3.101.10 \(\int \frac {18 x+(-2160+876 x-90 x^2) \log (\frac {24-5 x}{-5+x})+(1440-588 x+60 x^2) \log ^2(\frac {24-5 x}{-5+x})+(-240+98 x-10 x^2+120 x^3-49 x^4+5 x^5) \log ^3(\frac {24-5 x}{-5+x})}{(120 x^3-49 x^4+5 x^5) \log ^3(\frac {24-5 x}{-5+x})} \, dx\)

Optimal. Leaf size=27 \[ \frac {19}{4}+x+\frac {\left (1-\frac {3}{\log \left (-5+\frac {1}{5-x}\right )}\right )^2}{x^2} \]

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Rubi [F]  time = 0.82, 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 {18 x+\left (-2160+876 x-90 x^2\right ) \log \left (\frac {24-5 x}{-5+x}\right )+\left (1440-588 x+60 x^2\right ) \log ^2\left (\frac {24-5 x}{-5+x}\right )+\left (-240+98 x-10 x^2+120 x^3-49 x^4+5 x^5\right ) \log ^3\left (\frac {24-5 x}{-5+x}\right )}{\left (120 x^3-49 x^4+5 x^5\right ) \log ^3\left (\frac {24-5 x}{-5+x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(18*x + (-2160 + 876*x - 90*x^2)*Log[(24 - 5*x)/(-5 + x)] + (1440 - 588*x + 60*x^2)*Log[(24 - 5*x)/(-5 + x
)]^2 + (-240 + 98*x - 10*x^2 + 120*x^3 - 49*x^4 + 5*x^5)*Log[(24 - 5*x)/(-5 + x)]^3)/((120*x^3 - 49*x^4 + 5*x^
5)*Log[(24 - 5*x)/(-5 + x)]^3),x]

[Out]

x^(-2) + x + 18*Defer[Int][1/((-5 + x)*x^2*(-24 + 5*x)*Log[(24 - 5*x)/(-5 + x)]^3), x] - 6*Defer[Int][(360 - 1
46*x + 15*x^2)/((-5 + x)*x^3*(-24 + 5*x)*Log[(24 - 5*x)/(-5 + x)]^2), x] + 12*Defer[Int][1/(x^3*Log[(24 - 5*x)
/(-5 + x)]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {18 x+\left (-2160+876 x-90 x^2\right ) \log \left (\frac {24-5 x}{-5+x}\right )+\left (1440-588 x+60 x^2\right ) \log ^2\left (\frac {24-5 x}{-5+x}\right )+\left (-240+98 x-10 x^2+120 x^3-49 x^4+5 x^5\right ) \log ^3\left (\frac {24-5 x}{-5+x}\right )}{x^3 \left (120-49 x+5 x^2\right ) \log ^3\left (\frac {24-5 x}{-5+x}\right )} \, dx\\ &=\int \left (\frac {-2+x^3}{x^3}+\frac {18}{(-5+x) x^2 (-24+5 x) \log ^3\left (\frac {24-5 x}{-5+x}\right )}-\frac {6 \left (360-146 x+15 x^2\right )}{(-5+x) x^3 (-24+5 x) \log ^2\left (\frac {24-5 x}{-5+x}\right )}+\frac {12}{x^3 \log \left (\frac {24-5 x}{-5+x}\right )}\right ) \, dx\\ &=-\left (6 \int \frac {360-146 x+15 x^2}{(-5+x) x^3 (-24+5 x) \log ^2\left (\frac {24-5 x}{-5+x}\right )} \, dx\right )+12 \int \frac {1}{x^3 \log \left (\frac {24-5 x}{-5+x}\right )} \, dx+18 \int \frac {1}{(-5+x) x^2 (-24+5 x) \log ^3\left (\frac {24-5 x}{-5+x}\right )} \, dx+\int \frac {-2+x^3}{x^3} \, dx\\ &=-\left (6 \int \frac {360-146 x+15 x^2}{(-5+x) x^3 (-24+5 x) \log ^2\left (\frac {24-5 x}{-5+x}\right )} \, dx\right )+12 \int \frac {1}{x^3 \log \left (\frac {24-5 x}{-5+x}\right )} \, dx+18 \int \frac {1}{(-5+x) x^2 (-24+5 x) \log ^3\left (\frac {24-5 x}{-5+x}\right )} \, dx+\int \left (1-\frac {2}{x^3}\right ) \, dx\\ &=\frac {1}{x^2}+x-6 \int \frac {360-146 x+15 x^2}{(-5+x) x^3 (-24+5 x) \log ^2\left (\frac {24-5 x}{-5+x}\right )} \, dx+12 \int \frac {1}{x^3 \log \left (\frac {24-5 x}{-5+x}\right )} \, dx+18 \int \frac {1}{(-5+x) x^2 (-24+5 x) \log ^3\left (\frac {24-5 x}{-5+x}\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 43, normalized size = 1.59 \begin {gather*} \frac {1}{x^2}+x+\frac {9}{x^2 \log ^2\left (\frac {24-5 x}{-5+x}\right )}-\frac {6}{x^2 \log \left (\frac {24-5 x}{-5+x}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(18*x + (-2160 + 876*x - 90*x^2)*Log[(24 - 5*x)/(-5 + x)] + (1440 - 588*x + 60*x^2)*Log[(24 - 5*x)/(
-5 + x)]^2 + (-240 + 98*x - 10*x^2 + 120*x^3 - 49*x^4 + 5*x^5)*Log[(24 - 5*x)/(-5 + x)]^3)/((120*x^3 - 49*x^4
+ 5*x^5)*Log[(24 - 5*x)/(-5 + x)]^3),x]

