3.29.12 \(\int \frac {108 x+126 x^2+36 x^3+50 x^5+(100 x^3+150 x^4+50 x^5) \log (3)+e^{3 x} (-50 x^2+(-100-150 x-50 x^2) \log (3))+e^{2 x} (150 x^3+(300 x+450 x^2+150 x^3) \log (3))+e^x (-36-126 x-90 x^2-18 x^3-150 x^4+(-300 x^2-450 x^3-150 x^4) \log (3))}{-200 x^5-300 x^6-150 x^7-25 x^8+e^{3 x} (200 x^2+300 x^3+150 x^4+25 x^5)+e^{2 x} (-600 x^3-900 x^4-450 x^5-75 x^6)+e^x (600 x^4+900 x^5+450 x^6+75 x^7)} \, dx\)

Optimal. Leaf size=34 \[ 3+\frac {\frac {x}{2+x}+\frac {9}{25 \left (-e^x+x\right )^2}+\log (3)}{x (2+x)} \]

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Rubi [F]  time = 3.60, 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 {108 x+126 x^2+36 x^3+50 x^5+\left (100 x^3+150 x^4+50 x^5\right ) \log (3)+e^{3 x} \left (-50 x^2+\left (-100-150 x-50 x^2\right ) \log (3)\right )+e^{2 x} \left (150 x^3+\left (300 x+450 x^2+150 x^3\right ) \log (3)\right )+e^x \left (-36-126 x-90 x^2-18 x^3-150 x^4+\left (-300 x^2-450 x^3-150 x^4\right ) \log (3)\right )}{-200 x^5-300 x^6-150 x^7-25 x^8+e^{3 x} \left (200 x^2+300 x^3+150 x^4+25 x^5\right )+e^{2 x} \left (-600 x^3-900 x^4-450 x^5-75 x^6\right )+e^x \left (600 x^4+900 x^5+450 x^6+75 x^7\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(108*x + 126*x^2 + 36*x^3 + 50*x^5 + (100*x^3 + 150*x^4 + 50*x^5)*Log[3] + E^(3*x)*(-50*x^2 + (-100 - 150*
x - 50*x^2)*Log[3]) + E^(2*x)*(150*x^3 + (300*x + 450*x^2 + 150*x^3)*Log[3]) + E^x*(-36 - 126*x - 90*x^2 - 18*
x^3 - 150*x^4 + (-300*x^2 - 450*x^3 - 150*x^4)*Log[3]))/(-200*x^5 - 300*x^6 - 150*x^7 - 25*x^8 + E^(3*x)*(200*
x^2 + 300*x^3 + 150*x^4 + 25*x^5) + E^(2*x)*(-600*x^3 - 900*x^4 - 450*x^5 - 75*x^6) + E^x*(600*x^4 + 900*x^5 +
450*x^6 + 75*x^7)),x]

[Out]

(2 + x)^(-2) - Log[3]/(2*(2 + x)) + Log[9]/(4*x) - (9*Defer[Int][1/((E^x - x)^2*x^2), x])/50 - (9*Defer[Int][1
/((E^x - x)^2*x), x])/25 + (9*Defer[Int][1/((E^x - x)^2*(2 + x)^2), x])/50 + (9*Defer[Int][1/((E^x - x)^2*(2 +
x)), x])/25 - (9*Defer[Int][1/(x*(-E^x + x)^3), x])/25 + (27*Defer[Int][1/((2 + x)*(-E^x + x)^3), x])/25

