3.23.76 \(\int \frac {288 x^2+56 x^3+(-864 x^2-112 x^3) \log (x)+(-2592-1008 x-98 x^2) \log ^2(x)}{16 x^6+(288 x^4+56 x^5) \log (x)+(1296 x^2+504 x^3+49 x^4) \log ^2(x)} \, dx\)

Optimal. Leaf size=26 \[ \frac {2}{x \left (1+\frac {x^2}{\left (9+\frac {7 x}{4}\right ) \log (x)}\right )} \]

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Rubi [F]  time = 0.73, 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 {288 x^2+56 x^3+\left (-864 x^2-112 x^3\right ) \log (x)+\left (-2592-1008 x-98 x^2\right ) \log ^2(x)}{16 x^6+\left (288 x^4+56 x^5\right ) \log (x)+\left (1296 x^2+504 x^3+49 x^4\right ) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(288*x^2 + 56*x^3 + (-864*x^2 - 112*x^3)*Log[x] + (-2592 - 1008*x - 98*x^2)*Log[x]^2)/(16*x^6 + (288*x^4 +
56*x^5)*Log[x] + (1296*x^2 + 504*x^3 + 49*x^4)*Log[x]^2),x]

[Out]

2/x - (27360*Defer[Int][(4*x^2 + 36*Log[x] + 7*x*Log[x])^(-2), x])/49 + (1544*Defer[Int][x/(4*x^2 + 36*Log[x]
+ 7*x*Log[x])^2, x])/7 + 32*Defer[Int][x^2/(4*x^2 + 36*Log[x] + 7*x*Log[x])^2, x] + (1492992*Defer[Int][1/((36
+ 7*x)*(4*x^2 + 36*Log[x] + 7*x*Log[x])^2), x])/49 - 288*Defer[Int][1/((36 + 7*x)*(4*x^2 + 36*Log[x] + 7*x*Lo
g[x])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (4 x^2 (36+7 x)-8 x^2 (54+7 x) \log (x)-(36+7 x)^2 \log ^2(x)\right )}{\left (4 x^3+x (36+7 x) \log (x)\right )^2} \, dx\\ &=2 \int \frac {4 x^2 (36+7 x)-8 x^2 (54+7 x) \log (x)-(36+7 x)^2 \log ^2(x)}{\left (4 x^3+x (36+7 x) \log (x)\right )^2} \, dx\\ &=2 \int \left (-\frac {1}{x^2}+\frac {4 \left (1296+504 x+337 x^2+28 x^3\right )}{(36+7 x) \left (4 x^2+36 \log (x)+7 x \log (x)\right )^2}-\frac {144}{(36+7 x) \left (4 x^2+36 \log (x)+7 x \log (x)\right )}\right ) \, dx\\ &=\frac {2}{x}+8 \int \frac {1296+504 x+337 x^2+28 x^3}{(36+7 x) \left (4 x^2+36 \log (x)+7 x \log (x)\right )^2} \, dx-288 \int \frac {1}{(36+7 x) \left (4 x^2+36 \log (x)+7 x \log (x)\right )} \, dx\\ &=\frac {2}{x}+8 \int \left (-\frac {3420}{49 \left (4 x^2+36 \log (x)+7 x \log (x)\right )^2}+\frac {193 x}{7 \left (4 x^2+36 \log (x)+7 x \log (x)\right )^2}+\frac {4 x^2}{\left (4 x^2+36 \log (x)+7 x \log (x)\right )^2}+\frac {186624}{49 (36+7 x) \left (4 x^2+36 \log (x)+7 x \log (x)\right )^2}\right ) \, dx-288 \int \frac {1}{(36+7 x) \left (4 x^2+36 \log (x)+7 x \log (x)\right )} \, dx\\ &=\frac {2}{x}+32 \int \frac {x^2}{\left (4 x^2+36 \log (x)+7 x \log (x)\right )^2} \, dx+\frac {1544}{7} \int \frac {x}{\left (4 x^2+36 \log (x)+7 x \log (x)\right )^2} \, dx-288 \int \frac {1}{(36+7 x) \left (4 x^2+36 \log (x)+7 x \log (x)\right )} \, dx-\frac {27360}{49} \int \frac {1}{\left (4 x^2+36 \log (x)+7 x \log (x)\right )^2} \, dx+\frac {1492992}{49} \int \frac {1}{(36+7 x) \left (4 x^2+36 \log (x)+7 x \log (x)\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.21, size = 28, normalized size = 1.08 \begin {gather*} -2 \left (-\frac {1}{x}+\frac {4 x}{4 x^2+36 \log (x)+7 x \log (x)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(288*x^2 + 56*x^3 + (-864*x^2 - 112*x^3)*Log[x] + (-2592 - 1008*x - 98*x^2)*Log[x]^2)/(16*x^6 + (288
*x^4 + 56*x^5)*Log[x] + (1296*x^2 + 504*x^3 + 49*x^4)*Log[x]^2),x]

