3.43.15 \(\int \frac {-5 x^3+e^{\frac {4+4 x+x^2}{x^2}} (8+4 x-4 x^2)}{2 e^{\frac {3 (4+4 x+x^2)}{x^2}} x^{11}+6 e^{\frac {2 (4+4 x+x^2)}{x^2}} x^{12}+6 e^{\frac {4+4 x+x^2}{x^2}} x^{13}+2 x^{14}} \, dx\)

Optimal. Leaf size=22 \[ \frac {1}{4 x^8 \left (e^{\frac {(2+x)^2}{x^2}}+x\right )^2} \]

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Rubi [F]  time = 1.91, 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^3+e^{\frac {4+4 x+x^2}{x^2}} \left (8+4 x-4 x^2\right )}{2 e^{\frac {3 \left (4+4 x+x^2\right )}{x^2}} x^{11}+6 e^{\frac {2 \left (4+4 x+x^2\right )}{x^2}} x^{12}+6 e^{\frac {4+4 x+x^2}{x^2}} x^{13}+2 x^{14}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-5*x^3 + E^((4 + 4*x + x^2)/x^2)*(8 + 4*x - 4*x^2))/(2*E^((3*(4 + 4*x + x^2))/x^2)*x^11 + 6*E^((2*(4 + 4*
x + x^2))/x^2)*x^12 + 6*E^((4 + 4*x + x^2)/x^2)*x^13 + 2*x^14),x]

[Out]

-4*Defer[Int][1/(x^10*(E^((2 + x)^2/x^2) + x)^3), x] - 2*Defer[Int][1/(x^9*(E^((2 + x)^2/x^2) + x)^3), x] - De
fer[Int][1/(x^8*(E^((2 + x)^2/x^2) + x)^3), x]/2 + 4*Defer[Int][1/(x^11*(E^((2 + x)^2/x^2) + x)^2), x] + 2*Def
er[Int][1/(x^10*(E^((2 + x)^2/x^2) + x)^2), x] - 2*Defer[Int][1/(x^9*(E^((2 + x)^2/x^2) + x)^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-5 x^3+e^{\frac {(2+x)^2}{x^2}} \left (8+4 x-4 x^2\right )}{2 x^{11} \left (e^{\frac {(2+x)^2}{x^2}}+x\right )^3} \, dx\\ &=\frac {1}{2} \int \frac {-5 x^3+e^{\frac {(2+x)^2}{x^2}} \left (8+4 x-4 x^2\right )}{x^{11} \left (e^{\frac {(2+x)^2}{x^2}}+x\right )^3} \, dx\\ &=\frac {1}{2} \int \left (-\frac {4 (-2+x) (1+x)}{x^{11} \left (e^{\frac {(2+x)^2}{x^2}}+x\right )^2}-\frac {8+4 x+x^2}{x^{10} \left (e^{\frac {(2+x)^2}{x^2}}+x\right )^3}\right ) \, dx\\ &=-\left (\frac {1}{2} \int \frac {8+4 x+x^2}{x^{10} \left (e^{\frac {(2+x)^2}{x^2}}+x\right )^3} \, dx\right )-2 \int \frac {(-2+x) (1+x)}{x^{11} \left (e^{\frac {(2+x)^2}{x^2}}+x\right )^2} \, dx\\ &=-\left (\frac {1}{2} \int \left (\frac {8}{x^{10} \left (e^{\frac {(2+x)^2}{x^2}}+x\right )^3}+\frac {4}{x^9 \left (e^{\frac {(2+x)^2}{x^2}}+x\right )^3}+\frac {1}{x^8 \left (e^{\frac {(2+x)^2}{x^2}}+x\right )^3}\right ) \, dx\right )-2 \int \left (-\frac {2}{x^{11} \left (e^{\frac {(2+x)^2}{x^2}}+x\right )^2}-\frac {1}{x^{10} \left (e^{\frac {(2+x)^2}{x^2}}+x\right )^2}+\frac {1}{x^9 \left (e^{\frac {(2+x)^2}{x^2}}+x\right )^2}\right ) \, dx\\ &=-\left (\frac {1}{2} \int \frac {1}{x^8 \left (e^{\frac {(2+x)^2}{x^2}}+x\right )^3} \, dx\right )-2 \int \frac {1}{x^9 \left (e^{\frac {(2+x)^2}{x^2}}+x\right )^3} \, dx+2 \int \frac {1}{x^{10} \left (e^{\frac {(2+x)^2}{x^2}}+x\right )^2} \, dx-2 \int \frac {1}{x^9 \left (e^{\frac {(2+x)^2}{x^2}}+x\right )^2} \, dx-4 \int \frac {1}{x^{10} \left (e^{\frac {(2+x)^2}{x^2}}+x\right )^3} \, dx+4 \int \frac {1}{x^{11} \left (e^{\frac {(2+x)^2}{x^2}}+x\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.35, size = 22, normalized size = 1.00 \begin {gather*} \frac {1}{4 x^8 \left (e^{\frac {(2+x)^2}{x^2}}+x\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-5*x^3 + E^((4 + 4*x + x^2)/x^2)*(8 + 4*x - 4*x^2))/(2*E^((3*(4 + 4*x + x^2))/x^2)*x^11 + 6*E^((2*(
4 + 4*x + x^2))/x^2)*x^12 + 6*E^((4 + 4*x + x^2)/x^2)*x^13 + 2*x^14),x]

