3.86.67 \(\int \frac {e^{\frac {1}{18} (225-30 x+x^2)} (e^{3 x} (60-4 x)+15 x^3-x^4+e^{2 x} (36+84 x-8 x^2)+e^x (18 x+57 x^2-5 x^3))}{225 e^{3 x}+675 e^{2 x} x+675 e^x x^2+225 x^3} \, dx\)

Optimal. Leaf size=36 \[ 5-\frac {1}{25} e^{\frac {1}{2} \left (-5+\frac {x}{3}\right )^2} \left (1+\frac {e^x}{e^x+x}\right )^2 \]

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Rubi [F]  time = 3.58, 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 {e^{\frac {1}{18} \left (225-30 x+x^2\right )} \left (e^{3 x} (60-4 x)+15 x^3-x^4+e^{2 x} \left (36+84 x-8 x^2\right )+e^x \left (18 x+57 x^2-5 x^3\right )\right )}{225 e^{3 x}+675 e^{2 x} x+675 e^x x^2+225 x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((225 - 30*x + x^2)/18)*(E^(3*x)*(60 - 4*x) + 15*x^3 - x^4 + E^(2*x)*(36 + 84*x - 8*x^2) + E^x*(18*x +
57*x^2 - 5*x^3)))/(225*E^(3*x) + 675*E^(2*x)*x + 675*E^x*x^2 + 225*x^3),x]

[Out]

(-4*E^((15 - x)^2/18))/25 + (2*Defer[Int][(E^((-15 + x)^2/18)*x^2)/(E^x + x)^3, x])/25 - (2*Defer[Int][(E^((-1
5 + x)^2/18)*x^3)/(E^x + x)^3, x])/25 - (6*Defer[Int][(E^((-15 + x)^2/18)*x)/(E^x + x)^2, x])/25 + (23*Defer[I
nt][(E^((-15 + x)^2/18)*x^2)/(E^x + x)^2, x])/75 - Defer[Int][(E^((-15 + x)^2/18)*x^3)/(E^x + x)^2, x]/225 + (
4*Defer[Int][E^((-15 + x)^2/18)/(E^x + x), x])/25 - (32*Defer[Int][(E^((-15 + x)^2/18)*x)/(E^x + x), x])/75 +
(4*Defer[Int][(E^((-15 + x)^2/18)*x^2)/(E^x + x), x])/225

