\(\int \frac {600-165 x+554 x^2-168 x^3-6 x^4+3 x^5+(-240+18 x-226 x^2+19 x^3+6 x^4) \log (8+x)+(24+3 x+24 x^2+3 x^3) \log ^2(8+x)}{600 x^2-165 x^3-6 x^4+3 x^5+(-240 x^2+18 x^3+6 x^4) \log (8+x)+(24 x^2+3 x^3) \log ^2(8+x)} \, dx\) [7858]

Optimal result

Integrand size = 135, antiderivative size = 26 \[ \int \frac {600-165 x+554 x^2-168 x^3-6 x^4+3 x^5+\left (-240+18 x-226 x^2+19 x^3+6 x^4\right ) \log (8+x)+\left (24+3 x+24 x^2+3 x^3\right ) \log ^2(8+x)}{600 x^2-165 x^3-6 x^4+3 x^5+\left (-240 x^2+18 x^3+6 x^4\right ) \log (8+x)+\left (24 x^2+3 x^3\right ) \log ^2(8+x)} \, dx=4-\frac {1}{x}+x-\frac {x}{3 (5-x-\log (8+x))} \]

[Out]

4-1/3*x/(5-ln(x+8)-x)+x-1/x

Rubi [F]

\[ \int \frac {600-165 x+554 x^2-168 x^3-6 x^4+3 x^5+\left (-240+18 x-226 x^2+19 x^3+6 x^4\right ) \log (8+x)+\left (24+3 x+24 x^2+3 x^3\right ) \log ^2(8+x)}{600 x^2-165 x^3-6 x^4+3 x^5+\left (-240 x^2+18 x^3+6 x^4\right ) \log (8+x)+\left (24 x^2+3 x^3\right ) \log ^2(8+x)} \, dx=\int \frac {600-165 x+554 x^2-168 x^3-6 x^4+3 x^5+\left (-240+18 x-226 x^2+19 x^3+6 x^4\right ) \log (8+x)+\left (24+3 x+24 x^2+3 x^3\right ) \log ^2(8+x)}{600 x^2-165 x^3-6 x^4+3 x^5+\left (-240 x^2+18 x^3+6 x^4\right ) \log (8+x)+\left (24 x^2+3 x^3\right ) \log ^2(8+x)} \, dx \]

[In]

Int[(600 - 165*x + 554*x^2 - 168*x^3 - 6*x^4 + 3*x^5 + (-240 + 18*x - 226*x^2 + 19*x^3 + 6*x^4)*Log[8 + x] + (
24 + 3*x + 24*x^2 + 3*x^3)*Log[8 + x]^2)/(600*x^2 - 165*x^3 - 6*x^4 + 3*x^5 + (-240*x^2 + 18*x^3 + 6*x^4)*Log[
8 + x] + (24*x^2 + 3*x^3)*Log[8 + x]^2),x]

[Out]

-x^(-1) + x - Defer[Int][(-5 + x + Log[8 + x])^(-2), x]/3 - Defer[Int][x/(-5 + x + Log[8 + x])^2, x]/3 + (8*De
fer[Int][1/((8 + x)*(-5 + x + Log[8 + x])^2), x])/3 + Defer[Int][(-5 + x + Log[8 + x])^(-1), x]/3

