3.91.12 \(\int \frac {9-6 x+x^2+18 x^3+16 x^4+3 x^5}{9 x-6 x^2-3 x^3+6 x^4+5 x^5+x^6+(9 x+6 x^2+x^3) \log (5 x)} \, dx\)

Optimal. Leaf size=28 \[ \log \left (-1+x+\frac {x-x^2 (1+x)^2}{3+x}-\log (5 x)\right ) \]

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Rubi [F]  time = 1.04, 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 {9-6 x+x^2+18 x^3+16 x^4+3 x^5}{9 x-6 x^2-3 x^3+6 x^4+5 x^5+x^6+\left (9 x+6 x^2+x^3\right ) \log (5 x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(9 - 6*x + x^2 + 18*x^3 + 16*x^4 + 3*x^5)/(9*x - 6*x^2 - 3*x^3 + 6*x^4 + 5*x^5 + x^6 + (9*x + 6*x^2 + x^3)
*Log[5*x]),x]

[Out]

10*Defer[Int][(3 - 3*x + 2*x^3 + x^4 + 3*Log[5*x] + x*Log[5*x])^(-1), x] + 3*Defer[Int][1/(x*(3 - 3*x + 2*x^3
+ x^4 + 3*Log[5*x] + x*Log[5*x])), x] - 3*Defer[Int][x/(3 - 3*x + 2*x^3 + x^4 + 3*Log[5*x] + x*Log[5*x]), x] +
7*Defer[Int][x^2/(3 - 3*x + 2*x^3 + x^4 + 3*Log[5*x] + x*Log[5*x]), x] + 3*Defer[Int][x^3/(3 - 3*x + 2*x^3 +
x^4 + 3*Log[5*x] + x*Log[5*x]), x] - 39*Defer[Int][1/((3 + x)*(3 - 3*x + 2*x^3 + x^4 + 3*Log[5*x] + x*Log[5*x]
)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {9-6 x+x^2+18 x^3+16 x^4+3 x^5}{x (3+x) \left (3-3 x+2 x^3+x^4+(3+x) \log (5 x)\right )} \, dx\\ &=\int \left (\frac {10}{3-3 x+2 x^3+x^4+3 \log (5 x)+x \log (5 x)}+\frac {3}{x \left (3-3 x+2 x^3+x^4+3 \log (5 x)+x \log (5 x)\right )}-\frac {3 x}{3-3 x+2 x^3+x^4+3 \log (5 x)+x \log (5 x)}+\frac {7 x^2}{3-3 x+2 x^3+x^4+3 \log (5 x)+x \log (5 x)}+\frac {3 x^3}{3-3 x+2 x^3+x^4+3 \log (5 x)+x \log (5 x)}-\frac {39}{(3+x) \left (3-3 x+2 x^3+x^4+3 \log (5 x)+x \log (5 x)\right )}\right ) \, dx\\ &=3 \int \frac {1}{x \left (3-3 x+2 x^3+x^4+3 \log (5 x)+x \log (5 x)\right )} \, dx-3 \int \frac {x}{3-3 x+2 x^3+x^4+3 \log (5 x)+x \log (5 x)} \, dx+3 \int \frac {x^3}{3-3 x+2 x^3+x^4+3 \log (5 x)+x \log (5 x)} \, dx+7 \int \frac {x^2}{3-3 x+2 x^3+x^4+3 \log (5 x)+x \log (5 x)} \, dx+10 \int \frac {1}{3-3 x+2 x^3+x^4+3 \log (5 x)+x \log (5 x)} \, dx-39 \int \frac {1}{(3+x) \left (3-3 x+2 x^3+x^4+3 \log (5 x)+x \log (5 x)\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.43, size = 33, normalized size = 1.18 \begin {gather*} -\log (3+x)+\log \left (3-3 x+2 x^3+x^4+3 \log (5 x)+x \log (5 x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(9 - 6*x + x^2 + 18*x^3 + 16*x^4 + 3*x^5)/(9*x - 6*x^2 - 3*x^3 + 6*x^4 + 5*x^5 + x^6 + (9*x + 6*x^2
+ x^3)*Log[5*x]),x]

