3.64.76 \(\int \frac {400-1200 x+900 x^2+e^{\frac {x^2}{-2+3 x}} (20 x^3-15 x^4)}{1600-3200 x-800 x^2+2400 x^3+900 x^4+e^{2 e^5} (4 x^2-12 x^3+9 x^4)+e^{\frac {2 x^2}{-2+3 x}} (4 x^2-12 x^3+9 x^4)+e^{\frac {x^2}{-2+3 x}} (160 x-400 x^2+120 x^3+180 x^4)+e^{e^5} (160 x-400 x^2+120 x^3+180 x^4+e^{\frac {x^2}{-2+3 x}} (8 x^2-24 x^3+18 x^4))} \, dx\)

Optimal. Leaf size=33 \[ \frac {x}{4+2 x+\frac {1}{5} \left (e^{e^5}+e^{\frac {x^2}{-2+3 x}}\right ) x} \]

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Rubi [F]  time = 5.19, 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 {400-1200 x+900 x^2+e^{\frac {x^2}{-2+3 x}} \left (20 x^3-15 x^4\right )}{1600-3200 x-800 x^2+2400 x^3+900 x^4+e^{2 e^5} \left (4 x^2-12 x^3+9 x^4\right )+e^{\frac {2 x^2}{-2+3 x}} \left (4 x^2-12 x^3+9 x^4\right )+e^{\frac {x^2}{-2+3 x}} \left (160 x-400 x^2+120 x^3+180 x^4\right )+e^{e^5} \left (160 x-400 x^2+120 x^3+180 x^4+e^{\frac {x^2}{-2+3 x}} \left (8 x^2-24 x^3+18 x^4\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(400 - 1200*x + 900*x^2 + E^(x^2/(-2 + 3*x))*(20*x^3 - 15*x^4))/(1600 - 3200*x - 800*x^2 + 2400*x^3 + 900*
x^4 + E^(2*E^5)*(4*x^2 - 12*x^3 + 9*x^4) + E^((2*x^2)/(-2 + 3*x))*(4*x^2 - 12*x^3 + 9*x^4) + E^(x^2/(-2 + 3*x)
)*(160*x - 400*x^2 + 120*x^3 + 180*x^4) + E^E^5*(160*x - 400*x^2 + 120*x^3 + 180*x^4 + E^(x^2/(-2 + 3*x))*(8*x
^2 - 24*x^3 + 18*x^4))),x]

[Out]

(5*Defer[Int][x/(-20 - E^(x^2/(-2 + 3*x))*x - 10*(1 + E^E^5/10)*x), x])/3 + (20*(125 - E^E^5)*Defer[Int][(20 +
E^(x^2/(-2 + 3*x))*x + 10*(1 + E^E^5/10)*x)^(-2), x])/27 - (80*(40 + E^E^5)*Defer[Int][1/((2 - 3*x)^2*(20 + E
^(x^2/(-2 + 3*x))*x + 10*(1 + E^E^5/10)*x)^2), x])/27 + (80*(25 + E^E^5)*Defer[Int][1/((2 - 3*x)*(20 + E^(x^2/
(-2 + 3*x))*x + 10*(1 + E^E^5/10)*x)^2), x])/27 + (100*Defer[Int][x/(20 + E^(x^2/(-2 + 3*x))*x + 10*(1 + E^E^5
/10)*x)^2, x])/3 + (5*(10 + E^E^5)*Defer[Int][x^2/(20 + E^(x^2/(-2 + 3*x))*x + 10*(1 + E^E^5/10)*x)^2, x])/3 +
(40*Defer[Int][1/((2 - 3*x)^2*(20 + E^(x^2/(-2 + 3*x))*x + 10*(1 + E^E^5/10)*x)), x])/9 + (20*Defer[Int][1/((
-2 + 3*x)*(20 + E^(x^2/(-2 + 3*x))*x + 10*(1 + E^E^5/10)*x)), x])/9

