3.94.43 \(\int \frac {-7 x-7 \log (3)+(2+9 x+4 x^2) \log (\frac {1+4 x}{2+x})}{(2+9 x+4 x^2) \log ^2(\frac {1+4 x}{2+x})} \, dx\)

Optimal. Leaf size=17 \[ \frac {x+\log (3)}{\log \left (4-\frac {7}{2+x}\right )} \]

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Rubi [F]  time = 0.30, 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 {-7 x-7 \log (3)+\left (2+9 x+4 x^2\right ) \log \left (\frac {1+4 x}{2+x}\right )}{\left (2+9 x+4 x^2\right ) \log ^2\left (\frac {1+4 x}{2+x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-7*x - 7*Log[3] + (2 + 9*x + 4*x^2)*Log[(1 + 4*x)/(2 + x)])/((2 + 9*x + 4*x^2)*Log[(1 + 4*x)/(2 + x)]^2),
x]

[Out]

-7*Defer[Int][(x + Log[3])/((2 + x)*(1 + 4*x)*Log[(1 + 4*x)/(2 + x)]^2), x] + Defer[Int][Log[(1 + 4*x)/(2 + x)
]^(-1), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-\frac {7 (x+\log (3))}{2+9 x+4 x^2}+\log \left (\frac {1+4 x}{2+x}\right )}{\log ^2\left (\frac {1+4 x}{2+x}\right )} \, dx\\ &=\int \left (-\frac {7 (x+\log (3))}{(2+x) (1+4 x) \log ^2\left (\frac {1+4 x}{2+x}\right )}+\frac {1}{\log \left (\frac {1+4 x}{2+x}\right )}\right ) \, dx\\ &=-\left (7 \int \frac {x+\log (3)}{(2+x) (1+4 x) \log ^2\left (\frac {1+4 x}{2+x}\right )} \, dx\right )+\int \frac {1}{\log \left (\frac {1+4 x}{2+x}\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.09, size = 19, normalized size = 1.12 \begin {gather*} \frac {x+\log (3)}{\log \left (\frac {1+4 x}{2+x}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-7*x - 7*Log[3] + (2 + 9*x + 4*x^2)*Log[(1 + 4*x)/(2 + x)])/((2 + 9*x + 4*x^2)*Log[(1 + 4*x)/(2 + x
)]^2),x]

[Out]

(x + Log[3])/Log[(1 + 4*x)/(2 + x)]

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fricas [A]  time = 0.75, size = 19, normalized size = 1.12 \begin {gather*} \frac {x + \log \relax (3)}{\log \left (\frac {4 \, x + 1}{x + 2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^2+9*x+2)*log((4*x+1)/(2+x))-7*log(3)-7*x)/(4*x^2+9*x+2)/log((4*x+1)/(2+x))^2,x, algorithm="fri
cas")

[Out]

(x + log(3))/log((4*x + 1)/(x + 2))

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giac [B]  time = 0.24, size = 72, normalized size = 4.24 \begin {gather*} \frac {\frac {{\left (4 \, x + 1\right )} \log \relax (3)}{x + 2} - \frac {2 \, {\left (4 \, x + 1\right )}}{x + 2} - 4 \, \log \relax (3) + 1}{\frac {{\left (4 \, x + 1\right )} \log \left (\frac {4 \, x + 1}{x + 2}\right )}{x + 2} - 4 \, \log \left (\frac {4 \, x + 1}{x + 2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^2+9*x+2)*log((4*x+1)/(2+x))-7*log(3)-7*x)/(4*x^2+9*x+2)/log((4*x+1)/(2+x))^2,x, algorithm="gia
c")

[Out]

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

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maple [A]  time = 0.63, size = 20, normalized size = 1.18

method	result	size
norman	\(\frac {\ln \relax (3)+x}{\ln \left (\frac {4 x +1}{2+x}\right )}\)	\(20\)
risch	\(\frac {\ln \relax (3)+x}{\ln \left (\frac {4 x +1}{2+x}\right )}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*x^2+9*x+2)*ln((4*x+1)/(2+x))-7*ln(3)-7*x)/(4*x^2+9*x+2)/ln((4*x+1)/(2+x))^2,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [B]  time = 0.40, size = 36, normalized size = 2.12 \begin {gather*} \frac {x}{\log \left (4 \, x + 1\right ) - \log \left (x + 2\right )} + \frac {\log \relax (3)}{\log \left (4 \, x + 1\right ) - \log \left (x + 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^2+9*x+2)*log((4*x+1)/(2+x))-7*log(3)-7*x)/(4*x^2+9*x+2)/log((4*x+1)/(2+x))^2,x, algorithm="max
ima")

[Out]

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

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mupad [B]  time = 0.32, size = 19, normalized size = 1.12 \begin {gather*} \frac {x+\ln \relax (3)}{\ln \left (\frac {4\,x+1}{x+2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(7*x + 7*log(3) - log((4*x + 1)/(x + 2))*(9*x + 4*x^2 + 2))/(log((4*x + 1)/(x + 2))^2*(9*x + 4*x^2 + 2)),
x)

[Out]

(x + log(3))/log((4*x + 1)/(x + 2))

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sympy [A]  time = 0.14, size = 14, normalized size = 0.82 \begin {gather*} \frac {x + \log {\relax (3 )}}{\log {\left (\frac {4 x + 1}{x + 2} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x**2+9*x+2)*ln((4*x+1)/(2+x))-7*ln(3)-7*x)/(4*x**2+9*x+2)/ln((4*x+1)/(2+x))**2,x)

[Out]

(x + log(3))/log((4*x + 1)/(x + 2))

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