3.60.60 \(\int \frac {e^{5+x^2} (-3 x^2-4 x^3-4 x^4-4 x^5-4 x^6+6 x^7)}{1+4 x+8 x^2+2 x^3-8 x^4-12 x^5+9 x^6} \, dx\)

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

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Rubi [F]  time = 1.58, 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^{5+x^2} \left (-3 x^2-4 x^3-4 x^4-4 x^5-4 x^6+6 x^7\right )}{1+4 x+8 x^2+2 x^3-8 x^4-12 x^5+9 x^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(5 + x^2)*(-3*x^2 - 4*x^3 - 4*x^4 - 4*x^5 - 4*x^6 + 6*x^7))/(1 + 4*x + 8*x^2 + 2*x^3 - 8*x^4 - 12*x^5 +
9*x^6),x]

[Out]

E^(5 + x^2)/3 + (2*E^5*Sqrt[Pi]*Erfi[x])/9 - (16*Defer[Int][E^(5 + x^2)/(-1 - 2*x - 2*x^2 + 3*x^3)^2, x])/9 -
(38*Defer[Int][(E^(5 + x^2)*x)/(-1 - 2*x - 2*x^2 + 3*x^3)^2, x])/9 - (71*Defer[Int][(E^(5 + x^2)*x^2)/(-1 - 2*
x - 2*x^2 + 3*x^3)^2, x])/9 - (4*Defer[Int][E^(5 + x^2)/(-1 - 2*x - 2*x^2 + 3*x^3), x])/3 + (8*Defer[Int][(E^(
5 + x^2)*x)/(-1 - 2*x - 2*x^2 + 3*x^3), x])/9 + (20*Defer[Int][(E^(5 + x^2)*x^2)/(-1 - 2*x - 2*x^2 + 3*x^3), x
])/9

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{5+x^2} x^2 \left (-3-4 x-4 x^2-4 x^3-4 x^4+6 x^5\right )}{\left (1+2 x+2 x^2-3 x^3\right )^2} \, dx\\ &=\int \left (\frac {4 e^{5+x^2}}{9}+\frac {2}{3} e^{5+x^2} x+\frac {e^{5+x^2} \left (-16-38 x-71 x^2\right )}{9 \left (-1-2 x-2 x^2+3 x^3\right )^2}+\frac {4 e^{5+x^2} \left (-3+2 x+5 x^2\right )}{9 \left (-1-2 x-2 x^2+3 x^3\right )}\right ) \, dx\\ &=\frac {1}{9} \int \frac {e^{5+x^2} \left (-16-38 x-71 x^2\right )}{\left (-1-2 x-2 x^2+3 x^3\right )^2} \, dx+\frac {4}{9} \int e^{5+x^2} \, dx+\frac {4}{9} \int \frac {e^{5+x^2} \left (-3+2 x+5 x^2\right )}{-1-2 x-2 x^2+3 x^3} \, dx+\frac {2}{3} \int e^{5+x^2} x \, dx\\ &=\frac {e^{5+x^2}}{3}+\frac {2}{9} e^5 \sqrt {\pi } \text {erfi}(x)+\frac {1}{9} \int \left (-\frac {16 e^{5+x^2}}{\left (-1-2 x-2 x^2+3 x^3\right )^2}-\frac {38 e^{5+x^2} x}{\left (-1-2 x-2 x^2+3 x^3\right )^2}-\frac {71 e^{5+x^2} x^2}{\left (-1-2 x-2 x^2+3 x^3\right )^2}\right ) \, dx+\frac {4}{9} \int \left (-\frac {3 e^{5+x^2}}{-1-2 x-2 x^2+3 x^3}+\frac {2 e^{5+x^2} x}{-1-2 x-2 x^2+3 x^3}+\frac {5 e^{5+x^2} x^2}{-1-2 x-2 x^2+3 x^3}\right ) \, dx\\ &=\frac {e^{5+x^2}}{3}+\frac {2}{9} e^5 \sqrt {\pi } \text {erfi}(x)+\frac {8}{9} \int \frac {e^{5+x^2} x}{-1-2 x-2 x^2+3 x^3} \, dx-\frac {4}{3} \int \frac {e^{5+x^2}}{-1-2 x-2 x^2+3 x^3} \, dx-\frac {16}{9} \int \frac {e^{5+x^2}}{\left (-1-2 x-2 x^2+3 x^3\right )^2} \, dx+\frac {20}{9} \int \frac {e^{5+x^2} x^2}{-1-2 x-2 x^2+3 x^3} \, dx-\frac {38}{9} \int \frac {e^{5+x^2} x}{\left (-1-2 x-2 x^2+3 x^3\right )^2} \, dx-\frac {71}{9} \int \frac {e^{5+x^2} x^2}{\left (-1-2 x-2 x^2+3 x^3\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.32, size = 29, normalized size = 1.00 \begin {gather*} -\frac {e^{5+x^2} x^3}{1+2 x+2 x^2-3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(5 + x^2)*(-3*x^2 - 4*x^3 - 4*x^4 - 4*x^5 - 4*x^6 + 6*x^7))/(1 + 4*x + 8*x^2 + 2*x^3 - 8*x^4 - 12
*x^5 + 9*x^6),x]