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {5 \left (80-240 x+180 x^2+4 e^{\frac {x^2}{-2+3 x}} x^3-3 e^{\frac {x^2}{-2+3 x}} x^4\right )}{(2-3 x)^2 \left (20+\left (10+e^{e^5}+e^{\frac {x^2}{-2+3 x}}\right ) x\right )^2} \, dx\\ &=5 \int \frac {80-240 x+180 x^2+4 e^{\frac {x^2}{-2+3 x}} x^3-3 e^{\frac {x^2}{-2+3 x}} x^4}{(2-3 x)^2 \left (20+\left (10+e^{e^5}+e^{\frac {x^2}{-2+3 x}}\right ) x\right )^2} \, dx\\ &=5 \int \left (\frac {(4-3 x) x^2}{(2-3 x)^2 \left (20+e^{\frac {x^2}{-2+3 x}} x+10 \left (1+\frac {e^{e^5}}{10}\right ) x\right )}+\frac {80-240 x+100 x^2+4 \left (5-e^{e^5}\right ) x^3+3 \left (10+e^{e^5}\right ) x^4}{(2-3 x)^2 \left (20+e^{\frac {x^2}{-2+3 x}} x+10 \left (1+\frac {e^{e^5}}{10}\right ) x\right )^2}\right ) \, dx\\ &=5 \int \frac {(4-3 x) x^2}{(2-3 x)^2 \left (20+e^{\frac {x^2}{-2+3 x}} x+10 \left (1+\frac {e^{e^5}}{10}\right ) x\right )} \, dx+5 \int \frac {80-240 x+100 x^2+4 \left (5-e^{e^5}\right ) x^3+3 \left (10+e^{e^5}\right ) x^4}{(2-3 x)^2 \left (20+e^{\frac {x^2}{-2+3 x}} x+10 \left (1+\frac {e^{e^5}}{10}\right ) x\right )^2} \, dx\\ &=5 \int \left (\frac {4 \left (125-e^{e^5}\right )}{27 \left (20+e^{\frac {x^2}{-2+3 x}} x+10 \left (1+\frac {e^{e^5}}{10}\right ) x\right )^2}+\frac {16 \left (-40-e^{e^5}\right )}{27 (2-3 x)^2 \left (20+e^{\frac {x^2}{-2+3 x}} x+10 \left (1+\frac {e^{e^5}}{10}\right ) x\right )^2}+\frac {16 \left (25+e^{e^5}\right )}{27 (2-3 x) \left (20+e^{\frac {x^2}{-2+3 x}} x+10 \left (1+\frac {e^{e^5}}{10}\right ) x\right )^2}+\frac {20 x}{3 \left (20+e^{\frac {x^2}{-2+3 x}} x+10 \left (1+\frac {e^{e^5}}{10}\right ) x\right )^2}+\frac {\left (10+e^{e^5}\right ) x^2}{3 \left (20+e^{\frac {x^2}{-2+3 x}} x+10 \left (1+\frac {e^{e^5}}{10}\right ) x\right )^2}\right ) \, dx+5 \int \left (\frac {x}{3 \left (-20-e^{\frac {x^2}{-2+3 x}} x-10 \left (1+\frac {e^{e^5}}{10}\right ) x\right )}+\frac {8}{9 (2-3 x)^2 \left (20+e^{\frac {x^2}{-2+3 x}} x+10 \left (1+\frac {e^{e^5}}{10}\right ) x\right )}+\frac {4}{9 (-2+3 x) \left (20+e^{\frac {x^2}{-2+3 x}} x+10 \left (1+\frac {e^{e^5}}{10}\right ) x\right )}\right ) \, dx\\ &=\frac {5}{3} \int \frac {x}{-20-e^{\frac {x^2}{-2+3 x}} x-10 \left (1+\frac {e^{e^5}}{10}\right ) x} \, dx+\frac {20}{9} \int \frac {1}{(-2+3 x) \left (20+e^{\frac {x^2}{-2+3 x}} x+10 \left (1+\frac {e^{e^5}}{10}\right ) x\right )} \, dx+\frac {40}{9} \int \frac {1}{(2-3 x)^2 \left (20+e^{\frac {x^2}{-2+3 x}} x+10 \left (1+\frac {e^{e^5}}{10}\right ) x\right )} \, dx+\frac {100}{3} \int \frac {x}{\left (20+e^{\frac {x^2}{-2+3 x}} x+10 \left (1+\frac {e^{e^5}}{10}\right ) x\right )^2} \, dx+\frac {1}{27} \left (20 \left (125-e^{e^5}\right )\right ) \int \frac {1}{\left (20+e^{\frac {x^2}{-2+3 x}} x+10 \left (1+\frac {e^{e^5}}{10}\right ) x\right )^2} \, dx+\frac {1}{3} \left (5 \left (10+e^{e^5}\right )\right ) \int \frac {x^2}{\left (20+e^{\frac {x^2}{-2+3 x}} x+10 \left (1+\frac {e^{e^5}}{10}\right ) x\right )^2} \, dx+\frac {1}{27} \left (80 \left (25+e^{e^5}\right )\right ) \int \frac {1}{(2-3 x) \left (20+e^{\frac {x^2}{-2+3 x}} x+10 \left (1+\frac {e^{e^5}}{10}\right ) x\right )^2} \, dx-\frac {1}{27} \left (80 \left (40+e^{e^5}\right )\right ) \int \frac {1}{(2-3 x)^2 \left (20+e^{\frac {x^2}{-2+3 x}} x+10 \left (1+\frac {e^{e^5}}{10}\right ) x\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.13, size = 29, normalized size = 0.88 \begin {gather*} \frac {5 x}{20+\left (10+e^{e^5}+e^{\frac {x^2}{-2+3 x}}\right ) x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(400 - 1200*x + 900*x^2 + E^(x^2/(-2 + 3*x))*(20*x^3 - 15*x^4))/(1600 - 3200*x - 800*x^2 + 2400*x^3
+ 900*x^4 + E^(2*E^5)*(4*x^2 - 12*x^3 + 9*x^4) + E^((2*x^2)/(-2 + 3*x))*(4*x^2 - 12*x^3 + 9*x^4) + E^(x^2/(-2
+ 3*x))*(160*x - 400*x^2 + 120*x^3 + 180*x^4) + E^E^5*(160*x - 400*x^2 + 120*x^3 + 180*x^4 + E^(x^2/(-2 + 3*x)
)*(8*x^2 - 24*x^3 + 18*x^4))),x]

