3.3.87 \(\int \frac {15 x^2-15 x^3+e^{4+4 e^{3/x} x+e^{6/x} x^2} (3 x^2-3 x^3+e^{3/x} (12 x^2-4 x^3)+e^{6/x} (6 x^3-2 x^4))}{25 e^{3 x}+10 e^{4+3 x+4 e^{3/x} x+e^{6/x} x^2}+e^{8+3 x+8 e^{3/x} x+2 e^{6/x} x^2}} \, dx\)

Optimal. Leaf size=28 \[ \frac {e^{-3 x} x^3}{5+e^{\left (2+e^{3/x} x\right )^2}} \]

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Rubi [F]  time = 25.37, 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 {15 x^2-15 x^3+e^{4+4 e^{3/x} x+e^{6/x} x^2} \left (3 x^2-3 x^3+e^{3/x} \left (12 x^2-4 x^3\right )+e^{6/x} \left (6 x^3-2 x^4\right )\right )}{25 e^{3 x}+10 e^{4+3 x+4 e^{3/x} x+e^{6/x} x^2}+e^{8+3 x+8 e^{3/x} x+2 e^{6/x} x^2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(15*x^2 - 15*x^3 + E^(4 + 4*E^(3/x)*x + E^(6/x)*x^2)*(3*x^2 - 3*x^3 + E^(3/x)*(12*x^2 - 4*x^3) + E^(6/x)*(
6*x^3 - 2*x^4)))/(25*E^(3*x) + 10*E^(4 + 3*x + 4*E^(3/x)*x + E^(6/x)*x^2) + E^(8 + 3*x + 8*E^(3/x)*x + 2*E^(6/
x)*x^2)),x]

[Out]

