3.7.36 \(\int \frac {e^{\frac {3}{50 x-25 x^3+e^x (-10+5 x^2)}} (-90+135 x^2+e^x (18-18 x-9 x^2))}{500 x^2-500 x^4+125 x^6+e^{2 x} (20-20 x^2+5 x^4)+e^x (-200 x+200 x^3-50 x^5)} \, dx\)

Optimal. Leaf size=30 \[ -4+3 e^{\frac {3}{5 \left (-e^x+5 x\right ) \left (2-x^2\right )}} \]

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Rubi [F]  time = 7.61, 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 {3}{50 x-25 x^3+e^x \left (-10+5 x^2\right )}} \left (-90+135 x^2+e^x \left (18-18 x-9 x^2\right )\right )}{500 x^2-500 x^4+125 x^6+e^{2 x} \left (20-20 x^2+5 x^4\right )+e^x \left (-200 x+200 x^3-50 x^5\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(3/(50*x - 25*x^3 + E^x*(-10 + 5*x^2)))*(-90 + 135*x^2 + E^x*(18 - 18*x - 9*x^2)))/(500*x^2 - 500*x^4 +
125*x^6 + E^(2*x)*(20 - 20*x^2 + 5*x^4) + E^x*(-200*x + 200*x^3 - 50*x^5)),x]

[Out]

(9*Defer[Int][E^(3/(5*(E^x - 5*x)*(-2 + x^2)))/((E^x - 5*x)^2*(Sqrt[2] - x)), x])/2 - (9*Defer[Int][E^(3/(5*(E
^x - 5*x)*(-2 + x^2)))/((E^x - 5*x)^2*(Sqrt[2] - x)), x])/(2*Sqrt[2]) + (9*Defer[Int][E^(3/(5*(E^x - 5*x)*(-2
+ x^2)))/((E^x - 5*x)*(Sqrt[2] - x)), x])/(10*Sqrt[2]) - (9*Defer[Int][E^(3/(5*(E^x - 5*x)*(-2 + x^2)))/((E^x
- 5*x)^2*(Sqrt[2] + x)), x])/2 - (9*Defer[Int][E^(3/(5*(E^x - 5*x)*(-2 + x^2)))/((E^x - 5*x)^2*(Sqrt[2] + x)),
x])/(2*Sqrt[2]) + (9*Defer[Int][E^(3/(5*(E^x - 5*x)*(-2 + x^2)))/((E^x - 5*x)*(Sqrt[2] + x)), x])/(10*Sqrt[2]
) - (18*Defer[Int][(E^(3/(5*(E^x - 5*x)*(-2 + x^2)))*x)/((E^x - 5*x)*(-2 + x^2)^2), x])/5

