3.92.58 \(\int \frac {45 x^4-27 x^5+(180 x^3-144 x^4+27 x^5) \log (-2+x)}{(2000-4600 x+3960 x^2-1512 x^3+216 x^4) \log ^3(-2+x)} \, dx\)

Optimal. Leaf size=22 \[ 15+\frac {x^4}{16 \left (-\frac {5}{3}+x\right )^2 \log ^2(-2+x)} \]

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Rubi [F]  time = 0.71, 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 {45 x^4-27 x^5+\left (180 x^3-144 x^4+27 x^5\right ) \log (-2+x)}{\left (2000-4600 x+3960 x^2-1512 x^3+216 x^4\right ) \log ^3(-2+x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(45*x^4 - 27*x^5 + (180*x^3 - 144*x^4 + 27*x^5)*Log[-2 + x])/((2000 - 4600*x + 3960*x^2 - 1512*x^3 + 216*x
^4)*Log[-2 + x]^3),x]

[Out]

9/Log[-2 + x]^2 - (2 - x)/(3*Log[-2 + x]^2) - ((2 - x)*x)/(16*Log[-2 + x]^2) + (625*Defer[Int][1/((-5 + 3*x)^2
*Log[-2 + x]^3), x])/24 + (375*Defer[Int][1/((-5 + 3*x)*Log[-2 + x]^3), x])/8 - (625*Defer[Int][1/((-5 + 3*x)^
3*Log[-2 + x]^2), x])/24 - (125*Defer[Int][1/((-5 + 3*x)^2*Log[-2 + x]^2), x])/12