-60*Defer[Int][x^2/(E^((3*(-1 + x^2))/x)*(5 + E^(2 + E^(3/x)*x)^2)^2), x] + 3*Defer[Int][x^2/(E^(3*x)*(5 + E^(
2 + E^(3/x)*x)^2)), x] + 12*Defer[Int][x^2/(E^((3*(-1 + x^2))/x)*(5 + E^(2 + E^(3/x)*x)^2)), x] - 30*Defer[Int
][x^3/(E^((3*(-2 + x^2))/x)*(5 + E^(2 + E^(3/x)*x)^2)^2), x] + 20*Defer[Int][x^3/(E^((3*(-1 + x^2))/x)*(5 + E^
(2 + E^(3/x)*x)^2)^2), x] - 3*Defer[Int][x^3/(E^(3*x)*(5 + E^(2 + E^(3/x)*x)^2)), x] + 6*Defer[Int][x^3/(E^((3
*(-2 + x^2))/x)*(5 + E^(2 + E^(3/x)*x)^2)), x] - 4*Defer[Int][x^3/(E^((3*(-1 + x^2))/x)*(5 + E^(2 + E^(3/x)*x)
^2)), x] + 10*Defer[Int][x^4/(E^((3*(-2 + x^2))/x)*(5 + E^(2 + E^(3/x)*x)^2)^2), x] - 2*Defer[Int][x^4/(E^((3*
(-2 + x^2))/x)*(5 + E^(2 + E^(3/x)*x)^2)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-3 x} \left (15 x^2-15 x^3+e^{4+4 e^{3/x} x+e^{6/x} x^2} \left (3 x^2-3 x^3+e^{3/x} \left (12 x^2-4 x^3\right )+e^{6/x} \left (6 x^3-2 x^4\right )\right )\right )}{\left (5+e^{\left (2+e^{3/x} x\right )^2}\right )^2} \, dx\\ &=\int \frac {e^{-3 x} x^2 \left (15-15 x+e^{\left (2+e^{3/x} x\right )^2} \left (3-4 e^{3/x} (-3+x)-3 x-2 e^{6/x} (-3+x) x\right )\right )}{\left (5+e^{\left (2+e^{3/x} x\right )^2}\right )^2} \, dx\\ &=\int \left (\frac {10 e^{\frac {3}{x}-3 x} (-3+x) x^2 \left (2+e^{3/x} x\right )}{\left (5+e^{\left (2+e^{3/x} x\right )^2}\right )^2}-\frac {e^{-3 x} x^2 \left (-3-12 e^{3/x}+3 x+4 e^{3/x} x-6 e^{6/x} x+2 e^{6/x} x^2\right )}{5+e^{\left (2+e^{3/x} x\right )^2}}\right ) \, dx\\ &=10 \int \frac {e^{\frac {3}{x}-3 x} (-3+x) x^2 \left (2+e^{3/x} x\right )}{\left (5+e^{\left (2+e^{3/x} x\right )^2}\right )^2} \, dx-\int \frac {e^{-3 x} x^2 \left (-3-12 e^{3/x}+3 x+4 e^{3/x} x-6 e^{6/x} x+2 e^{6/x} x^2\right )}{5+e^{\left (2+e^{3/x} x\right )^2}} \, dx\\ &=10 \int \frac {e^{-\frac {3 \left (-1+x^2\right )}{x}} (3-x) x^2 \left (-2-e^{3/x} x\right )}{\left (5+e^{\left (2+e^{3/x} x\right )^2}\right )^2} \, dx-\int \frac {e^{-3 x} x^2 \left (4 e^{3/x} (-3+x)+3 (-1+x)+2 e^{6/x} (-3+x) x\right )}{5+e^{\left (2+e^{3/x} x\right )^2}} \, dx\\ &=10 \int \left (-\frac {3 e^{-\frac {3 \left (-1+x^2\right )}{x}} x^2 \left (2+e^{3/x} x\right )}{\left (5+e^{\left (2+e^{3/x} x\right )^2}\right )^2}+\frac {e^{-\frac {3 \left (-1+x^2\right )}{x}} x^3 \left (2+e^{3/x} x\right )}{\left (5+e^{\left (2+e^{3/x} x\right )^2}\right )^2}\right ) \, dx-\int \left (-\frac {12 e^{\frac {3}{x}-3 x} x^2}{5+e^{\left (2+e^{3/x} x\right )^2}}-\frac {3 e^{-3 x} x^2}{5+e^{\left (2+e^{3/x} x\right )^2}}+\frac {4 e^{\frac {3}{x}-3 x} x^3}{5+e^{\left (2+e^{3/x} x\right )^2}}-\frac {6 e^{\frac {6}{x}-3 x} x^3}{5+e^{\left (2+e^{3/x} x\right )^2}}+\frac {3 e^{-3 x} x^3}{5+e^{\left (2+e^{3/x} x\right )^2}}+\frac {2 e^{\frac {6}{x}-3 x} x^4}{5+e^{\left (2+e^{3/x} x\right )^2}}\right ) \, dx\\ &=-\left (2 \int \frac {e^{\frac {6}{x}-3 x} x^4}{5+e^{\left (2+e^{3/x} x\right )^2}} \, dx\right )+3 \int \frac {e^{-3 x} x^2}{5+e^{\left (2+e^{3/x} x\right )^2}} \, dx-3 \int \frac {e^{-3 x} x^3}{5+e^{\left (2+e^{3/x} x\right )^2}} \, dx-4 \int \frac {e^{\frac {3}{x}-3 x} x^3}{5+e^{\left (2+e^{3/x} x\right )^2}} \, dx+6 \int \frac {e^{\frac {6}{x}-3 x} x^3}{5+e^{\left (2+e^{3/x} x\right )^2}} \, dx+10 \int \frac {e^{-\frac {3 \left (-1+x^2\right )}{x}} x^3 \left (2+e^{3/x} x\right )}{\left (5+e^{\left (2+e^{3/x} x\right )^2}\right )^2} \, dx+12 \int \frac {e^{\frac {3}{x}-3 x} x^2}{5+e^{\left (2+e^{3/x} x\right )^2}} \, dx-30 \int \frac {e^{-\frac {3 \left (-1+x^2\right )}{x}} x^2 \left (2+e^{3/x} x\right )}{\left (5+e^{\left (2+e^{3/x} x\right )^2}\right )^2} \, dx\\ &=-\left (2 \int \frac {e^{-\frac {3 \left (-2+x^2\right )}{x}} x^4}{5+e^{\left (2+e^{3/x} x\right )^2}} \, dx\right )+3 \int \frac {e^{-3 x} x^2}{5+e^{\left (2+e^{3/x} x\right )^2}} \, dx-3 \int \frac {e^{-3 x} x^3}{5+e^{\left (2+e^{3/x} x\right )^2}} \, dx-4 \int \frac {e^{-\frac {3 \left (-1+x^2\right )}{x}} x^3}{5+e^{\left (2+e^{3/x} x\right )^2}} \, dx+6 \int \frac {e^{-\frac {3 \left (-2+x^2\right )}{x}} x^3}{5+e^{\left (2+e^{3/x} x\right )^2}} \, dx+10 \int \left (\frac {2 e^{-\frac {3 \left (-1+x^2\right )}{x}} x^3}{\left (5+e^{\left (2+e^{3/x} x\right )^2}\right )^2}+\frac {e^{\frac {3}{x}-\frac {3 \left (-1+x^2\right )}{x}} x^4}{\left (5+e^{\left (2+e^{3/x} x\right )^2}\right )^2}\right ) \, dx+12 \int \frac {e^{-\frac {3 \left (-1+x^2\right )}{x}} x^2}{5+e^{\left (2+e^{3/x} x\right )^2}} \, dx-30 \int \left (\frac {2 e^{-\frac {3 \left (-1+x^2\right )}{x}} x^2}{\left (5+e^{\left (2+e^{3/x} x\right )^2}\right )^2}+\frac {e^{\frac {3}{x}-\frac {3 \left (-1+x^2\right )}{x}} x^3}{\left (5+e^{\left (2+e^{3/x} x\right )^2}\right )^2}\right ) \, dx\\ &=-\left (2 \int \frac {e^{-\frac {3 \left (-2+x^2\right )}{x}} x^4}{5+e^{\left (2+e^{3/x} x\right )^2}} \, dx\right )+3 \int \frac {e^{-3 x} x^2}{5+e^{\left (2+e^{3/x} x\right )^2}} \, dx-3 \int \frac {e^{-3 x} x^3}{5+e^{\left (2+e^{3/x} x\right )^2}} \, dx-4 \int \frac {e^{-\frac {3 \left (-1+x^2\right )}{x}} x^3}{5+e^{\left (2+e^{3/x} x\right )^2}} \, dx+6 \int \frac {e^{-\frac {3 \left (-2+x^2\right )}{x}} x^3}{5+e^{\left (2+e^{3/x} x\right )^2}} \, dx+10 \int \frac {e^{\frac {3}{x}-\frac {3 \left (-1+x^2\right )}{x}} x^4}{\left (5+e^{\left (2+e^{3/x} x\right )^2}\right )^2} \, dx+12 \int \frac {e^{-\frac {3 \left (-1+x^2\right )}{x}} x^2}{5+e^{\left (2+e^{3/x} x\right )^2}} \, dx+20 \int \frac {e^{-\frac {3 \left (-1+x^2\right )}{x}} x^3}{\left (5+e^{\left (2+e^{3/x} x\right )^2}\right )^2} \, dx-30 \int \frac {e^{\frac {3}{x}-\frac {3 \left (-1+x^2\right )}{x}} x^3}{\left (5+e^{\left (2+e^{3/x} x\right )^2}\right )^2} \, dx-60 \int \frac {e^{-\frac {3 \left (-1+x^2\right )}{x}} x^2}{\left (5+e^{\left (2+e^{3/x} x\right )^2}\right )^2} \, dx\\ &=-\left (2 \int \frac {e^{-\frac {3 \left (-2+x^2\right )}{x}} x^4}{5+e^{\left (2+e^{3/x} x\right )^2}} \, dx\right )+3 \int \frac {e^{-3 x} x^2}{5+e^{\left (2+e^{3/x} x\right )^2}} \, dx-3 \int \frac {e^{-3 x} x^3}{5+e^{\left (2+e^{3/x} x\right )^2}} \, dx-4 \int \frac {e^{-\frac {3 \left (-1+x^2\right )}{x}} x^3}{5+e^{\left (2+e^{3/x} x\right )^2}} \, dx+6 \int \frac {e^{-\frac {3 \left (-2+x^2\right )}{x}} x^3}{5+e^{\left (2+e^{3/x} x\right )^2}} \, dx+10 \int \frac {e^{-\frac {3 \left (-2+x^2\right )}{x}} x^4}{\left (5+e^{\left (2+e^{3/x} x\right )^2}\right )^2} \, dx+12 \int \frac {e^{-\frac {3 \left (-1+x^2\right )}{x}} x^2}{5+e^{\left (2+e^{3/x} x\right )^2}} \, dx+20 \int \frac {e^{-\frac {3 \left (-1+x^2\right )}{x}} x^3}{\left (5+e^{\left (2+e^{3/x} x\right )^2}\right )^2} \, dx-30 \int \frac {e^{-\frac {3 \left (-2+x^2\right )}{x}} x^3}{\left (5+e^{\left (2+e^{3/x} x\right )^2}\right )^2} \, dx-60 \int \frac {e^{-\frac {3 \left (-1+x^2\right )}{x}} x^2}{\left (5+e^{\left (2+e^{3/x} x\right )^2}\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.24, size = 28, normalized size = 1.00 \begin {gather*} \frac {e^{-3 x} x^3}{5+e^{\left (2+e^{3/x} x\right )^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(15*x^2 - 15*x^3 + E^(4 + 4*E^(3/x)*x + E^(6/x)*x^2)*(3*x^2 - 3*x^3 + E^(3/x)*(12*x^2 - 4*x^3) + E^(
6/x)*(6*x^3 - 2*x^4)))/(25*E^(3*x) + 10*E^(4 + 3*x + 4*E^(3/x)*x + E^(6/x)*x^2) + E^(8 + 3*x + 8*E^(3/x)*x + 2
*E^(6/x)*x^2)),x]

