3.21.9 \(\int \frac {7+4 e^x+4 x}{36+4 e^x+7 x+2 x^2} \, dx\)

Optimal. Leaf size=23 \[ \log \left (3+\frac {1}{3} \left (e^x+\frac {7 x}{4}+\frac {x^2}{2}\right )\right ) \]

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Rubi [A]  time = 0.04, antiderivative size = 16, normalized size of antiderivative = 0.70, number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {6684} \begin {gather*} \log \left (2 x^2+7 x+4 e^x+36\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(7 + 4*E^x + 4*x)/(36 + 4*E^x + 7*x + 2*x^2),x]

[Out]

Log[36 + 4*E^x + 7*x + 2*x^2]

Rule 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\log \left (36+4 e^x+7 x+2 x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.10, size = 16, normalized size = 0.70 \begin {gather*} \log \left (36+4 e^x+7 x+2 x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(7 + 4*E^x + 4*x)/(36 + 4*E^x + 7*x + 2*x^2),x]

[Out]

Log[36 + 4*E^x + 7*x + 2*x^2]

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fricas [A]  time = 0.97, size = 15, normalized size = 0.65 \begin {gather*} \log \left (2 \, x^{2} + 7 \, x + 4 \, e^{x} + 36\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*exp(x)+4*x+7)/(4*exp(x)+2*x^2+7*x+36),x, algorithm="fricas")

[Out]

log(2*x^2 + 7*x + 4*e^x + 36)

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giac [A]  time = 0.23, size = 15, normalized size = 0.65 \begin {gather*} \log \left (2 \, x^{2} + 7 \, x + 4 \, e^{x} + 36\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*exp(x)+4*x+7)/(4*exp(x)+2*x^2+7*x+36),x, algorithm="giac")

[Out]

log(2*x^2 + 7*x + 4*e^x + 36)

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maple [A]  time = 0.02, size = 14, normalized size = 0.61

method	result	size
risch	\(\ln \left (\frac {x^{2}}{2}+\frac {7 x}{4}+{\mathrm e}^{x}+9\right )\)	\(14\)
derivativedivides	\(\ln \left (4 \,{\mathrm e}^{x}+2 x^{2}+7 x +36\right )\)	\(16\)
default	\(\ln \left (4 \,{\mathrm e}^{x}+2 x^{2}+7 x +36\right )\)	\(16\)
norman	\(\ln \left (4 \,{\mathrm e}^{x}+2 x^{2}+7 x +36\right )\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*exp(x)+4*x+7)/(4*exp(x)+2*x^2+7*x+36),x,method=_RETURNVERBOSE)

[Out]

ln(1/2*x^2+7/4*x+exp(x)+9)

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maxima [A]  time = 0.36, size = 15, normalized size = 0.65 \begin {gather*} \log \left (2 \, x^{2} + 7 \, x + 4 \, e^{x} + 36\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*exp(x)+4*x+7)/(4*exp(x)+2*x^2+7*x+36),x, algorithm="maxima")

[Out]

log(2*x^2 + 7*x + 4*e^x + 36)

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mupad [B]  time = 1.20, size = 13, normalized size = 0.57 \begin {gather*} \ln \left (\frac {7\,x}{2}+2\,{\mathrm {e}}^x+x^2+18\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x + 4*exp(x) + 7)/(7*x + 4*exp(x) + 2*x^2 + 36),x)

[Out]

log((7*x)/2 + 2*exp(x) + x^2 + 18)

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sympy [A]  time = 0.13, size = 15, normalized size = 0.65 \begin {gather*} \log {\left (\frac {x^{2}}{2} + \frac {7 x}{4} + e^{x} + 9 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*exp(x)+4*x+7)/(4*exp(x)+2*x**2+7*x+36),x)

[Out]

log(x**2/2 + 7*x/4 + exp(x) + 9)

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