3.71.88 \(\int \frac {-7+6 x+6 x^2-6 x^3+x^4+(6 x+28 x^2-4 x^3-2 x^4) \log (\frac {2+10 x+2 x^2}{x})}{(x+5 x^2+x^3) \log ^2(\frac {2+10 x+2 x^2}{x})} \, dx\)

Optimal. Leaf size=31 \[ \frac {5+(3-x) (-4+x)-x}{\log \left (\frac {2 (1+x+x (4+x))}{x}\right )} \]

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Rubi [F]  time = 1.59, 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+6 x+6 x^2-6 x^3+x^4+\left (6 x+28 x^2-4 x^3-2 x^4\right ) \log \left (\frac {2+10 x+2 x^2}{x}\right )}{\left (x+5 x^2+x^3\right ) \log ^2\left (\frac {2+10 x+2 x^2}{x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-7 + 6*x + 6*x^2 - 6*x^3 + x^4 + (6*x + 28*x^2 - 4*x^3 - 2*x^4)*Log[(2 + 10*x + 2*x^2)/x])/((x + 5*x^2 +
x^3)*Log[(2 + 10*x + 2*x^2)/x]^2),x]

[Out]

-11*Defer[Int][Log[2*(5 + x^(-1) + x)]^(-2), x] - (104*Defer[Int][1/((-5 + Sqrt[21] - 2*x)*Log[2*(5 + x^(-1) +
x)]^2), x])/Sqrt[21] - 7*Defer[Int][1/(x*Log[2*(5 + x^(-1) + x)]^2), x] + Defer[Int][x/Log[2*(5 + x^(-1) + x)
]^2, x] + (67*(21 - 5*Sqrt[21])*Defer[Int][1/((5 - Sqrt[21] + 2*x)*Log[2*(5 + x^(-1) + x)]^2), x])/21 - (104*D
efer[Int][1/((5 + Sqrt[21] + 2*x)*Log[2*(5 + x^(-1) + x)]^2), x])/Sqrt[21] + (67*(21 + 5*Sqrt[21])*Defer[Int][
1/((5 + Sqrt[21] + 2*x)*Log[2*(5 + x^(-1) + x)]^2), x])/21 + 6*Defer[Int][Log[2*(5 + x^(-1) + x)]^(-1), x] - 2
*Defer[Int][x/Log[2*(5 + x^(-1) + x)], x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-7+6 x+6 x^2-6 x^3+x^4+\left (6 x+28 x^2-4 x^3-2 x^4\right ) \log \left (\frac {2+10 x+2 x^2}{x}\right )}{x \left (1+5 x+x^2\right ) \log ^2\left (\frac {2+10 x+2 x^2}{x}\right )} \, dx\\ &=\int \left (\frac {-7+6 x+6 x^2-6 x^3+x^4}{x \left (1+5 x+x^2\right ) \log ^2\left (2 \left (5+\frac {1}{x}+x\right )\right )}-\frac {2 (-3+x)}{\log \left (2 \left (5+\frac {1}{x}+x\right )\right )}\right ) \, dx\\ &=-\left (2 \int \frac {-3+x}{\log \left (2 \left (5+\frac {1}{x}+x\right )\right )} \, dx\right )+\int \frac {-7+6 x+6 x^2-6 x^3+x^4}{x \left (1+5 x+x^2\right ) \log ^2\left (2 \left (5+\frac {1}{x}+x\right )\right )} \, dx\\ &=-\left (2 \int \left (-\frac {3}{\log \left (2 \left (5+\frac {1}{x}+x\right )\right )}+\frac {x}{\log \left (2 \left (5+\frac {1}{x}+x\right )\right )}\right ) \, dx\right )+\int \left (-\frac {11}{\log ^2\left (2 \left (5+\frac {1}{x}+x\right )\right )}-\frac {7}{x \log ^2\left (2 \left (5+\frac {1}{x}+x\right )\right )}+\frac {x}{\log ^2\left (2 \left (5+\frac {1}{x}+x\right )\right )}+\frac {52+67 x}{\left (1+5 x+x^2\right ) \log ^2\left (2 \left (5+\frac {1}{x}+x\right )\right )}\right ) \, dx\\ &=-\left (2 \int \frac {x}{\log \left (2 \left (5+\frac {1}{x}+x\right )\right )} \, dx\right )+6 \int \frac {1}{\log \left (2 \left (5+\frac {1}{x}+x\right )\right )} \, dx-7 \int \frac {1}{x \log ^2\left (2 \left (5+\frac {1}{x}+x\right )\right )} \, dx-11 \int \frac {1}{\log ^2\left (2 \left (5+\frac {1}{x}+x\right )\right )} \, dx+\int \frac {x}{\log ^2\left (2 \left (5+\frac {1}{x}+x\right )\right )} \, dx+\int \frac {52+67 x}{\left (1+5 x+x^2\right ) \log ^2\left (2 \left (5+\frac {1}{x}+x\right )\right )} \, dx\\ &=-\left (2 \int \frac {x}{\log \left (2 \left (5+\frac {1}{x}+x\right )\right )} \, dx\right )+6 \int \frac {1}{\log \left (2 \left (5+\frac {1}{x}+x\right )\right )} \, dx-7 \int \frac {1}{x \log ^2\left (2 \left (5+\frac {1}{x}+x\right )\right )} \, dx-11 \int \frac {1}{\log ^2\left (2 \left (5+\frac {1}{x}+x\right )\right )} \, dx+\int \left (\frac {52}{\left (1+5 x+x^2\right ) \log ^2\left (2 \left (5+\frac {1}{x}+x\right )\right )}+\frac {67 x}{\left (1+5 x+x^2\right ) \log ^2\left (2 \left (5+\frac {1}{x}+x\right )\right )}\right ) \, dx+\int \frac {x}{\log ^2\left (2 \left (5+\frac {1}{x}+x\right )\right )} \, dx\\ &=-\left (2 \int \frac {x}{\log \left (2 \left (5+\frac {1}{x}+x\right )\right )} \, dx\right )+6 \int \frac {1}{\log \left (2 \left (5+\frac {1}{x}+x\right )\right )} \, dx-7 \int \frac {1}{x \log ^2\left (2 \left (5+\frac {1}{x}+x\right )\right )} \, dx-11 \int \frac {1}{\log ^2\left (2 \left (5+\frac {1}{x}+x\right )\right )} \, dx+52 \int \frac {1}{\left (1+5 x+x^2\right ) \log ^2\left (2 \left (5+\frac {1}{x}+x\right )\right )} \, dx+67 \int \frac {x}{\left (1+5 x+x^2\right ) \log ^2\left (2 \left (5+\frac {1}{x}+x\right )\right )} \, dx+\int \frac {x}{\log ^2\left (2 \left (5+\frac {1}{x}+x\right )\right )} \, dx\\ &=-\left (2 \int \frac {x}{\log \left (2 \left (5+\frac {1}{x}+x\right )\right )} \, dx\right )+6 \int \frac {1}{\log \left (2 \left (5+\frac {1}{x}+x\right )\right )} \, dx-7 \int \frac {1}{x \log ^2\left (2 \left (5+\frac {1}{x}+x\right )\right )} \, dx-11 \int \frac {1}{\log ^2\left (2 \left (5+\frac {1}{x}+x\right )\right )} \, dx+52 \int \left (-\frac {2}{\sqrt {21} \left (-5+\sqrt {21}-2 x\right ) \log ^2\left (2 \left (5+\frac {1}{x}+x\right )\right )}-\frac {2}{\sqrt {21} \left (5+\sqrt {21}+2 x\right ) \log ^2\left (2 \left (5+\frac {1}{x}+x\right )\right )}\right ) \, dx+67 \int \left (\frac {1-\frac {5}{\sqrt {21}}}{\left (5-\sqrt {21}+2 x\right ) \log ^2\left (2 \left (5+\frac {1}{x}+x\right )\right )}+\frac {1+\frac {5}{\sqrt {21}}}{\left (5+\sqrt {21}+2 x\right ) \log ^2\left (2 \left (5+\frac {1}{x}+x\right )\right )}\right ) \, dx+\int \frac {x}{\log ^2\left (2 \left (5+\frac {1}{x}+x\right )\right )} \, dx\\ &=-\left (2 \int \frac {x}{\log \left (2 \left (5+\frac {1}{x}+x\right )\right )} \, dx\right )+6 \int \frac {1}{\log \left (2 \left (5+\frac {1}{x}+x\right )\right )} \, dx-7 \int \frac {1}{x \log ^2\left (2 \left (5+\frac {1}{x}+x\right )\right )} \, dx-11 \int \frac {1}{\log ^2\left (2 \left (5+\frac {1}{x}+x\right )\right )} \, dx-\frac {104 \int \frac {1}{\left (-5+\sqrt {21}-2 x\right ) \log ^2\left (2 \left (5+\frac {1}{x}+x\right )\right )} \, dx}{\sqrt {21}}-\frac {104 \int \frac {1}{\left (5+\sqrt {21}+2 x\right ) \log ^2\left (2 \left (5+\frac {1}{x}+x\right )\right )} \, dx}{\sqrt {21}}+\frac {1}{21} \left (67 \left (21-5 \sqrt {21}\right )\right ) \int \frac {1}{\left (5-\sqrt {21}+2 x\right ) \log ^2\left (2 \left (5+\frac {1}{x}+x\right )\right )} \, dx+\frac {1}{21} \left (67 \left (21+5 \sqrt {21}\right )\right ) \int \frac {1}{\left (5+\sqrt {21}+2 x\right ) \log ^2\left (2 \left (5+\frac {1}{x}+x\right )\right )} \, dx+\int \frac {x}{\log ^2\left (2 \left (5+\frac {1}{x}+x\right )\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.49, size = 22, normalized size = 0.71 \begin {gather*} \frac {-7+6 x-x^2}{\log \left (2 \left (5+\frac {1}{x}+x\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-7 + 6*x + 6*x^2 - 6*x^3 + x^4 + (6*x + 28*x^2 - 4*x^3 - 2*x^4)*Log[(2 + 10*x + 2*x^2)/x])/((x + 5*
x^2 + x^3)*Log[(2 + 10*x + 2*x^2)/x]^2),x]

