3.43.23 \(\int \frac {15 x^2-3 x^3+(45 x-45 x^2+10 x^3) \log (\frac {-3+2 x}{-3+x})}{(-2250+3600 x-2120 x^2+588 x^3-78 x^4+4 x^5) \log ^3(\frac {-3+2 x}{-3+x})} \, dx\)

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

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Rubi [F]  time = 0.64, 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 {15 x^2-3 x^3+\left (45 x-45 x^2+10 x^3\right ) \log \left (\frac {-3+2 x}{-3+x}\right )}{\left (-2250+3600 x-2120 x^2+588 x^3-78 x^4+4 x^5\right ) \log ^3\left (\frac {-3+2 x}{-3+x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(15*x^2 - 3*x^3 + (45*x - 45*x^2 + 10*x^3)*Log[(-3 + 2*x)/(-3 + x)])/((-2250 + 3600*x - 2120*x^2 + 588*x^3
- 78*x^4 + 4*x^5)*Log[(-3 + 2*x)/(-3 + x)]^3),x]

[Out]

(-3*Defer[Int][x^2/((-5 + x)^2*(-3 + x)*(-3 + 2*x)*Log[(-3 + 2*x)/(-3 + x)]^3), x])/2 + (5*Defer[Int][x/((-5 +
x)^3*Log[(-3 + 2*x)/(-3 + x)]^2), x])/2

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x \left (\frac {3 (-5+x) x}{9-9 x+2 x^2}-5 \log \left (\frac {-3+2 x}{-3+x}\right )\right )}{2 (5-x)^3 \log ^3\left (\frac {-3+2 x}{-3+x}\right )} \, dx\\ &=\frac {1}{2} \int \frac {x \left (\frac {3 (-5+x) x}{9-9 x+2 x^2}-5 \log \left (\frac {-3+2 x}{-3+x}\right )\right )}{(5-x)^3 \log ^3\left (\frac {-3+2 x}{-3+x}\right )} \, dx\\ &=\frac {1}{2} \int \left (-\frac {3 x^2}{(-5+x)^2 (-3+x) (-3+2 x) \log ^3\left (\frac {-3+2 x}{-3+x}\right )}+\frac {5 x}{(-5+x)^3 \log ^2\left (\frac {-3+2 x}{-3+x}\right )}\right ) \, dx\\ &=-\left (\frac {3}{2} \int \frac {x^2}{(-5+x)^2 (-3+x) (-3+2 x) \log ^3\left (\frac {-3+2 x}{-3+x}\right )} \, dx\right )+\frac {5}{2} \int \frac {x}{(-5+x)^3 \log ^2\left (\frac {-3+2 x}{-3+x}\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.19, size = 26, normalized size = 0.84 \begin {gather*} -\frac {x^2}{4 (-5+x)^2 \log ^2\left (\frac {-3+2 x}{-3+x}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(15*x^2 - 3*x^3 + (45*x - 45*x^2 + 10*x^3)*Log[(-3 + 2*x)/(-3 + x)])/((-2250 + 3600*x - 2120*x^2 + 5
88*x^3 - 78*x^4 + 4*x^5)*Log[(-3 + 2*x)/(-3 + x)]^3),x]

[Out]

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

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fricas [A]  time = 0.64, size = 29, normalized size = 0.94 \begin {gather*} -\frac {x^{2}}{4 \, {\left (x^{2} - 10 \, x + 25\right )} \log \left (\frac {2 \, x - 3}{x - 3}\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x^3-45*x^2+45*x)*log((2*x-3)/(x-3))-3*x^3+15*x^2)/(4*x^5-78*x^4+588*x^3-2120*x^2+3600*x-2250)/l
og((2*x-3)/(x-3))^3,x, algorithm="fricas")

[Out]

-1/4*x^2/((x^2 - 10*x + 25)*log((2*x - 3)/(x - 3))^2)

