3.15.31 \(\int \frac {9 x^2+2 e^3 x^2+e^{2 x} x^2+e^{\frac {2 (1-3 x+x \log (5))}{x}} x^2+6 x^3+x^4+e^x (-6 x^2-2 e^3 x^2-2 x^3)+e^{\frac {1-3 x+x \log (5)}{x}} (2 e^3-6 x^2+2 e^x x^2-2 x^3)}{9 x^2+e^{2 x} x^2+e^{\frac {2 (1-3 x+x \log (5))}{x}} x^2+6 x^3+x^4+e^x (-6 x^2-2 x^3)+e^{\frac {1-3 x+x \log (5)}{x}} (-6 x^2+2 e^x x^2-2 x^3)} \, dx\)

Optimal. Leaf size=26 \[ x-\frac {2 e^3}{3-5 e^{-3+\frac {1}{x}}-e^x+x} \]

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Rubi [F]  time = 3.17, 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 {9 x^2+2 e^3 x^2+e^{2 x} x^2+e^{\frac {2 (1-3 x+x \log (5))}{x}} x^2+6 x^3+x^4+e^x \left (-6 x^2-2 e^3 x^2-2 x^3\right )+e^{\frac {1-3 x+x \log (5)}{x}} \left (2 e^3-6 x^2+2 e^x x^2-2 x^3\right )}{9 x^2+e^{2 x} x^2+e^{\frac {2 (1-3 x+x \log (5))}{x}} x^2+6 x^3+x^4+e^x \left (-6 x^2-2 x^3\right )+e^{\frac {1-3 x+x \log (5)}{x}} \left (-6 x^2+2 e^x x^2-2 x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(9*x^2 + 2*E^3*x^2 + E^(2*x)*x^2 + E^((2*(1 - 3*x + x*Log[5]))/x)*x^2 + 6*x^3 + x^4 + E^x*(-6*x^2 - 2*E^3*
x^2 - 2*x^3) + E^((1 - 3*x + x*Log[5])/x)*(2*E^3 - 6*x^2 + 2*E^x*x^2 - 2*x^3))/(9*x^2 + E^(2*x)*x^2 + E^((2*(1
- 3*x + x*Log[5]))/x)*x^2 + 6*x^3 + x^4 + E^x*(-6*x^2 - 2*x^3) + E^((1 - 3*x + x*Log[5])/x)*(-6*x^2 + 2*E^x*x
^2 - 2*x^3)),x]

[Out]