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {9 e^{\frac {3}{5 \left (e^x-5 x\right ) \left (-2+x^2\right )}} \left (-10+15 x^2-e^x \left (-2+2 x+x^2\right )\right )}{5 \left (e^x-5 x\right )^2 \left (2-x^2\right )^2} \, dx\\ &=\frac {9}{5} \int \frac {e^{\frac {3}{5 \left (e^x-5 x\right ) \left (-2+x^2\right )}} \left (-10+15 x^2-e^x \left (-2+2 x+x^2\right )\right )}{\left (e^x-5 x\right )^2 \left (2-x^2\right )^2} \, dx\\ &=\frac {9}{5} \int \left (-\frac {5 e^{\frac {3}{5 \left (e^x-5 x\right ) \left (-2+x^2\right )}} (-1+x)}{\left (e^x-5 x\right )^2 \left (-2+x^2\right )}-\frac {e^{\frac {3}{5 \left (e^x-5 x\right ) \left (-2+x^2\right )}} \left (-2+2 x+x^2\right )}{\left (e^x-5 x\right ) \left (-2+x^2\right )^2}\right ) \, dx\\ &=-\left (\frac {9}{5} \int \frac {e^{\frac {3}{5 \left (e^x-5 x\right ) \left (-2+x^2\right )}} \left (-2+2 x+x^2\right )}{\left (e^x-5 x\right ) \left (-2+x^2\right )^2} \, dx\right )-9 \int \frac {e^{\frac {3}{5 \left (e^x-5 x\right ) \left (-2+x^2\right )}} (-1+x)}{\left (e^x-5 x\right )^2 \left (-2+x^2\right )} \, dx\\ &=-\left (\frac {9}{5} \int \left (\frac {2 e^{\frac {3}{5 \left (e^x-5 x\right ) \left (-2+x^2\right )}} x}{\left (e^x-5 x\right ) \left (-2+x^2\right )^2}+\frac {e^{\frac {3}{5 \left (e^x-5 x\right ) \left (-2+x^2\right )}}}{\left (e^x-5 x\right ) \left (-2+x^2\right )}\right ) \, dx\right )-9 \int \left (-\frac {e^{\frac {3}{5 \left (e^x-5 x\right ) \left (-2+x^2\right )}}}{\left (e^x-5 x\right )^2 \left (-2+x^2\right )}+\frac {e^{\frac {3}{5 \left (e^x-5 x\right ) \left (-2+x^2\right )}} x}{\left (e^x-5 x\right )^2 \left (-2+x^2\right )}\right ) \, dx\\ &=-\left (\frac {9}{5} \int \frac {e^{\frac {3}{5 \left (e^x-5 x\right ) \left (-2+x^2\right )}}}{\left (e^x-5 x\right ) \left (-2+x^2\right )} \, dx\right )-\frac {18}{5} \int \frac {e^{\frac {3}{5 \left (e^x-5 x\right ) \left (-2+x^2\right )}} x}{\left (e^x-5 x\right ) \left (-2+x^2\right )^2} \, dx+9 \int \frac {e^{\frac {3}{5 \left (e^x-5 x\right ) \left (-2+x^2\right )}}}{\left (e^x-5 x\right )^2 \left (-2+x^2\right )} \, dx-9 \int \frac {e^{\frac {3}{5 \left (e^x-5 x\right ) \left (-2+x^2\right )}} x}{\left (e^x-5 x\right )^2 \left (-2+x^2\right )} \, dx\\ &=-\left (\frac {9}{5} \int \left (-\frac {e^{\frac {3}{5 \left (e^x-5 x\right ) \left (-2+x^2\right )}}}{2 \sqrt {2} \left (e^x-5 x\right ) \left (\sqrt {2}-x\right )}-\frac {e^{\frac {3}{5 \left (e^x-5 x\right ) \left (-2+x^2\right )}}}{2 \sqrt {2} \left (e^x-5 x\right ) \left (\sqrt {2}+x\right )}\right ) \, dx\right )-\frac {18}{5} \int \frac {e^{\frac {3}{5 \left (e^x-5 x\right ) \left (-2+x^2\right )}} x}{\left (e^x-5 x\right ) \left (-2+x^2\right )^2} \, dx-9 \int \left (-\frac {e^{\frac {3}{5 \left (e^x-5 x\right ) \left (-2+x^2\right )}}}{2 \left (e^x-5 x\right )^2 \left (\sqrt {2}-x\right )}+\frac {e^{\frac {3}{5 \left (e^x-5 x\right ) \left (-2+x^2\right )}}}{2 \left (e^x-5 x\right )^2 \left (\sqrt {2}+x\right )}\right ) \, dx+9 \int \left (-\frac {e^{\frac {3}{5 \left (e^x-5 x\right ) \left (-2+x^2\right )}}}{2 \sqrt {2} \left (e^x-5 x\right )^2 \left (\sqrt {2}-x\right )}-\frac {e^{\frac {3}{5 \left (e^x-5 x\right ) \left (-2+x^2\right )}}}{2 \sqrt {2} \left (e^x-5 x\right )^2 \left (\sqrt {2}+x\right )}\right ) \, dx\\ &=-\left (\frac {18}{5} \int \frac {e^{\frac {3}{5 \left (e^x-5 x\right ) \left (-2+x^2\right )}} x}{\left (e^x-5 x\right ) \left (-2+x^2\right )^2} \, dx\right )+\frac {9}{2} \int \frac {e^{\frac {3}{5 \left (e^x-5 x\right ) \left (-2+x^2\right )}}}{\left (e^x-5 x\right )^2 \left (\sqrt {2}-x\right )} \, dx-\frac {9}{2} \int \frac {e^{\frac {3}{5 \left (e^x-5 x\right ) \left (-2+x^2\right )}}}{\left (e^x-5 x\right )^2 \left (\sqrt {2}+x\right )} \, dx+\frac {9 \int \frac {e^{\frac {3}{5 \left (e^x-5 x\right ) \left (-2+x^2\right )}}}{\left (e^x-5 x\right ) \left (\sqrt {2}-x\right )} \, dx}{10 \sqrt {2}}+\frac {9 \int \frac {e^{\frac {3}{5 \left (e^x-5 x\right ) \left (-2+x^2\right )}}}{\left (e^x-5 x\right ) \left (\sqrt {2}+x\right )} \, dx}{10 \sqrt {2}}-\frac {9 \int \frac {e^{\frac {3}{5 \left (e^x-5 x\right ) \left (-2+x^2\right )}}}{\left (e^x-5 x\right )^2 \left (\sqrt {2}-x\right )} \, dx}{2 \sqrt {2}}-\frac {9 \int \frac {e^{\frac {3}{5 \left (e^x-5 x\right ) \left (-2+x^2\right )}}}{\left (e^x-5 x\right )^2 \left (\sqrt {2}+x\right )} \, dx}{2 \sqrt {2}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.96, size = 24, normalized size = 0.80 \begin {gather*} 3 e^{\frac {3}{5 \left (e^x-5 x\right ) \left (-2+x^2\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(3/(50*x - 25*x^3 + E^x*(-10 + 5*x^2)))*(-90 + 135*x^2 + E^x*(18 - 18*x - 9*x^2)))/(500*x^2 - 500
*x^4 + 125*x^6 + E^(2*x)*(20 - 20*x^2 + 5*x^4) + E^x*(-200*x + 200*x^3 - 50*x^5)),x]

