3.88.37 \(\int \frac {5 x^2+e^x x^2+e^{x-2 x^2+x^3} (-3-7 x-7 x^2-11 x^3+15 x^4+e^x (-1-4 x^2+3 x^3))}{e^{2 x-4 x^2+2 x^3}-2 e^{x-2 x^2+x^3} x+x^2} \, dx\)

Optimal. Leaf size=33 \[ \frac {x \left (2+5 x+\frac {x+e^x x}{x}\right )}{-e^{(-1+x)^2 x}+x} \]

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Rubi [F]  time = 22.87, 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 {5 x^2+e^x x^2+e^{x-2 x^2+x^3} \left (-3-7 x-7 x^2-11 x^3+15 x^4+e^x \left (-1-4 x^2+3 x^3\right )\right )}{e^{2 x-4 x^2+2 x^3}-2 e^{x-2 x^2+x^3} x+x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(5*x^2 + E^x*x^2 + E^(x - 2*x^2 + x^3)*(-3 - 7*x - 7*x^2 - 11*x^3 + 15*x^4 + E^x*(-1 - 4*x^2 + 3*x^3)))/(E
^(2*x - 4*x^2 + 2*x^3) - 2*E^(x - 2*x^2 + x^3)*x + x^2),x]

[Out]

-Defer[Int][(E^(x + 4*x^2)*x)/(E^(x + x^3) - E^(2*x^2)*x)^2, x] + Defer[Int][(E^(x + 4*x^2)*x^2)/(E^(x + x^3)
- E^(2*x^2)*x)^2, x] - 4*Defer[Int][(E^(x + 4*x^2)*x^3)/(E^(x + x^3) - E^(2*x^2)*x)^2, x] + 3*Defer[Int][(E^(x
+ 4*x^2)*x^4)/(E^(x + x^3) - E^(2*x^2)*x)^2, x] - 3*Defer[Int][(E^(4*x^2)*x)/(-E^(x + x^3) + E^(2*x^2)*x)^2,
x] - 2*Defer[Int][(E^(4*x^2)*x^2)/(-E^(x + x^3) + E^(2*x^2)*x)^2, x] - 7*Defer[Int][(E^(4*x^2)*x^3)/(-E^(x + x
^3) + E^(2*x^2)*x)^2, x] - 11*Defer[Int][(E^(4*x^2)*x^4)/(-E^(x + x^3) + E^(2*x^2)*x)^2, x] + 15*Defer[Int][(E
^(4*x^2)*x^5)/(-E^(x + x^3) + E^(2*x^2)*x)^2, x] - 3*Defer[Int][E^(4*x^2)/(E^(x*(1 + x)^2) - E^(4*x^2)*x), x]
- Defer[Int][E^(x + 4*x^2)/(E^(x*(1 + x)^2) - E^(4*x^2)*x), x] + 7*Defer[Int][(E^(4*x^2)*x)/(-E^(x*(1 + x)^2)
+ E^(4*x^2)*x), x] + 7*Defer[Int][(E^(4*x^2)*x^2)/(-E^(x*(1 + x)^2) + E^(4*x^2)*x), x] + 4*Defer[Int][(E^(x +
4*x^2)*x^2)/(-E^(x*(1 + x)^2) + E^(4*x^2)*x), x] + 11*Defer[Int][(E^(4*x^2)*x^3)/(-E^(x*(1 + x)^2) + E^(4*x^2)
*x), x] - 3*Defer[Int][(E^(x + 4*x^2)*x^3)/(-E^(x*(1 + x)^2) + E^(4*x^2)*x), x] - 15*Defer[Int][(E^(4*x^2)*x^4
)/(-E^(x*(1 + x)^2) + E^(4*x^2)*x), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{4 x^2} \left (5 x^2+e^x x^2+e^{x-2 x^2+x^3} \left (-3-7 x-7 x^2-11 x^3+15 x^4+e^x \left (-1-4 x^2+3 x^3\right )\right )\right )}{\left (e^{x+x^3}-e^{2 x^2} x\right )^2} \, dx\\ &=\int \left (\frac {e^{4 x^2} x \left (3+e^x+5 x\right ) \left (-1+x-4 x^2+3 x^3\right )}{\left (-e^{x+x^3}+e^{2 x^2} x\right )^2}+\frac {e^{4 x^2} \left (-3-e^x-7 x-7 x^2-4 e^x x^2-11 x^3+3 e^x x^3+15 x^4\right )}{e^{x (1+x)^2}-e^{4 x^2} x}\right ) \, dx\\ &=\int \frac {e^{4 x^2} x \left (3+e^x+5 x\right ) \left (-1+x-4 x^2+3 x^3\right )}{\left (-e^{x+x^3}+e^{2 x^2} x\right )^2} \, dx+\int \frac {e^{4 x^2} \left (-3-e^x-7 x-7 x^2-4 e^x x^2-11 x^3+3 e^x x^3+15 x^4\right )}{e^{x (1+x)^2}-e^{4 x^2} x} \, dx\\ &=\int \left (-\frac {e^{4 x^2} x \left (3+e^x+5 x\right )}{\left (-e^{x+x^3}+e^{2 x^2} x\right )^2}+\frac {e^{4 x^2} x^2 \left (3+e^x+5 x\right )}{\left (-e^{x+x^3}+e^{2 x^2} x\right )^2}-\frac {4 e^{4 x^2} x^3 \left (3+e^x+5 x\right )}{\left (-e^{x+x^3}+e^{2 x^2} x\right )^2}+\frac {3 e^{4 x^2} x^4 \left (3+e^x+5 x\right )}{\left (-e^{x+x^3}+e^{2 x^2} x\right )^2}\right ) \, dx+\int \left (-\frac {3 e^{4 x^2}}{e^{x (1+x)^2}-e^{4 x^2} x}-\frac {e^{x+4 x^2}}{e^{x (1+x)^2}-e^{4 x^2} x}+\frac {7 e^{4 x^2} x}{-e^{x (1+x)^2}+e^{4 x^2} x}+\frac {7 e^{4 x^2} x^2}{-e^{x (1+x)^2}+e^{4 x^2} x}+\frac {4 e^{x+4 x^2} x^2}{-e^{x (1+x)^2}+e^{4 x^2} x}+\frac {11 e^{4 x^2} x^3}{-e^{x (1+x)^2}+e^{4 x^2} x}-\frac {3 e^{x+4 x^2} x^3}{-e^{x (1+x)^2}+e^{4 x^2} x}-\frac {15 e^{4 x^2} x^4}{-e^{x (1+x)^2}+e^{4 x^2} x}\right ) \, dx\\ &=3 \int \frac {e^{4 x^2} x^4 \left (3+e^x+5 x\right )}{\left (-e^{x+x^3}+e^{2 x^2} x\right )^2} \, dx-3 \int \frac {e^{4 x^2}}{e^{x (1+x)^2}-e^{4 x^2} x} \, dx-3 \int \frac {e^{x+4 x^2} x^3}{-e^{x (1+x)^2}+e^{4 x^2} x} \, dx-4 \int \frac {e^{4 x^2} x^3 \left (3+e^x+5 x\right )}{\left (-e^{x+x^3}+e^{2 x^2} x\right )^2} \, dx+4 \int \frac {e^{x+4 x^2} x^2}{-e^{x (1+x)^2}+e^{4 x^2} x} \, dx+7 \int \frac {e^{4 x^2} x}{-e^{x (1+x)^2}+e^{4 x^2} x} \, dx+7 \int \frac {e^{4 x^2} x^2}{-e^{x (1+x)^2}+e^{4 x^2} x} \, dx+11 \int \frac {e^{4 x^2} x^3}{-e^{x (1+x)^2}+e^{4 x^2} x} \, dx-15 \int \frac {e^{4 x^2} x^4}{-e^{x (1+x)^2}+e^{4 x^2} x} \, dx-\int \frac {e^{4 x^2} x \left (3+e^x+5 x\right )}{\left (-e^{x+x^3}+e^{2 x^2} x\right )^2} \, dx+\int \frac {e^{4 x^2} x^2 \left (3+e^x+5 x\right )}{\left (-e^{x+x^3}+e^{2 x^2} x\right )^2} \, dx-\int \frac {e^{x+4 x^2}}{e^{x (1+x)^2}-e^{4 x^2} x} \, dx\\ &=-\left (3 \int \frac {e^{4 x^2}}{e^{x (1+x)^2}-e^{4 x^2} x} \, dx\right )-3 \int \frac {e^{x+4 x^2} x^3}{-e^{x (1+x)^2}+e^{4 x^2} x} \, dx+3 \int \left (\frac {e^{x+4 x^2} x^4}{\left (e^{x+x^3}-e^{2 x^2} x\right )^2}+\frac {3 e^{4 x^2} x^4}{\left (-e^{x+x^3}+e^{2 x^2} x\right )^2}+\frac {5 e^{4 x^2} x^5}{\left (-e^{x+x^3}+e^{2 x^2} x\right )^2}\right ) \, dx+4 \int \frac {e^{x+4 x^2} x^2}{-e^{x (1+x)^2}+e^{4 x^2} x} \, dx-4 \int \left (\frac {e^{x+4 x^2} x^3}{\left (e^{x+x^3}-e^{2 x^2} x\right )^2}+\frac {3 e^{4 x^2} x^3}{\left (-e^{x+x^3}+e^{2 x^2} x\right )^2}+\frac {5 e^{4 x^2} x^4}{\left (-e^{x+x^3}+e^{2 x^2} x\right )^2}\right ) \, dx+7 \int \frac {e^{4 