3.42.1 \(\int \frac {-40 x-90 x^2-20 x^3+(60+220 x-50 x^2-10 x^3) \log (9-6 x+x^2)+(-93 x^2+31 x^3) \log ^2(9-6 x+x^2)}{(-3 x^2-11 x^3-8 x^4+4 x^5) \log ^2(9-6 x+x^2)} \, dx\)

Optimal. Leaf size=29 \[ \frac {x+5 \left (-3+\frac {4+x}{x \log \left ((-3+x)^2\right )}\right )}{1+2 x} \]

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Rubi [F]  time = 1.94, 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 {-40 x-90 x^2-20 x^3+\left (60+220 x-50 x^2-10 x^3\right ) \log \left (9-6 x+x^2\right )+\left (-93 x^2+31 x^3\right ) \log ^2\left (9-6 x+x^2\right )}{\left (-3 x^2-11 x^3-8 x^4+4 x^5\right ) \log ^2\left (9-6 x+x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-40*x - 90*x^2 - 20*x^3 + (60 + 220*x - 50*x^2 - 10*x^3)*Log[9 - 6*x + x^2] + (-93*x^2 + 31*x^3)*Log[9 -
6*x + x^2]^2)/((-3*x^2 - 11*x^3 - 8*x^4 + 4*x^5)*Log[9 - 6*x + x^2]^2),x]

[Out]

-31/(2*(1 + 2*x)) + 5/(3*Log[(-3 + x)^2]) + (40*Defer[Int][1/(x*Log[(-3 + x)^2]^2), x])/3 - 20*Defer[Int][1/((
1 + 2*x)*Log[(-3 + x)^2]^2), x] - 20*Defer[Int][1/(x^2*Log[(-3 + x)^2]), x] + 70*Defer[Int][1/((1 + 2*x)^2*Log
[(-3 + x)^2]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {40 x+90 x^2+20 x^3-\left (60+220 x-50 x^2-10 x^3\right ) \log \left (9-6 x+x^2\right )-\left (-93 x^2+31 x^3\right ) \log ^2\left (9-6 x+x^2\right )}{(3-x) x^2 (1+2 x)^2 \log ^2\left (9-6 x+x^2\right )} \, dx\\ &=\int \left (\frac {31}{(1+2 x)^2}-\frac {10 (4+x)}{(-3+x) x (1+2 x) \log ^2\left ((-3+x)^2\right )}-\frac {10 \left (2+8 x+x^2\right )}{x^2 (1+2 x)^2 \log \left ((-3+x)^2\right )}\right ) \, dx\\ &=-\frac {31}{2 (1+2 x)}-10 \int \frac {4+x}{(-3+x) x (1+2 x) \log ^2\left ((-3+x)^2\right )} \, dx-10 \int \frac {2+8 x+x^2}{x^2 (1+2 x)^2 \log \left ((-3+x)^2\right )} \, dx\\ &=-\frac {31}{2 (1+2 x)}-10 \int \left (\frac {1}{3 (-3+x) \log ^2\left ((-3+x)^2\right )}-\frac {4}{3 x \log ^2\left ((-3+x)^2\right )}+\frac {2}{(1+2 x) \log ^2\left ((-3+x)^2\right )}\right ) \, dx-10 \int \left (\frac {2}{x^2 \log \left ((-3+x)^2\right )}-\frac {7}{(1+2 x)^2 \log \left ((-3+x)^2\right )}\right ) \, dx\\ &=-\frac {31}{2 (1+2 x)}-\frac {10}{3} \int \frac {1}{(-3+x) \log ^2\left ((-3+x)^2\right )} \, dx+\frac {40}{3} \int \frac {1}{x \log ^2\left ((-3+x)^2\right )} \, dx-20 \int \frac {1}{(1+2 x) \log ^2\left ((-3+x)^2\right )} \, dx-20 \int \frac {1}{x^2 \log \left ((-3+x)^2\right )} \, dx+70 \int \frac {1}{(1+2 x)^2 \log \left ((-3+x)^2\right )} \, dx\\ &=-\frac {31}{2 (1+2 x)}-\frac {10}{3} \operatorname {Subst}\left (\int \frac {1}{x \log ^2\left (x^2\right )} \, dx,x,-3+x\right )+\frac {40}{3} \int \frac {1}{x \log ^2\left ((-3+x)^2\right )} \, dx-20 \int \frac {1}{(1+2 x) \log ^2\left ((-3+x)^2\right )} \, dx-20 \int \frac {1}{x^2 \log \left ((-3+x)^2\right )} \, dx+70 \int \frac {1}{(1+2 x)^2 \log \left ((-3+x)^2\right )} \, dx\\ &=-\frac {31}{2 (1+2 x)}-\frac {5}{3} \operatorname {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log \left ((-3+x)^2\right )\right )+\frac {40}{3} \int \frac {1}{x \log ^2\left ((-3+x)^2\right )} \, dx-20 \int \frac {1}{(1+2 x) \log ^2\left ((-3+x)^2\right )} \, dx-20 \int \frac {1}{x^2 \log \left ((-3+x)^2\right )} \, dx+70 \int \frac {1}{(1+2 x)^2 \log \left ((-3+x)^2\right )} \, dx\\ &=-\frac {31}{2 (1+2 x)}+\frac {5}{3 \log \left ((-3+x)^2\right )}+\frac {40}{3} \int \frac {1}{x \log ^2\left ((-3+x)^2\right )} \, dx-20 \int \frac {1}{(1+2 x) \log ^2\left ((-3+x)^2\right )} \, dx-20 \int \frac {1}{x^2 \log \left ((-3+x)^2\right )} \, dx+70 \int \frac {1}{(1+2 x)^2 \log \left ((-3+x)^2\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.18, size = 29, normalized size = 1.00 \begin {gather*} \frac {-31+\frac {10 (4+x)}{x \log \left ((-3+x)^2\right )}}{2 (1+2 x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-40*x - 90*x^2 - 20*x^3 + (60 + 220*x - 50*x^2 - 10*x^3)*Log[9 - 6*x + x^2] + (-93*x^2 + 31*x^3)*Lo
g[9 - 6*x + x^2]^2)/((-3*x^2 - 11*x^3 - 8*x^4 + 4*x^5)*Log[9 - 6*x + x^2]^2),x]