[Out]

3*E^(3/(5*(E^x - 5*x)*(-2 + x^2)))

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fricas [A]  time = 0.83, size = 25, normalized size = 0.83 \begin {gather*} 3 \, e^{\left (-\frac {3}{5 \, {\left (5 \, x^{3} - {\left (x^{2} - 2\right )} e^{x} - 10 \, x\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-9*x^2-18*x+18)*exp(x)+135*x^2-90)*exp(3/((5*x^2-10)*exp(x)-25*x^3+50*x))/((5*x^4-20*x^2+20)*exp(x
)^2+(-50*x^5+200*x^3-200*x)*exp(x)+125*x^6-500*x^4+500*x^2),x, algorithm="fricas")

[Out]

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

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giac [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(((-9*x^2-18*x+18)*exp(x)+135*x^2-90)*exp(3/((5*x^2-10)*exp(x)-25*x^3+50*x))/((5*x^4-20*x^2+20)*exp(x
)^2+(-50*x^5+200*x^3-200*x)*exp(x)+125*x^6-500*x^4+500*x^2),x, algorithm="giac")

[Out]

Exception raised: RuntimeError >> An error occurred running a Giac command:INPUT:sage2OUTPUT:Unable to divide,
perhaps due to rounding error%%%{-375,[1,15]%%%}+%%%{375,[1,14]%%%}+%%%{5625,[1,13]%%%}+%%%{-7875,[1,12]%%%}+
%%%{-30750,[1,

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maple [A]  time = 0.09, size = 21, normalized size = 0.70

method	result	size
risch	\(3 \,{\mathrm e}^{\frac {3}{5 \left (x^{2}-2\right ) \left ({\mathrm e}^{x}-5 x \right )}}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-9*x^2-18*x+18)*exp(x)+135*x^2-90)*exp(3/((5*x^2-10)*exp(x)-25*x^3+50*x))/((5*x^4-20*x^2+20)*exp(x)^2+(-
50*x^5+200*x^3-200*x)*exp(x)+125*x^6-500*x^4+500*x^2),x,method=_RETURNVERBOSE)

[Out]

3*exp(3/5/(x^2-2)/(exp(x)-5*x))

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maxima [B]  time = 0.95, size = 76, normalized size = 2.53 \begin {gather*} 3 \, e^{\left (-\frac {3 \, x}{50 \, x^{2} - {\left (x^{2} - 2\right )} e^{\left (2 \, x\right )} - 100} - \frac {3 \, e^{x}}{5 \, {\left (50 \, x^{2} - {\left (x^{2} - 2\right )} e^{\left (2 \, x\right )} - 100\right )}} - \frac {15}{5 \, x e^{\left (2 \, x\right )} - 250 \, x - e^{\left (3 \, x\right )} + 50 \, e^{x}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-9*x^2-18*x+18)*exp(x)+135*x^2-90)*exp(3/((5*x^2-10)*exp(x)-25*x^3+50*x))/((5*x^4-20*x^2+20)*exp(x
)^2+(-50*x^5+200*x^3-200*x)*exp(x)+125*x^6-500*x^4+500*x^2),x, algorithm="maxima")

[Out]

3*e^(-3*x/(50*x^2 - (x^2 - 2)*e^(2*x) - 100) - 3/5*e^x/(50*x^2 - (x^2 - 2)*e^(2*x) - 100) - 15/(5*x*e^(2*x) -
250*x - e^(3*x) + 50*e^x))

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mupad [B]  time = 1.07, size = 27, normalized size = 0.90 \begin {gather*} 3\,{\mathrm {e}}^{\frac {3}{5\,\left (10\,x-2\,{\mathrm {e}}^x+x^2\,{\mathrm {e}}^x-5\,x^3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(3/(50*x + exp(x)*(5*x^2 - 10) - 25*x^3))*(exp(x)*(18*x + 9*x^2 - 18) - 135*x^2 + 90))/(exp(2*x)*(5*x
^4 - 20*x^2 + 20) + 500*x^2 - 500*x^4 + 125*x^6 - exp(x)*(200*x - 200*x^3 + 50*x^5)),x)

[Out]

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

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sympy [A]  time = 0.48, size = 22, normalized size = 0.73 \begin {gather*} 3 e^{\frac {3}{- 25 x^{3} + 50 x + \left (5 x^{2} - 10\right ) e^{x}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-9*x**2-18*x+18)*exp(x)+135*x**2-90)*exp(3/((5*x**2-10)*exp(x)-25*x**3+50*x))/((5*x**4-20*x**2+20)
*exp(x)**2+(-50*x**5+200*x**3-200*x)*exp(x)+125*x**6-500*x**4+500*x**2),x)

[Out]

3*exp(3/(-25*x**3 + 50*x + (5*x**2 - 10)*exp(x)))

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