x^2} x}{-e^{x (1+x)^2}+e^{4 x^2} x} \, dx+7 \int \frac {e^{4 x^2} x^2}{-e^{x (1+x)^2}+e^{4 x^2} x} \, dx+11 \int \frac {e^{4 x^2} x^3}{-e^{x (1+x)^2}+e^{4 x^2} x} \, dx-15 \int \frac {e^{4 x^2} x^4}{-e^{x (1+x)^2}+e^{4 x^2} x} \, dx-\int \frac {e^{x+4 x^2}}{e^{x (1+x)^2}-e^{4 x^2} x} \, dx-\int \left (\frac {e^{x+4 x^2} x}{\left (e^{x+x^3}-e^{2 x^2} x\right )^2}+\frac {3 e^{4 x^2} x}{\left (-e^{x+x^3}+e^{2 x^2} x\right )^2}+\frac {5 e^{4 x^2} x^2}{\left (-e^{x+x^3}+e^{2 x^2} x\right )^2}\right ) \, dx+\int \left (\frac {e^{x+4 x^2} x^2}{\left (e^{x+x^3}-e^{2 x^2} x\right )^2}+\frac {3 e^{4 x^2} x^2}{\left (-e^{x+x^3}+e^{2 x^2} x\right )^2}+\frac {5 e^{4 x^2} x^3}{\left (-e^{x+x^3}+e^{2 x^2} x\right )^2}\right ) \, dx\\ &=3 \int \frac {e^{x+4 x^2} x^4}{\left (e^{x+x^3}-e^{2 x^2} x\right )^2} \, dx-3 \int \frac {e^{4 x^2} x}{\left (-e^{x+x^3}+e^{2 x^2} x\right )^2} \, dx+3 \int \frac {e^{4 x^2} x^2}{\left (-e^{x+x^3}+e^{2 x^2} x\right )^2} \, dx-3 \int \frac {e^{4 x^2}}{e^{x (1+x)^2}-e^{4 x^2} x} \, dx-3 \int \frac {e^{x+4 x^2} x^3}{-e^{x (1+x)^2}+e^{4 x^2} x} \, dx-4 \int \frac {e^{x+4 x^2} x^3}{\left (e^{x+x^3}-e^{2 x^2} x\right )^2} \, dx+4 \int \frac {e^{x+4 x^2} x^2}{-e^{x (1+x)^2}+e^{4 x^2} x} \, dx-5 \int \frac {e^{4 x^2} x^2}{\left (-e^{x+x^3}+e^{2 x^2} x\right )^2} \, dx+5 \int \frac {e^{4 x^2} x^3}{\left (-e^{x+x^3}+e^{2 x^2} x\right )^2} \, dx+7 \int \frac {e^{4 x^2} x}{-e^{x (1+x)^2}+e^{4 x^2} x} \, dx+7 \int \frac {e^{4 x^2} x^2}{-e^{x (1+x)^2}+e^{4 x^2} x} \, dx+9 \int \frac {e^{4 x^2} x^4}{\left (-e^{x+x^3}+e^{2 x^2} x\right )^2} \, dx+11 \int \frac {e^{4 x^2} x^3}{-e^{x (1+x)^2}+e^{4 x^2} x} \, dx-12 \int \frac {e^{4 x^2} x^3}{\left (-e^{x+x^3}+e^{2 x^2} x\right )^2} \, dx+15 \int \frac {e^{4 x^2} x^5}{\left (-e^{x+x^3}+e^{2 x^2} x\right )^2} \, dx-15 \int \frac {e^{4 x^2} x^4}{-e^{x (1+x)^2}+e^{4 x^2} x} \, dx-20 \int \frac {e^{4 x^2} x^4}{\left (-e^{x+x^3}+e^{2 x^2} x\right )^2} \, dx-\int \frac {e^{x+4 x^2} x}{\left (e^{x+x^3}-e^{2 x^2} x\right )^2} \, dx+\int \frac {e^{x+4 x^2} x^2}{\left (e^{x+x^3}-e^{2 x^2} x\right )^2} \, dx-\int \frac {e^{x+4 x^2}}{e^{x (1+x)^2}-e^{4 x^2} x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.13, size = 38, normalized size = 1.15 \begin {gather*} -\frac {e^{2 x^2} x \left (3+e^x+5 x\right )}{e^{x+x^3}-e^{2 x^2} x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(5*x^2 + E^x*x^2 + E^(x - 2*x^2 + x^3)*(-3 - 7*x - 7*x^2 - 11*x^3 + 15*x^4 + E^x*(-1 - 4*x^2 + 3*x^3
)))/(E^(2*x - 4*x^2 + 2*x^3) - 2*E^(x - 2*x^2 + x^3)*x + x^2),x]