[Out]

x^(-2) + x + 9/(x^2*Log[(24 - 5*x)/(-5 + x)]^2) - 6/(x^2*Log[(24 - 5*x)/(-5 + x)])

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fricas [B]  time = 0.71, size = 57, normalized size = 2.11 \begin {gather*} \frac {{\left (x^{3} + 1\right )} \log \left (-\frac {5 \, x - 24}{x - 5}\right )^{2} - 6 \, \log \left (-\frac {5 \, x - 24}{x - 5}\right ) + 9}{x^{2} \log \left (-\frac {5 \, x - 24}{x - 5}\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x^5-49*x^4+120*x^3-10*x^2+98*x-240)*log((-5*x+24)/(x-5))^3+(60*x^2-588*x+1440)*log((-5*x+24)/(x-
5))^2+(-90*x^2+876*x-2160)*log((-5*x+24)/(x-5))+18*x)/(5*x^5-49*x^4+120*x^3)/log((-5*x+24)/(x-5))^3,x, algorit
hm="fricas")

[Out]

((x^3 + 1)*log(-(5*x - 24)/(x - 5))^2 - 6*log(-(5*x - 24)/(x - 5)) + 9)/(x^2*log(-(5*x - 24)/(x - 5))^2)

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giac [B]  time = 0.32, size = 235, normalized size = 8.70 \begin {gather*} -\frac {3 \, {\left (\frac {2 \, {\left (5 \, x - 24\right )}^{2} \log \left (-\frac {5 \, x - 24}{x - 5}\right )}{{\left (x - 5\right )}^{2}} - \frac {20 \, {\left (5 \, x - 24\right )} \log \left (-\frac {5 \, x - 24}{x - 5}\right )}{x - 5} - \frac {3 \, {\left (5 \, x - 24\right )}^{2}}{{\left (x - 5\right )}^{2}} + \frac {30 \, {\left (5 \, x - 24\right )}}{x - 5} + 50 \, \log \left (-\frac {5 \, x - 24}{x - 5}\right ) - 75\right )}}{\frac {25 \, {\left (5 \, x - 24\right )}^{2} \log \left (-\frac {5 \, x - 24}{x - 5}\right )^{2}}{{\left (x - 5\right )}^{2}} - \frac {240 \, {\left (5 \, x - 24\right )} \log \left (-\frac {5 \, x - 24}{x - 5}\right )^{2}}{x - 5} + 576 \, \log \left (-\frac {5 \, x - 24}{x - 5}\right )^{2}} - \frac {\frac {10 \, {\left (5 \, x - 24\right )}}{x - 5} - 49}{25 \, {\left (\frac {25 \, {\left (5 \, x - 24\right )}^{2}}{{\left (x - 5\right )}^{2}} - \frac {240 \, {\left (5 \, x - 24\right )}}{x - 5} + 576\right )}} + \frac {1}{\frac {5 \, x - 24}{x - 5} - 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x^5-49*x^4+120*x^3-10*x^2+98*x-240)*log((-5*x+24)/(x-5))^3+(60*x^2-588*x+1440)*log((-5*x+24)/(x-
5))^2+(-90*x^2+876*x-2160)*log((-5*x+24)/(x-5))+18*x)/(5*x^5-49*x^4+120*x^3)/log((-5*x+24)/(x-5))^3,x, algorit
hm="giac")

[Out]

-3*(2*(5*x - 24)^2*log(-(5*x - 24)/(x - 5))/(x - 5)^2 - 20*(5*x - 24)*log(-(5*x - 24)/(x - 5))/(x - 5) - 3*(5*
x - 24)^2/(x - 5)^2 + 30*(5*x - 24)/(x - 5) + 50*log(-(5*x - 24)/(x - 5)) - 75)/(25*(5*x - 24)^2*log(-(5*x - 2
4)/(x - 5))^2/(x - 5)^2 - 240*(5*x - 24)*log(-(5*x - 24)/(x - 5))^2/(x - 5) + 576*log(-(5*x - 24)/(x - 5))^2)
- 1/25*(10*(5*x - 24)/(x - 5) - 49)/(25*(5*x - 24)^2/(x - 5)^2 - 240*(5*x - 24)/(x - 5) + 576) + 1/((5*x - 24)
/(x - 5) - 5)