[Out]

1/(4*x^8*(E^((2 + x)^2/x^2) + x)^2)

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fricas [B]  time = 0.78, size = 44, normalized size = 2.00 \begin {gather*} \frac {1}{4 \, {\left (x^{10} + 2 \, x^{9} e^{\left (\frac {x^{2} + 4 \, x + 4}{x^{2}}\right )} + x^{8} e^{\left (\frac {2 \, {\left (x^{2} + 4 \, x + 4\right )}}{x^{2}}\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^2+4*x+8)*exp((x^2+4*x+4)/x^2)-5*x^3)/(2*x^11*exp((x^2+4*x+4)/x^2)^3+6*x^12*exp((x^2+4*x+4)/x^
2)^2+6*x^13*exp((x^2+4*x+4)/x^2)+2*x^14),x, algorithm="fricas")

[Out]

1/4/(x^10 + 2*x^9*e^((x^2 + 4*x + 4)/x^2) + x^8*e^(2*(x^2 + 4*x + 4)/x^2))

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giac [B]  time = 0.35, size = 44, normalized size = 2.00 \begin {gather*} \frac {1}{4 \, {\left (x^{10} + 2 \, x^{9} e^{\left (\frac {x^{2} + 4 \, x + 4}{x^{2}}\right )} + x^{8} e^{\left (\frac {2 \, {\left (x^{2} + 4 \, x + 4\right )}}{x^{2}}\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^2+4*x+8)*exp((x^2+4*x+4)/x^2)-5*x^3)/(2*x^11*exp((x^2+4*x+4)/x^2)^3+6*x^12*exp((x^2+4*x+4)/x^
2)^2+6*x^13*exp((x^2+4*x+4)/x^2)+2*x^14),x, algorithm="giac")

[Out]

1/4/(x^10 + 2*x^9*e^((x^2 + 4*x + 4)/x^2) + x^8*e^(2*(x^2 + 4*x + 4)/x^2))

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maple [A]  time = 0.09, size = 20, normalized size = 0.91

method	result	size
risch	\(\frac {1}{4 \left (x +{\mathrm e}^{\frac {\left (2+x \right )^{2}}{x^{2}}}\right )^{2} x^{8}}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-4*x^2+4*x+8)*exp((x^2+4*x+4)/x^2)-5*x^3)/(2*x^11*exp((x^2+4*x+4)/x^2)^3+6*x^12*exp((x^2+4*x+4)/x^2)^2+6
*x^13*exp((x^2+4*x+4)/x^2)+2*x^14),x,method=_RETURNVERBOSE)

[Out]

1/4/(x+exp((2+x)^2/x^2))^2/x^8

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maxima [B]  time = 0.39, size = 43, normalized size = 1.95 \begin {gather*} \frac {1}{4 \, {\left (x^{10} + 2 \, x^{9} e^{\left (\frac {4}{x} + \frac {4}{x^{2}} + 1\right )} + x^{8} e^{\left (\frac {8}{x} + \frac {8}{x^{2}} + 2\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^2+4*x+8)*exp((x^2+4*x+4)/x^2)-5*x^3)/(2*x^11*exp((x^2+4*x+4)/x^2)^3+6*x^12*exp((x^2+4*x+4)/x^
2)^2+6*x^13*exp((x^2+4*x+4)/x^2)+2*x^14),x, algorithm="maxima")

[Out]

1/4/(x^10 + 2*x^9*e^(4/x + 4/x^2 + 1) + x^8*e^(8/x + 8/x^2 + 2))

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mupad [B]  time = 3.31, size = 50, normalized size = 2.27 \begin {gather*} \frac {x^3}{2\,\left (2\,x^{13}+4\,x^{12}\,\mathrm {e}\,{\mathrm {e}}^{4/x}\,{\mathrm {e}}^{\frac {4}{x^2}}+2\,x^{11}\,{\mathrm {e}}^2\,{\mathrm {e}}^{8/x}\,{\mathrm {e}}^{\frac {8}{x^2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((4*x + x^2 + 4)/x^2)*(4*x - 4*x^2 + 8) - 5*x^3)/(2*x^14 + 6*x^13*exp((4*x + x^2 + 4)/x^2) + 2*x^11*ex
p((3*(4*x + x^2 + 4))/x^2) + 6*x^12*exp((2*(4*x + x^2 + 4))/x^2)),x)

[Out]

x^3/(2*(2*x^13 + 4*x^12*exp(1)*exp(4/x)*exp(4/x^2) + 2*x^11*exp(2)*exp(8/x)*exp(8/x^2)))

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sympy [B]  time = 0.18, size = 44, normalized size = 2.00 \begin {gather*} \frac {1}{4 x^{10} + 8 x^{9} e^{\frac {x^{2} + 4 x + 4}{x^{2}}} + 4 x^{8} e^{\frac {2 \left (x^{2} + 4 x + 4\right )}{x^{2}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x**2+4*x+8)*exp((x**2+4*x+4)/x**2)-5*x**3)/(2*x**11*exp((x**2+4*x+4)/x**2)**3+6*x**12*exp((x**2
+4*x+4)/x**2)**2+6*x**13*exp((x**2+4*x+4)/x**2)+2*x**14),x)

[Out]

1/(4*x**10 + 8*x**9*exp((x**2 + 4*x + 4)/x**2) + 4*x**8*exp(2*(x**2 + 4*x + 4)/x**2))

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