[Out]

(-31 + (10*(4 + x))/(x*Log[(-3 + x)^2]))/(2*(1 + 2*x))

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fricas [A]  time = 0.62, size = 39, normalized size = 1.34 \begin {gather*} -\frac {31 \, x \log \left (x^{2} - 6 \, x + 9\right ) - 10 \, x - 40}{2 \, {\left (2 \, x^{2} + x\right )} \log \left (x^{2} - 6 \, x + 9\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((31*x^3-93*x^2)*log(x^2-6*x+9)^2+(-10*x^3-50*x^2+220*x+60)*log(x^2-6*x+9)-20*x^3-90*x^2-40*x)/(4*x^
5-8*x^4-11*x^3-3*x^2)/log(x^2-6*x+9)^2,x, algorithm="fricas")

[Out]

-1/2*(31*x*log(x^2 - 6*x + 9) - 10*x - 40)/((2*x^2 + x)*log(x^2 - 6*x + 9))

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giac [A]  time = 0.29, size = 43, normalized size = 1.48 \begin {gather*} \frac {5 \, {\left (x + 4\right )}}{2 \, x^{2} \log \left (x^{2} - 6 \, x + 9\right ) + x \log \left (x^{2} - 6 \, x + 9\right )} - \frac {31}{2 \, {\left (2 \, x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((31*x^3-93*x^2)*log(x^2-6*x+9)^2+(-10*x^3-50*x^2+220*x+60)*log(x^2-6*x+9)-20*x^3-90*x^2-40*x)/(4*x^
5-8*x^4-11*x^3-3*x^2)/log(x^2-6*x+9)^2,x, algorithm="giac")

[Out]

5*(x + 4)/(2*x^2*log(x^2 - 6*x + 9) + x*log(x^2 - 6*x + 9)) - 31/2/(2*x + 1)

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maple [A]  time = 0.06, size = 37, normalized size = 1.28

method	result	size
risch	\(-\frac {31}{2 \left (2 x +1\right )}+\frac {20+5 x}{x \left (2 x +1\right ) \ln \left (x^{2}-6 x +9\right )}\)	\(37\)
norman	\(\frac {20-\frac {31 x \ln \left (x^{2}-6 x +9\right )}{2}+5 x}{x \left (2 x +1\right ) \ln \left (x^{2}-6 x +9\right )}\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((31*x^3-93*x^2)*ln(x^2-6*x+9)^2+(-10*x^3-50*x^2+220*x+60)*ln(x^2-6*x+9)-20*x^3-90*x^2-40*x)/(4*x^5-8*x^4-
11*x^3-3*x^2)/ln(x^2-6*x+9)^2,x,method=_RETURNVERBOSE)

[Out]

-31/2/(2*x+1)+5*(4+x)/(2*x+1)/x/ln(x^2-6*x+9)

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maxima [A]  time = 0.42, size = 29, normalized size = 1.00 \begin {gather*} -\frac {31 \, x \log \left (x - 3\right ) - 5 \, x - 20}{2 \, {\left (2 \, x^{2} + x\right )} \log \left (x - 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((31*x^3-93*x^2)*log(x^2-6*x+9)^2+(-10*x^3-50*x^2+220*x+60)*log(x^2-6*x+9)-20*x^3-90*x^2-40*x)/(4*x^
5-8*x^4-11*x^3-3*x^2)/log(x^2-6*x+9)^2,x, algorithm="maxima")

[Out]

-1/2*(31*x*log(x - 3) - 5*x - 20)/((2*x^2 + x)*log(x - 3))

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mupad [B]  time = 0.30, size = 38, normalized size = 1.31 \begin {gather*} \frac {31\,x}{2\,x+1}+\frac {5\,x+20}{x\,\ln \left (x^2-6\,x+9\right )\,\left (2\,x+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((40*x + log(x^2 - 6*x + 9)^2*(93*x^2 - 31*x^3) - log(x^2 - 6*x + 9)*(220*x - 50*x^2 - 10*x^3 + 60) + 90*x^
2 + 20*x^3)/(log(x^2 - 6*x + 9)^2*(3*x^2 + 11*x^3 + 8*x^4 - 4*x^5)),x)

[Out]

(31*x)/(2*x + 1) + (5*x + 20)/(x*log(x^2 - 6*x + 9)*(2*x + 1))

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sympy [A]  time = 0.18, size = 27, normalized size = 0.93 \begin {gather*} \frac {5 x + 20}{\left (2 x^{2} + x\right ) \log {\left (x^{2} - 6 x + 9 \right )}} - \frac {31}{4 x + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((31*x**3-93*x**2)*ln(x**2-6*x+9)**2+(-10*x**3-50*x**2+220*x+60)*ln(x**2-6*x+9)-20*x**3-90*x**2-40*x
)/(4*x**5-8*x**4-11*x**3-3*x**2)/ln(x**2-6*x+9)**2,x)

[Out]

(5*x + 20)/((2*x**2 + x)*log(x**2 - 6*x + 9)) - 31/(4*x + 2)

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