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 x \left (54+63 x+75 x^3 \log (3)+25 x^4 (1+\log (3))+2 x^2 (9+25 \log (3))\right )-6 e^x \left (6+21 x+25 x^4 (1+\log (3))+5 x^2 (3+10 \log (3))+x^3 (3+75 \log (3))\right )-50 e^{3 x} \left (x^2 (1+\log (3))+\log (9)+x \log (27)\right )+150 e^{2 x} x \left (x^2 (1+\log (3))+\log (9)+x \log (27)\right )}{25 \left (e^x-x\right )^3 x^2 (2+x)^3} \, dx\\ &=\frac {1}{25} \int \frac {2 x \left (54+63 x+75 x^3 \log (3)+25 x^4 (1+\log (3))+2 x^2 (9+25 \log (3))\right )-6 e^x \left (6+21 x+25 x^4 (1+\log (3))+5 x^2 (3+10 \log (3))+x^3 (3+75 \log (3))\right )-50 e^{3 x} \left (x^2 (1+\log (3))+\log (9)+x \log (27)\right )+150 e^{2 x} x \left (x^2 (1+\log (3))+\log (9)+x \log (27)\right )}{\left (e^x-x\right )^3 x^2 (2+x)^3} \, dx\\ &=\frac {1}{25} \int \left (\frac {18 (-1+x)}{x (2+x) \left (-e^x+x\right )^3}-\frac {18 \left (1+3 x+x^2\right )}{\left (e^x-x\right )^2 x^2 (2+x)^2}+\frac {50 \left (-x^2 (1+\log (3))-\log (9)-x \log (27)\right )}{x^2 (2+x)^3}\right ) \, dx\\ &=\frac {18}{25} \int \frac {-1+x}{x (2+x) \left (-e^x+x\right )^3} \, dx-\frac {18}{25} \int \frac {1+3 x+x^2}{\left (e^x-x\right )^2 x^2 (2+x)^2} \, dx+2 \int \frac {-x^2 (1+\log (3))-\log (9)-x \log (27)}{x^2 (2+x)^3} \, dx\\ &=-\left (\frac {18}{25} \int \left (\frac {1}{4 \left (e^x-x\right )^2 x^2}+\frac {1}{2 \left (e^x-x\right )^2 x}-\frac {1}{4 \left (e^x-x\right )^2 (2+x)^2}-\frac {1}{2 \left (e^x-x\right )^2 (2+x)}\right ) \, dx\right )+\frac {18}{25} \int \left (-\frac {1}{2 x \left (-e^x+x\right )^3}+\frac {3}{2 (2+x) \left (-e^x+x\right )^3}\right ) \, dx+2 \int \left (-\frac {1}{(2+x)^3}+\frac {\log (3)}{4 (2+x)^2}-\frac {\log (9)}{8 x^2}\right ) \, dx\\ &=\frac {1}{(2+x)^2}-\frac {\log (3)}{2 (2+x)}+\frac {\log (9)}{4 x}-\frac {9}{50} \int \frac {1}{\left (e^x-x\right )^2 x^2} \, dx+\frac {9}{50} \int \frac {1}{\left (e^x-x\right )^2 (2+x)^2} \, dx-\frac {9}{25} \int \frac {1}{\left (e^x-x\right )^2 x} \, dx+\frac {9}{25} \int \frac {1}{\left (e^x-x\right )^2 (2+x)} \, dx-\frac {9}{25} \int \frac {1}{x \left (-e^x+x\right )^3} \, dx+\frac {27}{25} \int \frac {1}{(2+x) \left (-e^x+x\right )^3} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.25, size = 45, normalized size = 1.32 \begin {gather*} \frac {1}{100} \left (-\frac {50 (-2+x \log (3)+\log (9))}{(2+x)^2}+\frac {\frac {36}{\left (e^x-x\right )^2 (2+x)}+25 \log (9)}{x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(108*x + 126*x^2 + 36*x^3 + 50*x^5 + (100*x^3 + 150*x^4 + 50*x^5)*Log[3] + E^(3*x)*(-50*x^2 + (-100
- 150*x - 50*x^2)*Log[3]) + E^(2*x)*(150*x^3 + (300*x + 450*x^2 + 150*x^3)*Log[3]) + E^x*(-36 - 126*x - 90*x^2
- 18*x^3 - 150*x^4 + (-300*x^2 - 450*x^3 - 150*x^4)*Log[3]))/(-200*x^5 - 300*x^6 - 150*x^7 - 25*x^8 + E^(3*x)
*(200*x^2 + 300*x^3 + 150*x^4 + 25*x^5) + E^(2*x)*(-600*x^3 - 900*x^4 - 450*x^5 - 75*x^6) + E^x*(600*x^4 + 900
*x^5 + 450*x^6 + 75*x^7)),x]