[Out]

x^3/(E^(3*x)*(5 + E^(2 + E^(3/x)*x)^2))

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fricas [A]  time = 0.67, size = 38, normalized size = 1.36 \begin {gather*} \frac {x^{3}}{e^{\left (x^{2} e^{\frac {6}{x}} + 4 \, x e^{\frac {3}{x}} + 3 \, x + 4\right )} + 5 \, e^{\left (3 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x^4+6*x^3)*exp(3/x)^2+(-4*x^3+12*x^2)*exp(3/x)-3*x^3+3*x^2)*exp(x^2*exp(3/x)^2+4*x*exp(3/x)+4)
-15*x^3+15*x^2)/(exp(x)^3*exp(x^2*exp(3/x)^2+4*x*exp(3/x)+4)^2+10*exp(x)^3*exp(x^2*exp(3/x)^2+4*x*exp(3/x)+4)+
25*exp(x)^3),x, algorithm="fricas")

[Out]

x^3/(e^(x^2*e^(6/x) + 4*x*e^(3/x) + 3*x + 4) + 5*e^(3*x))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {15 \, x^{3} - 15 \, x^{2} + {\left (3 \, x^{3} - 3 \, x^{2} + 2 \, {\left (x^{4} - 3 \, x^{3}\right )} e^{\frac {6}{x}} + 4 \, {\left (x^{3} - 3 \, x^{2}\right )} e^{\frac {3}{x}}\right )} e^{\left (x^{2} e^{\frac {6}{x}} + 4 \, x e^{\frac {3}{x}} + 4\right )}}{e^{\left (2 \, x^{2} e^{\frac {6}{x}} + 8 \, x e^{\frac {3}{x}} + 3 \, x + 8\right )} + 10 \, e^{\left (x^{2} e^{\frac {6}{x}} + 4 \, x e^{\frac {3}{x}} + 3 \, x + 4\right )} + 25 \, e^{\left (3 \, x\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x^4+6*x^3)*exp(3/x)^2+(-4*x^3+12*x^2)*exp(3/x)-3*x^3+3*x^2)*exp(x^2*exp(3/x)^2+4*x*exp(3/x)+4)
-15*x^3+15*x^2)/(exp(x)^3*exp(x^2*exp(3/x)^2+4*x*exp(3/x)+4)^2+10*exp(x)^3*exp(x^2*exp(3/x)^2+4*x*exp(3/x)+4)+
25*exp(x)^3),x, algorithm="giac")

[Out]

integrate(-(15*x^3 - 15*x^2 + (3*x^3 - 3*x^2 + 2*(x^4 - 3*x^3)*e^(6/x) + 4*(x^3 - 3*x^2)*e^(3/x))*e^(x^2*e^(6/
x) + 4*x*e^(3/x) + 4))/(e^(2*x^2*e^(6/x) + 8*x*e^(3/x) + 3*x + 8) + 10*e^(x^2*e^(6/x) + 4*x*e^(3/x) + 3*x + 4)
+ 25*e^(3*x)), x)

