3.64.68 \(\int \frac {-75-150 e^{2 x}-150 x^2+e^{2 x/3} (-15+10 x)+e^x (75+5 e^{2 x/3}+300 x)}{100+4 e^{4 x/3}-400 x^2+400 e^{2 x} x^2+400 x^4+e^{2 x/3} (40-80 x^2)+e^x (400 x+80 e^{2 x/3} x-800 x^3)} \, dx\)

Optimal. Leaf size=32 \[ \frac {3}{4 \left (\frac {5+e^{2 x/3}}{5 \left (e^x-x\right )}+2 x\right )} \]

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Rubi [F]  time = 8.77, 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 {-75-150 e^{2 x}-150 x^2+e^{2 x/3} (-15+10 x)+e^x \left (75+5 e^{2 x/3}+300 x\right )}{100+4 e^{4 x/3}-400 x^2+400 e^{2 x} x^2+400 x^4+e^{2 x/3} \left (40-80 x^2\right )+e^x \left (400 x+80 e^{2 x/3} x-800 x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-75 - 150*E^(2*x) - 150*x^2 + E^((2*x)/3)*(-15 + 10*x) + E^x*(75 + 5*E^((2*x)/3) + 300*x))/(100 + 4*E^((4
*x)/3) - 400*x^2 + 400*E^(2*x)*x^2 + 400*x^4 + E^((2*x)/3)*(40 - 80*x^2) + E^x*(400*x + 80*E^((2*x)/3)*x - 800
*x^3)),x]

[Out]

3/(8*x) - (75*Defer[Int][(5 + E^((2*x)/3) + 10*E^x*x - 10*x^2)^(-2), x])/4 - Defer[Int][E^(x/3)/(5 + E^((2*x)/
3) + 10*E^x*x - 10*x^2)^2, x]/8 - (15*Defer[Int][E^((2*x)/3)/(5 + E^((2*x)/3) + 10*E^x*x - 10*x^2)^2, x])/4 -
(3*Defer[Int][E^((2*x)/3)/(x^4*(5 + E^((2*x)/3) + 10*E^x*x - 10*x^2)^2), x])/800 + (3*Defer[Int][E^(x/3)/(x^3*
(5 + E^((2*x)/3) + 10*E^x*x - 10*x^2)^2), x])/16 - Defer[Int][E^((2*x)/3)/(x^3*(5 + E^((2*x)/3) + 10*E^x*x - 1
0*x^2)^2), x]/800 + Defer[Int][E^(x/3)/(x^2*(5 + E^((2*x)/3) + 10*E^x*x - 10*x^2)^2), x]/16 - (15*Defer[Int][E
^((2*x)/3)/(x^2*(5 + E^((2*x)/3) + 10*E^x*x - 10*x^2)^2), x])/4 - (3*Defer[Int][E^(x/3)/(x*(5 + E^((2*x)/3) +
10*E^x*x - 10*x^2)^2), x])/8 - (5*Defer[Int][E^((2*x)/3)/(x*(5 + E^((2*x)/3) + 10*E^x*x - 10*x^2)^2), x])/2 +
(15*Defer[Int][(E^((2*x)/3)*x)/(5 + E^((2*x)/3) + 10*E^x*x - 10*x^2)^2, x])/4 - (3*Defer[Int][E^(x/3)/(x^3*(5
+ E^((2*x)/3) + 10*E^x*x - 10*x^2)), x])/80 - Defer[Int][E^(x/3)/(x^2*(5 + E^((2*x)/3) + 10*E^x*x - 10*x^2)),
x]/80 + (3*Defer[Int][E^((2*x)/3)/(x^2*(5 + E^((2*x)/3) + 10*E^x*x - 10*x^2)), x])/4 + Defer[Int][E^((2*x)/3)/
(x*(5 + E^((2*x)/3) + 10*E^x*x - 10*x^2)), x]/8 - (3*Defer[Int][1/(x^4*(-5 - E^((2*x)/3) - 10*E^x*x + 10*x^2)^
2), x])/160 - Defer[Int][1/(x^3*(-5 - E^((2*x)/3) - 10*E^x*x + 10*x^2)^2), x]/160 - (747*Defer[Int][1/(x^2*(-5
- E^((2*x)/3) - 10*E^x*x + 10*x^2)^2), x])/80 - (749*Defer[Int][1/(x*(-5 - E^((2*x)/3) - 10*E^x*x + 10*x^2)^2
), x])/80 + (75*Defer[Int][x/(-5 - E^((2*x)/3) - 10*E^x*x + 10*x^2)^2, x])/4 - (3*Defer[Int][1/(x^4*(-5 - E^((
2*x)/3) - 10*E^x*x + 10*x^2)), x])/800 - Defer[Int][1/(x^3*(-5 - E^((2*x)/3) - 10*E^x*x + 10*x^2)), x]/800 - (
15*Defer[Int][1/(x^2*(-5 - E^((2*x)/3) - 10*E^x*x + 10*x^2)), x])/4 - (15*Defer[Int][1/(x*(-5 - E^((2*x)/3) -
10*E^x*x + 10*x^2)), x])/8

