3.40.10 \(\int \frac {e^{-x+\frac {e^{-x} (45 x^3+e^x (15-10 x-x^3))}{-15+x^3}} (-2025 x^2+675 x^3-45 x^6+e^x (150+20 x^3))}{225-30 x^3+x^6} \, dx\)

Optimal. Leaf size=29 \[ e^{-1+\frac {x \left (-2+9 e^{-x} x^2\right )}{-3+\frac {x^3}{5}}} \]

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Rubi [F]  time = 10.55, 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 {\exp \left (-x+\frac {e^{-x} \left (45 x^3+e^x \left (15-10 x-x^3\right )\right )}{-15+x^3}\right ) \left (-2025 x^2+675 x^3-45 x^6+e^x \left (150+20 x^3\right )\right )}{225-30 x^3+x^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(-x + (45*x^3 + E^x*(15 - 10*x - x^3))/(E^x*(-15 + x^3)))*(-2025*x^2 + 675*x^3 - 45*x^6 + E^x*(150 + 20
*x^3)))/(225 - 30*x^3 + x^6),x]

[Out]

-45*Defer[Int][E^(-x + (45*x^3 + E^x*(15 - 10*x - x^3))/(E^x*(-15 + x^3))), x] + (2*5^(2/3)*Defer[Int][E^((45*
x^3 + E^x*(15 - 10*x - x^3))/(E^x*(-15 + x^3)))/(15^(1/3) - x)^2, x])/(3*3^(1/3)) - (4*5^(1/3)*Defer[Int][E^((
45*x^3 + E^x*(15 - 10*x - x^3))/(E^x*(-15 + x^3)))/(15^(1/3) - x), x])/(3*3^(2/3)) + 15*15^(1/3)*Defer[Int][E^
(-x + (45*x^3 + E^x*(15 - 10*x - x^3))/(E^x*(-15 + x^3)))/(15^(1/3) - x), x] + (2*5^(2/3)*Defer[Int][E^((45*x^
3 + E^x*(15 - 10*x - x^3))/(E^x*(-15 + x^3)))/(15^(1/3) + (-1)^(1/3)*x)^2, x])/(3*3^(1/3)) - (4*5^(1/3)*Defer[
Int][E^((45*x^3 + E^x*(15 - 10*x - x^3))/(E^x*(-15 + x^3)))/(15^(1/3) + (-1)^(1/3)*x), x])/(3*3^(2/3)) + 15*15
^(1/3)*Defer[Int][E^(-x + (45*x^3 + E^x*(15 - 10*x - x^3))/(E^x*(-15 + x^3)))/(15^(1/3) + (-1)^(1/3)*x), x] -
(4*5^(1/3)*Defer[Int][E^((45*x^3 + E^x*(15 - 10*x - x^3))/(E^x*(-15 + x^3)))/(15^(1/3) - (-1)^(2/3)*x), x])/(3
*3^(2/3)) + 15*15^(1/3)*Defer[Int][E^(-x + (45*x^3 + E^x*(15 - 10*x - x^3))/(E^x*(-15 + x^3)))/(15^(1/3) - (-1
)^(2/3)*x), x] - (20*Defer[Int][E^((45*x^3 + E^x*(15 - 10*x - x^3))/(E^x*(-15 + x^3)))/(-15 + 15^(2/3)*x), x])
/3 + (2*15^(2/3)*Defer[Int][E^((45*x^3 + E^x*(15 - 10*x - x^3))/(E^x*(-15 + x^3)))/(15^(1/3)*(1 - (-1)^(1/3))
- (1 + (-1)^(2/3))*x)^2, x])/(1 + (-1)^(1/3))^4 + (40*Defer[Int][E^((45*x^3 + E^x*(15 - 10*x - x^3))/(E^x*(-15
+ x^3)))/(30 + 15^(2/3)*(1 - I*Sqrt[3])*x), x])/3 + (40*Defer[Int][E^((45*x^3 + E^x*(15 - 10*x - x^3))/(E^x*(
-15 + x^3)))/(30 + 15^(2/3)*(1 + I*Sqrt[3])*x), x])/3 - 2025*Defer[Int][(E^(-x + (45*x^3 + E^x*(15 - 10*x - x^
3))/(E^x*(-15 + x^3)))*x^2)/(-15 + x^3)^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (-x+\frac {e^{-x} \left (45 x^3+e^x \left (15-10 x-x^3\right )\right )}{-15+x^3}\right ) \left (-2025 x^2+675 x^3-45 x^6+e^x \left (150+20 x^3\right )\right )}{\left (-15+x^3\right )^2} \, dx\\ &=\int \left (\frac {10 \exp \left (\frac {e^{-x} \left (45 x^3+e^x \left (15-10 x-x^3\right )\right )}{-15+x^3}\right ) \left (15+2 x^3\right )}{\left (-15+x^3\right )^2}-\frac {45 \exp \left (-x+\frac {e^{-x} \left (45 x^3+e^x \left (15-10 x-x^3\right )\right )}{-15+x^3}\right ) x^2 \left (45-15 x+x^4\right )}{\left (-15+x^3\right )^2}\right ) \, dx\\ &=10 \int \frac {\exp \left (\frac {e^{-x} \left (45 x^3+e^x \left (15-10 x-x^3\right )\right )}{-15+x^3}\right ) \left (15+2 x^3\right )}{\left (-15+x^3\right )^2} \, dx-45 \int \frac {\exp \left (-x+\frac {e^{-x} \left (45 x^3+e^x \left (15-10 x-x^3\right )\right )}{-15+x^3}\right ) x^2 \left (45-15 x+x^4\right )}{\left (-15+x^3\right )^2} \, dx\\ &=10 \int \left (\frac {45 \exp \left (\frac {e^{-x} \left (45 x^3+e^x \left (15-10 x-x^3\right )\right )}{-15+x^3}\right )}{\left (-15+x^3\right )^2}+\frac {2 \exp \left (\frac {e^{-x} \left (45 x^3+e^x \left (15-10 x-x^3\right )\right )}{-15+x^3}\right )}{-15+x^3}\right ) \, dx-45 \int \left (\exp \left (-x+\frac {e^{-x} \left (45 x^3+e^x \left (15-10 x-x^3\right )\right )}{-15+x^3}\right )+\frac {45 \exp \left (-x+\frac {e^{-x} \left (45 x^3+e^x \left (15-10 x-x^3\right )\right )}{-15+x^3}\right ) x^2}{\left (-15+x^3\right )^2}+\frac {15 \exp \left (-x+\frac {e^{-x} \left (45 x^3+e^x \left (15-10 x-x^3\right )\right )}{-15+x^3}\right )}{-15+x^3}\right ) \, dx\\ &=20 \int \frac {\exp \left (\frac {e^{-x} \left (45 x^3+e^x \left (15-10 x-x^3\right )\right )}{-15+x^3}\right )}{-15+x^3} \, dx-45 \int \exp \left (-x+\frac {e^{-x} \left (45 x^3+e^x \left (15-10 x-x^3\right )\right )}{-15+x^3}\right ) \, dx+450 \int \frac {\exp \left (\frac {e^{-x} \left (45 x^3+e^x \left (15-10 x-x^3\right )\right )}{-15+x^3}\right )}{\left (-15+x^3\right )^2} \, dx-675 \int \frac {\exp \left (-x+\frac {e^{-x} \left (45 x^3+e^x \left (15-10 x-x^3\right )\right )}{-15+x^3}\right )}{-15+x^3} \, dx-2025 \int \frac {\exp \left (-x+\frac {e^{-x} \left (45 x^3+e^x \left (15-10 x-x^3\right )\right )}{-15+x^3}\right ) x^2}{\left (-15+x^3\right )^2} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.78, size = 31, normalized size = 1.07 \begin {gather*} e^{-1-\frac {10 x}{-15+x^3}+\frac {45 e^{-x} x^3}{-15+x^3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-x + (45*x^3 + E^x*(15 - 10*x - x^3))/(E^x*(-15 + x^3)))*(-2025*x^2 + 675*x^3 - 45*x^6 + E^x*(15
0 + 20*x^3)))/(225 - 30*x^3 + x^6),x]

[Out]

E^(-1 - (10*x)/(-15 + x^3) + (45*x^3)/(E^x*(-15 + x^3)))

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fricas [A]  time = 0.70, size = 36, normalized size = 1.24 \begin {gather*} e^{\left (x + \frac {{\left (45 \, x^{3} - {\left (x^{4} + x^{3} - 5 \, x - 15\right )} e^{x}\right )} e^{\left (-x\right )}}{x^{3} - 15}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((20*x^3+150)*exp(x)-45*x^6+675*x^3-2025*x^2)*exp(((-x^3-10*x+15)*exp(x)+45*x^3)/(x^3-15)/exp(x))/(x
^6-30*x^3+225)/exp(x),x, algorithm="fricas")

