3.2.34 \(\int \frac {-27-27 x-9 x^2-x^3+e^{\frac {35+27 x+5 x^2+x^3}{9+6 x+x^2}} (11+3 x+9 x^2+x^3)}{27+27 x+9 x^2+x^3} \, dx\)

Optimal. Leaf size=25 \[ e^{3+x-\frac {4 \left (-2+x^2\right )}{(3+x)^2}}-x+5 \log (5) \]

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Rubi [F]  time = 0.80, 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 {-27-27 x-9 x^2-x^3+e^{\frac {35+27 x+5 x^2+x^3}{9+6 x+x^2}} \left (11+3 x+9 x^2+x^3\right )}{27+27 x+9 x^2+x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-27 - 27*x - 9*x^2 - x^3 + E^((35 + 27*x + 5*x^2 + x^3)/(9 + 6*x + x^2))*(11 + 3*x + 9*x^2 + x^3))/(27 +
27*x + 9*x^2 + x^3),x]

[Out]

-x + 81/(2*(3 + x)^2) - (9*x^2)/(2*(3 + x)^2) - 27/(3 + x) + Defer[Int][E^((35 + 27*x + 5*x^2 + x^3)/(3 + x)^2
), x] + 56*Defer[Int][E^((35 + 27*x + 5*x^2 + x^3)/(3 + x)^2)/(3 + x)^3, x] - 24*Defer[Int][E^((35 + 27*x + 5*
x^2 + x^3)/(3 + x)^2)/(3 + x)^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {27}{(3+x)^3}-\frac {27 x}{(3+x)^3}-\frac {9 x^2}{(3+x)^3}-\frac {x^3}{(3+x)^3}+\frac {e^{\frac {35+27 x+5 x^2+x^3}{(3+x)^2}} \left (11+3 x+9 x^2+x^3\right )}{(3+x)^3}\right ) \, dx\\ &=\frac {27}{2 (3+x)^2}-9 \int \frac {x^2}{(3+x)^3} \, dx-27 \int \frac {x}{(3+x)^3} \, dx-\int \frac {x^3}{(3+x)^3} \, dx+\int \frac {e^{\frac {35+27 x+5 x^2+x^3}{(3+x)^2}} \left (11+3 x+9 x^2+x^3\right )}{(3+x)^3} \, dx\\ &=\frac {27}{2 (3+x)^2}-\frac {9 x^2}{2 (3+x)^2}-9 \int \left (\frac {9}{(3+x)^3}-\frac {6}{(3+x)^2}+\frac {1}{3+x}\right ) \, dx+\int \left (e^{\frac {35+27 x+5 x^2+x^3}{(3+x)^2}}+\frac {56 e^{\frac {35+27 x+5 x^2+x^3}{(3+x)^2}}}{(3+x)^3}-\frac {24 e^{\frac {35+27 x+5 x^2+x^3}{(3+x)^2}}}{(3+x)^2}\right ) \, dx-\int \left (1-\frac {27}{(3+x)^3}+\frac {27}{(3+x)^2}-\frac {9}{3+x}\right ) \, dx\\ &=-x+\frac {81}{2 (3+x)^2}-\frac {9 x^2}{2 (3+x)^2}-\frac {27}{3+x}-24 \int \frac {e^{\frac {35+27 x+5 x^2+x^3}{(3+x)^2}}}{(3+x)^2} \, dx+56 \int \frac {e^{\frac {35+27 x+5 x^2+x^3}{(3+x)^2}}}{(3+x)^3} \, dx+\int e^{\frac {35+27 x+5 x^2+x^3}{(3+x)^2}} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.32, size = 23, normalized size = 0.92 \begin {gather*} e^{-1+x-\frac {28}{(3+x)^2}+\frac {24}{3+x}}-x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-27 - 27*x - 9*x^2 - x^3 + E^((35 + 27*x + 5*x^2 + x^3)/(9 + 6*x + x^2))*(11 + 3*x + 9*x^2 + x^3))/
(27 + 27*x + 9*x^2 + x^3),x]

[Out]

E^(-1 + x - 28/(3 + x)^2 + 24/(3 + x)) - x

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fricas [A]  time = 0.57, size = 29, normalized size = 1.16 \begin {gather*} -x + e^{\left (\frac {x^{3} + 5 \, x^{2} + 27 \, x + 35}{x^{2} + 6 \, x + 9}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^3+9*x^2+3*x+11)*exp((x^3+5*x^2+27*x+35)/(x^2+6*x+9))-x^3-9*x^2-27*x-27)/(x^3+9*x^2+27*x+27),x, a
lgorithm="fricas")

[Out]

-x + e^((x^3 + 5*x^2 + 27*x + 35)/(x^2 + 6*x + 9))