[Out]

(-7 + 6*x - x^2)/Log[2*(5 + x^(-1) + x)]

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fricas [A]  time = 0.85, size = 26, normalized size = 0.84 \begin {gather*} -\frac {x^{2} - 6 \, x + 7}{\log \left (\frac {2 \, {\left (x^{2} + 5 \, x + 1\right )}}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^4-4*x^3+28*x^2+6*x)*log((2*x^2+10*x+2)/x)+x^4-6*x^3+6*x^2+6*x-7)/(x^3+5*x^2+x)/log((2*x^2+10*
x+2)/x)^2,x, algorithm="fricas")

[Out]

-(x^2 - 6*x + 7)/log(2*(x^2 + 5*x + 1)/x)

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giac [A]  time = 0.24, size = 26, normalized size = 0.84 \begin {gather*} -\frac {x^{2} - 6 \, x + 7}{\log \left (\frac {2 \, {\left (x^{2} + 5 \, x + 1\right )}}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^4-4*x^3+28*x^2+6*x)*log((2*x^2+10*x+2)/x)+x^4-6*x^3+6*x^2+6*x-7)/(x^3+5*x^2+x)/log((2*x^2+10*
x+2)/x)^2,x, algorithm="giac")

[Out]

-(x^2 - 6*x + 7)/log(2*(x^2 + 5*x + 1)/x)

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maple [A]  time = 0.04, size = 28, normalized size = 0.90

method	result	size
risch	\(-\frac {x^{2}-6 x +7}{\ln \left (\frac {2 x^{2}+10 x +2}{x}\right )}\)	\(28\)
norman	\(\frac {-x^{2}+6 x -7}{\ln \left (\frac {2 x^{2}+10 x +2}{x}\right )}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x^4-4*x^3+28*x^2+6*x)*ln((2*x^2+10*x+2)/x)+x^4-6*x^3+6*x^2+6*x-7)/(x^3+5*x^2+x)/ln((2*x^2+10*x+2)/x)^
2,x,method=_RETURNVERBOSE)

[Out]

-(x^2-6*x+7)/ln((2*x^2+10*x+2)/x)

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maxima [A]  time = 0.50, size = 28, normalized size = 0.90 \begin {gather*} -\frac {x^{2} - 6 \, x + 7}{\log \relax (2) + \log \left (x^{2} + 5 \, x + 1\right ) - \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^4-4*x^3+28*x^2+6*x)*log((2*x^2+10*x+2)/x)+x^4-6*x^3+6*x^2+6*x-7)/(x^3+5*x^2+x)/log((2*x^2+10*
x+2)/x)^2,x, algorithm="maxima")

[Out]

-(x^2 - 6*x + 7)/(log(2) + log(x^2 + 5*x + 1) - log(x))

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mupad [B]  time = 4.68, size = 27, normalized size = 0.87 \begin {gather*} -\frac {x^2-6\,x+7}{\ln \left (\frac {2\,x^2+10\,x+2}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((6*x + log((10*x + 2*x^2 + 2)/x)*(6*x + 28*x^2 - 4*x^3 - 2*x^4) + 6*x^2 - 6*x^3 + x^4 - 7)/(log((10*x + 2*
x^2 + 2)/x)^2*(x + 5*x^2 + x^3)),x)

[Out]

-(x^2 - 6*x + 7)/log((10*x + 2*x^2 + 2)/x)

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sympy [A]  time = 0.18, size = 20, normalized size = 0.65 \begin {gather*} \frac {- x^{2} + 6 x - 7}{\log {\left (\frac {2 x^{2} + 10 x + 2}{x} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x**4-4*x**3+28*x**2+6*x)*ln((2*x**2+10*x+2)/x)+x**4-6*x**3+6*x**2+6*x-7)/(x**3+5*x**2+x)/ln((2*
x**2+10*x+2)/x)**2,x)

[Out]

(-x**2 + 6*x - 7)/log((2*x**2 + 10*x + 2)/x)

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