Rubi steps \begin{align*} \text {integral}& = \int \frac {600-165 x+554 x^2-168 x^3-6 x^4+3 x^5+\left (-240+18 x-226 x^2+19 x^3+6 x^4\right ) \log (8+x)+3 \left (8+x+8 x^2+x^3\right ) \log ^2(8+x)}{3 x^2 (8+x) (5-x-\log (8+x))^2} \, dx \\ & = \frac {1}{3} \int \frac {600-165 x+554 x^2-168 x^3-6 x^4+3 x^5+\left (-240+18 x-226 x^2+19 x^3+6 x^4\right ) \log (8+x)+3 \left (8+x+8 x^2+x^3\right ) \log ^2(8+x)}{x^2 (8+x) (5-x-\log (8+x))^2} \, dx \\ & = \frac {1}{3} \int \left (\frac {3 \left (1+x^2\right )}{x^2}-\frac {x (9+x)}{(8+x) (-5+x+\log (8+x))^2}+\frac {1}{-5+x+\log (8+x)}\right ) \, dx \\ & = -\left (\frac {1}{3} \int \frac {x (9+x)}{(8+x) (-5+x+\log (8+x))^2} \, dx\right )+\frac {1}{3} \int \frac {1}{-5+x+\log (8+x)} \, dx+\int \frac {1+x^2}{x^2} \, dx \\ & = \frac {1}{3} \int \frac {1}{-5+x+\log (8+x)} \, dx-\frac {1}{3} \int \left (\frac {1}{(-5+x+\log (8+x))^2}+\frac {x}{(-5+x+\log (8+x))^2}-\frac {8}{(8+x) (-5+x+\log (8+x))^2}\right ) \, dx+\int \left (1+\frac {1}{x^2}\right ) \, dx \\ & = -\frac {1}{x}+x-\frac {1}{3} \int \frac {1}{(-5+x+\log (8+x))^2} \, dx-\frac {1}{3} \int \frac {x}{(-5+x+\log (8+x))^2} \, dx+\frac {1}{3} \int \frac {1}{-5+x+\log (8+x)} \, dx+\frac {8}{3} \int \frac {1}{(8+x) (-5+x+\log (8+x))^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {600-165 x+554 x^2-168 x^3-6 x^4+3 x^5+\left (-240+18 x-226 x^2+19 x^3+6 x^4\right ) \log (8+x)+\left (24+3 x+24 x^2+3 x^3\right ) \log ^2(8+x)}{600 x^2-165 x^3-6 x^4+3 x^5+\left (-240 x^2+18 x^3+6 x^4\right ) \log (8+x)+\left (24 x^2+3 x^3\right ) \log ^2(8+x)} \, dx=\frac {1}{3} \left (-\frac {3}{x}+3 x+\frac {x}{-5+x+\log (8+x)}\right ) \]

[In]

Integrate[(600 - 165*x + 554*x^2 - 168*x^3 - 6*x^4 + 3*x^5 + (-240 + 18*x - 226*x^2 + 19*x^3 + 6*x^4)*Log[8 +
x] + (24 + 3*x + 24*x^2 + 3*x^3)*Log[8 + x]^2)/(600*x^2 - 165*x^3 - 6*x^4 + 3*x^5 + (-240*x^2 + 18*x^3 + 6*x^4
)*Log[8 + x] + (24*x^2 + 3*x^3)*Log[8 + x]^2),x]

[Out]

(-3/x + 3*x + x/(-5 + x + Log[8 + x]))/3

Maple [A] (verified)

Time = 0.34 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88

method	result	size
risch	\(\frac {x^{2}-1}{x}+\frac {x}{3 \ln \left (x +8\right )+3 x -15}\)	\(23\)
norman	\(\frac {5+x^{3}-\frac {73 x}{3}+x^{2} \ln \left (x +8\right )+\frac {14 x \ln \left (x +8\right )}{3}-\ln \left (x +8\right )}{x \left (\ln \left (x +8\right )+x -5\right )}\)	\(43\)
derivativedivides	\(\frac {-211 x +911-195 \ln \left (x +8\right )-38 \left (x +8\right )^{2}+3 \left (x +8\right )^{3}+3 \ln \left (x +8\right ) \left (x +8\right )^{2}}{3 x \left (\ln \left (x +8\right )+x -5\right )}\)	\(51\)
default	\(\frac {-211 x +911-195 \ln \left (x +8\right )-38 \left (x +8\right )^{2}+3 \left (x +8\right )^{3}+3 \ln \left (x +8\right ) \left (x +8\right )^{2}}{3 x \left (\ln \left (x +8\right )+x -5\right )}\)	\(51\)
parallelrisch	\(\frac {15+3 x^{3}+3 x^{2} \ln \left (x +8\right )-62 x^{2}-48 x \ln \left (x +8\right )+237 x -3 \ln \left (x +8\right )}{3 x \left (\ln \left (x +8\right )+x -5\right )}\)	\(52\)