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giac [A]  time = 0.40, size = 33, normalized size = 1.32 \begin {gather*} -x + e^{\left (\frac {9 \, x^{3} + 10 \, x^{2} + 33 \, x}{9 \, {\left (x^{2} + 6 \, x + 9\right )}} + \frac {35}{9}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^3+9*x^2+3*x+11)*exp((x^3+5*x^2+27*x+35)/(x^2+6*x+9))-x^3-9*x^2-27*x-27)/(x^3+9*x^2+27*x+27),x, a
lgorithm="giac")

[Out]

-x + e^(1/9*(9*x^3 + 10*x^2 + 33*x)/(x^2 + 6*x + 9) + 35/9)

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maple [A]  time = 0.14, size = 25, normalized size = 1.00

method	result	size
risch	\(-x +{\mathrm e}^{\frac {x^{3}+5 x^{2}+27 x +35}{\left (3+x \right )^{2}}}\)	\(25\)
norman	\(\frac {27 x +{\mathrm e}^{\frac {x^{3}+5 x^{2}+27 x +35}{x^{2}+6 x +9}} x^{2}-x^{3}+6 \,{\mathrm e}^{\frac {x^{3}+5 x^{2}+27 x +35}{x^{2}+6 x +9}} x +9 \,{\mathrm e}^{\frac {x^{3}+5 x^{2}+27 x +35}{x^{2}+6 x +9}}+54}{\left (3+x \right )^{2}}\)	\(101\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^3+9*x^2+3*x+11)*exp((x^3+5*x^2+27*x+35)/(x^2+6*x+9))-x^3-9*x^2-27*x-27)/(x^3+9*x^2+27*x+27),x,method=_
RETURNVERBOSE)

[Out]

-x+exp((x^3+5*x^2+27*x+35)/(3+x)^2)

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maxima [B]  time = 1.07, size = 90, normalized size = 3.60 \begin {gather*} -x - \frac {27 \, {\left (4 \, x + 9\right )}}{2 \, {\left (x^{2} + 6 \, x + 9\right )}} + \frac {27 \, {\left (2 \, x + 5\right )}}{2 \, {\left (x^{2} + 6 \, x + 9\right )}} + \frac {27 \, {\left (2 \, x + 3\right )}}{2 \, {\left (x^{2} + 6 \, x + 9\right )}} + \frac {27}{2 \, {\left (x^{2} + 6 \, x + 9\right )}} + e^{\left (x - \frac {28}{x^{2} + 6 \, x + 9} + \frac {24}{x + 3} - 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^3+9*x^2+3*x+11)*exp((x^3+5*x^2+27*x+35)/(x^2+6*x+9))-x^3-9*x^2-27*x-27)/(x^3+9*x^2+27*x+27),x, a
lgorithm="maxima")

[Out]

-x - 27/2*(4*x + 9)/(x^2 + 6*x + 9) + 27/2*(2*x + 5)/(x^2 + 6*x + 9) + 27/2*(2*x + 3)/(x^2 + 6*x + 9) + 27/2/(
x^2 + 6*x + 9) + e^(x - 28/(x^2 + 6*x + 9) + 24/(x + 3) - 1)

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mupad [B]  time = 0.38, size = 63, normalized size = 2.52 \begin {gather*} {\mathrm {e}}^{\frac {x^3}{x^2+6\,x+9}}\,{\mathrm {e}}^{\frac {5\,x^2}{x^2+6\,x+9}}\,{\mathrm {e}}^{\frac {35}{x^2+6\,x+9}}\,{\mathrm {e}}^{\frac {27\,x}{x^2+6\,x+9}}-x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(27*x - exp((27*x + 5*x^2 + x^3 + 35)/(6*x + x^2 + 9))*(3*x + 9*x^2 + x^3 + 11) + 9*x^2 + x^3 + 27)/(27*x
+ 9*x^2 + x^3 + 27),x)

[Out]

exp(x^3/(6*x + x^2 + 9))*exp((5*x^2)/(6*x + x^2 + 9))*exp(35/(6*x + x^2 + 9))*exp((27*x)/(6*x + x^2 + 9)) - x

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sympy [A]  time = 0.20, size = 24, normalized size = 0.96 \begin {gather*} - x + e^{\frac {x^{3} + 5 x^{2} + 27 x + 35}{x^{2} + 6 x + 9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**3+9*x**2+3*x+11)*exp((x**3+5*x**2+27*x+35)/(x**2+6*x+9))-x**3-9*x**2-27*x-27)/(x**3+9*x**2+27*x
+27),x)

[Out]

-x + exp((x**3 + 5*x**2 + 27*x + 35)/(x**2 + 6*x + 9))

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