x - 4*E^9*Defer[Int][(-3*E^3 + 5*E^x^(-1) + E^(3 + x) - E^3*x)^(-2), x] + 10*E^6*Defer[Int][E^x^(-1)/(-3*E^3 +
5*E^x^(-1) + E^(3 + x) - E^3*x)^2, x] + 10*E^6*Defer[Int][E^x^(-1)/(x^2*(-3*E^3 + 5*E^x^(-1) + E^(3 + x) - E^
3*x)^2), x] - 2*E^6*Defer[Int][(-3*E^3 + 5*E^x^(-1) + E^(3 + x) - E^3*x)^(-1), x] - 2*E^9*Defer[Int][x/(3*E^3
- 5*E^x^(-1) - E^(3 + x) + E^3*x)^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{2 x} x^2+e^{\frac {2 (1-3 x+x \log (5))}{x}} x^2+\left (9+2 e^3\right ) x^2+6 x^3+x^4+e^x \left (-6 x^2-2 e^3 x^2-2 x^3\right )+e^{\frac {1-3 x+x \log (5)}{x}} \left (2 e^3-6 x^2+2 e^x x^2-2 x^3\right )}{9 x^2+e^{2 x} x^2+e^{\frac {2 (1-3 x+x \log (5))}{x}} x^2+6 x^3+x^4+e^x \left (-6 x^2-2 x^3\right )+e^{\frac {1-3 x+x \log (5)}{x}} \left (-6 x^2+2 e^x x^2-2 x^3\right )} \, dx\\ &=\int \frac {10 e^{6+\frac {1}{x}}+2 e^9 x^2+25 e^{2/x} x^2-2 e^{9+x} x^2+10 e^{3+\frac {1}{x}+x} x^2+e^{6+2 x} x^2-10 e^{3+\frac {1}{x}} x^2 (3+x)-2 e^{6+x} x^2 (3+x)+e^6 x^2 (3+x)^2}{x^2 \left (5 e^{\frac {1}{x}}+e^{3+x}-e^3 (3+x)\right )^2} \, dx\\ &=\int \left (1-\frac {2 e^6}{-3 e^3+5 e^{\frac {1}{x}}+e^{3+x}-e^3 x}-\frac {2 e^6 \left (-5 e^{\frac {1}{x}}+2 e^3 x^2-5 e^{\frac {1}{x}} x^2+e^3 x^3\right )}{x^2 \left (3 e^3-5 e^{\frac {1}{x}}-e^{3+x}+e^3 x\right )^2}\right ) \, dx\\ &=x-\left (2 e^6\right ) \int \frac {1}{-3 e^3+5 e^{\frac {1}{x}}+e^{3+x}-e^3 x} \, dx-\left (2 e^6\right ) \int \frac {-5 e^{\frac {1}{x}}+2 e^3 x^2-5 e^{\frac {1}{x}} x^2+e^3 x^3}{x^2 \left (3 e^3-5 e^{\frac {1}{x}}-e^{3+x}+e^3 x\right )^2} \, dx\\ &=x-\left (2 e^6\right ) \int \frac {1}{-3 e^3+5 e^{\frac {1}{x}}+e^{3+x}-e^3 x} \, dx-\left (2 e^6\right ) \int \left (\frac {2 e^3}{\left (-3 e^3+5 e^{\frac {1}{x}}+e^{3+x}-e^3 x\right )^2}-\frac {5 e^{\frac {1}{x}}}{\left (-3 e^3+5 e^{\frac {1}{x}}+e^{3+x}-e^3 x\right )^2}-\frac {5 e^{\frac {1}{x}}}{x^2 \left (-3 e^3+5 e^{\frac {1}{x}}+e^{3+x}-e^3 x\right )^2}+\frac {e^3 x}{\left (3 e^3-5 e^{\frac {1}{x}}-e^{3+x}+e^3 x\right )^2}\right ) \, dx\\ &=x-\left (2 e^6\right ) \int \frac {1}{-3 e^3+5 e^{\frac {1}{x}}+e^{3+x}-e^3 x} \, dx+\left (10 e^6\right ) \int \frac {e^{\frac {1}{x}}}{\left (-3 e^3+5 e^{\frac {1}{x}}+e^{3+x}-e^3 x\right )^2} \, dx+\left (10 e^6\right ) \int \frac {e^{\frac {1}{x}}}{x^2 \left (-3 e^3+5 e^{\frac {1}{x}}+e^{3+x}-e^3 x\right )^2} \, dx-\left (2 e^9\right ) \int \frac {x}{\left (3 e^3-5 e^{\frac {1}{x}}-e^{3+x}+e^3 x\right )^2} \, dx-\left (4 e^9\right ) \int \frac {1}{\left (-3 e^3+5 e^{\frac {1}{x}}+e^{3+x}-e^3 x\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.17, size = 30, normalized size = 1.15 \begin {gather*} x+\frac {2 e^6}{5 e^{\frac {1}{x}}+e^{3+x}-e^3 (3+x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(9*x^2 + 2*E^3*x^2 + E^(2*x)*x^2 + E^((2*(1 - 3*x + x*Log[5]))/x)*x^2 + 6*x^3 + x^4 + E^x*(-6*x^2 -
2*E^3*x^2 - 2*x^3) + E^((1 - 3*x + x*Log[5])/x)*(2*E^3 - 6*x^2 + 2*E^x*x^2 - 2*x^3))/(9*x^2 + E^(2*x)*x^2 + E^
((2*(1 - 3*x + x*Log[5]))/x)*x^2 + 6*x^3 + x^4 + E^x*(-6*x^2 - 2*x^3) + E^((1 - 3*x + x*Log[5])/x)*(-6*x^2 + 2
*E^x*x^2 - 2*x^3)),x]

[Out]

x + (2*E^6)/(5*E^x^(-1) + E^(3 + x) - E^3*(3 + x))