[Out]

((-50*(-2 + x*Log[3] + Log[9]))/(2 + x)^2 + (36/((E^x - x)^2*(2 + x)) + 25*Log[9])/x)/100

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fricas [B]  time = 2.78, size = 108, normalized size = 3.18 \begin {gather*} \frac {25 \, x^{3} + 25 \, {\left ({\left (x + 2\right )} \log \relax (3) + x\right )} e^{\left (2 \, x\right )} - 50 \, {\left (x^{2} + {\left (x^{2} + 2 \, x\right )} \log \relax (3)\right )} e^{x} + 25 \, {\left (x^{3} + 2 \, x^{2}\right )} \log \relax (3) + 9 \, x + 18}{25 \, {\left (x^{5} + 4 \, x^{4} + 4 \, x^{3} + {\left (x^{3} + 4 \, x^{2} + 4 \, x\right )} e^{\left (2 \, x\right )} - 2 \, {\left (x^{4} + 4 \, x^{3} + 4 \, x^{2}\right )} e^{x}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-50*x^2-150*x-100)*log(3)-50*x^2)*exp(x)^3+((150*x^3+450*x^2+300*x)*log(3)+150*x^3)*exp(x)^2+((-1
50*x^4-450*x^3-300*x^2)*log(3)-150*x^4-18*x^3-90*x^2-126*x-36)*exp(x)+(50*x^5+150*x^4+100*x^3)*log(3)+50*x^5+3
6*x^3+126*x^2+108*x)/((25*x^5+150*x^4+300*x^3+200*x^2)*exp(x)^3+(-75*x^6-450*x^5-900*x^4-600*x^3)*exp(x)^2+(75
*x^7+450*x^6+900*x^5+600*x^4)*exp(x)-25*x^8-150*x^7-300*x^6-200*x^5),x, algorithm="fricas")

[Out]

1/25*(25*x^3 + 25*((x + 2)*log(3) + x)*e^(2*x) - 50*(x^2 + (x^2 + 2*x)*log(3))*e^x + 25*(x^3 + 2*x^2)*log(3) +
9*x + 18)/(x^5 + 4*x^4 + 4*x^3 + (x^3 + 4*x^2 + 4*x)*e^(2*x) - 2*(x^4 + 4*x^3 + 4*x^2)*e^x)

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giac [B]  time = 0.34, size = 134, normalized size = 3.94 \begin {gather*} \frac {25 \, x^{3} \log \relax (3) - 50 \, x^{2} e^{x} \log \relax (3) + 25 \, x^{3} - 50 \, x^{2} e^{x} + 50 \, x^{2} \log \relax (3) + 25 \, x e^{\left (2 \, x\right )} \log \relax (3) - 100 \, x e^{x} \log \relax (3) + 25 \, x e^{\left (2 \, x\right )} + 50 \, e^{\left (2 \, x\right )} \log \relax (3) + 9 \, x + 18}{25 \, {\left (x^{5} - 2 \, x^{4} e^{x} + 4 \, x^{4} + x^{3} e^{\left (2 \, x\right )} - 8 \, x^{3} e^{x} + 4 \, x^{3} + 4 \, x^{2} e^{\left (2 \, x\right )} - 8 \, x^{2} e^{x} + 4 \, x e^{\left (2 \, x\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-50*x^2-150*x-100)*log(3)-50*x^2)*exp(x)^3+((150*x^3+450*x^2+300*x)*log(3)+150*x^3)*exp(x)^2+((-1
50*x^4-450*x^3-300*x^2)*log(3)-150*x^4-18*x^3-90*x^2-126*x-36)*exp(x)+(50*x^5+150*x^4+100*x^3)*log(3)+50*x^5+3
6*x^3+126*x^2+108*x)/((25*x^5+150*x^4+300*x^3+200*x^2)*exp(x)^3+(-75*x^6-450*x^5-900*x^4-600*x^3)*exp(x)^2+(75
*x^7+450*x^6+900*x^5+600*x^4)*exp(x)-25*x^8-150*x^7-300*x^6-200*x^5),x, algorithm="giac")

