3.92.14 \(\int \frac {e^{\frac {6+6 x-15 x^2-5 x^3}{3 x^2+x^3}} (-180-216 x-96 x^2-21 x^3-6 x^4-x^5)}{7875 x^3+8400 x^4+3290 x^5+560 x^6+35 x^7} \, dx\) [9114]

Optimal. Leaf size=28 \[ \frac {e^{-5+\frac {2}{x^2}+\frac {4}{x (3+x)}}}{35 (5+x)} \]

[Out]

1/35*exp(2/x^2+4/(3+x)/x-5)/(5+x)

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Rubi [F]
time = 2.09, 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 {6+6 x-15 x^2-5 x^3}{3 x^2+x^3}} \left (-180-216 x-96 x^2-21 x^3-6 x^4-x^5\right )}{7875 x^3+8400 x^4+3290 x^5+560 x^6+35 x^7} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((6 + 6*x - 15*x^2 - 5*x^3)/(3*x^2 + x^3))*(-180 - 216*x - 96*x^2 - 21*x^3 - 6*x^4 - x^5))/(7875*x^3 +
8400*x^4 + 3290*x^5 + 560*x^6 + 35*x^7),x]

[Out]

(-4*Defer[Int][E^((6 + 6*x - 15*x^2 - 5*x^3)/(x^2*(3 + x)))/x^3, x])/175 - (8*Defer[Int][E^((6 + 6*x - 15*x^2
- 5*x^3)/(x^2*(3 + x)))/x^2, x])/2625 + (8*Defer[Int][E^((6 + 6*x - 15*x^2 - 5*x^3)/(x^2*(3 + x)))/x, x])/1312
5 + (2*Defer[Int][E^((6 + 6*x - 15*x^2 - 5*x^3)/(x^2*(3 + x)))/(3 + x)^2, x])/105 - Defer[Int][E^((6 + 6*x - 1
5*x^2 - 5*x^3)/(x^2*(3 + x)))/(3 + x), x]/105 - Defer[Int][E^((6 + 6*x - 15*x^2 - 5*x^3)/(x^2*(3 + x)))/(5 + x
)^2, x]/35 + (39*Defer[Int][E^((6 + 6*x - 15*x^2 - 5*x^3)/(x^2*(3 + x)))/(5 + x), x])/4375

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{\frac {6+6 x-15 x^2-5 x^3}{x^2 (3+x)}} \left (-180-216 x-96 x^2-21 x^3-6 x^4-x^5\right )}{35 x^3 \left (15+8 x+x^2\right )^2} \, dx\\ &=\frac {1}{35} \int \frac {e^{\frac {6+6 x-15 x^2-5 x^3}{x^2 (3+x)}} \left (-180-216 x-96 x^2-21 x^3-6 x^4-x^5\right )}{x^3 \left (15+8 x+x^2\right )^2} \, dx\\ &=\frac {1}{35} \int \left (-\frac {4 e^{\frac {6+6 x-15 x^2-5 x^3}{x^2 (3+x)}}}{5 x^3}-\frac {8 e^{\frac {6+6 x-15 x^2-5 x^3}{x^2 (3+x)}}}{75 x^2}+\frac {8 e^{\frac {6+6 x-15 x^2-5 x^3}{x^2 (3+x)}}}{375 x}+\frac {2 e^{\frac {6+6 x-15 x^2-5 x^3}{x^2 (3+x)}}}{3 (3+x)^2}-\frac {e^{\frac {6+6 x-15 x^2-5 x^3}{x^2 (3+x)}}}{3 (3+x)}-\frac {e^{\frac {6+6 x-15 x^2-5 x^3}{x^2 (3+x)}}}{(5+x)^2}+\frac {39 e^{\frac {6+6 x-15 x^2-5 x^3}{x^2 (3+x)}}}{125 (5+x)}\right ) \, dx\\ &=\frac {8 \int \frac {e^{\frac {6+6 x-15 x^2-5 x^3}{x^2 (3+x)}}}{x} \, dx}{13125}-\frac {8 \int \frac {e^{\frac {6+6 x-15 x^2-5 x^3}{x^2 (3+x)}}}{x^2} \, dx}{2625}+\frac {39 \int \frac {e^{\frac {6+6 x-15 x^2-5 x^3}{x^2 (3+x)}}}{5+x} \, dx}{4375}-\frac {1}{105} \int \frac {e^{\frac {6+6 x-15 x^2-5 x^3}{x^2 (3+x)}}}{3+x} \, dx+\frac {2}{105} \int \frac {e^{\frac {6+6 x-15 x^2-5 x^3}{x^2 (3+x)}}}{(3+x)^2} \, dx-\frac {4}{175} \int \frac {e^{\frac {6+6 x-15 x^2-5 x^3}{x^2 (3+x)}}}{x^3} \, dx-\frac {1}{35} \int \frac {e^{\frac {6+6 x-15 x^2-5 x^3}{x^2 (3+x)}}}{(5+x)^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.39, size = 34, normalized size = 1.21 \begin {gather*} \frac {e^{-5+\frac {2}{x^2}+\frac {4}{3 x}-\frac {4}{3 (3+x)}}}{35 (5+x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((6 + 6*x - 15*x^2 - 5*x^3)/(3*x^2 + x^3))*(-180 - 216*x - 96*x^2 - 21*x^3 - 6*x^4 - x^5))/(7875*
x^3 + 8400*x^4 + 3290*x^5 + 560*x^6 + 35*x^7),x]