[Out]

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

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fricas [A]  time = 0.56, size = 27, normalized size = 0.93 \begin {gather*} \frac {x^{3} e^{\left (x^{2} + 5\right )}}{3 \, x^{3} - 2 \, x^{2} - 2 \, x - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x^7-4*x^6-4*x^5-4*x^4-4*x^3-3*x^2)*exp(x^2+5)/(9*x^6-12*x^5-8*x^4+2*x^3+8*x^2+4*x+1),x, algorithm
="fricas")

[Out]

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

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giac [A]  time = 0.15, size = 27, normalized size = 0.93 \begin {gather*} \frac {x^{3} e^{\left (x^{2} + 5\right )}}{3 \, x^{3} - 2 \, x^{2} - 2 \, x - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x^7-4*x^6-4*x^5-4*x^4-4*x^3-3*x^2)*exp(x^2+5)/(9*x^6-12*x^5-8*x^4+2*x^3+8*x^2+4*x+1),x, algorithm
="giac")

[Out]

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

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maple [A]  time = 0.11, size = 28, normalized size = 0.97

method	result	size
gosper	\(\frac {x^{3} {\mathrm e}^{x^{2}+5}}{3 x^{3}-2 x^{2}-2 x -1}\)	\(28\)
norman	\(\frac {x^{3} {\mathrm e}^{x^{2}+5}}{3 x^{3}-2 x^{2}-2 x -1}\)	\(28\)
risch	\(\frac {x^{3} {\mathrm e}^{x^{2}+5}}{3 x^{3}-2 x^{2}-2 x -1}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((6*x^7-4*x^6-4*x^5-4*x^4-4*x^3-3*x^2)*exp(x^2+5)/(9*x^6-12*x^5-8*x^4+2*x^3+8*x^2+4*x+1),x,method=_RETURNVE
RBOSE)

[Out]

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

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maxima [A]  time = 0.40, size = 27, normalized size = 0.93 \begin {gather*} \frac {x^{3} e^{\left (x^{2} + 5\right )}}{3 \, x^{3} - 2 \, x^{2} - 2 \, x - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x^7-4*x^6-4*x^5-4*x^4-4*x^3-3*x^2)*exp(x^2+5)/(9*x^6-12*x^5-8*x^4+2*x^3+8*x^2+4*x+1),x, algorithm
="maxima")

[Out]

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

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mupad [B]  time = 4.38, size = 28, normalized size = 0.97 \begin {gather*} -\frac {x^3\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^5}{-3\,x^3+2\,x^2+2\,x+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(x^2 + 5)*(3*x^2 + 4*x^3 + 4*x^4 + 4*x^5 + 4*x^6 - 6*x^7))/(4*x + 8*x^2 + 2*x^3 - 8*x^4 - 12*x^5 + 9*
x^6 + 1),x)

[Out]

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

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sympy [A]  time = 0.12, size = 24, normalized size = 0.83 \begin {gather*} \frac {x^{3} e^{x^{2} + 5}}{3 x^{3} - 2 x^{2} - 2 x - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x**7-4*x**6-4*x**5-4*x**4-4*x**3-3*x**2)*exp(x**2+5)/(9*x**6-12*x**5-8*x**4+2*x**3+8*x**2+4*x+1),
x)

[Out]

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

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