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {5 \left (-15+e^{5 x/3}-30 e^{2 x}-e^{2 x/3} (3-2 x)-30 x^2+15 e^x (1+4 x)\right )}{4 \left (5+e^{2 x/3}+10 e^x x-10 x^2\right )^2} \, dx\\ &=\frac {5}{4} \int \frac {-15+e^{5 x/3}-30 e^{2 x}-e^{2 x/3} (3-2 x)-30 x^2+15 e^x (1+4 x)}{\left (5+e^{2 x/3}+10 e^x x-10 x^2\right )^2} \, dx\\ &=\frac {5}{4} \int \left (-\frac {3}{10 x^2}-\frac {3+x-30 e^{x/3} x+3000 x^2-10 e^{x/3} x^2+600 e^{2 x/3} x^2+1500 x^3+100 e^{2 x/3} x^3}{1000 x^4 \left (-5-e^{2 x/3}-10 e^x x+10 x^2\right )}+\frac {-15-3 e^{2 x/3}-5 x+150 e^{x/3} x-e^{2 x/3} x-7470 x^2+50 e^{x/3} x^2-3000 e^{2 x/3} x^2-7490 x^3-300 e^{x/3} x^3-2000 e^{2 x/3} x^3-15000 x^4-100 e^{x/3} x^4-3000 e^{2 x/3} x^4+15000 x^5+3000 e^{2 x/3} x^5}{1000 x^4 \left (-5-e^{2 x/3}-10 e^x x+10 x^2\right )^2}\right ) \, dx\\ &=\frac {3}{8 x}-\frac {1}{800} \int \frac {3+x-30 e^{x/3} x+3000 x^2-10 e^{x/3} x^2+600 e^{2 x/3} x^2+1500 x^3+100 e^{2 x/3} x^3}{x^4 \left (-5-e^{2 x/3}-10 e^x x+10 x^2\right )} \, dx+\frac {1}{800} \int \frac {-15-3 e^{2 x/3}-5 x+150 e^{x/3} x-e^{2 x/3} x-7470 x^2+50 e^{x/3} x^2-3000 e^{2 x/3} x^2-7490 x^3-300 e^{x/3} x^3-2000 e^{2 x/3} x^3-15000 x^4-100 e^{x/3} x^4-3000 e^{2 x/3} x^4+15000 x^5+3000 e^{2 x/3} x^5}{x^4 \left (-5-e^{2 x/3}-10 e^x x+10 x^2\right )^2} \, dx\\ &=\frac {3}{8 x}-\frac {1}{800} \int \left (\frac {30 e^{x/3}}{x^3 \left (5+e^{2 x/3}+10 e^x x-10 x^2\right )}+\frac {10 e^{x/3}}{x^2 \left (5+e^{2 x/3}+10 e^x x-10 x^2\right )}-\frac {600 e^{2 x/3}}{x^2 \left (5+e^{2 x/3}+10 e^x x-10 x^2\right )}-\frac {100 e^{2 x/3}}{x \left (5+e^{2 x/3}+10 e^x x-10 x^2\right )}+\frac {3}{x^4 \left (-5-e^{2 x/3}-10 e^x x+10 x^2\right )}+\frac {1}{x^3 \left (-5-e^{2 x/3}-10 e^x x+10 x^2\right )}+\frac {3000}{x^2 \left (-5-e^{2 x/3}-10 e^x x+10 x^2\right )}+\frac {1500}{x \left (-5-e^{2 x/3}-10 e^x x+10 x^2\right )}\right ) \, dx+\frac {1}{800} \int \frac {-50 e^{x/3} x \left (-3-x+6 x^2+2 x^3\right )+e^{2 x/3} \left (-3-x-3000 x^2-2000 x^3-3000 x^4+3000 x^5\right )+5 \left (-3-x-1494 x^2-1498 x^3-3000 x^4+3000 x^5\right )}{x^4 \left (5+e^{2 x/3}+10 e^x x-10 x^2\right )^2} \, dx\\ &=\frac {3}{8 x}-\frac {1}{800} \int \frac {1}{x^3 \left (-5-e^{2 