[Out]

E^(-5 + 2/x^2 + 4/(3*x) - 4/(3*(3 + x)))/(35*(5 + x))

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Maple [A]
time = 0.56, size = 34, normalized size = 1.21

method	result	size
gosper	\(\frac {{\mathrm e}^{-\frac {5 x^{3}+15 x^{2}-6 x -6}{x^{2} \left (3+x \right )}}}{175+35 x}\)	\(34\)
risch	\(\frac {{\mathrm e}^{-\frac {5 x^{3}+15 x^{2}-6 x -6}{x^{2} \left (3+x \right )}}}{175+35 x}\)	\(34\)
norman	\(\frac {\frac {3 x^{2} {\mathrm e}^{\frac {-5 x^{3}-15 x^{2}+6 x +6}{x^{3}+3 x^{2}}}}{35}+\frac {x^{3} {\mathrm e}^{\frac {-5 x^{3}-15 x^{2}+6 x +6}{x^{3}+3 x^{2}}}}{35}}{x^{2} \left (x^{2}+8 x +15\right )}\)	\(82\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x^5-6*x^4-21*x^3-96*x^2-216*x-180)*exp((-5*x^3-15*x^2+6*x+6)/(x^3+3*x^2))/(35*x^7+560*x^6+3290*x^5+8400*
x^4+7875*x^3),x,method=_RETURNVERBOSE)

[Out]

1/35*exp(-(5*x^3+15*x^2-6*x-6)/x^2/(3+x))/(5+x)

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Maxima [A]
time = 0.38, size = 32, normalized size = 1.14 \begin {gather*} \frac {e^{\left (-\frac {4}{3 \, {\left (x + 3\right )}} + \frac {4}{3 \, x} + \frac {2}{x^{2}}\right )}}{35 \, {\left (x e^{5} + 5 \, e^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^5-6*x^4-21*x^3-96*x^2-216*x-180)*exp((-5*x^3-15*x^2+6*x+6)/(x^3+3*x^2))/(35*x^7+560*x^6+3290*x^5
+8400*x^4+7875*x^3),x, algorithm="maxima")

[Out]

1/35*e^(-4/3/(x + 3) + 4/3/x + 2/x^2)/(x*e^5 + 5*e^5)

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Fricas [A]
time = 0.37, size = 36, normalized size = 1.29 \begin {gather*} \frac {e^{\left (-\frac {5 \, x^{3} + 15 \, x^{2} - 6 \, x - 6}{x^{3} + 3 \, x^{2}}\right )}}{35 \, {\left (x + 5\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^5-6*x^4-21*x^3-96*x^2-216*x-180)*exp((-5*x^3-15*x^2+6*x+6)/(x^3+3*x^2))/(35*x^7+560*x^6+3290*x^5
+8400*x^4+7875*x^3),x, algorithm="fricas")

[Out]

1/35*e^(-(5*x^3 + 15*x^2 - 6*x - 6)/(x^3 + 3*x^2))/(x + 5)

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Sympy [A]
time = 0.10, size = 29, normalized size = 1.04 \begin {gather*} \frac {e^{\frac {- 5 x^{3} - 15 x^{2} + 6 x + 6}{x^{3} + 3 x^{2}}}}{35 x + 175} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x**5-6*x**4-21*x**3-96*x**2-216*x-180)*exp((-5*x**3-15*x**2+6*x+6)/(x**3+3*x**2))/(35*x**7+560*x**
6+3290*x**5+8400*x**4+7875*x**3),x)

[Out]

exp((-5*x**3 - 15*x**2 + 6*x + 6)/(x**3 + 3*x**2))/(35*x + 175)

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Giac [A]
time = 0.45, size = 36, normalized size = 1.29 \begin {gather*} \frac {e^{\left (-\frac {5 \, x^{3} + 15 \, x^{2} - 6 \, x - 6}{x^{3} + 3 \, x^{2}}\right )}}{35 \, {\left (x + 5\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^5-6*x^4-21*x^3-96*x^2-216*x-180)*exp((-5*x^3-15*x^2+6*x+6)/(x^3+3*x^2))/(35*x^7+560*x^6+3290*x^5
+8400*x^4+7875*x^3),x, algorithm="giac")

[Out]

1/35*e^(-(5*x^3 + 15*x^2 - 6*x - 6)/(x^3 + 3*x^2))/(x + 5)

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Mupad [B]
time = 7.48, size = 51, normalized size = 1.82 \begin {gather*} \frac {{\mathrm {e}}^{\frac {6}{x^2+3\,x}}\,{\mathrm {e}}^{-\frac {5\,x}{x+3}}\,{\mathrm {e}}^{\frac {6}{x^3+3\,x^2}}\,{\mathrm {e}}^{-\frac {15}{x+3}}}{35\,\left (x+5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((6*x - 15*x^2 - 5*x^3 + 6)/(3*x^2 + x^3))*(216*x + 96*x^2 + 21*x^3 + 6*x^4 + x^5 + 180))/(7875*x^3 +
8400*x^4 + 3290*x^5 + 560*x^6 + 35*x^7),x)

[Out]

(exp(6/(3*x + x^2))*exp(-(5*x)/(x + 3))*exp(6/(3*x^2 + x^3))*exp(-15/(x + 3)))/(35*(x + 5))

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