3.49.52 \(\int \frac {3-11 x-3 x^2+72 x^5+24 x^6+(6 x+2 x^2) \log (3+x)}{3 x-5 x^2-2 x^3+12 x^6+4 x^7+(3 x^2+x^3) \log (3+x)} \, dx\)

Optimal. Leaf size=23 \[ \log \left (x-x^2 \left (2-4 x^4\right )+x^2 \log (3+x)\right ) \]

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Rubi [A]  time = 0.33, antiderivative size = 19, normalized size of antiderivative = 0.83, number of steps used = 2, number of rules used = 2, integrand size = 75, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.027, Rules used = {6741, 6685} \begin {gather*} \log \left (x \left (4 x^5-2 x+x \log (x+3)+1\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3 - 11*x - 3*x^2 + 72*x^5 + 24*x^6 + (6*x + 2*x^2)*Log[3 + x])/(3*x - 5*x^2 - 2*x^3 + 12*x^6 + 4*x^7 + (3
*x^2 + x^3)*Log[3 + x]),x]

[Out]

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

Rule 6685

Int[(u_)/((w_)*(y_)), x_Symbol] :> With[{q = DerivativeDivides[y*w, u, x]}, Simp[q*Log[RemoveContent[y*w, x]],
x] /;  !FalseQ[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 {3-11 x-3 x^2+72 x^5+24 x^6+\left (6 x+2 x^2\right ) \log (3+x)}{x (3+x) \left (1-2 x+4 x^5+x \log (3+x)\right )} \, dx\\ &=\log \left (x \left (1-2 x+4 x^5+x \log (3+x)\right )\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.41, size = 20, normalized size = 0.87 \begin {gather*} \log (x)+\log \left (1-2 x+4 x^5+x \log (3+x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3 - 11*x - 3*x^2 + 72*x^5 + 24*x^6 + (6*x + 2*x^2)*Log[3 + x])/(3*x - 5*x^2 - 2*x^3 + 12*x^6 + 4*x^
7 + (3*x^2 + x^3)*Log[3 + x]),x]

[Out]

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

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fricas [A]  time = 1.14, size = 26, normalized size = 1.13 \begin {gather*} 2 \, \log \relax (x) + \log \left (\frac {4 \, x^{5} + x \log \left (x + 3\right ) - 2 \, x + 1}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2+6*x)*log(3+x)+24*x^6+72*x^5-3*x^2-11*x+3)/((x^3+3*x^2)*log(3+x)+4*x^7+12*x^6-2*x^3-5*x^2+3*x
),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.22, size = 20, normalized size = 0.87 \begin {gather*} \log \left (4 \, x^{5} + x \log \left (x + 3\right ) - 2 \, x + 1\right ) + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2+6*x)*log(3+x)+24*x^6+72*x^5-3*x^2-11*x+3)/((x^3+3*x^2)*log(3+x)+4*x^7+12*x^6-2*x^3-5*x^2+3*x
),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.07, size = 21, normalized size = 0.91

method	result	size
norman	\(\ln \relax (x )+\ln \left (4 x^{5}+x \ln \left (3+x \right )-2 x +1\right )\)	\(21\)
risch	\(2 \ln \relax (x )+\ln \left (\ln \left (3+x \right )+\frac {4 x^{5}-2 x +1}{x}\right )\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^2+6*x)*ln(3+x)+24*x^6+72*x^5-3*x^2-11*x+3)/((x^3+3*x^2)*ln(3+x)+4*x^7+12*x^6-2*x^3-5*x^2+3*x),x,meth
od=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.40, size = 26, normalized size = 1.13 \begin {gather*} 2 \, \log \relax (x) + \log \left (\frac {4 \, x^{5} + x \log \left (x + 3\right ) - 2 \, x + 1}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2+6*x)*log(3+x)+24*x^6+72*x^5-3*x^2-11*x+3)/((x^3+3*x^2)*log(3+x)+4*x^7+12*x^6-2*x^3-5*x^2+3*x
),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 3.35, size = 20, normalized size = 0.87 \begin {gather*} \ln \left (\ln \left (x+3\right )+\frac {1}{x}+4\,x^4-2\right )+2\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x + 3)*(6*x + 2*x^2) - 11*x - 3*x^2 + 72*x^5 + 24*x^6 + 3)/(3*x + log(x + 3)*(3*x^2 + x^3) - 5*x^2 -
2*x^3 + 12*x^6 + 4*x^7),x)

[Out]

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

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sympy [A]  time = 0.31, size = 22, normalized size = 0.96 \begin {gather*} 2 \log {\relax (x )} + \log {\left (\log {\left (x + 3 \right )} + \frac {4 x^{5} - 2 x + 1}{x} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**2+6*x)*ln(3+x)+24*x**6+72*x**5-3*x**2-11*x+3)/((x**3+3*x**2)*ln(3+x)+4*x**7+12*x**6-2*x**3-5*
x**2+3*x),x)

[Out]

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

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