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fricas [B]  time = 0.75, size = 59, normalized size = 2.27 \begin {gather*} \frac {x^{2} - x e^{x} - x e^{\left (\frac {x \log \relax (5) - 3 \, x + 1}{x}\right )} + 3 \, x - 2 \, e^{3}}{x - e^{x} - e^{\left (\frac {x \log \relax (5) - 3 \, x + 1}{x}\right )} + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2*exp((x*log(5)-3*x+1)/x)^2+(2*exp(x)*x^2+2*exp(3)-2*x^3-6*x^2)*exp((x*log(5)-3*x+1)/x)+exp(x)^2*
x^2+(-2*x^2*exp(3)-2*x^3-6*x^2)*exp(x)+2*x^2*exp(3)+x^4+6*x^3+9*x^2)/(x^2*exp((x*log(5)-3*x+1)/x)^2+(2*exp(x)*
x^2-2*x^3-6*x^2)*exp((x*log(5)-3*x+1)/x)+exp(x)^2*x^2+(-2*x^3-6*x^2)*exp(x)+x^4+6*x^3+9*x^2),x, algorithm="fri
cas")

[Out]

(x^2 - x*e^x - x*e^((x*log(5) - 3*x + 1)/x) + 3*x - 2*e^3)/(x - e^x - e^((x*log(5) - 3*x + 1)/x) + 3)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} + 6 \, x^{3} + 2 \, x^{2} e^{3} + x^{2} e^{\left (2 \, x\right )} + x^{2} e^{\left (\frac {2 \, {\left (x \log \relax (5) - 3 \, x + 1\right )}}{x}\right )} + 9 \, x^{2} - 2 \, {\left (x^{3} + x^{2} e^{3} + 3 \, x^{2}\right )} e^{x} - 2 \, {\left (x^{3} - x^{2} e^{x} + 3 \, x^{2} - e^{3}\right )} e^{\left (\frac {x \log \relax (5) - 3 \, x + 1}{x}\right )}}{x^{4} + 6 \, x^{3} + x^{2} e^{\left (2 \, x\right )} + x^{2} e^{\left (\frac {2 \, {\left (x \log \relax (5) - 3 \, x + 1\right )}}{x}\right )} + 9 \, x^{2} - 2 \, {\left (x^{3} + 3 \, x^{2}\right )} e^{x} - 2 \, {\left (x^{3} - x^{2} e^{x} + 3 \, x^{2}\right )} e^{\left (\frac {x \log \relax (5) - 3 \, x + 1}{x}\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2*exp((x*log(5)-3*x+1)/x)^2+(2*exp(x)*x^2+2*exp(3)-2*x^3-6*x^2)*exp((x*log(5)-3*x+1)/x)+exp(x)^2*
x^2+(-2*x^2*exp(3)-2*x^3-6*x^2)*exp(x)+2*x^2*exp(3)+x^4+6*x^3+9*x^2)/(x^2*exp((x*log(5)-3*x+1)/x)^2+(2*exp(x)*
x^2-2*x^3-6*x^2)*exp((x*log(5)-3*x+1)/x)+exp(x)^2*x^2+(-2*x^3-6*x^2)*exp(x)+x^4+6*x^3+9*x^2),x, algorithm="gia
c")

[Out]

integrate((x^4 + 6*x^3 + 2*x^2*e^3 + x^2*e^(2*x) + x^2*e^(2*(x*log(5) - 3*x + 1)/x) + 9*x^2 - 2*(x^3 + x^2*e^3
+ 3*x^2)*e^x - 2*(x^3 - x^2*e^x + 3*x^2 - e^3)*e^((x*log(5) - 3*x + 1)/x))/(x^4 + 6*x^3 + x^2*e^(2*x) + x^2*e
^(2*(x*log(5) - 3*x + 1)/x) + 9*x^2 - 2*(x^3 + 3*x^2)*e^x - 2*(x^3 - x^2*e^x + 3*x^2)*e^((x*log(5) - 3*x + 1)/
x)), x)