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maple [A]  time = 0.16, size = 46, normalized size = 1.70

method	result	size
risch	\(\frac {x^{3}+1}{x^{2}}-\frac {3 \left (2 \ln \left (\frac {-5 x +24}{x -5}\right )-3\right )}{x^{2} \ln \left (\frac {-5 x +24}{x -5}\right )^{2}}\)	\(46\)
norman	\(\frac {9+\ln \left (\frac {-5 x +24}{x -5}\right )^{2}+x^{3} \ln \left (\frac {-5 x +24}{x -5}\right )^{2}-6 \ln \left (\frac {-5 x +24}{x -5}\right )}{\ln \left (\frac {-5 x +24}{x -5}\right )^{2} x^{2}}\)	\(67\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((5*x^5-49*x^4+120*x^3-10*x^2+98*x-240)*ln((-5*x+24)/(x-5))^3+(60*x^2-588*x+1440)*ln((-5*x+24)/(x-5))^2+(-
90*x^2+876*x-2160)*ln((-5*x+24)/(x-5))+18*x)/(5*x^5-49*x^4+120*x^3)/ln((-5*x+24)/(x-5))^3,x,method=_RETURNVERB
OSE)

[Out]

(x^3+1)/x^2-3/x^2*(2*ln((-5*x+24)/(x-5))-3)/ln((-5*x+24)/(x-5))^2

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maxima [B]  time = 0.55, size = 95, normalized size = 3.52 \begin {gather*} \frac {{\left (x^{3} + 1\right )} \log \left (x - 5\right )^{2} + {\left (x^{3} + 1\right )} \log \left (-5 \, x + 24\right )^{2} - 2 \, {\left ({\left (x^{3} + 1\right )} \log \left (x - 5\right ) + 3\right )} \log \left (-5 \, x + 24\right ) + 6 \, \log \left (x - 5\right ) + 9}{x^{2} \log \left (x - 5\right )^{2} - 2 \, x^{2} \log \left (x - 5\right ) \log \left (-5 \, x + 24\right ) + x^{2} \log \left (-5 \, x + 24\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x^5-49*x^4+120*x^3-10*x^2+98*x-240)*log((-5*x+24)/(x-5))^3+(60*x^2-588*x+1440)*log((-5*x+24)/(x-
5))^2+(-90*x^2+876*x-2160)*log((-5*x+24)/(x-5))+18*x)/(5*x^5-49*x^4+120*x^3)/log((-5*x+24)/(x-5))^3,x, algorit
hm="maxima")

[Out]

((x^3 + 1)*log(x - 5)^2 + (x^3 + 1)*log(-5*x + 24)^2 - 2*((x^3 + 1)*log(x - 5) + 3)*log(-5*x + 24) + 6*log(x -
5) + 9)/(x^2*log(x - 5)^2 - 2*x^2*log(x - 5)*log(-5*x + 24) + x^2*log(-5*x + 24)^2)

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mupad [B]  time = 8.82, size = 70, normalized size = 2.59 \begin {gather*} x+150\,\ln \left (\frac {5\,x-24}{x-5}\right )-\frac {6}{x^2\,\ln \left (-\frac {5\,x-24}{x-5}\right )}+\frac {9}{x^2\,{\ln \left (-\frac {5\,x-24}{x-5}\right )}^2}+\frac {1}{x^2}+\mathrm {atan}\left (x\,10{}\mathrm {i}-49{}\mathrm {i}\right )\,300{}\mathrm {i} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((18*x - log(-(5*x - 24)/(x - 5))*(90*x^2 - 876*x + 2160) + log(-(5*x - 24)/(x - 5))^3*(98*x - 10*x^2 + 120
*x^3 - 49*x^4 + 5*x^5 - 240) + log(-(5*x - 24)/(x - 5))^2*(60*x^2 - 588*x + 1440))/(log(-(5*x - 24)/(x - 5))^3
*(120*x^3 - 49*x^4 + 5*x^5)),x)

[Out]

x + 150*log((5*x - 24)/(x - 5)) + atan(x*10i - 49i)*300i - 6/(x^2*log(-(5*x - 24)/(x - 5))) + 9/(x^2*log(-(5*x
- 24)/(x - 5))^2) + 1/x^2

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sympy [A]  time = 0.19, size = 34, normalized size = 1.26 \begin {gather*} x + \frac {9 - 6 \log {\left (\frac {24 - 5 x}{x - 5} \right )}}{x^{2} \log {\left (\frac {24 - 5 x}{x - 5} \right )}^{2}} + \frac {1}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x**5-49*x**4+120*x**3-10*x**2+98*x-240)*ln((-5*x+24)/(x-5))**3+(60*x**2-588*x+1440)*ln((-5*x+24)
/(x-5))**2+(-90*x**2+876*x-2160)*ln((-5*x+24)/(x-5))+18*x)/(5*x**5-49*x**4+120*x**3)/ln((-5*x+24)/(x-5))**3,x)

[Out]

x + (9 - 6*log((24 - 5*x)/(x - 5)))/(x**2*log((24 - 5*x)/(x - 5))**2) + x**(-2)

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