[Out]

1/25*(25*x^3*log(3) - 50*x^2*e^x*log(3) + 25*x^3 - 50*x^2*e^x + 50*x^2*log(3) + 25*x*e^(2*x)*log(3) - 100*x*e^
x*log(3) + 25*x*e^(2*x) + 50*e^(2*x)*log(3) + 9*x + 18)/(x^5 - 2*x^4*e^x + 4*x^4 + x^3*e^(2*x) - 8*x^3*e^x + 4
*x^3 + 4*x^2*e^(2*x) - 8*x^2*e^x + 4*x*e^(2*x))

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maple [A]  time = 0.06, size = 45, normalized size = 1.32

method	result	size
risch	\(\frac {\left (\ln \relax (3)+1\right ) x +2 \ln \relax (3)}{x \left (x^{2}+4 x +4\right )}+\frac {9}{25 x \left (2+x \right ) \left (x -{\mathrm e}^{x}\right )^{2}}\)	\(45\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-50*x^2-150*x-100)*ln(3)-50*x^2)*exp(x)^3+((150*x^3+450*x^2+300*x)*ln(3)+150*x^3)*exp(x)^2+((-150*x^4-4
50*x^3-300*x^2)*ln(3)-150*x^4-18*x^3-90*x^2-126*x-36)*exp(x)+(50*x^5+150*x^4+100*x^3)*ln(3)+50*x^5+36*x^3+126*
x^2+108*x)/((25*x^5+150*x^4+300*x^3+200*x^2)*exp(x)^3+(-75*x^6-450*x^5-900*x^4-600*x^3)*exp(x)^2+(75*x^7+450*x
^6+900*x^5+600*x^4)*exp(x)-25*x^8-150*x^7-300*x^6-200*x^5),x,method=_RETURNVERBOSE)

[Out]

((ln(3)+1)*x+2*ln(3))/x/(x^2+4*x+4)+9/25/x/(2+x)/(x-exp(x))^2

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maxima [B]  time = 1.13, size = 109, normalized size = 3.21 \begin {gather*} \frac {25 \, x^{3} {\left (\log \relax (3) + 1\right )} + 50 \, x^{2} \log \relax (3) + 25 \, {\left (x {\left (\log \relax (3) + 1\right )} + 2 \, \log \relax (3)\right )} e^{\left (2 \, x\right )} - 50 \, {\left (x^{2} {\left (\log \relax (3) + 1\right )} + 2 \, x \log \relax (3)\right )} e^{x} + 9 \, x + 18}{25 \, {\left (x^{5} + 4 \, x^{4} + 4 \, x^{3} + {\left (x^{3} + 4 \, x^{2} + 4 \, x\right )} e^{\left (2 \, x\right )} - 2 \, {\left (x^{4} + 4 \, x^{3} + 4 \, x^{2}\right )} e^{x}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-50*x^2-150*x-100)*log(3)-50*x^2)*exp(x)^3+((150*x^3+450*x^2+300*x)*log(3)+150*x^3)*exp(x)^2+((-1
50*x^4-450*x^3-300*x^2)*log(3)-150*x^4-18*x^3-90*x^2-126*x-36)*exp(x)+(50*x^5+150*x^4+100*x^3)*log(3)+50*x^5+3
6*x^3+126*x^2+108*x)/((25*x^5+150*x^4+300*x^3+200*x^2)*exp(x)^3+(-75*x^6-450*x^5-900*x^4-600*x^3)*exp(x)^2+(75
*x^7+450*x^6+900*x^5+600*x^4)*exp(x)-25*x^8-150*x^7-300*x^6-200*x^5),x, algorithm="maxima")

[Out]