[Out]

-Log[3 + x] + Log[3 - 3*x + 2*x^3 + x^4 + 3*Log[5*x] + x*Log[5*x]]

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fricas [A]  time = 0.47, size = 28, normalized size = 1.00 \begin {gather*} \log \left (\frac {x^{4} + 2 \, x^{3} + {\left (x + 3\right )} \log \left (5 \, x\right ) - 3 \, x + 3}{x + 3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^5+16*x^4+18*x^3+x^2-6*x+9)/((x^3+6*x^2+9*x)*log(5*x)+x^6+5*x^5+6*x^4-3*x^3-6*x^2+9*x),x, algori
thm="fricas")

[Out]

log((x^4 + 2*x^3 + (x + 3)*log(5*x) - 3*x + 3)/(x + 3))

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giac [A]  time = 0.15, size = 33, normalized size = 1.18 \begin {gather*} \log \left (x^{4} + 2 \, x^{3} + x \log \left (5 \, x\right ) - 3 \, x + 3 \, \log \left (5 \, x\right ) + 3\right ) - \log \left (x + 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^5+16*x^4+18*x^3+x^2-6*x+9)/((x^3+6*x^2+9*x)*log(5*x)+x^6+5*x^5+6*x^4-3*x^3-6*x^2+9*x),x, algori
thm="giac")

[Out]

log(x^4 + 2*x^3 + x*log(5*x) - 3*x + 3*log(5*x) + 3) - log(x + 3)

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maple [A]  time = 0.07, size = 26, normalized size = 0.93

method	result	size
risch	\(\ln \left (\ln \left (5 x \right )+\frac {x^{4}+2 x^{3}-3 x +3}{3+x}\right )\)	\(26\)
norman	\(-\ln \left (3+x \right )+\ln \left (x^{4}+2 x^{3}+x \ln \left (5 x \right )-3 x +3 \ln \left (5 x \right )+3\right )\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^5+16*x^4+18*x^3+x^2-6*x+9)/((x^3+6*x^2+9*x)*ln(5*x)+x^6+5*x^5+6*x^4-3*x^3-6*x^2+9*x),x,method=_RETURN
VERBOSE)

[Out]

ln(ln(5*x)+(x^4+2*x^3-3*x+3)/(3+x))

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maxima [A]  time = 0.48, size = 33, normalized size = 1.18 \begin {gather*} \log \left (\frac {x^{4} + 2 \, x^{3} + x {\left (\log \relax (5) - 3\right )} + {\left (x + 3\right )} \log \relax (x) + 3 \, \log \relax (5) + 3}{x + 3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^5+16*x^4+18*x^3+x^2-6*x+9)/((x^3+6*x^2+9*x)*log(5*x)+x^6+5*x^5+6*x^4-3*x^3-6*x^2+9*x),x, algori
thm="maxima")

[Out]

log((x^4 + 2*x^3 + x*(log(5) - 3) + (x + 3)*log(x) + 3*log(5) + 3)/(x + 3))

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mupad [B]  time = 7.62, size = 25, normalized size = 0.89 \begin {gather*} \ln \left (3\,x+\ln \left (5\,x\right )+\frac {39}{x+3}-x^2+x^3-12\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2 - 6*x + 18*x^3 + 16*x^4 + 3*x^5 + 9)/(9*x - 6*x^2 - 3*x^3 + 6*x^4 + 5*x^5 + x^6 + log(5*x)*(9*x + 6*x
^2 + x^3)),x)

[Out]

log(3*x + log(5*x) + 39/(x + 3) - x^2 + x^3 - 12)

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sympy [A]  time = 0.35, size = 22, normalized size = 0.79 \begin {gather*} \log {\left (\log {\left (5 x \right )} + \frac {x^{4} + 2 x^{3} - 3 x + 3}{x + 3} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x**5+16*x**4+18*x**3+x**2-6*x+9)/((x**3+6*x**2+9*x)*ln(5*x)+x**6+5*x**5+6*x**4-3*x**3-6*x**2+9*x)
,x)

[Out]

log(log(5*x) + (x**4 + 2*x**3 - 3*x + 3)/(x + 3))

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