Integrand size = 109, antiderivative size = 21 \[ \int \frac {\left (216+324 x+72 x^2-80 x^3-32 x^4\right ) \log ^3\left (\frac {-24-25 x-6 x^2}{4+4 x+x^2}\right )+\left (3456+9936 x+11304 x^2+6352 x^3+1760 x^4+192 x^5\right ) \log ^4\left (\frac {-24-25 x-6 x^2}{4+4 x+x^2}\right )}{16+14 x+3 x^2} \, dx=(3+2 x)^4 \log ^4\left (-6-\frac {x}{(2+x)^2}\right ) \] Output:
ln(-6-x/(2+x)^2)^4*(3+2*x)^4
Time = 0.04 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.33 \[ \int \frac {\left (216+324 x+72 x^2-80 x^3-32 x^4\right ) \log ^3\left (\frac {-24-25 x-6 x^2}{4+4 x+x^2}\right )+\left (3456+9936 x+11304 x^2+6352 x^3+1760 x^4+192 x^5\right ) \log ^4\left (\frac {-24-25 x-6 x^2}{4+4 x+x^2}\right )}{16+14 x+3 x^2} \, dx=(3+2 x)^4 \log ^4\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right ) \] Input:
Integrate[((216 + 324*x + 72*x^2 - 80*x^3 - 32*x^4)*Log[(-24 - 25*x - 6*x^ 2)/(4 + 4*x + x^2)]^3 + (3456 + 9936*x + 11304*x^2 + 6352*x^3 + 1760*x^4 + 192*x^5)*Log[(-24 - 25*x - 6*x^2)/(4 + 4*x + x^2)]^4)/(16 + 14*x + 3*x^2) ,x]
Output:
(3 + 2*x)^4*Log[-((24 + 25*x + 6*x^2)/(2 + x)^2)]^4
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\left (-32 x^4-80 x^3+72 x^2+324 x+216\right ) \log ^3\left (\frac {-6 x^2-25 x-24}{x^2+4 x+4}\right )+\left (192 x^5+1760 x^4+6352 x^3+11304 x^2+9936 x+3456\right ) \log ^4\left (\frac {-6 x^2-25 x-24}{x^2+4 x+4}\right )}{3 x^2+14 x+16} \, dx\) |
\(\Big \downarrow \) 7279 |
\(\displaystyle \int \left (8 (2 x+3)^3 \log ^4\left (-\frac {6 x^2+25 x+24}{(x+2)^2}\right )-\frac {4 (x-2) (2 x+3)^3 \log ^3\left (-\frac {6 x^2+25 x+24}{(x+2)^2}\right )}{(x+2) (3 x+8)}\right )dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \left (8 (2 x+3)^3 \log ^4\left (-\frac {6 x^2+25 x+24}{(x+2)^2}\right )-\frac {4 (x-2) (2 x+3)^3 \log ^3\left (-\frac {6 x^2+25 x+24}{(x+2)^2}\right )}{(x+2) (3 x+8)}\right )dx\) |
Input:
Int[((216 + 324*x + 72*x^2 - 80*x^3 - 32*x^4)*Log[(-24 - 25*x - 6*x^2)/(4 + 4*x + x^2)]^3 + (3456 + 9936*x + 11304*x^2 + 6352*x^3 + 1760*x^4 + 192*x ^5)*Log[(-24 - 25*x - 6*x^2)/(4 + 4*x + x^2)]^4)/(16 + 14*x + 3*x^2),x]
Output:
$Aborted
Time = 11.53 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.57
method | result | size |
risch | \(\left (3+2 x \right )^{4} \ln \left (\frac {-6 x^{2}-25 x -24}{x^{2}+4 x +4}\right )^{4}\) | \(33\) |
parallelrisch | \(16 \ln \left (-\frac {6 x^{2}+25 x +24}{x^{2}+4 x +4}\right )^{4} x^{4}+96 \ln \left (-\frac {6 x^{2}+25 x +24}{x^{2}+4 x +4}\right )^{4} x^{3}+216 \ln \left (-\frac {6 x^{2}+25 x +24}{x^{2}+4 x +4}\right )^{4} x^{2}+216 \ln \left (-\frac {6 x^{2}+25 x +24}{x^{2}+4 x +4}\right )^{4} x +81 \ln \left (-\frac {6 x^{2}+25 x +24}{x^{2}+4 x +4}\right )^{4}\) | \(147\) |
orering | \(\text {Expression too large to display}\) | \(28706\) |
Input:
int(((192*x^5+1760*x^4+6352*x^3+11304*x^2+9936*x+3456)*ln((-6*x^2-25*x-24) /(x^2+4*x+4))^4+(-32*x^4-80*x^3+72*x^2+324*x+216)*ln((-6*x^2-25*x-24)/(x^2 +4*x+4))^3)/(3*x^2+14*x+16),x,method=_RETURNVERBOSE)
Output:
(3+2*x)^4*ln((-6*x^2-25*x-24)/(x^2+4*x+4))^4
Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (21) = 42\).