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {9 x^4}{8 (-2+x) (-5+3 x)^2 \log ^3(-2+x)}+\frac {9 x^3 (-10+3 x)}{8 (-5+3 x)^3 \log ^2(-2+x)}\right ) \, dx\\ &=-\left (\frac {9}{8} \int \frac {x^4}{(-2+x) (-5+3 x)^2 \log ^3(-2+x)} \, dx\right )+\frac {9}{8} \int \frac {x^3 (-10+3 x)}{(-5+3 x)^3 \log ^2(-2+x)} \, dx\\ &=-\left (\frac {9}{8} \int \left (\frac {16}{27 \log ^3(-2+x)}+\frac {16}{(-2+x) \log ^3(-2+x)}+\frac {x}{9 \log ^3(-2+x)}-\frac {625}{27 (-5+3 x)^2 \log ^3(-2+x)}-\frac {125}{3 (-5+3 x) \log ^3(-2+x)}\right ) \, dx\right )+\frac {9}{8} \int \left (\frac {5}{27 \log ^2(-2+x)}+\frac {x}{9 \log ^2(-2+x)}-\frac {625}{27 (-5+3 x)^3 \log ^2(-2+x)}-\frac {250}{27 (-5+3 x)^2 \log ^2(-2+x)}\right ) \, dx\\ &=-\left (\frac {1}{8} \int \frac {x}{\log ^3(-2+x)} \, dx\right )+\frac {1}{8} \int \frac {x}{\log ^2(-2+x)} \, dx+\frac {5}{24} \int \frac {1}{\log ^2(-2+x)} \, dx-\frac {2}{3} \int \frac {1}{\log ^3(-2+x)} \, dx-\frac {125}{12} \int \frac {1}{(-5+3 x)^2 \log ^2(-2+x)} \, dx-18 \int \frac {1}{(-2+x) \log ^3(-2+x)} \, dx+\frac {625}{24} \int \frac {1}{(-5+3 x)^2 \log ^3(-2+x)} \, dx-\frac {625}{24} \int \frac {1}{(-5+3 x)^3 \log ^2(-2+x)} \, dx+\frac {375}{8} \int \frac {1}{(-5+3 x) \log ^3(-2+x)} \, dx\\ &=-\frac {(2-x) x}{16 \log ^2(-2+x)}+\frac {(2-x) x}{8 \log (-2+x)}+\frac {1}{8} \int \frac {1}{\log ^2(-2+x)} \, dx-\frac {1}{8} \int \frac {x}{\log ^2(-2+x)} \, dx+\frac {5}{24} \operatorname {Subst}\left (\int \frac {1}{\log ^2(x)} \, dx,x,-2+x\right )-\frac {1}{4} \int \frac {1}{\log (-2+x)} \, dx+\frac {1}{4} \int \frac {x}{\log (-2+x)} \, dx-\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{\log ^3(x)} \, dx,x,-2+x\right )-\frac {125}{12} \int \frac {1}{(-5+3 x)^2 \log ^2(-2+x)} \, dx-18 \operatorname {Subst}\left (\int \frac {1}{x \log ^3(x)} \, dx,x,-2+x\right )+\frac {625}{24} \int \frac {1}{(-5+3 x)^2 \log ^3(-2+x)} \, dx-\frac {625}{24} \int \frac {1}{(-5+3 x)^3 \log ^2(-2+x)} \, dx+\frac {375}{8} \int \frac {1}{(-5+3 x) \log ^3(-2+x)} \, dx\\ &=-\frac {2-x}{3 \log ^2(-2+x)}-\frac {(2-x) x}{16 \log ^2(-2+x)}+\frac {5 (2-x)}{24 \log (-2+x)}+\frac {1}{8} \operatorname {Subst}\left (\int \frac {1}{\log ^2(x)} \, dx,x,-2+x\right )+\frac {5}{24} \operatorname {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,-2+x\right )+\frac {1}{4} \int \left (\frac {2}{\log (-2+x)}+\frac {-2+x}{\log (-2+x)}\right ) \, dx+\frac {1}{4} \int \frac {1}{\log (-2+x)} \, dx-\frac {1}{4} \int \frac {x}{\log (-2+x)} \, dx-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,-2+x\right )-\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{\log ^2(x)} \, dx,x,-2+x\right )-\frac {125}{12} \int \frac {1}{(-5+3 x)^2 \log ^2(-2+x)} \, dx-18 \operatorname {Subst}\left (\int \frac {1}{x^3} \, dx,x,\log (-2+x)\right )+\frac {625}{24} \int \frac {1}{(-5+3 x)^2 \log ^3(-2+x)} \, dx-\frac {625}{24} \int \frac {1}{(-5+3 x)^3 \log ^2(-2+x)} \, dx+\frac {375}{8} \int \frac {1}{(-5+3 x) \log ^3(-2+x)} \, dx\\ &=\frac {9}{\log ^2(-2+x)}-\frac {2-x}{3 \log ^2(-2+x)}-\frac {(2-x) x}{16 \log ^2(-2+x)}-\frac {\text {li}(-2+x)}{24}+\frac {1}{8} \operatorname {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,-2+x\right )-\frac {1}{4} \int \left (\frac {2}{\log (-2+x)}+\frac {-2+x}{\log (-2+x)}\right ) \, dx+\frac {1}{4} \int \frac {-2+x}{\log (-2+x)} \, dx+\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,-2+x\right )-\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,-2+x\right )+\frac {1}{2} \int \frac {1}{\log (-2+x)} \, dx-\frac {125}{12} \int \frac {1}{(-5+3 x)^2 \log ^2(-2+x)} \, dx+\frac {625}{24} \int \frac {1}{(-5+3 x)^2 \log ^3(-2+x)} \, dx-\frac {625}{24} \int \frac {1}{(-5+3 x)^3 \log ^2(-2+x)} \, dx+\frac {375}{8} \int \frac {1}{(-5+3 x) \log ^3(-2+x)} \, dx\\ &=\frac {9}{\log ^2(-2+x)}-\frac {2-x}{3 \log ^2(-2+x)}-\frac {(2-x) x}{16 \log ^2(-2+x)}-\frac {1}{4} \int \frac {-2+x}{\log (-2+x)} \, dx+\frac {1}{4} \operatorname {Subst}\left (\int \frac {x}{\log (x)} \, dx,x,-2+x\right )-\frac {1}{2} \int \frac {1}{\log (-2+x)} \, dx+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,-2+x\right )-\frac {125}{12} \int \frac {1}{(-5+3 x)^2 \log ^2(-2+x)} \, dx+\frac {625}{24} \int \frac {1}{(-5+3 x)^2 \log ^3(-2+x)} \, dx-\frac {625}{24} \int \frac {1}{(-5+3 x)^3 \log ^2(-2+x)} \, dx+\frac {375}{8} \int \frac {1}{(-5+3 x) \log ^3(-2+x)} \, dx\\ &=\frac {9}{\log ^2(-2+x)}-\frac {2-x}{3 \log ^2(-2+x)}-\frac {(2-x) x}{16 \log ^2(-2+x)}+\frac {\text {li}(-2+x)}{2}+\frac {1}{4} \operatorname {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (-2+x)\right )-\frac {1}{4} \operatorname {Subst}\left (\int \frac {x}{\log (x)} \, dx,x,-2+x\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,-2+x\right )-\frac {125}{12} \int \frac {1}{(-5+3 x)^2 \log ^2(-2+x)} \, dx+\frac {625}{24} \int \frac {1}{(-5+3 x)^2 \log ^3(-2+x)} \, dx-\frac {625}{24} \int \frac {1}{(-5+3 x)^3 \log ^2(-2+x)} \, dx+\frac {375}{8} \int \frac {1}{(-5+3 x) \log ^3(-2+x)} \, dx\\ &=\frac {1}{4} \text {Ei}(2 \log (-2+x))+\frac {9}{\log ^2(-2+x)}-\frac {2-x}{3 \log ^2(-2+x)}-\frac {(2-x) x}{16 \log ^2(-2+x)}-\frac {1}{4} \operatorname {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (-2+x)\right )-\frac {125}{12} \int \frac {1}{(-5+3 x)^2 \log ^2(-2+x)} \, dx+\frac {625}{24} \int \frac {1}{(-5+3 x)^2 \log ^3(-2+x)} \, dx-\frac {625}{24} \int \frac {1}{(-5+3 x)^3 \log ^2(-2+x)} \, dx+\frac {375}{8} \int \frac {1}{(-5+3 x) \log ^3(-2+x)} \, dx\\ &=\frac {9}{\log ^2(-2+x)}-\frac {2-x}{3 \log ^2(-2+x)}-\frac {(2-x) x}{16 \log ^2(-2+x)}-\frac {125}{12} \int \frac {1}{(-5+3 x)^2 \log ^2(-2+x)} \, dx+\frac {625}{24} \int \frac {1}{(-5+3 x)^2 \log ^3(-2+x)} \, dx-\frac {625}{24} \int \frac {1}{(-5+3 x)^3 \log ^2(-2+x)} \, dx+\frac {375}{8} \int \frac {1}{(-5+3 x) \log ^3(-2+x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.32, size = 22, normalized size = 1.00 \begin {gather*} \frac {9 x^4}{16 (1+3 (-2+x))^2 \log ^2(-2+x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(45*x^4 - 27*x^5 + (180*x^3 - 144*x^4 + 27*x^5)*Log[-2 + x])/((2000 - 4600*x + 3960*x^2 - 1512*x^3 +
216*x^4)*Log[-2 + x]^3),x]

