x^2-450*x)*exp(x/(3*x-1))+3600*x^8+28200
*x^7+18025*x^6-136700*x^5+131350*x^4-46500*x^3+5625*x^2),x, algorithm="fricas")

[Out]

25*x/(20*x^3 + 85*x^2 - (4*x - 3)*e^(x/(3*x - 1)) - 75*x)

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giac [B]  time = 3.06, size = 149, normalized size = 4.66 \begin {gather*} \frac {25 \, {\left (\frac {x}{3 \, x - 1} - \frac {6 \, x^{2}}{{\left (3 \, x - 1\right )}^{2}} + \frac {9 \, x^{3}}{{\left (3 \, x - 1\right )}^{3}}\right )}}{\frac {23 \, x e^{\left (\frac {x}{3 \, x - 1}\right )}}{3 \, x - 1} - \frac {57 \, x^{2} e^{\left (\frac {x}{3 \, x - 1}\right )}}{{\left (3 \, x - 1\right )}^{2}} + \frac {45 \, x^{3} e^{\left (\frac {x}{3 \, x - 1}\right )}}{{\left (3 \, x - 1\right )}^{3}} - \frac {75 \, x}{3 \, x - 1} + \frac {365 \, x^{2}}{{\left (3 \, x - 1\right )}^{2}} - \frac {400 \, x^{3}}{{\left (3 \, x - 1\right )}^{3}} - 3 \, e^{\left (\frac {x}{3 \, x - 1}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((575*x^2-375*x+75)*exp(x/(3*x-1))-9000*x^5-13125*x^4+11750*x^3-2125*x^2)/((144*x^4-312*x^3+241*x^2-
78*x+9)*exp(x/(3*x-1))^2+(-1440*x^6-4080*x^5+13190*x^4-11270*x^3+3810*x^2-450*x)*exp(x/(3*x-1))+3600*x^8+28200
*x^7+18025*x^6-136700*x^5+131350*x^4-46500*x^3+5625*x^2),x, algorithm="giac")

[Out]

25*(x/(3*x - 1) - 6*x^2/(3*x - 1)^2 + 9*x^3/(3*x - 1)^3)/(23*x*e^(x/(3*x - 1))/(3*x - 1) - 57*x^2*e^(x/(3*x -
1))/(3*x - 1)^2 + 45*x^3*e^(x/(3*x - 1))/(3*x - 1)^3 - 75*x/(3*x - 1) + 365*x^2/(3*x - 1)^2 - 400*x^3/(3*x - 1
)^3 - 3*e^(x/(3*x - 1)))

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maple [A]  time = 0.98, size = 34, normalized size = 1.06

method	result	size
risch	\(\frac {25 x}{\left (4 x -3\right ) \left (5 x^{2}+25 x -{\mathrm e}^{\frac {x}{3 x -1}}\right )}\)	\(34\)
norman	\(\frac {75 x^{2}-25 x}{60 x^{4}+235 x^{3}-12 \,{\mathrm e}^{\frac {x}{3 x -1}} x^{2}-310 x^{2}+13 x \,{\mathrm e}^{\frac {x}{3 x -1}}+75 x -3 \,{\mathrm e}^{\frac {x}{3 x -1}}}\)	\(72\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((575*x^2-375*x+75)*exp(x/(3*x-1))-9000*x^5-13125*x^4+11750*x^3-2125*x^2)/((144*x^4-312*x^3+241*x^2-78*x+9
)*exp(x/(3*x-1))^2+(-1440*x^6-4080*x^5+13190*x^4-11270*x^3+3810*x^2-450*x)*exp(x/(3*x-1))+3600*x^8+28200*x^7+1
8025*x^6-136700*x^5+131350*x^4-46500*x^3+5625*x^2),x,method=_RETURNVERBOSE)

[Out]

25*x/(4*x-3)/(5*x^2+25*x-exp(x/(3*x-1)))

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((575*x^2-375*x+75)*exp(x/(3*x-1))-9000*x^5-13125*x^4+11750*x^3-2125*x^2)/((144*x^4-312*x^3+241*x^2-
78*x+9)*exp(x/(3*x-1))^2+(-1440*x^6-4080*x^5+13190*x^4-11270*x^3+3810*x^2-450*x)*exp(x/(3*x-1))+3600*x^8+28200
*x^7+18025*x^6-136700*x^5+131350*x^4-46500*x^3+5625*x^2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int -\frac {2125\,x^2-{\mathrm {e}}^{\frac {x}{3\,x-1}}\,\left (575\,x^2-375\,x+75\right )-11750\,x^3+13125\,x^4+9000\,x^5}{{\mathrm {e}}^{\frac {2\,x}{3\,x-1}}\,\left (144\,x^4-312\,x^3+241\,x^2-78\,x+9\right )-{\mathrm {e}}^{\frac {x}{3\,x-1}}\,\left (1440\,x^6+4080\,x^5-13190\,x^4+11270\,x^3-3810\,x^2+450\,x\right )+5625\,x^2-46500\,x^3+131350\,x^4-136700\,x^5+18025\,x^6+28200\,x^7+3600\,x^8} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2125*x^2 - exp(x/(3*x - 1))*(575*x^2 - 375*x + 75) - 11750*x^3 + 13125*x^4 + 9000*x^5)/(exp((2*x)/(3*x -
1))*(241*x^2 - 78*x - 312*x^3 + 144*x^4 + 9) - exp(x/(3*x - 1))*(450*x - 3810*x^2 + 11270*x^3 - 13190*x^4 + 4
080*x^5 + 1440*x^6) + 5625*x^2 - 46500*x^3 + 131350*x^4 - 136700*x^5 + 18025*x^6 + 28200*x^7 + 3600*x^8),x)

[Out]

int(-(2125*x^2 - exp(x/(3*x - 1))*(575*x^2 - 375*x + 75) - 11750*x^3 + 13125*x^4 + 9000*x^5)/(exp((2*x)/(3*x -
1))*(241*x^2 - 78*x - 312*x^3 + 144*x^4 + 9) - exp(x/(3*x - 1))*(450*x - 3810*x^2 + 11270*x^3 - 13190*x^4 + 4
080*x^5 + 1440*x^6) + 5625*x^2 - 46500*x^3 + 131350*x^4 - 136700*x^5 + 18025*x^6 + 28200*x^7 + 3600*x^8), x)

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sympy [A]  time = 0.22, size = 31, normalized size = 0.97 \begin {gather*} - \frac {25 x}{- 20 x^{3} - 85 x^{2} + 75 x + \left (4 x - 3\right ) e^{\frac {x}{3 x - 1}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((575*x**2-375*x+75)*exp(x/(3*x-1))-9000*x**5-13125*x**4+11750*x**3-2125*x**2)/((144*x**4-312*x**3+2
41*x**2-78*x+9)*exp(x/(3*x-1))**2+(-1440*x**6-4080*x**5+13190*x**4-11270*x**3+3810*x**2-450*x)*exp(x/(3*x-1))+
3600*x**8+28200*x**7+18025*x**6-136700*x**5+131350*x**4-46500*x**3+5625*x**2),x)

[Out]

-25*x/(-20*x**3 - 85*x**2 + 75*x + (4*x - 3)*exp(x/(3*x - 1)))

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