3.80.68 \(\int \frac {e^x (6+6 x+3 x^2)}{32+32 x+8 x^2+e^x (6 x+3 x^2)} \, dx\)

Optimal. Leaf size=16 \[ \log \left (4+\frac {3 e^x x}{2 (2+x)}\right ) \]

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Rubi [F]  time = 0.71, 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^x \left (6+6 x+3 x^2\right )}{32+32 x+8 x^2+e^x \left (6 x+3 x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^x*(6 + 6*x + 3*x^2))/(32 + 32*x + 8*x^2 + E^x*(6*x + 3*x^2)),x]

[Out]

3*Defer[Int][(E^x*x)/(16 + 8*x + 3*E^x*x), x] + 6*Defer[Int][E^x/((2 + x)*(16 + 8*x + 3*E^x*x)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^x \left (6+6 x+3 x^2\right )}{(2+x) \left (16+8 x+3 e^x x\right )} \, dx\\ &=\int \left (\frac {3 e^x x}{16+8 x+3 e^x x}+\frac {6 e^x}{(2+x) \left (16+8 x+3 e^x x\right )}\right ) \, dx\\ &=3 \int \frac {e^x x}{16+8 x+3 e^x x} \, dx+6 \int \frac {e^x}{(2+x) \left (16+8 x+3 e^x x\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.16, size = 27, normalized size = 1.69 \begin {gather*} 3 \left (-\frac {1}{3} \log (2+x)+\frac {1}{3} \log \left (16+8 x+3 e^x x\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^x*(6 + 6*x + 3*x^2))/(32 + 32*x + 8*x^2 + E^x*(6*x + 3*x^2)),x]

[Out]

3*(-1/3*Log[2 + x] + Log[16 + 8*x + 3*E^x*x]/3)

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fricas [A]  time = 0.64, size = 24, normalized size = 1.50 \begin {gather*} -\log \left (x + 2\right ) + \log \relax (x) + \log \left (\frac {3 \, x e^{x} + 8 \, x + 16}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2+6*x+6)*exp(x)/((3*x^2+6*x)*exp(x)+8*x^2+32*x+32),x, algorithm="fricas")

[Out]

-log(x + 2) + log(x) + log((3*x*e^x + 8*x + 16)/x)

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giac [A]  time = 0.13, size = 18, normalized size = 1.12 \begin {gather*} \log \left (3 \, x e^{x} + 8 \, x + 16\right ) - \log \left (x + 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2+6*x+6)*exp(x)/((3*x^2+6*x)*exp(x)+8*x^2+32*x+32),x, algorithm="giac")

[Out]

log(3*x*e^x + 8*x + 16) - log(x + 2)

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maple [A]  time = 0.04, size = 19, normalized size = 1.19

method	result	size
norman	\(-\ln \left (2+x \right )+\ln \left (3 \,{\mathrm e}^{x} x +8 x +16\right )\)	\(19\)
risch	\(\ln \relax (x )-\ln \left (2+x \right )+\ln \left ({\mathrm e}^{x}+\frac {\frac {8 x}{3}+\frac {16}{3}}{x}\right )\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^2+6*x+6)*exp(x)/((3*x^2+6*x)*exp(x)+8*x^2+32*x+32),x,method=_RETURNVERBOSE)

[Out]

-ln(2+x)+ln(3*exp(x)*x+8*x+16)

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maxima [A]  time = 0.40, size = 25, normalized size = 1.56 \begin {gather*} -\log \left (x + 2\right ) + \log \relax (x) + \log \left (\frac {3 \, x e^{x} + 8 \, x + 16}{3 \, x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2+6*x+6)*exp(x)/((3*x^2+6*x)*exp(x)+8*x^2+32*x+32),x, algorithm="maxima")

[Out]

-log(x + 2) + log(x) + log(1/3*(3*x*e^x + 8*x + 16)/x)

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mupad [B]  time = 0.16, size = 18, normalized size = 1.12 \begin {gather*} \ln \left (8\,x+3\,x\,{\mathrm {e}}^x+16\right )-\ln \left (x+2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x)*(6*x + 3*x^2 + 6))/(32*x + exp(x)*(6*x + 3*x^2) + 8*x^2 + 32),x)

[Out]

log(8*x + 3*x*exp(x) + 16) - log(x + 2)

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sympy [A]  time = 0.23, size = 20, normalized size = 1.25 \begin {gather*} \log {\relax (x )} - \log {\left (x + 2 \right )} + \log {\left (e^{x} + \frac {8 x + 16}{3 x} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x**2+6*x+6)*exp(x)/((3*x**2+6*x)*exp(x)+8*x**2+32*x+32),x)

[Out]

log(x) - log(x + 2) + log(exp(x) + (8*x + 16)/(3*x))

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