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{\frac {1}{18} (-15+x)^2} \left (-4 e^{3 x} (-15+x)-(-15+x) x^3-4 e^{2 x} \left (-9-21 x+2 x^2\right )-e^x x \left (-18-57 x+5 x^2\right )\right )}{225 \left (e^x+x\right )^3} \, dx\\ &=\frac {1}{225} \int \frac {e^{\frac {1}{18} (-15+x)^2} \left (-4 e^{3 x} (-15+x)-(-15+x) x^3-4 e^{2 x} \left (-9-21 x+2 x^2\right )-e^x x \left (-18-57 x+5 x^2\right )\right )}{\left (e^x+x\right )^3} \, dx\\ &=\frac {1}{225} \int \left (-4 e^{\frac {1}{18} (-15+x)^2} (-15+x)-\frac {18 e^{\frac {1}{18} (-15+x)^2} (-1+x) x^2}{\left (e^x+x\right )^3}-\frac {e^{\frac {1}{18} (-15+x)^2} x \left (54-69 x+x^2\right )}{\left (e^x+x\right )^2}+\frac {4 e^{\frac {1}{18} (-15+x)^2} \left (9-24 x+x^2\right )}{e^x+x}\right ) \, dx\\ &=-\left (\frac {1}{225} \int \frac {e^{\frac {1}{18} (-15+x)^2} x \left (54-69 x+x^2\right )}{\left (e^x+x\right )^2} \, dx\right )-\frac {4}{225} \int e^{\frac {1}{18} (-15+x)^2} (-15+x) \, dx+\frac {4}{225} \int \frac {e^{\frac {1}{18} (-15+x)^2} \left (9-24 x+x^2\right )}{e^x+x} \, dx-\frac {2}{25} \int \frac {e^{\frac {1}{18} (-15+x)^2} (-1+x) x^2}{\left (e^x+x\right )^3} \, dx\\ &=-\frac {4}{25} e^{\frac {1}{18} (15-x)^2}-\frac {1}{225} \int \left (\frac {54 e^{\frac {1}{18} (-15+x)^2} x}{\left (e^x+x\right )^2}-\frac {69 e^{\frac {1}{18} (-15+x)^2} x^2}{\left (e^x+x\right )^2}+\frac {e^{\frac {1}{18} (-15+x)^2} x^3}{\left (e^x+x\right )^2}\right ) \, dx+\frac {4}{225} \int \left (\frac {9 e^{\frac {1}{18} (-15+x)^2}}{e^x+x}-\frac {24 e^{\frac {1}{18} (-15+x)^2} x}{e^x+x}+\frac {e^{\frac {1}{18} (-15+x)^2} x^2}{e^x+x}\right ) \, dx-\frac {2}{25} \int \left (-\frac {e^{\frac {1}{18} (-15+x)^2} x^2}{\left (e^x+x\right )^3}+\frac {e^{\frac {1}{18} (-15+x)^2} x^3}{\left (e^x+x\right )^3}\right ) \, dx\\ &=-\frac {4}{25} e^{\frac {1}{18} (15-x)^2}-\frac {1}{225} \int \frac {e^{\frac {1}{18} (-15+x)^2} x^3}{\left (e^x+x\right )^2} \, dx+\frac {4}{225} \int \frac {e^{\frac {1}{18} (-15+x)^2} x^2}{e^x+x} \, dx+\frac {2}{25} \int \frac {e^{\frac {1}{18} (-15+x)^2} x^2}{\left (e^x+x\right )^3} \, dx-\frac {2}{25} \int \frac {e^{\frac {1}{18} (-15+x)^2} x^3}{\left (e^x+x\right )^3} \, dx+\frac {4}{25} \int \frac {e^{\frac {1}{18} (-15+x)^2}}{e^x+x} \, dx-\frac {6}{25} \int \frac {e^{\frac {1}{18} (-15+x)^2} x}{\left (e^x+x\right )^2} \, dx+\frac {23}{75} \int \frac {e^{\frac {1}{18} (-15+x)^2} x^2}{\left (e^x+x\right )^2} \, dx-\frac {32}{75} \int \frac {e^{\frac {1}{18} (-15+x)^2} x}{e^x+x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.71, size = 31, normalized size = 0.86 \begin {gather*} -\frac {e^{\frac {1}{18} (-15+x)^2} \left (2 e^x+x\right )^2}{25 \left (e^x+x\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((225 - 30*x + x^2)/18)*(E^(3*x)*(60 - 4*x) + 15*x^3 - x^4 + E^(2*x)*(36 + 84*x - 8*x^2) + E^x*(1
8*x + 57*x^2 - 5*x^3)))/(225*E^(3*x) + 675*E^(2*x)*x + 675*E^x*x^2 + 225*x^3),x]

[Out]

-1/25*(E^((-15 + x)^2/18)*(2*E^x + x)^2)/(E^x + x)^2

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fricas [A]  time = 0.93, size = 43, normalized size = 1.19 \begin {gather*} -\frac {{\left (x^{2} + 4 \, x e^{x} + 4 \, e^{\left (2 \, x\right )}\right )} e^{\left (\frac {1}{18} \, x^{2} - \frac {5}{3} \, x + \frac {25}{2}\right )}}{25 \, {\left (x^{2} + 2 \, x e^{x} + e^{\left (2 \, x\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x+60)*exp(x)^3+(-8*x^2+84*x+36)*exp(x)^2+(-5*x^3+57*x^2+18*x)*exp(x)-x^4+15*x^3)*exp(1/36*x^2-5
/6*x+25/4)^2/(225*exp(x)^3+675*x*exp(x)^2+675*exp(x)*x^2+225*x^3),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {sage}_{0} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x+60)*exp(x)^3+(-8*x^2+84*x+36)*exp(x)^2+(-5*x^3+57*x^2+18*x)*exp(x)-x^4+15*x^3)*exp(1/36*x^2-5
/6*x+25/4)^2/(225*exp(x)^3+675*x*exp(x)^2+675*exp(x)*x^2+225*x^3),x, algorithm="giac")