[Out]

-2*(-x^(-1) + (4*x)/(4*x^2 + 36*Log[x] + 7*x*Log[x]))

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fricas [A]  time = 0.72, size = 29, normalized size = 1.12 \begin {gather*} \frac {2 \, {\left (7 \, x + 36\right )} \log \relax (x)}{4 \, x^{3} + {\left (7 \, x^{2} + 36 \, x\right )} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-98*x^2-1008*x-2592)*log(x)^2+(-112*x^3-864*x^2)*log(x)+56*x^3+288*x^2)/((49*x^4+504*x^3+1296*x^2)
*log(x)^2+(56*x^5+288*x^4)*log(x)+16*x^6),x, algorithm="fricas")

[Out]

2*(7*x + 36)*log(x)/(4*x^3 + (7*x^2 + 36*x)*log(x))

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giac [A]  time = 0.45, size = 26, normalized size = 1.00 \begin {gather*} -\frac {8 \, x}{4 \, x^{2} + 7 \, x \log \relax (x) + 36 \, \log \relax (x)} + \frac {2}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-98*x^2-1008*x-2592)*log(x)^2+(-112*x^3-864*x^2)*log(x)+56*x^3+288*x^2)/((49*x^4+504*x^3+1296*x^2)
*log(x)^2+(56*x^5+288*x^4)*log(x)+16*x^6),x, algorithm="giac")

[Out]

-8*x/(4*x^2 + 7*x*log(x) + 36*log(x)) + 2/x

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maple [A]  time = 0.05, size = 27, normalized size = 1.04

method	result	size
risch	\(\frac {2}{x}-\frac {8 x}{7 x \ln \relax (x )+4 x^{2}+36 \ln \relax (x )}\)	\(27\)
norman	\(\frac {14 x \ln \relax (x )+72 \ln \relax (x )}{x \left (7 x \ln \relax (x )+4 x^{2}+36 \ln \relax (x )\right )}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-98*x^2-1008*x-2592)*ln(x)^2+(-112*x^3-864*x^2)*ln(x)+56*x^3+288*x^2)/((49*x^4+504*x^3+1296*x^2)*ln(x)^2
+(56*x^5+288*x^4)*ln(x)+16*x^6),x,method=_RETURNVERBOSE)

[Out]

2/x-8*x/(7*x*ln(x)+4*x^2+36*ln(x))

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maxima [A]  time = 0.48, size = 29, normalized size = 1.12 \begin {gather*} \frac {2 \, {\left (7 \, x + 36\right )} \log \relax (x)}{4 \, x^{3} + {\left (7 \, x^{2} + 36 \, x\right )} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-98*x^2-1008*x-2592)*log(x)^2+(-112*x^3-864*x^2)*log(x)+56*x^3+288*x^2)/((49*x^4+504*x^3+1296*x^2)
*log(x)^2+(56*x^5+288*x^4)*log(x)+16*x^6),x, algorithm="maxima")

[Out]

2*(7*x + 36)*log(x)/(4*x^3 + (7*x^2 + 36*x)*log(x))

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mupad [B]  time = 1.40, size = 29, normalized size = 1.12 \begin {gather*} \frac {2\,\ln \relax (x)\,\left (7\,x+36\right )}{x\,\left (36\,\ln \relax (x)+7\,x\,\ln \relax (x)+4\,x^2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x)*(864*x^2 + 112*x^3) + log(x)^2*(1008*x + 98*x^2 + 2592) - 288*x^2 - 56*x^3)/(log(x)*(288*x^4 + 56
*x^5) + log(x)^2*(1296*x^2 + 504*x^3 + 49*x^4) + 16*x^6),x)

[Out]

(2*log(x)*(7*x + 36))/(x*(36*log(x) + 7*x*log(x) + 4*x^2))

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sympy [A]  time = 0.18, size = 19, normalized size = 0.73 \begin {gather*} - \frac {8 x}{4 x^{2} + \left (7 x + 36\right ) \log {\relax (x )}} + \frac {2}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-98*x**2-1008*x-2592)*ln(x)**2+(-112*x**3-864*x**2)*ln(x)+56*x**3+288*x**2)/((49*x**4+504*x**3+129
6*x**2)*ln(x)**2+(56*x**5+288*x**4)*ln(x)+16*x**6),x)

[Out]

-8*x/(4*x**2 + (7*x + 36)*log(x)) + 2/x

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