3.54.38 \(\int \frac {e^{-14+\frac {4}{11 x}+\frac {3}{1+x}} (-4+14 x+7 x^2+22 x^3)}{22+44 x+22 x^2} \, dx\)

Optimal. Leaf size=25 \[ \frac {1}{2} e^{-14+\frac {4}{11 x}+\frac {3}{1+x}} x^2 \]

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Rubi [F]  time = 0.50, 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^{-14+\frac {4}{11 x}+\frac {3}{1+x}} \left (-4+14 x+7 x^2+22 x^3\right )}{22+44 x+22 x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(-14 + 4/(11*x) + 3/(1 + x))*(-4 + 14*x + 7*x^2 + 22*x^3))/(22 + 44*x + 22*x^2),x]

[Out]

(-37*Defer[Int][E^(-14 + 4/(11*x) + 3/(1 + x)), x])/22 + Defer[Int][E^(-14 + 4/(11*x) + 3/(1 + x))*x, x] - (3*
Defer[Int][E^(-14 + 4/(11*x) + 3/(1 + x))/(1 + x)^2, x])/2 + 3*Defer[Int][E^(-14 + 4/(11*x) + 3/(1 + x))/(1 +
x), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-14+\frac {4}{11 x}+\frac {3}{1+x}} \left (-4+14 x+7 x^2+22 x^3\right )}{22 (1+x)^2} \, dx\\ &=\frac {1}{22} \int \frac {e^{-14+\frac {4}{11 x}+\frac {3}{1+x}} \left (-4+14 x+7 x^2+22 x^3\right )}{(1+x)^2} \, dx\\ &=\frac {1}{22} \int \left (-37 e^{-14+\frac {4}{11 x}+\frac {3}{1+x}}+22 e^{-14+\frac {4}{11 x}+\frac {3}{1+x}} x-\frac {33 e^{-14+\frac {4}{11 x}+\frac {3}{1+x}}}{(1+x)^2}+\frac {66 e^{-14+\frac {4}{11 x}+\frac {3}{1+x}}}{1+x}\right ) \, dx\\ &=-\left (\frac {3}{2} \int \frac {e^{-14+\frac {4}{11 x}+\frac {3}{1+x}}}{(1+x)^2} \, dx\right )-\frac {37}{22} \int e^{-14+\frac {4}{11 x}+\frac {3}{1+x}} \, dx+3 \int \frac {e^{-14+\frac {4}{11 x}+\frac {3}{1+x}}}{1+x} \, dx+\int e^{-14+\frac {4}{11 x}+\frac {3}{1+x}} x \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.15, size = 25, normalized size = 1.00 \begin {gather*} \frac {1}{2} e^{-14+\frac {4}{11 x}+\frac {3}{1+x}} x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-14 + 4/(11*x) + 3/(1 + x))*(-4 + 14*x + 7*x^2 + 22*x^3))/(22 + 44*x + 22*x^2),x]

[Out]

(E^(-14 + 4/(11*x) + 3/(1 + x))*x^2)/2

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fricas [A]  time = 1.05, size = 25, normalized size = 1.00 \begin {gather*} \frac {1}{2} \, x^{2} e^{\left (-\frac {154 \, x^{2} + 117 \, x - 4}{11 \, {\left (x^{2} + x\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((22*x^3+7*x^2+14*x-4)*exp(2/11/x)^2*exp(3/(x+1))/(22*x^2+44*x+22)/exp(7)^2,x, algorithm="fricas")

[Out]

1/2*x^2*e^(-1/11*(154*x^2 + 117*x - 4)/(x^2 + x))

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giac [A]  time = 0.20, size = 25, normalized size = 1.00 \begin {gather*} \frac {1}{2} \, x^{2} e^{\left (-\frac {154 \, x^{2} + 117 \, x - 4}{11 \, {\left (x^{2} + x\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((22*x^3+7*x^2+14*x-4)*exp(2/11/x)^2*exp(3/(x+1))/(22*x^2+44*x+22)/exp(7)^2,x, algorithm="giac")

[Out]

1/2*x^2*e^(-1/11*(154*x^2 + 117*x - 4)/(x^2 + x))

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maple [A]  time = 0.25, size = 26, normalized size = 1.04

method	result	size
gosper	\(\frac {{\mathrm e}^{\frac {3}{x +1}} {\mathrm e}^{-14} {\mathrm e}^{\frac {4}{11 x}} x^{2}}{2}\)	\(26\)
risch	\(\frac {x^{2} {\mathrm e}^{-\frac {154 x^{2}+117 x -4}{11 \left (x +1\right ) x}}}{2}\)	\(27\)
norman	\(\frac {\left (\frac {{\mathrm e}^{-7} x^{2} {\mathrm e}^{\frac {4}{11 x}} {\mathrm e}^{\frac {3}{x +1}}}{2}+\frac {{\mathrm e}^{-7} x^{3} {\mathrm e}^{\frac {4}{11 x}} {\mathrm e}^{\frac {3}{x +1}}}{2}\right ) {\mathrm e}^{-7}}{x +1}\)	\(62\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((22*x^3+7*x^2+14*x-4)*exp(2/11/x)^2*exp(3/(x+1))/(22*x^2+44*x+22)/exp(7)^2,x,method=_RETURNVERBOSE)

[Out]

1/2*exp(3/(x+1))/exp(7)^2*exp(2/11/x)^2*x^2

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maxima [A]  time = 0.45, size = 20, normalized size = 0.80 \begin {gather*} \frac {1}{2} \, x^{2} e^{\left (\frac {3}{x + 1} + \frac {4}{11 \, x} - 14\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((22*x^3+7*x^2+14*x-4)*exp(2/11/x)^2*exp(3/(x+1))/(22*x^2+44*x+22)/exp(7)^2,x, algorithm="maxima")

[Out]

1/2*x^2*e^(3/(x + 1) + 4/11/x - 14)

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mupad [B]  time = 3.56, size = 21, normalized size = 0.84 \begin {gather*} \frac {x^2\,{\mathrm {e}}^{-14}\,{\mathrm {e}}^{\frac {4}{11\,x}}\,{\mathrm {e}}^{\frac {3}{x+1}}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-14)*exp(4/(11*x))*exp(3/(x + 1))*(14*x + 7*x^2 + 22*x^3 - 4))/(44*x + 22*x^2 + 22),x)

[Out]

(x^2*exp(-14)*exp(4/(11*x))*exp(3/(x + 1)))/2

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sympy [A]  time = 5.09, size = 20, normalized size = 0.80 \begin {gather*} \frac {x^{2} e^{\frac {4}{11 x}} e^{\frac {3}{x + 1}}}{2 e^{14}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((22*x**3+7*x**2+14*x-4)*exp(2/11/x)**2*exp(3/(x+1))/(22*x**2+44*x+22)/exp(7)**2,x)

[Out]

x**2*exp(-14)*exp(4/(11*x))*exp(3/(x + 1))/2

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