x/3}-10 e^x x+10 x^2\right )} \, dx+\frac {1}{800} \int \left (-\frac {15000}{\left (5+e^{2 x/3}+10 e^x x-10 x^2\right )^2}-\frac {100 e^{x/3}}{\left (5+e^{2 x/3}+10 e^x x-10 x^2\right )^2}-\frac {3000 e^{2 x/3}}{\left (5+e^{2 x/3}+10 e^x x-10 x^2\right )^2}-\frac {3 e^{2 x/3}}{x^4 \left (5+e^{2 x/3}+10 e^x x-10 x^2\right )^2}+\frac {150 e^{x/3}}{x^3 \left (5+e^{2 x/3}+10 e^x x-10 x^2\right )^2}-\frac {e^{2 x/3}}{x^3 \left (5+e^{2 x/3}+10 e^x x-10 x^2\right )^2}+\frac {50 e^{x/3}}{x^2 \left (5+e^{2 x/3}+10 e^x x-10 x^2\right )^2}-\frac {3000 e^{2 x/3}}{x^2 \left (5+e^{2 x/3}+10 e^x x-10 x^2\right )^2}-\frac {300 e^{x/3}}{x \left (5+e^{2 x/3}+10 e^x x-10 x^2\right )^2}-\frac {2000 e^{2 x/3}}{x \left (5+e^{2 x/3}+10 e^x x-10 x^2\right )^2}+\frac {3000 e^{2 x/3} x}{\left (5+e^{2 x/3}+10 e^x x-10 x^2\right )^2}-\frac {15}{x^4 \left (-5-e^{2 x/3}-10 e^x x+10 x^2\right )^2}-\frac {5}{x^3 \left (-5-e^{2 x/3}-10 e^x x+10 x^2\right )^2}-\frac {7470}{x^2 \left (-5-e^{2 x/3}-10 e^x x+10 x^2\right )^2}-\frac {7490}{x \left (-5-e^{2 x/3}-10 e^x x+10 x^2\right )^2}+\frac {15000 x}{\left (-5-e^{2 x/3}-10 e^x x+10 x^2\right )^2}\right ) \, dx-\frac {3}{800} \int \frac {1}{x^4 \left (-5-e^{2 x/3}-10 e^x x+10 x^2\right )} \, dx-\frac {1}{80} \int \frac {e^{x/3}}{x^2 \left (5+e^{2 x/3}+10 e^x x-10 x^2\right )} \, dx-\frac {3}{80} \int \frac {e^{x/3}}{x^3 \left (5+e^{2 x/3}+10 e^x x-10 x^2\right )} \, dx+\frac {1}{8} \int \frac {e^{2 x/3}}{x \left (5+e^{2 x/3}+10 e^x x-10 x^2\right )} \, dx+\frac {3}{4} \int \frac {e^{2 x/3}}{x^2 \left (5+e^{2 x/3}+10 e^x x-10 x^2\right )} \, dx-\frac {15}{8} \int \frac {1}{x \left (-5-e^{2 x/3}-10 e^x x+10 x^2\right )} \, dx-\frac {15}{4} \int \frac {1}{x^2 \left (-5-e^{2 x/3}-10 e^x x+10 x^2\right )} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.84, size = 33, normalized size = 1.03 \begin {gather*} \frac {15 \left (e^x-x\right )}{4 \left (5+e^{2 x/3}+10 e^x x-10 x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-75 - 150*E^(2*x) - 150*x^2 + E^((2*x)/3)*(-15 + 10*x) + E^x*(75 + 5*E^((2*x)/3) + 300*x))/(100 + 4
*E^((4*x)/3) - 400*x^2 + 400*E^(2*x)*x^2 + 400*x^4 + E^((2*x)/3)*(40 - 80*x^2) + E^x*(400*x + 80*E^((2*x)/3)*x
- 800*x^3)),x]