[Out]

sage0*x

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maple [A]  time = 0.07, size = 32, normalized size = 0.89

method	result	size
risch	\(-\frac {\left (x^{2}+4 \,{\mathrm e}^{x} x +4 \,{\mathrm e}^{2 x}\right ) {\mathrm e}^{\frac {\left (x -15\right )^{2}}{18}}}{25 \left ({\mathrm e}^{x}+x \right )^{2}}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-4*x+60)*exp(x)^3+(-8*x^2+84*x+36)*exp(x)^2+(-5*x^3+57*x^2+18*x)*exp(x)-x^4+15*x^3)*exp(1/36*x^2-5/6*x+2
5/4)^2/(225*exp(x)^3+675*x*exp(x)^2+675*exp(x)*x^2+225*x^3),x,method=_RETURNVERBOSE)

[Out]

-1/25*(x^2+4*exp(x)*x+4*exp(2*x))/(exp(x)+x)^2*exp(1/18*(x-15)^2)

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maxima [B]  time = 0.60, size = 52, normalized size = 1.44 \begin {gather*} -\frac {{\left (x^{2} e^{\frac {25}{2}} + 4 \, x e^{\left (x + \frac {25}{2}\right )} + 4 \, e^{\left (2 \, x + \frac {25}{2}\right )}\right )} e^{\left (\frac {1}{18} \, x^{2}\right )}}{25 \, {\left (x^{2} e^{\left (\frac {5}{3} \, x\right )} + 2 \, x e^{\left (\frac {8}{3} \, x\right )} + e^{\left (\frac {11}{3} \, x\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x+60)*exp(x)^3+(-8*x^2+84*x+36)*exp(x)^2+(-5*x^3+57*x^2+18*x)*exp(x)-x^4+15*x^3)*exp(1/36*x^2-5
/6*x+25/4)^2/(225*exp(x)^3+675*x*exp(x)^2+675*exp(x)*x^2+225*x^3),x, algorithm="maxima")

[Out]

-1/25*(x^2*e^(25/2) + 4*x*e^(x + 25/2) + 4*e^(2*x + 25/2))*e^(1/18*x^2)/(x^2*e^(5/3*x) + 2*x*e^(8/3*x) + e^(11
/3*x))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {{\mathrm {e}}^{\frac {x^2}{18}-\frac {5\,x}{3}+\frac {25}{2}}\,\left ({\mathrm {e}}^{2\,x}\,\left (-8\,x^2+84\,x+36\right )-{\mathrm {e}}^{3\,x}\,\left (4\,x-60\right )+15\,x^3-x^4+{\mathrm {e}}^x\,\left (-5\,x^3+57\,x^2+18\,x\right )\right )}{225\,{\mathrm {e}}^{3\,x}+675\,x\,{\mathrm {e}}^{2\,x}+675\,x^2\,{\mathrm {e}}^x+225\,x^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x^2/18 - (5*x)/3 + 25/2)*(exp(2*x)*(84*x - 8*x^2 + 36) - exp(3*x)*(4*x - 60) + 15*x^3 - x^4 + exp(x)*
(18*x + 57*x^2 - 5*x^3)))/(225*exp(3*x) + 675*x*exp(2*x) + 675*x^2*exp(x) + 225*x^3),x)

[Out]

int((exp(x^2/18 - (5*x)/3 + 25/2)*(exp(2*x)*(84*x - 8*x^2 + 36) - exp(3*x)*(4*x - 60) + 15*x^3 - x^4 + exp(x)*
(18*x + 57*x^2 - 5*x^3)))/(225*exp(3*x) + 675*x*exp(2*x) + 675*x^2*exp(x) + 225*x^3), x)

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sympy [B]  time = 0.28, size = 51, normalized size = 1.42 \begin {gather*} \frac {\left (- x^{2} - 4 x e^{x} - 4 e^{2 x}\right ) e^{\frac {x^{2}}{18} - \frac {5 x}{3} + \frac {25}{2}}}{25 x^{2} + 50 x e^{x} + 25 e^{2 x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x+60)*exp(x)**3+(-8*x**2+84*x+36)*exp(x)**2+(-5*x**3+57*x**2+18*x)*exp(x)-x**4+15*x**3)*exp(1/3
6*x**2-5/6*x+25/4)**2/(225*exp(x)**3+675*x*exp(x)**2+675*exp(x)*x**2+225*x**3),x)

[Out]

(-x**2 - 4*x*exp(x) - 4*exp(2*x))*exp(x**2/18 - 5*x/3 + 25/2)/(25*x**2 + 50*x*exp(x) + 25*exp(2*x))

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