1/25*(25*x^3*(log(3) + 1) + 50*x^2*log(3) + 25*(x*(log(3) + 1) + 2*log(3))*e^(2*x) - 50*(x^2*(log(3) + 1) + 2*
x*log(3))*e^x + 9*x + 18)/(x^5 + 4*x^4 + 4*x^3 + (x^3 + 4*x^2 + 4*x)*e^(2*x) - 2*(x^4 + 4*x^3 + 4*x^2)*e^x)

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mupad [B]  time = 2.23, size = 99, normalized size = 2.91 \begin {gather*} \frac {2\,\ln \relax (3)+x\,\left (\frac {\ln \relax (9)}{2}+1\right )}{x^3+4\,x^2+4\,x}+\frac {3\,\left (50\,x^2\,\ln \relax (3)+75\,x^3\,\ln \relax (3)-25\,x^2\,\ln \relax (9)-25\,x^3\,\ln \left (27\right )+9\,x^2+3\,x^3-12\right )}{25\,x\,\left (x-1\right )\,{\left (x+2\right )}^3\,\left ({\mathrm {e}}^{2\,x}-2\,x\,{\mathrm {e}}^x+x^2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(108*x - exp(3*x)*(log(3)*(150*x + 50*x^2 + 100) + 50*x^2) + exp(2*x)*(log(3)*(300*x + 450*x^2 + 150*x^3)
+ 150*x^3) + log(3)*(100*x^3 + 150*x^4 + 50*x^5) + 126*x^2 + 36*x^3 + 50*x^5 - exp(x)*(126*x + log(3)*(300*x^
2 + 450*x^3 + 150*x^4) + 90*x^2 + 18*x^3 + 150*x^4 + 36))/(exp(2*x)*(600*x^3 + 900*x^4 + 450*x^5 + 75*x^6) - e
xp(3*x)*(200*x^2 + 300*x^3 + 150*x^4 + 25*x^5) - exp(x)*(600*x^4 + 900*x^5 + 450*x^6 + 75*x^7) + 200*x^5 + 300
*x^6 + 150*x^7 + 25*x^8),x)

[Out]

(2*log(3) + x*(log(9)/2 + 1))/(4*x + 4*x^2 + x^3) + (3*(50*x^2*log(3) + 75*x^3*log(3) - 25*x^2*log(9) - 25*x^3
*log(27) + 9*x^2 + 3*x^3 - 12))/(25*x*(x - 1)*(x + 2)^3*(exp(2*x) - 2*x*exp(x) + x^2))

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sympy [B]  time = 0.54, size = 65, normalized size = 1.91 \begin {gather*} - \frac {x \left (- \log {\relax (3 )} - 1\right ) - 2 \log {\relax (3 )}}{x^{3} + 4 x^{2} + 4 x} + \frac {9}{25 x^{4} + 50 x^{3} + \left (25 x^{2} + 50 x\right ) e^{2 x} + \left (- 50 x^{3} - 100 x^{2}\right ) e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-50*x**2-150*x-100)*ln(3)-50*x**2)*exp(x)**3+((150*x**3+450*x**2+300*x)*ln(3)+150*x**3)*exp(x)**2
+((-150*x**4-450*x**3-300*x**2)*ln(3)-150*x**4-18*x**3-90*x**2-126*x-36)*exp(x)+(50*x**5+150*x**4+100*x**3)*ln
(3)+50*x**5+36*x**3+126*x**2+108*x)/((25*x**5+150*x**4+300*x**3+200*x**2)*exp(x)**3+(-75*x**6-450*x**5-900*x**
4-600*x**3)*exp(x)**2+(75*x**7+450*x**6+900*x**5+600*x**4)*exp(x)-25*x**8-150*x**7-300*x**6-200*x**5),x)

[Out]

-(x*(-log(3) - 1) - 2*log(3))/(x**3 + 4*x**2 + 4*x) + 9/(25*x**4 + 50*x**3 + (25*x**2 + 50*x)*exp(2*x) + (-50*
x**3 - 100*x**2)*exp(x))

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