[Out]

(5*x)/(20 + (10 + E^E^5 + E^(x^2/(-2 + 3*x)))*x)

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fricas [A]  time = 0.57, size = 29, normalized size = 0.88 \begin {gather*} \frac {5 \, x}{x e^{\left (\frac {x^{2}}{3 \, x - 2}\right )} + x e^{\left (e^{5}\right )} + 10 \, x + 20} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-15*x^4+20*x^3)*exp(x^2/(3*x-2))+900*x^2-1200*x+400)/((9*x^4-12*x^3+4*x^2)*exp(exp(5))^2+((18*x^4-
24*x^3+8*x^2)*exp(x^2/(3*x-2))+180*x^4+120*x^3-400*x^2+160*x)*exp(exp(5))+(9*x^4-12*x^3+4*x^2)*exp(x^2/(3*x-2)
)^2+(180*x^4+120*x^3-400*x^2+160*x)*exp(x^2/(3*x-2))+900*x^4+2400*x^3-800*x^2-3200*x+1600),x, algorithm="frica
s")

[Out]

5*x/(x*e^(x^2/(3*x - 2)) + x*e^(e^5) + 10*x + 20)

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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(((-15*x^4+20*x^3)*exp(x^2/(3*x-2))+900*x^2-1200*x+400)/((9*x^4-12*x^3+4*x^2)*exp(exp(5))^2+((18*x^4-
24*x^3+8*x^2)*exp(x^2/(3*x-2))+180*x^4+120*x^3-400*x^2+160*x)*exp(exp(5))+(9*x^4-12*x^3+4*x^2)*exp(x^2/(3*x-2)
)^2+(180*x^4+120*x^3-400*x^2+160*x)*exp(x^2/(3*x-2))+900*x^4+2400*x^3-800*x^2-3200*x+1600),x, algorithm="giac"
)

[Out]