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maple [A]  time = 0.24, size = 35, normalized size = 1.25

method	result	size
risch	\(\frac {x^{3} {\mathrm e}^{-3 x}}{{\mathrm e}^{x^{2} {\mathrm e}^{\frac {6}{x}}+4 x \,{\mathrm e}^{\frac {3}{x}}+4}+5}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-2*x^4+6*x^3)*exp(3/x)^2+(-4*x^3+12*x^2)*exp(3/x)-3*x^3+3*x^2)*exp(x^2*exp(3/x)^2+4*x*exp(3/x)+4)-15*x^
3+15*x^2)/(exp(x)^3*exp(x^2*exp(3/x)^2+4*x*exp(3/x)+4)^2+10*exp(x)^3*exp(x^2*exp(3/x)^2+4*x*exp(3/x)+4)+25*exp
(x)^3),x,method=_RETURNVERBOSE)

[Out]

x^3*exp(-3*x)/(exp(x^2*exp(6/x)+4*x*exp(3/x)+4)+5)

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maxima [A]  time = 0.58, size = 38, normalized size = 1.36 \begin {gather*} \frac {x^{3}}{e^{\left (x^{2} e^{\frac {6}{x}} + 4 \, x e^{\frac {3}{x}} + 3 \, x + 4\right )} + 5 \, e^{\left (3 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x^4+6*x^3)*exp(3/x)^2+(-4*x^3+12*x^2)*exp(3/x)-3*x^3+3*x^2)*exp(x^2*exp(3/x)^2+4*x*exp(3/x)+4)
-15*x^3+15*x^2)/(exp(x)^3*exp(x^2*exp(3/x)^2+4*x*exp(3/x)+4)^2+10*exp(x)^3*exp(x^2*exp(3/x)^2+4*x*exp(3/x)+4)+
25*exp(x)^3),x, algorithm="maxima")

[Out]

x^3/(e^(x^2*e^(6/x) + 4*x*e^(3/x) + 3*x + 4) + 5*e^(3*x))

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mupad [B]  time = 0.85, size = 41, normalized size = 1.46 \begin {gather*} \frac {x^3}{5\,{\mathrm {e}}^{3\,x}+{\mathrm {e}}^{x^2\,{\mathrm {e}}^{6/x}}\,{\mathrm {e}}^{3\,x}\,{\mathrm {e}}^4\,{\mathrm {e}}^{4\,x\,{\mathrm {e}}^{3/x}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(4*x*exp(3/x) + x^2*exp(6/x) + 4)*(exp(6/x)*(6*x^3 - 2*x^4) + exp(3/x)*(12*x^2 - 4*x^3) + 3*x^2 - 3*x^
3) + 15*x^2 - 15*x^3)/(25*exp(3*x) + 10*exp(4*x*exp(3/x) + x^2*exp(6/x) + 4)*exp(3*x) + exp(8*x*exp(3/x) + 2*x
^2*exp(6/x) + 8)*exp(3*x)),x)

[Out]

x^3/(5*exp(3*x) + exp(x^2*exp(6/x))*exp(3*x)*exp(4)*exp(4*x*exp(3/x)))

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sympy [A]  time = 0.45, size = 34, normalized size = 1.21 \begin {gather*} \frac {x^{3}}{e^{3 x} e^{x^{2} e^{\frac {6}{x}} + 4 x e^{\frac {3}{x}} + 4} + 5 e^{3 x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x**4+6*x**3)*exp(3/x)**2+(-4*x**3+12*x**2)*exp(3/x)-3*x**3+3*x**2)*exp(x**2*exp(3/x)**2+4*x*ex
p(3/x)+4)-15*x**3+15*x**2)/(exp(x)**3*exp(x**2*exp(3/x)**2+4*x*exp(3/x)+4)**2+10*exp(x)**3*exp(x**2*exp(3/x)**
2+4*x*exp(3/x)+4)+25*exp(x)**3),x)

[Out]

x**3/(exp(3*x)*exp(x**2*exp(6/x) + 4*x*exp(3/x) + 4) + 5*exp(3*x))

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