[Out]

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

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fricas [A]  time = 0.82, size = 31, normalized size = 0.94 \begin {gather*} \frac {5 \, x^{2} + x e^{x} + 3 \, x}{x - e^{\left (x^{3} - 2 \, x^{2} + x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3*x^3-4*x^2-1)*exp(x)+15*x^4-11*x^3-7*x^2-7*x-3)*exp(x^3-2*x^2+x)+exp(x)*x^2+5*x^2)/(exp(x^3-2*x^
2+x)^2-2*x*exp(x^3-2*x^2+x)+x^2),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.22, size = 31, normalized size = 0.94 \begin {gather*} \frac {5 \, x^{2} + x e^{x} + 3 \, x}{x - e^{\left (x^{3} - 2 \, x^{2} + x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3*x^3-4*x^2-1)*exp(x)+15*x^4-11*x^3-7*x^2-7*x-3)*exp(x^3-2*x^2+x)+exp(x)*x^2+5*x^2)/(exp(x^3-2*x^
2+x)^2-2*x*exp(x^3-2*x^2+x)+x^2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.07, size = 24, normalized size = 0.73

method	result	size
risch	\(\frac {\left (5 x +{\mathrm e}^{x}+3\right ) x}{x -{\mathrm e}^{x \left (x -1\right )^{2}}}\)	\(24\)
norman	\(\frac {3 \,{\mathrm e}^{x^{3}-2 x^{2}+x}+{\mathrm e}^{x} x +5 x^{2}}{x -{\mathrm e}^{x^{3}-2 x^{2}+x}}\)	\(42\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((3*x^3-4*x^2-1)*exp(x)+15*x^4-11*x^3-7*x^2-7*x-3)*exp(x^3-2*x^2+x)+exp(x)*x^2+5*x^2)/(exp(x^3-2*x^2+x)^2
-2*x*exp(x^3-2*x^2+x)+x^2),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.40, size = 39, normalized size = 1.18 \begin {gather*} \frac {{\left (5 \, x^{2} + x e^{x} + 3 \, x\right )} e^{\left (2 \, x^{2}\right )}}{x e^{\left (2 \, x^{2}\right )} - e^{\left (x^{3} + x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3*x^3-4*x^2-1)*exp(x)+15*x^4-11*x^3-7*x^2-7*x-3)*exp(x^3-2*x^2+x)+exp(x)*x^2+5*x^2)/(exp(x^3-2*x^
2+x)^2-2*x*exp(x^3-2*x^2+x)+x^2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 5.59, size = 31, normalized size = 0.94 \begin {gather*} \frac {3\,x+x\,{\mathrm {e}}^x+5\,x^2}{x-{\mathrm {e}}^{x^3-2\,x^2+x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2*exp(x) - exp(x - 2*x^2 + x^3)*(7*x + exp(x)*(4*x^2 - 3*x^3 + 1) + 7*x^2 + 11*x^3 - 15*x^4 + 3) + 5*x^
2)/(exp(2*x - 4*x^2 + 2*x^3) - 2*x*exp(x - 2*x^2 + x^3) + x^2),x)

[Out]

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

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sympy [A]  time = 0.19, size = 27, normalized size = 0.82 \begin {gather*} \frac {- 5 x^{2} - x e^{x} - 3 x}{- x + e^{x^{3} - 2 x^{2} + x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3*x**3-4*x**2-1)*exp(x)+15*x**4-11*x**3-7*x**2-7*x-3)*exp(x**3-2*x**2+x)+exp(x)*x**2+5*x**2)/(exp
(x**3-2*x**2+x)**2-2*x*exp(x**3-2*x**2+x)+x**2),x)

[Out]

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

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