[In]

int(((3*x^3+24*x^2+3*x+24)*ln(x+8)^2+(6*x^4+19*x^3-226*x^2+18*x-240)*ln(x+8)+3*x^5-6*x^4-168*x^3+554*x^2-165*x
+600)/((3*x^3+24*x^2)*ln(x+8)^2+(6*x^4+18*x^3-240*x^2)*ln(x+8)+3*x^5-6*x^4-165*x^3+600*x^2),x,method=_RETURNVE
RBOSE)

[Out]

(x^2-1)/x+1/3*x/(ln(x+8)+x-5)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 43 vs. \(2 (20) = 40\).

Time = 0.26 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.65 \[ \int \frac {600-165 x+554 x^2-168 x^3-6 x^4+3 x^5+\left (-240+18 x-226 x^2+19 x^3+6 x^4\right ) \log (8+x)+\left (24+3 x+24 x^2+3 x^3\right ) \log ^2(8+x)}{600 x^2-165 x^3-6 x^4+3 x^5+\left (-240 x^2+18 x^3+6 x^4\right ) \log (8+x)+\left (24 x^2+3 x^3\right ) \log ^2(8+x)} \, dx=\frac {3 \, x^{3} - 14 \, x^{2} + 3 \, {\left (x^{2} - 1\right )} \log \left (x + 8\right ) - 3 \, x + 15}{3 \, {\left (x^{2} + x \log \left (x + 8\right ) - 5 \, x\right )}} \]

[In]

integrate(((3*x^3+24*x^2+3*x+24)*log(x+8)^2+(6*x^4+19*x^3-226*x^2+18*x-240)*log(x+8)+3*x^5-6*x^4-168*x^3+554*x
^2-165*x+600)/((3*x^3+24*x^2)*log(x+8)^2+(6*x^4+18*x^3-240*x^2)*log(x+8)+3*x^5-6*x^4-165*x^3+600*x^2),x, algor
ithm="fricas")

[Out]

1/3*(3*x^3 - 14*x^2 + 3*(x^2 - 1)*log(x + 8) - 3*x + 15)/(x^2 + x*log(x + 8) - 5*x)

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.65 \[ \int \frac {600-165 x+554 x^2-168 x^3-6 x^4+3 x^5+\left (-240+18 x-226 x^2+19 x^3+6 x^4\right ) \log (8+x)+\left (24+3 x+24 x^2+3 x^3\right ) \log ^2(8+x)}{600 x^2-165 x^3-6 x^4+3 x^5+\left (-240 x^2+18 x^3+6 x^4\right ) \log (8+x)+\left (24 x^2+3 x^3\right ) \log ^2(8+x)} \, dx=x + \frac {x}{3 x + 3 \log {\left (x + 8 \right )} - 15} - \frac {1}{x} \]

[In]

integrate(((3*x**3+24*x**2+3*x+24)*ln(x+8)**2+(6*x**4+19*x**3-226*x**2+18*x-240)*ln(x+8)+3*x**5-6*x**4-168*x**
3+554*x**2-165*x+600)/((3*x**3+24*x**2)*ln(x+8)**2+(6*x**4+18*x**3-240*x**2)*ln(x+8)+3*x**5-6*x**4-165*x**3+60
0*x**2),x)

[Out]

x + x/(3*x + 3*log(x + 8) - 15) - 1/x

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 43 vs. \(2 (20) = 40\).

