3.55.89 \(\int \frac {5+10 x}{1+x+x^2+\log (3)} \, dx\)

Optimal. Leaf size=14 \[ 5 \log \left (-x+(1+x)^2+\log (3)\right ) \]

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Rubi [A]  time = 0.00, antiderivative size = 11, normalized size of antiderivative = 0.79, number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {628} \begin {gather*} 5 \log \left (x^2+x+1+\log (3)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(5 + 10*x)/(1 + x + x^2 + Log[3]),x]

[Out]

5*Log[1 + x + x^2 + Log[3]]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=5 \log \left (1+x+x^2+\log (3)\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 11, normalized size = 0.79 \begin {gather*} 5 \log \left (1+x+x^2+\log (3)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(5 + 10*x)/(1 + x + x^2 + Log[3]),x]

[Out]

5*Log[1 + x + x^2 + Log[3]]

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fricas [A]  time = 1.03, size = 11, normalized size = 0.79 \begin {gather*} 5 \, \log \left (x^{2} + x + \log \relax (3) + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((10*x+5)/(x^2+log(3)+x+1),x, algorithm="fricas")

[Out]

5*log(x^2 + x + log(3) + 1)

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giac [A]  time = 0.17, size = 11, normalized size = 0.79 \begin {gather*} 5 \, \log \left (x^{2} + x + \log \relax (3) + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((10*x+5)/(x^2+log(3)+x+1),x, algorithm="giac")

[Out]

5*log(x^2 + x + log(3) + 1)

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maple [A]  time = 0.63, size = 12, normalized size = 0.86

method	result	size
default	\(5 \ln \left (x^{2}+\ln \relax (3)+x +1\right )\)	\(12\)
norman	\(5 \ln \left (x^{2}+\ln \relax (3)+x +1\right )\)	\(12\)
risch	\(5 \ln \left (x^{2}+\ln \relax (3)+x +1\right )\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((10*x+5)/(x^2+ln(3)+x+1),x,method=_RETURNVERBOSE)

[Out]

5*ln(x^2+ln(3)+x+1)

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maxima [A]  time = 0.36, size = 11, normalized size = 0.79 \begin {gather*} 5 \, \log \left (x^{2} + x + \log \relax (3) + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((10*x+5)/(x^2+log(3)+x+1),x, algorithm="maxima")

[Out]

5*log(x^2 + x + log(3) + 1)

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mupad [B]  time = 0.05, size = 11, normalized size = 0.79 \begin {gather*} 5\,\ln \left (x^2+x+\ln \relax (3)+1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((10*x + 5)/(x + log(3) + x^2 + 1),x)

[Out]

5*log(x + log(3) + x^2 + 1)

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sympy [A]  time = 0.10, size = 12, normalized size = 0.86 \begin {gather*} 5 \log {\left (x^{2} + x + 1 + \log {\relax (3 )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((10*x+5)/(x**2+ln(3)+x+1),x)

[Out]

5*log(x**2 + x + 1 + log(3))

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