Timed out

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maple [A]  time = 0.48, size = 30, normalized size = 0.91

method	result	size
risch	\(\frac {5 x}{x \,{\mathrm e}^{\frac {x^{2}}{3 x -2}}+x \,{\mathrm e}^{{\mathrm e}^{5}}+10 x +20}\)	\(30\)
norman	\(\frac {15 x^{2}-10 x}{\left (3 x -2\right ) \left (x \,{\mathrm e}^{\frac {x^{2}}{3 x -2}}+x \,{\mathrm e}^{{\mathrm e}^{5}}+10 x +20\right )}\)	\(44\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-15*x^4+20*x^3)*exp(x^2/(3*x-2))+900*x^2-1200*x+400)/((9*x^4-12*x^3+4*x^2)*exp(exp(5))^2+((18*x^4-24*x^3
+8*x^2)*exp(x^2/(3*x-2))+180*x^4+120*x^3-400*x^2+160*x)*exp(exp(5))+(9*x^4-12*x^3+4*x^2)*exp(x^2/(3*x-2))^2+(1
80*x^4+120*x^3-400*x^2+160*x)*exp(x^2/(3*x-2))+900*x^4+2400*x^3-800*x^2-3200*x+1600),x,method=_RETURNVERBOSE)

[Out]

5*x/(x*exp(x^2/(3*x-2))+x*exp(exp(5))+10*x+20)

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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(((-15*x^4+20*x^3)*exp(x^2/(3*x-2))+900*x^2-1200*x+400)/((9*x^4-12*x^3+4*x^2)*exp(exp(5))^2+((18*x^4-
24*x^3+8*x^2)*exp(x^2/(3*x-2))+180*x^4+120*x^3-400*x^2+160*x)*exp(exp(5))+(9*x^4-12*x^3+4*x^2)*exp(x^2/(3*x-2)
)^2+(180*x^4+120*x^3-400*x^2+160*x)*exp(x^2/(3*x-2))+900*x^4+2400*x^3-800*x^2-3200*x+1600),x, algorithm="maxim
a")

[Out]

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

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mupad [B]  time = 4.52, size = 31, normalized size = 0.94 \begin {gather*} \frac {5}{{\mathrm {e}}^{\frac {x^2}{3\,x-2}}+\frac {10\,x+x\,{\mathrm {e}}^{{\mathrm {e}}^5}+20}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x^2/(3*x - 2))*(20*x^3 - 15*x^4) - 1200*x + 900*x^2 + 400)/(exp(x^2/(3*x - 2))*(160*x - 400*x^2 + 120
*x^3 + 180*x^4) - 3200*x + exp(2*exp(5))*(4*x^2 - 12*x^3 + 9*x^4) + exp((2*x^2)/(3*x - 2))*(4*x^2 - 12*x^3 + 9
*x^4) + exp(exp(5))*(160*x + exp(x^2/(3*x - 2))*(8*x^2 - 24*x^3 + 18*x^4) - 400*x^2 + 120*x^3 + 180*x^4) - 800
*x^2 + 2400*x^3 + 900*x^4 + 1600),x)

[Out]

5/(exp(x^2/(3*x - 2)) + (10*x + x*exp(exp(5)) + 20)/x)

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sympy [A]  time = 0.30, size = 26, normalized size = 0.79 \begin {gather*} \frac {5 x}{x e^{\frac {x^{2}}{3 x - 2}} + 10 x + x e^{e^{5}} + 20} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-15*x**4+20*x**3)*exp(x**2/(3*x-2))+900*x**2-1200*x+400)/((9*x**4-12*x**3+4*x**2)*exp(exp(5))**2+(
(18*x**4-24*x**3+8*x**2)*exp(x**2/(3*x-2))+180*x**4+120*x**3-400*x**2+160*x)*exp(exp(5))+(9*x**4-12*x**3+4*x**
2)*exp(x**2/(3*x-2))**2+(180*x**4+120*x**3-400*x**2+160*x)*exp(x**2/(3*x-2))+900*x**4+2400*x**3-800*x**2-3200*
x+1600),x)

[Out]

5*x/(x*exp(x**2/(3*x - 2)) + 10*x + x*exp(exp(5)) + 20)

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