[Out]

e^(x + (45*x^3 - (x^4 + x^3 - 5*x - 15)*e^x)*e^(-x)/(x^3 - 15))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {5 \, {\left (9 \, x^{6} - 135 \, x^{3} + 405 \, x^{2} - 2 \, {\left (2 \, x^{3} + 15\right )} e^{x}\right )} e^{\left (-x + \frac {{\left (45 \, x^{3} - {\left (x^{3} + 10 \, x - 15\right )} e^{x}\right )} e^{\left (-x\right )}}{x^{3} - 15}\right )}}{x^{6} - 30 \, x^{3} + 225}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((20*x^3+150)*exp(x)-45*x^6+675*x^3-2025*x^2)*exp(((-x^3-10*x+15)*exp(x)+45*x^3)/(x^3-15)/exp(x))/(x
^6-30*x^3+225)/exp(x),x, algorithm="giac")

[Out]

integrate(-5*(9*x^6 - 135*x^3 + 405*x^2 - 2*(2*x^3 + 15)*e^x)*e^(-x + (45*x^3 - (x^3 + 10*x - 15)*e^x)*e^(-x)/
(x^3 - 15))/(x^6 - 30*x^3 + 225), x)

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maple [A]  time = 0.11, size = 36, normalized size = 1.24

method	result	size
risch	\({\mathrm e}^{-\frac {\left ({\mathrm e}^{x} x^{3}-45 x^{3}+10 \,{\mathrm e}^{x} x -15 \,{\mathrm e}^{x}\right ) {\mathrm e}^{-x}}{x^{3}-15}}\)	\(36\)
norman	\(\frac {\left ({\mathrm e}^{x} x^{3} {\mathrm e}^{\frac {\left (\left (-x^{3}-10 x +15\right ) {\mathrm e}^{x}+45 x^{3}\right ) {\mathrm e}^{-x}}{x^{3}-15}}-15 \,{\mathrm e}^{x} {\mathrm e}^{\frac {\left (\left (-x^{3}-10 x +15\right ) {\mathrm e}^{x}+45 x^{3}\right ) {\mathrm e}^{-x}}{x^{3}-15}}\right ) {\mathrm e}^{-x}}{x^{3}-15}\)	\(88\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((20*x^3+150)*exp(x)-45*x^6+675*x^3-2025*x^2)*exp(((-x^3-10*x+15)*exp(x)+45*x^3)/(x^3-15)/exp(x))/(x^6-30*
x^3+225)/exp(x),x,method=_RETURNVERBOSE)

[Out]

exp(-(exp(x)*x^3-45*x^3+10*exp(x)*x-15*exp(x))*exp(-x)/(x^3-15))

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maxima [A]  time = 0.62, size = 32, normalized size = 1.10 \begin {gather*} e^{\left (-\frac {10 \, x}{x^{3} - 15} + \frac {675 \, e^{\left (-x\right )}}{x^{3} - 15} + 45 \, e^{\left (-x\right )} - 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((20*x^3+150)*exp(x)-45*x^6+675*x^3-2025*x^2)*exp(((-x^3-10*x+15)*exp(x)+45*x^3)/(x^3-15)/exp(x))/(x
^6-30*x^3+225)/exp(x),x, algorithm="maxima")

[Out]

e^(-10*x/(x^3 - 15) + 675*e^(-x)/(x^3 - 15) + 45*e^(-x) - 1)

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mupad [B]  time = 2.40, size = 52, normalized size = 1.79 \begin {gather*} {\mathrm {e}}^{-\frac {x^3}{x^3-15}}\,{\mathrm {e}}^{\frac {15}{x^3-15}}\,{\mathrm {e}}^{\frac {45\,x^3\,{\mathrm {e}}^{-x}}{x^3-15}}\,{\mathrm {e}}^{-\frac {10\,x}{x^3-15}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-x)*exp(-(exp(-x)*(exp(x)*(10*x + x^3 - 15) - 45*x^3))/(x^3 - 15))*(exp(x)*(20*x^3 + 150) - 2025*x^2
+ 675*x^3 - 45*x^6))/(x^6 - 30*x^3 + 225),x)

[Out]

exp(-x^3/(x^3 - 15))*exp(15/(x^3 - 15))*exp((45*x^3*exp(-x))/(x^3 - 15))*exp(-(10*x)/(x^3 - 15))

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sympy [A]  time = 0.39, size = 26, normalized size = 0.90 \begin {gather*} e^{\frac {\left (45 x^{3} + \left (- x^{3} - 10 x + 15\right ) e^{x}\right ) e^{- x}}{x^{3} - 15}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((20*x**3+150)*exp(x)-45*x**6+675*x**3-2025*x**2)*exp(((-x**3-10*x+15)*exp(x)+45*x**3)/(x**3-15)/exp
(x))/(x**6-30*x**3+225)/exp(x),x)

[Out]

exp((45*x**3 + (-x**3 - 10*x + 15)*exp(x))*exp(-x)/(x**3 - 15))

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