[Out]

(15*(E^x - x))/(4*(5 + E^((2*x)/3) + 10*E^x*x - 10*x^2))

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fricas [A]  time = 0.64, size = 28, normalized size = 0.88 \begin {gather*} \frac {15 \, {\left (x - e^{x}\right )}}{4 \, {\left (10 \, x^{2} - 10 \, x e^{x} - e^{\left (\frac {2}{3} \, x\right )} - 5\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-150*exp(x)^2+(5*exp(2/3*x)+300*x+75)*exp(x)+(10*x-15)*exp(2/3*x)-150*x^2-75)/(400*exp(x)^2*x^2+(80
*x*exp(2/3*x)-800*x^3+400*x)*exp(x)+4*exp(2/3*x)^2+(-80*x^2+40)*exp(2/3*x)+400*x^4-400*x^2+100),x, algorithm="
fricas")

[Out]

15/4*(x - e^x)/(10*x^2 - 10*x*e^x - e^(2/3*x) - 5)

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giac [A]  time = 1.14, size = 43, normalized size = 1.34 \begin {gather*} \frac {3 \, {\left (10 \, x^{2} - 10 \, x e^{x} + e^{\left (\frac {2}{3} \, x\right )} + 5\right )}}{8 \, {\left (10 \, x^{3} - 10 \, x^{2} e^{x} - x e^{\left (\frac {2}{3} \, x\right )} - 5 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-150*exp(x)^2+(5*exp(2/3*x)+300*x+75)*exp(x)+(10*x-15)*exp(2/3*x)-150*x^2-75)/(400*exp(x)^2*x^2+(80
*x*exp(2/3*x)-800*x^3+400*x)*exp(x)+4*exp(2/3*x)^2+(-80*x^2+40)*exp(2/3*x)+400*x^4-400*x^2+100),x, algorithm="
giac")

[Out]

3/8*(10*x^2 - 10*x*e^x + e^(2/3*x) + 5)/(10*x^3 - 10*x^2*e^x - x*e^(2/3*x) - 5*x)

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maple [A]  time = 0.09, size = 38, normalized size = 1.19

method	result	size
risch	\(\frac {3}{8 x}+\frac {\frac {3 \,{\mathrm e}^{\frac {2 x}{3}}}{8}+\frac {15}{8}}{x \left (-10 \,{\mathrm e}^{x} x +10 x^{2}-{\mathrm e}^{\frac {2 x}{3}}-5\right )}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-150*exp(x)^2+(5*exp(2/3*x)+300*x+75)*exp(x)+(10*x-15)*exp(2/3*x)-150*x^2-75)/(400*exp(x)^2*x^2+(80*x*exp
(2/3*x)-800*x^3+400*x)*exp(x)+4*exp(2/3*x)^2+(-80*x^2+40)*exp(2/3*x)+400*x^4-400*x^2+100),x,method=_RETURNVERB
OSE)

