3.8.5 \(\int \frac {9+6 e^4+6 x+(9+6 x) \log (x)}{-3-2 x+(3 x+2 x^2) \log (x)+e^4 (3+2 x) \log (\frac {1}{2} (3+2 x))} \, dx\)

Optimal. Leaf size=21 \[ 3 \left (5+\log \left (-1+x \log (x)+e^4 \log \left (\frac {3}{2}+x\right )\right )\right ) \]

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Rubi [A]  time = 0.21, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 57, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.035, Rules used = {6741, 6684} \begin {gather*} 3 \log \left (-x \log (x)-e^4 \log \left (x+\frac {3}{2}\right )+1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(9 + 6*E^4 + 6*x + (9 + 6*x)*Log[x])/(-3 - 2*x + (3*x + 2*x^2)*Log[x] + E^4*(3 + 2*x)*Log[(3 + 2*x)/2]),x]

[Out]

3*Log[1 - x*Log[x] - E^4*Log[3/2 + x]]

Rule 6684

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

Rule 6741

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-9 \left (1+\frac {2 e^4}{3}\right )-6 x-(9+6 x) \log (x)}{(3+2 x) \left (1-x \log (x)-e^4 \log \left (\frac {3}{2}+x\right )\right )} \, dx\\ &=3 \log \left (1-x \log (x)-e^4 \log \left (\frac {3}{2}+x\right )\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.48, size = 21, normalized size = 1.00 \begin {gather*} 3 \log \left (1-x \log (x)-e^4 \log \left (\frac {3}{2}+x\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(9 + 6*E^4 + 6*x + (9 + 6*x)*Log[x])/(-3 - 2*x + (3*x + 2*x^2)*Log[x] + E^4*(3 + 2*x)*Log[(3 + 2*x)/
2]),x]

[Out]

3*Log[1 - x*Log[x] - E^4*Log[3/2 + x]]

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fricas [A]  time = 0.80, size = 16, normalized size = 0.76 \begin {gather*} 3 \, \log \left (e^{4} \log \left (x + \frac {3}{2}\right ) + x \log \relax (x) - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x+9)*log(x)+6*exp(4)+6*x+9)/((2*x+3)*exp(4)*log(x+3/2)+(2*x^2+3*x)*log(x)-2*x-3),x, algorithm="f
ricas")

[Out]

3*log(e^4*log(x + 3/2) + x*log(x) - 1)

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giac [A]  time = 0.40, size = 24, normalized size = 1.14 \begin {gather*} 3 \, \log \left (-e^{4} \log \relax (2) + e^{4} \log \left (2 \, x + 3\right ) + x \log \relax (x) - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x+9)*log(x)+6*exp(4)+6*x+9)/((2*x+3)*exp(4)*log(x+3/2)+(2*x^2+3*x)*log(x)-2*x-3),x, algorithm="g
iac")

[Out]

3*log(-e^4*log(2) + e^4*log(2*x + 3) + x*log(x) - 1)

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maple [A]  time = 0.25, size = 17, normalized size = 0.81

method	result	size
norman	\(3 \ln \left ({\mathrm e}^{4} \ln \left (x +\frac {3}{2}\right )+x \ln \relax (x )-1\right )\)	\(17\)
risch	\(3 \ln \left (\ln \left (x +\frac {3}{2}\right )+\left (x \ln \relax (x )-1\right ) {\mathrm e}^{-4}\right )\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((6*x+9)*ln(x)+6*exp(4)+6*x+9)/((2*x+3)*exp(4)*ln(x+3/2)+(2*x^2+3*x)*ln(x)-2*x-3),x,method=_RETURNVERBOSE)

[Out]

3*ln(exp(4)*ln(x+3/2)+x*ln(x)-1)

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maxima [A]  time = 0.56, size = 29, normalized size = 1.38 \begin {gather*} 3 \, \log \left (-{\left (e^{4} \log \relax (2) - e^{4} \log \left (2 \, x + 3\right ) - x \log \relax (x) + 1\right )} e^{\left (-4\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x+9)*log(x)+6*exp(4)+6*x+9)/((2*x+3)*exp(4)*log(x+3/2)+(2*x^2+3*x)*log(x)-2*x-3),x, algorithm="m
axima")

[Out]

3*log(-(e^4*log(2) - e^4*log(2*x + 3) - x*log(x) + 1)*e^(-4))

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mupad [B]  time = 1.01, size = 16, normalized size = 0.76 \begin {gather*} 3\,\ln \left (\ln \left (x+\frac {3}{2}\right )\,{\mathrm {e}}^4+x\,\ln \relax (x)-1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(6*x + 6*exp(4) + log(x)*(6*x + 9) + 9)/(2*x - log(x)*(3*x + 2*x^2) - log(x + 3/2)*exp(4)*(2*x + 3) + 3),
x)

[Out]

3*log(log(x + 3/2)*exp(4) + x*log(x) - 1)

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sympy [A]  time = 0.41, size = 19, normalized size = 0.90 \begin {gather*} 3 \log {\left (\frac {x \log {\relax (x )} - 1}{e^{4}} + \log {\left (x + \frac {3}{2} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x+9)*ln(x)+6*exp(4)+6*x+9)/((2*x+3)*exp(4)*ln(x+3/2)+(2*x**2+3*x)*ln(x)-2*x-3),x)

[Out]

3*log((x*log(x) - 1)*exp(-4) + log(x + 3/2))

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