Time = 0.24 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.65 \[ \int \frac {600-165 x+554 x^2-168 x^3-6 x^4+3 x^5+\left (-240+18 x-226 x^2+19 x^3+6 x^4\right ) \log (8+x)+\left (24+3 x+24 x^2+3 x^3\right ) \log ^2(8+x)}{600 x^2-165 x^3-6 x^4+3 x^5+\left (-240 x^2+18 x^3+6 x^4\right ) \log (8+x)+\left (24 x^2+3 x^3\right ) \log ^2(8+x)} \, dx=\frac {3 \, x^{3} - 14 \, x^{2} + 3 \, {\left (x^{2} - 1\right )} \log \left (x + 8\right ) - 3 \, x + 15}{3 \, {\left (x^{2} + x \log \left (x + 8\right ) - 5 \, x\right )}} \]

[In]

integrate(((3*x^3+24*x^2+3*x+24)*log(x+8)^2+(6*x^4+19*x^3-226*x^2+18*x-240)*log(x+8)+3*x^5-6*x^4-168*x^3+554*x
^2-165*x+600)/((3*x^3+24*x^2)*log(x+8)^2+(6*x^4+18*x^3-240*x^2)*log(x+8)+3*x^5-6*x^4-165*x^3+600*x^2),x, algor
ithm="maxima")

[Out]

1/3*(3*x^3 - 14*x^2 + 3*(x^2 - 1)*log(x + 8) - 3*x + 15)/(x^2 + x*log(x + 8) - 5*x)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.73 \[ \int \frac {600-165 x+554 x^2-168 x^3-6 x^4+3 x^5+\left (-240+18 x-226 x^2+19 x^3+6 x^4\right ) \log (8+x)+\left (24+3 x+24 x^2+3 x^3\right ) \log ^2(8+x)}{600 x^2-165 x^3-6 x^4+3 x^5+\left (-240 x^2+18 x^3+6 x^4\right ) \log (8+x)+\left (24 x^2+3 x^3\right ) \log ^2(8+x)} \, dx=x + \frac {x}{3 \, {\left (x + \log \left (x + 8\right ) - 5\right )}} - \frac {1}{x} \]

[In]

integrate(((3*x^3+24*x^2+3*x+24)*log(x+8)^2+(6*x^4+19*x^3-226*x^2+18*x-240)*log(x+8)+3*x^5-6*x^4-168*x^3+554*x
^2-165*x+600)/((3*x^3+24*x^2)*log(x+8)^2+(6*x^4+18*x^3-240*x^2)*log(x+8)+3*x^5-6*x^4-165*x^3+600*x^2),x, algor
ithm="giac")

[Out]

x + 1/3*x/(x + log(x + 8) - 5) - 1/x

Mupad [B] (verification not implemented)

Time = 12.44 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.23 \[ \int \frac {600-165 x+554 x^2-168 x^3-6 x^4+3 x^5+\left (-240+18 x-226 x^2+19 x^3+6 x^4\right ) \log (8+x)+\left (24+3 x+24 x^2+3 x^3\right ) \log ^2(8+x)}{600 x^2-165 x^3-6 x^4+3 x^5+\left (-240 x^2+18 x^3+6 x^4\right ) \log (8+x)+\left (24 x^2+3 x^3\right ) \log ^2(8+x)} \, dx=x-\frac {\ln \left (x+8\right )+x\,\left (\frac {\ln \left (x+8\right )}{3}-\frac {2}{3}\right )-5}{x\,\left (x+\ln \left (x+8\right )-5\right )} \]

[In]

int((log(x + 8)*(18*x - 226*x^2 + 19*x^3 + 6*x^4 - 240) - 165*x + log(x + 8)^2*(3*x + 24*x^2 + 3*x^3 + 24) + 5
54*x^2 - 168*x^3 - 6*x^4 + 3*x^5 + 600)/(log(x + 8)*(18*x^3 - 240*x^2 + 6*x^4) + log(x + 8)^2*(24*x^2 + 3*x^3)
+ 600*x^2 - 165*x^3 - 6*x^4 + 3*x^5),x)

[Out]

x - (log(x + 8) + x*(log(x + 8)/3 - 2/3) - 5)/(x*(x + log(x + 8) - 5))