[Out]

(9*x^4)/(16*(1 + 3*(-2 + x))^2*Log[-2 + x]^2)

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fricas [A]  time = 0.67, size = 23, normalized size = 1.05 \begin {gather*} \frac {9 \, x^{4}}{16 \, {\left (9 \, x^{2} - 30 \, x + 25\right )} \log \left (x - 2\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((27*x^5-144*x^4+180*x^3)*log(x-2)-27*x^5+45*x^4)/(216*x^4-1512*x^3+3960*x^2-4600*x+2000)/log(x-2)^3
,x, algorithm="fricas")

[Out]

9/16*x^4/((9*x^2 - 30*x + 25)*log(x - 2)^2)

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giac [A]  time = 0.17, size = 36, normalized size = 1.64 \begin {gather*} \frac {9 \, x^{4}}{16 \, {\left (9 \, x^{2} \log \left (x - 2\right )^{2} - 30 \, x \log \left (x - 2\right )^{2} + 25 \, \log \left (x - 2\right )^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((27*x^5-144*x^4+180*x^3)*log(x-2)-27*x^5+45*x^4)/(216*x^4-1512*x^3+3960*x^2-4600*x+2000)/log(x-2)^3
,x, algorithm="giac")

[Out]

9/16*x^4/(9*x^2*log(x - 2)^2 - 30*x*log(x - 2)^2 + 25*log(x - 2)^2)

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maple [A]  time = 0.28, size = 19, normalized size = 0.86

method	result	size
norman	\(\frac {9 x^{4}}{16 \left (3 x -5\right )^{2} \ln \left (x -2\right )^{2}}\)	\(19\)
risch	\(\frac {9 x^{4}}{16 \left (3 x -5\right )^{2} \ln \left (x -2\right )^{2}}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((27*x^5-144*x^4+180*x^3)*ln(x-2)-27*x^5+45*x^4)/(216*x^4-1512*x^3+3960*x^2-4600*x+2000)/ln(x-2)^3,x,metho
d=_RETURNVERBOSE)

[Out]

9/16*x^4/(3*x-5)^2/ln(x-2)^2

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maxima [A]  time = 0.41, size = 23, normalized size = 1.05 \begin {gather*} \frac {9 \, x^{4}}{16 \, {\left (9 \, x^{2} - 30 \, x + 25\right )} \log \left (x - 2\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((27*x^5-144*x^4+180*x^3)*log(x-2)-27*x^5+45*x^4)/(216*x^4-1512*x^3+3960*x^2-4600*x+2000)/log(x-2)^3
,x, algorithm="maxima")

[Out]

9/16*x^4/((9*x^2 - 30*x + 25)*log(x - 2)^2)

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mupad [B]  time = 7.47, size = 158, normalized size = 7.18 \begin {gather*} \frac {\frac {9\,x^4}{16\,{\left (3\,x-5\right )}^2}-\frac {9\,x^3\,\ln \left (x-2\right )\,\left (3\,x-10\right )\,\left (x-2\right )}{16\,{\left (3\,x-5\right )}^3}}{{\ln \left (x-2\right )}^2}-\frac {\frac {9\,\left (10\,x^3-3\,x^4\right )\,\left (x-2\right )}{16\,{\left (3\,x-5\right )}^3}-\frac {9\,\ln \left (x-2\right )\,\left (x-2\right )\,\left (-18\,x^5+123\,x^4-320\,x^3+300\,x^2\right )}{16\,{\left (3\,x-5\right )}^4}}{\ln \left (x-2\right )}-\frac {13\,x}{48}+\frac {x^2}{8}+\frac {\frac {125\,x^3}{216}-\frac {1625\,x^2}{648}+\frac {11125\,x}{3888}-\frac {625}{1944}}{x^4-\frac {20\,x^3}{3}+\frac {50\,x^2}{3}-\frac {500\,x}{27}+\frac {625}{81}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x - 2)*(180*x^3 - 144*x^4 + 27*x^5) + 45*x^4 - 27*x^5)/(log(x - 2)^3*(3960*x^2 - 4600*x - 1512*x^3 +
216*x^4 + 2000)),x)

[Out]

((9*x^4)/(16*(3*x - 5)^2) - (9*x^3*log(x - 2)*(3*x - 10)*(x - 2))/(16*(3*x - 5)^3))/log(x - 2)^2 - ((9*(10*x^3
- 3*x^4)*(x - 2))/(16*(3*x - 5)^3) - (9*log(x - 2)*(x - 2)*(300*x^2 - 320*x^3 + 123*x^4 - 18*x^5))/(16*(3*x -
5)^4))/log(x - 2) - (13*x)/48 + x^2/8 + ((11125*x)/3888 - (1625*x^2)/648 + (125*x^3)/216 - 625/1944)/((50*x^2
)/3 - (500*x)/27 - (20*x^3)/3 + x^4 + 625/81)

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sympy [A]  time = 0.15, size = 20, normalized size = 0.91 \begin {gather*} \frac {9 x^{4}}{\left (144 x^{2} - 480 x + 400\right ) \log {\left (x - 2 \right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((27*x**5-144*x**4+180*x**3)*ln(x-2)-27*x**5+45*x**4)/(216*x**4-1512*x**3+3960*x**2-4600*x+2000)/ln(
x-2)**3,x)

[Out]

9*x**4/((144*x**2 - 480*x + 400)*log(x - 2)**2)

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