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giac [B]  time = 0.23, size = 102, normalized size = 3.29 \begin {gather*} -\frac {9 \, {\left (\frac {{\left (2 \, x - 3\right )}^{2}}{{\left (x - 3\right )}^{2}} - \frac {2 \, {\left (2 \, x - 3\right )}}{x - 3} + 1\right )}}{4 \, {\left (\frac {4 \, {\left (2 \, x - 3\right )}^{2} \log \left (\frac {2 \, x - 3}{x - 3}\right )^{2}}{{\left (x - 3\right )}^{2}} - \frac {28 \, {\left (2 \, x - 3\right )} \log \left (\frac {2 \, x - 3}{x - 3}\right )^{2}}{x - 3} + 49 \, \log \left (\frac {2 \, x - 3}{x - 3}\right )^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x^3-45*x^2+45*x)*log((2*x-3)/(x-3))-3*x^3+15*x^2)/(4*x^5-78*x^4+588*x^3-2120*x^2+3600*x-2250)/l
og((2*x-3)/(x-3))^3,x, algorithm="giac")

[Out]

-9/4*((2*x - 3)^2/(x - 3)^2 - 2*(2*x - 3)/(x - 3) + 1)/(4*(2*x - 3)^2*log((2*x - 3)/(x - 3))^2/(x - 3)^2 - 28*
(2*x - 3)*log((2*x - 3)/(x - 3))^2/(x - 3) + 49*log((2*x - 3)/(x - 3))^2)

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maple [A]  time = 0.29, size = 25, normalized size = 0.81

method	result	size
norman	\(-\frac {x^{2}}{4 \left (x -5\right )^{2} \ln \left (\frac {2 x -3}{x -3}\right )^{2}}\)	\(25\)
risch	\(-\frac {x^{2}}{4 \left (x^{2}-10 x +25\right ) \ln \left (\frac {2 x -3}{x -3}\right )^{2}}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((10*x^3-45*x^2+45*x)*ln((2*x-3)/(x-3))-3*x^3+15*x^2)/(4*x^5-78*x^4+588*x^3-2120*x^2+3600*x-2250)/ln((2*x-
3)/(x-3))^3,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [B]  time = 0.39, size = 60, normalized size = 1.94 \begin {gather*} -\frac {x^{2}}{4 \, {\left ({\left (x^{2} - 10 \, x + 25\right )} \log \left (2 \, x - 3\right )^{2} - 2 \, {\left (x^{2} - 10 \, x + 25\right )} \log \left (2 \, x - 3\right ) \log \left (x - 3\right ) + {\left (x^{2} - 10 \, x + 25\right )} \log \left (x - 3\right )^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x^3-45*x^2+45*x)*log((2*x-3)/(x-3))-3*x^3+15*x^2)/(4*x^5-78*x^4+588*x^3-2120*x^2+3600*x-2250)/l
og((2*x-3)/(x-3))^3,x, algorithm="maxima")

[Out]

-1/4*x^2/((x^2 - 10*x + 25)*log(2*x - 3)^2 - 2*(x^2 - 10*x + 25)*log(2*x - 3)*log(x - 3) + (x^2 - 10*x + 25)*l
og(x - 3)^2)

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mupad [B]  time = 0.37, size = 50, normalized size = 1.61 \begin {gather*} \frac {875}{6\,{\left (x-5\right )}^2}-\frac {175\,x}{3\,{\left (x-5\right )}^2}+\frac {35\,x^2}{6\,{\left (x-5\right )}^2}-\frac {x^2}{4\,{\ln \left (\frac {2\,x-3}{x-3}\right )}^2\,{\left (x-5\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log((2*x - 3)/(x - 3))*(45*x - 45*x^2 + 10*x^3) + 15*x^2 - 3*x^3)/(log((2*x - 3)/(x - 3))^3*(3600*x - 212
0*x^2 + 588*x^3 - 78*x^4 + 4*x^5 - 2250)),x)

[Out]

875/(6*(x - 5)^2) - (175*x)/(3*(x - 5)^2) + (35*x^2)/(6*(x - 5)^2) - x^2/(4*log((2*x - 3)/(x - 3))^2*(x - 5)^2
)

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sympy [A]  time = 0.18, size = 26, normalized size = 0.84 \begin {gather*} - \frac {x^{2}}{\left (4 x^{2} - 40 x + 100\right ) \log {\left (\frac {2 x - 3}{x - 3} \right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x**3-45*x**2+45*x)*ln((2*x-3)/(x-3))-3*x**3+15*x**2)/(4*x**5-78*x**4+588*x**3-2120*x**2+3600*x-
2250)/ln((2*x-3)/(x-3))**3,x)

[Out]

-x**2/((4*x**2 - 40*x + 100)*log((2*x - 3)/(x - 3))**2)

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