3.39.95 \(\int \frac {10+80 x+90 x^2}{-x-4 x^2-3 x^3+(x+4 x^2+3 x^3) \log ((2 x+8 x^2+6 x^3) \log (5))} \, dx\)

Optimal. Leaf size=20 \[ 10 \log (-1+\log (x (2+2 x (4+3 x)) \log (5))) \]

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Rubi [A]  time = 0.20, antiderivative size = 23, normalized size of antiderivative = 1.15, number of steps used = 2, number of rules used = 2, integrand size = 58, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {6741, 6684} \begin {gather*} 10 \log \left (1-\log \left (2 x \left (3 x^2+4 x+1\right ) \log (5)\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(10 + 80*x + 90*x^2)/(-x - 4*x^2 - 3*x^3 + (x + 4*x^2 + 3*x^3)*Log[(2*x + 8*x^2 + 6*x^3)*Log[5]]),x]

[Out]

10*Log[1 - Log[2*x*(1 + 4*x + 3*x^2)*Log[5]]]

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 {-10-80 x-90 x^2}{x \left (1+4 x+3 x^2\right ) \left (1-\log \left (2 x \left (1+4 x+3 x^2\right ) \log (5)\right )\right )} \, dx\\ &=10 \log \left (1-\log \left (2 x \left (1+4 x+3 x^2\right ) \log (5)\right )\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.08, size = 21, normalized size = 1.05 \begin {gather*} 10 \log \left (-1+\log \left (2 x \left (1+4 x+3 x^2\right ) \log (5)\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(10 + 80*x + 90*x^2)/(-x - 4*x^2 - 3*x^3 + (x + 4*x^2 + 3*x^3)*Log[(2*x + 8*x^2 + 6*x^3)*Log[5]]),x]

[Out]

10*Log[-1 + Log[2*x*(1 + 4*x + 3*x^2)*Log[5]]]

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fricas [A]  time = 0.44, size = 22, normalized size = 1.10 \begin {gather*} 10 \, \log \left (\log \left (2 \, {\left (3 \, x^{3} + 4 \, x^{2} + x\right )} \log \relax (5)\right ) - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((90*x^2+80*x+10)/((3*x^3+4*x^2+x)*log((6*x^3+8*x^2+2*x)*log(5))-3*x^3-4*x^2-x),x, algorithm="fricas"
)

[Out]

10*log(log(2*(3*x^3 + 4*x^2 + x)*log(5)) - 1)

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giac [A]  time = 0.24, size = 26, normalized size = 1.30 \begin {gather*} 10 \, \log \left (\log \left (6 \, x^{3} \log \relax (5) + 8 \, x^{2} \log \relax (5) + 2 \, x \log \relax (5)\right ) - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((90*x^2+80*x+10)/((3*x^3+4*x^2+x)*log((6*x^3+8*x^2+2*x)*log(5))-3*x^3-4*x^2-x),x, algorithm="giac")

[Out]

10*log(log(6*x^3*log(5) + 8*x^2*log(5) + 2*x*log(5)) - 1)

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maple [A]  time = 0.04, size = 24, normalized size = 1.20

method	result	size
norman	\(10 \ln \left (\ln \left (\left (6 x^{3}+8 x^{2}+2 x \right ) \ln \relax (5)\right )-1\right )\)	\(24\)
risch	\(10 \ln \left (\ln \left (\left (6 x^{3}+8 x^{2}+2 x \right ) \ln \relax (5)\right )-1\right )\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((90*x^2+80*x+10)/((3*x^3+4*x^2+x)*ln((6*x^3+8*x^2+2*x)*ln(5))-3*x^3-4*x^2-x),x,method=_RETURNVERBOSE)

[Out]

10*ln(ln((6*x^3+8*x^2+2*x)*ln(5))-1)

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maxima [A]  time = 0.48, size = 22, normalized size = 1.10 \begin {gather*} 10 \, \log \left (\log \relax (2) + \log \left (3 \, x + 1\right ) + \log \left (x + 1\right ) + \log \relax (x) + \log \left (\log \relax (5)\right ) - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((90*x^2+80*x+10)/((3*x^3+4*x^2+x)*log((6*x^3+8*x^2+2*x)*log(5))-3*x^3-4*x^2-x),x, algorithm="maxima"
)

[Out]

10*log(log(2) + log(3*x + 1) + log(x + 1) + log(x) + log(log(5)) - 1)

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mupad [B]  time = 5.28, size = 23, normalized size = 1.15 \begin {gather*} 10\,\ln \left (\ln \left (\ln \relax (5)\,\left (6\,x^3+8\,x^2+2\,x\right )\right )-1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(80*x + 90*x^2 + 10)/(x - log(log(5)*(2*x + 8*x^2 + 6*x^3))*(x + 4*x^2 + 3*x^3) + 4*x^2 + 3*x^3),x)

[Out]

10*log(log(log(5)*(2*x + 8*x^2 + 6*x^3)) - 1)

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sympy [A]  time = 0.26, size = 22, normalized size = 1.10 \begin {gather*} 10 \log {\left (\log {\left (\left (6 x^{3} + 8 x^{2} + 2 x\right ) \log {\relax (5 )} \right )} - 1 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((90*x**2+80*x+10)/((3*x**3+4*x**2+x)*ln((6*x**3+8*x**2+2*x)*ln(5))-3*x**3-4*x**2-x),x)

[Out]

10*log(log((6*x**3 + 8*x**2 + 2*x)*log(5)) - 1)

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