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maple [A]  time = 0.21, size = 29, normalized size = 1.12

method	result	size
risch	\(x -\frac {2 \,{\mathrm e}^{3}}{x -{\mathrm e}^{x}-5 \,{\mathrm e}^{-\frac {3 x -1}{x}}+3}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2*exp((x*ln(5)-3*x+1)/x)^2+(2*exp(x)*x^2+2*exp(3)-2*x^3-6*x^2)*exp((x*ln(5)-3*x+1)/x)+exp(x)^2*x^2+(-2*
x^2*exp(3)-2*x^3-6*x^2)*exp(x)+2*x^2*exp(3)+x^4+6*x^3+9*x^2)/(x^2*exp((x*ln(5)-3*x+1)/x)^2+(2*exp(x)*x^2-2*x^3
-6*x^2)*exp((x*ln(5)-3*x+1)/x)+exp(x)^2*x^2+(-2*x^3-6*x^2)*exp(x)+x^4+6*x^3+9*x^2),x,method=_RETURNVERBOSE)

[Out]

x-2*exp(3)/(x-exp(x)-5*exp(-(3*x-1)/x)+3)

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maxima [B]  time = 0.54, size = 54, normalized size = 2.08 \begin {gather*} \frac {x^{2} e^{3} + 3 \, x e^{3} - x e^{\left (x + 3\right )} - 5 \, x e^{\frac {1}{x}} - 2 \, e^{6}}{x e^{3} + 3 \, e^{3} - e^{\left (x + 3\right )} - 5 \, e^{\frac {1}{x}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2*exp((x*log(5)-3*x+1)/x)^2+(2*exp(x)*x^2+2*exp(3)-2*x^3-6*x^2)*exp((x*log(5)-3*x+1)/x)+exp(x)^2*
x^2+(-2*x^2*exp(3)-2*x^3-6*x^2)*exp(x)+2*x^2*exp(3)+x^4+6*x^3+9*x^2)/(x^2*exp((x*log(5)-3*x+1)/x)^2+(2*exp(x)*
x^2-2*x^3-6*x^2)*exp((x*log(5)-3*x+1)/x)+exp(x)^2*x^2+(-2*x^3-6*x^2)*exp(x)+x^4+6*x^3+9*x^2),x, algorithm="max
ima")

[Out]

(x^2*e^3 + 3*x*e^3 - x*e^(x + 3) - 5*x*e^(1/x) - 2*e^6)/(x*e^3 + 3*e^3 - e^(x + 3) - 5*e^(1/x))

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mupad [B]  time = 1.10, size = 23, normalized size = 0.88 \begin {gather*} x-\frac {2\,{\mathrm {e}}^3}{x-{\mathrm {e}}^x-5\,{\mathrm {e}}^{1/x}\,{\mathrm {e}}^{-3}+3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((x*log(5) - 3*x + 1)/x)*(2*exp(3) + 2*x^2*exp(x) - 6*x^2 - 2*x^3) + x^2*exp((2*(x*log(5) - 3*x + 1))/
x) + x^2*exp(2*x) + 2*x^2*exp(3) - exp(x)*(2*x^2*exp(3) + 6*x^2 + 2*x^3) + 9*x^2 + 6*x^3 + x^4)/(x^2*exp((2*(x
*log(5) - 3*x + 1))/x) - exp(x)*(6*x^2 + 2*x^3) - exp((x*log(5) - 3*x + 1)/x)*(6*x^2 - 2*x^2*exp(x) + 2*x^3) +
x^2*exp(2*x) + 9*x^2 + 6*x^3 + x^4),x)

[Out]

x - (2*exp(3))/(x - exp(x) - 5*exp(1/x)*exp(-3) + 3)

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sympy [A]  time = 0.23, size = 26, normalized size = 1.00 \begin {gather*} x + \frac {2 e^{3}}{- x + e^{x} + e^{\frac {- 3 x + x \log {\relax (5 )} + 1}{x}} - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2*exp((x*ln(5)-3*x+1)/x)**2+(2*exp(x)*x**2+2*exp(3)-2*x**3-6*x**2)*exp((x*ln(5)-3*x+1)/x)+exp(x)
**2*x**2+(-2*x**2*exp(3)-2*x**3-6*x**2)*exp(x)+2*x**2*exp(3)+x**4+6*x**3+9*x**2)/(x**2*exp((x*ln(5)-3*x+1)/x)*
*2+(2*exp(x)*x**2-2*x**3-6*x**2)*exp((x*ln(5)-3*x+1)/x)+exp(x)**2*x**2+(-2*x**3-6*x**2)*exp(x)+x**4+6*x**3+9*x
**2),x)

[Out]

x + 2*exp(3)/(-x + exp(x) + exp((-3*x + x*log(5) + 1)/x) - 3)

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