3.36.19 \(\int \frac {241+8 e^x+12 x+4 x^2}{948+16 e^x+16 x+16 x^2} \, dx\)

Optimal. Leaf size=19 \[ \frac {1}{4} \left (-1+x+\log \left (\frac {237}{4}+e^x+x+x^2\right )\right ) \]

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Rubi [F]  time = 0.36, 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 {241+8 e^x+12 x+4 x^2}{948+16 e^x+16 x+16 x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(241 + 8*E^x + 12*x + 4*x^2)/(948 + 16*E^x + 16*x + 16*x^2),x]

[Out]

x/2 - (233*Defer[Int][(237 + 4*E^x + 4*x + 4*x^2)^(-1), x])/4 + Defer[Int][x/(237 + 4*E^x + 4*x + 4*x^2), x] -
Defer[Int][x^2/(237 + 4*E^x + 4*x + 4*x^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {241+8 e^x+12 x+4 x^2}{4 \left (237+4 e^x+4 x+4 x^2\right )} \, dx\\ &=\frac {1}{4} \int \frac {241+8 e^x+12 x+4 x^2}{237+4 e^x+4 x+4 x^2} \, dx\\ &=\frac {1}{4} \int \left (2-\frac {233-4 x+4 x^2}{237+4 e^x+4 x+4 x^2}\right ) \, dx\\ &=\frac {x}{2}-\frac {1}{4} \int \frac {233-4 x+4 x^2}{237+4 e^x+4 x+4 x^2} \, dx\\ &=\frac {x}{2}-\frac {1}{4} \int \left (\frac {233}{237+4 e^x+4 x+4 x^2}-\frac {4 x}{237+4 e^x+4 x+4 x^2}+\frac {4 x^2}{237+4 e^x+4 x+4 x^2}\right ) \, dx\\ &=\frac {x}{2}-\frac {233}{4} \int \frac {1}{237+4 e^x+4 x+4 x^2} \, dx+\int \frac {x}{237+4 e^x+4 x+4 x^2} \, dx-\int \frac {x^2}{237+4 e^x+4 x+4 x^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.13, size = 26, normalized size = 1.37 \begin {gather*} \frac {x}{4}+\frac {1}{4} \log \left (237+4 e^x+4 x+4 x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(241 + 8*E^x + 12*x + 4*x^2)/(948 + 16*E^x + 16*x + 16*x^2),x]

[Out]

x/4 + Log[237 + 4*E^x + 4*x + 4*x^2]/4

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fricas [A]  time = 0.82, size = 21, normalized size = 1.11 \begin {gather*} \frac {1}{4} \, x + \frac {1}{4} \, \log \left (4 \, x^{2} + 4 \, x + 4 \, e^{x} + 237\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*exp(x)+4*x^2+12*x+241)/(16*exp(x)+16*x^2+16*x+948),x, algorithm="fricas")

[Out]

1/4*x + 1/4*log(4*x^2 + 4*x + 4*e^x + 237)

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giac [A]  time = 0.24, size = 21, normalized size = 1.11 \begin {gather*} \frac {1}{4} \, x + \frac {1}{4} \, \log \left (4 \, x^{2} + 4 \, x + 4 \, e^{x} + 237\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*exp(x)+4*x^2+12*x+241)/(16*exp(x)+16*x^2+16*x+948),x, algorithm="giac")

[Out]

1/4*x + 1/4*log(4*x^2 + 4*x + 4*e^x + 237)

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maple [A]  time = 0.02, size = 16, normalized size = 0.84

method	result	size
risch	\(\frac {x}{4}+\frac {\ln \left ({\mathrm e}^{x}+x +\frac {237}{4}+x^{2}\right )}{4}\)	\(16\)
norman	\(\frac {x}{4}+\frac {\ln \left (16 \,{\mathrm e}^{x}+16 x^{2}+16 x +948\right )}{4}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((8*exp(x)+4*x^2+12*x+241)/(16*exp(x)+16*x^2+16*x+948),x,method=_RETURNVERBOSE)

[Out]

1/4*x+1/4*ln(exp(x)+x+237/4+x^2)

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maxima [A]  time = 0.55, size = 15, normalized size = 0.79 \begin {gather*} \frac {1}{4} \, x + \frac {1}{4} \, \log \left (x^{2} + x + e^{x} + \frac {237}{4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*exp(x)+4*x^2+12*x+241)/(16*exp(x)+16*x^2+16*x+948),x, algorithm="maxima")

[Out]

1/4*x + 1/4*log(x^2 + x + e^x + 237/4)

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mupad [B]  time = 0.15, size = 15, normalized size = 0.79 \begin {gather*} \frac {x}{4}+\frac {\ln \left (x+{\mathrm {e}}^x+x^2+\frac {237}{4}\right )}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((12*x + 8*exp(x) + 4*x^2 + 241)/(16*x + 16*exp(x) + 16*x^2 + 948),x)

[Out]

x/4 + log(x + exp(x) + x^2 + 237/4)/4

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sympy [A]  time = 0.13, size = 17, normalized size = 0.89 \begin {gather*} \frac {x}{4} + \frac {\log {\left (x^{2} + x + e^{x} + \frac {237}{4} \right )}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*exp(x)+4*x**2+12*x+241)/(16*exp(x)+16*x**2+16*x+948),x)

[Out]

x/4 + log(x**2 + x + exp(x) + 237/4)/4

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