[Out]

3/8/x+3/8*(exp(2/3*x)+5)/x/(-10*exp(x)*x+10*x^2-exp(2/3*x)-5)

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maxima [A]  time = 0.52, size = 28, normalized size = 0.88 \begin {gather*} \frac {15 \, {\left (x - e^{x}\right )}}{4 \, {\left (10 \, x^{2} - 10 \, x e^{x} - e^{\left (\frac {2}{3} \, x\right )} - 5\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-150*exp(x)^2+(5*exp(2/3*x)+300*x+75)*exp(x)+(10*x-15)*exp(2/3*x)-150*x^2-75)/(400*exp(x)^2*x^2+(80
*x*exp(2/3*x)-800*x^3+400*x)*exp(x)+4*exp(2/3*x)^2+(-80*x^2+40)*exp(2/3*x)+400*x^4-400*x^2+100),x, algorithm="
maxima")

[Out]

15/4*(x - e^x)/(10*x^2 - 10*x*e^x - e^(2/3*x) - 5)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int -\frac {150\,{\mathrm {e}}^{2\,x}-{\mathrm {e}}^x\,\left (300\,x+5\,{\mathrm {e}}^{\frac {2\,x}{3}}+75\right )-{\mathrm {e}}^{\frac {2\,x}{3}}\,\left (10\,x-15\right )+150\,x^2+75}{4\,{\mathrm {e}}^{\frac {4\,x}{3}}-{\mathrm {e}}^{\frac {2\,x}{3}}\,\left (80\,x^2-40\right )+400\,x^2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^x\,\left (400\,x+80\,x\,{\mathrm {e}}^{\frac {2\,x}{3}}-800\,x^3\right )-400\,x^2+400\,x^4+100} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(150*exp(2*x) - exp(x)*(300*x + 5*exp((2*x)/3) + 75) - exp((2*x)/3)*(10*x - 15) + 150*x^2 + 75)/(4*exp((4
*x)/3) - exp((2*x)/3)*(80*x^2 - 40) + 400*x^2*exp(2*x) + exp(x)*(400*x + 80*x*exp((2*x)/3) - 800*x^3) - 400*x^
2 + 400*x^4 + 100),x)

[Out]

int(-(150*exp(2*x) - exp(x)*(300*x + 5*exp((2*x)/3) + 75) - exp((2*x)/3)*(10*x - 15) + 150*x^2 + 75)/(4*exp((4
*x)/3) - exp((2*x)/3)*(80*x^2 - 40) + 400*x^2*exp(2*x) + exp(x)*(400*x + 80*x*exp((2*x)/3) - 800*x^3) - 400*x^
2 + 400*x^4 + 100), x)

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sympy [A]  time = 0.27, size = 42, normalized size = 1.31 \begin {gather*} \frac {- 3 e^{\frac {2 x}{3}} - 15}{- 80 x^{3} + 80 x^{2} e^{x} + 8 x e^{\frac {2 x}{3}} + 40 x} + \frac {3}{8 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-150*exp(x)**2+(5*exp(2/3*x)+300*x+75)*exp(x)+(10*x-15)*exp(2/3*x)-150*x**2-75)/(400*exp(x)**2*x**2
+(80*x*exp(2/3*x)-800*x**3+400*x)*exp(x)+4*exp(2/3*x)**2+(-80*x**2+40)*exp(2/3*x)+400*x**4-400*x**2+100),x)

[Out]

(-3*exp(2*x/3) - 15)/(-80*x**3 + 80*x**2*exp(x) + 8*x*exp(2*x/3) + 40*x) + 3/(8*x)

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