3.101.24 \(\int \frac {e^{\frac {4+x+x^2}{x}} (32-4 x-7 x^2+x^3+(-8+2 x^2) \log (2))}{2 x^2} \, dx\)

Optimal. Leaf size=20 \[ e^{1+\frac {4}{x}+x} \left (-4+\frac {x}{2}+\log (2)\right ) \]

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Rubi [F]  time = 0.61, 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 {4+x+x^2}{x}} \left (32-4 x-7 x^2+x^3+\left (-8+2 x^2\right ) \log (2)\right )}{2 x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((4 + x + x^2)/x)*(32 - 4*x - 7*x^2 + x^3 + (-8 + 2*x^2)*Log[2]))/(2*x^2),x]

[Out]

-1/2*((7 - Log[4])*Defer[Int][E^(1 + 4/x + x), x]) + 4*(4 - Log[2])*Defer[Int][E^(1 + 4/x + x)/x^2, x] - 2*Def
er[Int][E^(1 + 4/x + x)/x, x] + Defer[Int][E^(1 + 4/x + x)*x, x]/2

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{2} \int \frac {e^{\frac {4+x+x^2}{x}} \left (32-4 x-7 x^2+x^3+\left (-8+2 x^2\right ) \log (2)\right )}{x^2} \, dx\\ &=\frac {1}{2} \int \frac {e^{1+\frac {4}{x}+x} \left (-4 x+x^3+8 (4-\log (2))-x^2 (7-\log (4))\right )}{x^2} \, dx\\ &=\frac {1}{2} \int \left (-\frac {4 e^{1+\frac {4}{x}+x}}{x}+e^{1+\frac {4}{x}+x} x-7 e^{1+\frac {4}{x}+x} \left (1-\frac {2 \log (2)}{7}\right )-\frac {8 e^{1+\frac {4}{x}+x} (-4+\log (2))}{x^2}\right ) \, dx\\ &=\frac {1}{2} \int e^{1+\frac {4}{x}+x} x \, dx-2 \int \frac {e^{1+\frac {4}{x}+x}}{x} \, dx+(4 (4-\log (2))) \int \frac {e^{1+\frac {4}{x}+x}}{x^2} \, dx+\frac {1}{2} (-7+\log (4)) \int e^{1+\frac {4}{x}+x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [F]  time = 0.27, size = 45, normalized size = 2.25 \begin {gather*} \frac {1}{2} \int \frac {e^{\frac {4+x+x^2}{x}} \left (32-4 x-7 x^2+x^3+\left (-8+2 x^2\right ) \log (2)\right )}{x^2} \, dx \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((4 + x + x^2)/x)*(32 - 4*x - 7*x^2 + x^3 + (-8 + 2*x^2)*Log[2]))/(2*x^2),x]

[Out]

Integrate[(E^((4 + x + x^2)/x)*(32 - 4*x - 7*x^2 + x^3 + (-8 + 2*x^2)*Log[2]))/x^2, x]/2

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fricas [A]  time = 0.85, size = 20, normalized size = 1.00 \begin {gather*} \frac {1}{2} \, {\left (x + 2 \, \log \relax (2) - 8\right )} e^{\left (\frac {x^{2} + x + 4}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((2*x^2-8)*log(2)+x^3-7*x^2-4*x+32)*exp((x^2+x+4)/x)/x^2,x, algorithm="fricas")

[Out]

1/2*(x + 2*log(2) - 8)*e^((x^2 + x + 4)/x)

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giac [B]  time = 0.20, size = 42, normalized size = 2.10 \begin {gather*} \frac {1}{2} \, x e^{\left (\frac {x^{2} + x + 4}{x}\right )} + e^{\left (\frac {x^{2} + x + 4}{x}\right )} \log \relax (2) - 4 \, e^{\left (\frac {x^{2} + x + 4}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((2*x^2-8)*log(2)+x^3-7*x^2-4*x+32)*exp((x^2+x+4)/x)/x^2,x, algorithm="giac")

[Out]

1/2*x*e^((x^2 + x + 4)/x) + e^((x^2 + x + 4)/x)*log(2) - 4*e^((x^2 + x + 4)/x)

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maple [A]  time = 0.15, size = 21, normalized size = 1.05

method	result	size
gosper	\(\frac {{\mathrm e}^{\frac {x^{2}+x +4}{x}} \left (x +2 \ln \relax (2)-8\right )}{2}\)	\(21\)
risch	\(\frac {{\mathrm e}^{\frac {x^{2}+x +4}{x}} \left (x +2 \ln \relax (2)-8\right )}{2}\)	\(21\)
norman	\(\frac {x \left (\ln \relax (2)-4\right ) {\mathrm e}^{\frac {x^{2}+x +4}{x}}+\frac {x^{2} {\mathrm e}^{\frac {x^{2}+x +4}{x}}}{2}}{x}\)	\(39\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2*((2*x^2-8)*ln(2)+x^3-7*x^2-4*x+32)*exp((x^2+x+4)/x)/x^2,x,method=_RETURNVERBOSE)

[Out]

1/2*exp((x^2+x+4)/x)*(x+2*ln(2)-8)

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maxima [A]  time = 0.67, size = 23, normalized size = 1.15 \begin {gather*} \frac {1}{2} \, {\left (x e + 2 \, {\left (\log \relax (2) - 4\right )} e\right )} e^{\left (x + \frac {4}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((2*x^2-8)*log(2)+x^3-7*x^2-4*x+32)*exp((x^2+x+4)/x)/x^2,x, algorithm="maxima")

[Out]

1/2*(x*e + 2*(log(2) - 4)*e)*e^(x + 4/x)

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mupad [B]  time = 6.67, size = 17, normalized size = 0.85 \begin {gather*} {\mathrm {e}}^{x+\frac {4}{x}+1}\,\left (\frac {x}{2}+\ln \relax (2)-4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((x + x^2 + 4)/x)*(log(2)*(2*x^2 - 8) - 4*x - 7*x^2 + x^3 + 32))/(2*x^2),x)

[Out]

exp(x + 4/x + 1)*(x/2 + log(2) - 4)

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sympy [A]  time = 0.22, size = 19, normalized size = 0.95 \begin {gather*} \frac {\left (x - 8 + 2 \log {\relax (2 )}\right ) e^{\frac {x^{2} + x + 4}{x}}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((2*x**2-8)*ln(2)+x**3-7*x**2-4*x+32)*exp((x**2+x+4)/x)/x**2,x)

[Out]

(x - 8 + 2*log(2))*exp((x**2 + x + 4)/x)/2

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