Time = 0.08 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.19 \[ \int \frac {\left (216+324 x+72 x^2-80 x^3-32 x^4\right ) \log ^3\left (\frac {-24-25 x-6 x^2}{4+4 x+x^2}\right )+\left (3456+9936 x+11304 x^2+6352 x^3+1760 x^4+192 x^5\right ) \log ^4\left (\frac {-24-25 x-6 x^2}{4+4 x+x^2}\right )}{16+14 x+3 x^2} \, dx={\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (-\frac {6 \, x^{2} + 25 \, x + 24}{x^{2} + 4 \, x + 4}\right )^{4} \] Input:
integrate(((192*x^5+1760*x^4+6352*x^3+11304*x^2+9936*x+3456)*log((-6*x^2-2 5*x-24)/(x^2+4*x+4))^4+(-32*x^4-80*x^3+72*x^2+324*x+216)*log((-6*x^2-25*x- 24)/(x^2+4*x+4))^3)/(3*x^2+14*x+16),x, algorithm="fricas")
Output:
(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)*log(-(6*x^2 + 25*x + 24)/(x^2 + 4 *x + 4))^4
Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (19) = 38\).
Time = 0.14 (sec) , antiderivative size = 42, normalized size of antiderivative = 2.00 \[ \int \frac {\left (216+324 x+72 x^2-80 x^3-32 x^4\right ) \log ^3\left (\frac {-24-25 x-6 x^2}{4+4 x+x^2}\right )+\left (3456+9936 x+11304 x^2+6352 x^3+1760 x^4+192 x^5\right ) \log ^4\left (\frac {-24-25 x-6 x^2}{4+4 x+x^2}\right )}{16+14 x+3 x^2} \, dx=\left (16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81\right ) \log {\left (\frac {- 6 x^{2} - 25 x - 24}{x^{2} + 4 x + 4} \right )}^{4} \] Input:
integrate(((192*x**5+1760*x**4+6352*x**3+11304*x**2+9936*x+3456)*ln((-6*x* *2-25*x-24)/(x**2+4*x+4))**4+(-32*x**4-80*x**3+72*x**2+324*x+216)*ln((-6*x **2-25*x-24)/(x**2+4*x+4))**3)/(3*x**2+14*x+16),x)
Output:
(16*x**4 + 96*x**3 + 216*x**2 + 216*x + 81)*log((-6*x**2 - 25*x - 24)/(x** 2 + 4*x + 4))**4
Leaf count of result is larger than twice the leaf count of optimal. 491 vs. \(2 (21) = 42\).
Time = 0.13 (sec) , antiderivative size = 491, normalized size of antiderivative = 23.38 \[ \int \frac {\left (216+324 x+72 x^2-80 x^3-32 x^4\right ) \log ^3\left (\frac {-24-25 x-6 x^2}{4+4 x+x^2}\right )+\left (3456+9936 x+11304 x^2+6352 x^3+1760 x^4+192 x^5\right ) \log ^4\left (\frac {-24-25 x-6 x^2}{4+4 x+x^2}\right )}{16+14 x+3 x^2} \, dx={\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (2 \, x + 3\right )^{4} + 16 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (x + 2\right )^{4} - 32 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (x + 2\right )^{3} \log \left (-3 \, x - 8\right ) + 24 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (x + 2\right )^{2} \log \left (-3 \, x - 8\right )^{2} - 8 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (x + 2\right ) \log \left (-3 \, x - 8\right )^{3} + {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (-3 \, x - 8\right )^{4} - 4 \, {\left (2 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (x + 2\right ) - {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (-3 \, x - 8\right )\right )} \log \left (2 \, x + 3\right )^{3} + 6 \, {\left (4 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (x + 2\right )^{2} - 4 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (x + 2\right ) \log \left (-3 \, x - 8\right ) + {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (-3 \, x - 8\right )^{2}\right )} \log \left (2 \, x + 3\right )^{2} - 4 \, {\left (8 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (x + 2\right )^{3} - 12 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (x + 2\right )^{2} \log \left (-3 \, x - 8\right ) + 6 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (x + 2\right ) \log \left (-3 \, x - 8\right )^{2} - {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (-3 \, x - 8\right )^{3}\right )} \log \left (2 \, x + 3\right ) \] Input:
integrate(((192*x^5+1760*x^4+6352*x^3+11304*x^2+9936*x+3456)*log((-6*x^2-2 5*x-24)/(x^2+4*x+4))^4+(-32*x^4-80*x^3+72*x^2+324*x+216)*log((-6*x^2-25*x- 24)/(x^2+4*x+4))^3)/(3*x^2+14*x+16),x, algorithm="maxima")
Output:
(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)*log(2*x + 3)^4 + 16*(16*x^4 + 96* x^3 + 216*x^2 + 216*x + 81)*log(x + 2)^4 - 32*(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)*log(x + 2)^3*log(-3*x - 8) + 24*(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)*log(x + 2)^2*log(-3*x - 8)^2 - 8*(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)*log(x + 2)*log(-3*x - 8)^3 + (16*x^4 + 96*x^3 + 216*x^2 + 216* x + 81)*log(-3*x - 8)^4 - 4*(2*(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)*lo g(x + 2) - (16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)*log(-3*x - 8))*log(2*x + 3)^3 + 6*(4*(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)*log(x + 2)^2 - 4*( 16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)*log(x + 2)*log(-3*x - 8) + (16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)*log(-3*x - 8)^2)*log(2*x + 3)^2 - 4*(8*( 16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)*log(x + 2)^3 - 12*(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)*log(x + 2)^2*log(-3*x - 8) + 6*(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)*log(x + 2)*log(-3*x - 8)^2 - (16*x^4 + 96*x^3 + 216 *x^2 + 216*x + 81)*log(-3*x - 8)^3)*log(2*x + 3)
Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (21) = 42\).
Time = 4.21 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.19 \[ \int \frac {\left (216+324 x+72 x^2-80 x^3-32 x^4\right ) \log ^3\left (\frac {-24-25 x-6 x^2}{4+4 x+x^2}\right )+\left (3456+9936 x+11304 x^2+6352 x^3+1760 x^4+192 x^5\right ) \log ^4\left (\frac {-24-25 x-6 x^2}{4+4 x+x^2}\right )}{16+14 x+3 x^2} \, dx={\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (-\frac {6 \, x^{2} + 25 \, x + 24}{x^{2} + 4 \, x + 4}\right )^{4} \] Input:
integrate(((192*x^5+1760*x^4+6352*x^3+11304*x^2+9936*x+3456)*log((-6*x^2-2 5*x-24)/(x^2+4*x+4))^4+(-32*x^4-80*x^3+72*x^2+324*x+216)*log((-6*x^2-25*x- 24)/(x^2+4*x+4))^3)/(3*x^2+14*x+16),x, algorithm="giac")
Output:
(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)*log(-(6*x^2 + 25*x + 24)/(x^2 + 4 *x + 4))^4
Time = 3.49 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.57 \[ \int \frac {\left (216+324 x+72 x^2-80 x^3-32 x^4\right ) \log ^3\left (\frac {-24-25 x-6 x^2}{4+4 x+x^2}\right )+\left (3456+9936 x+11304 x^2+6352 x^3+1760 x^4+192 x^5\right ) \log ^4\left (\frac {-24-25 x-6 x^2}{4+4 x+x^2}\right )}{16+14 x+3 x^2} \, dx={\ln \left (-\frac {6\,x^2+25\,x+24}{x^2+4\,x+4}\right )}^4\,{\left (2\,x+3\right )}^4 \] Input:
int((log(-(25*x + 6*x^2 + 24)/(4*x + x^2 + 4))^3*(324*x + 72*x^2 - 80*x^3 - 32*x^4 + 216) + log(-(25*x + 6*x^2 + 24)/(4*x + x^2 + 4))^4*(9936*x + 11 304*x^2 + 6352*x^3 + 1760*x^4 + 192*x^5 + 3456))/(14*x + 3*x^2 + 16),x)
Output:
log(-(25*x + 6*x^2 + 24)/(4*x + x^2 + 4))^4*(2*x + 3)^4
Time = 0.17 (sec) , antiderivative size = 45, normalized size of antiderivative = 2.14 \[ \int \frac {\left (216+324 x+72 x^2-80 x^3-32 x^4\right ) \log ^3\left (\frac {-24-25 x-6 x^2}{4+4 x+x^2}\right )+\left (3456+9936 x+11304 x^2+6352 x^3+1760 x^4+192 x^5\right ) \log ^4\left (\frac {-24-25 x-6 x^2}{4+4 x+x^2}\right )}{16+14 x+3 x^2} \, dx=\mathrm {log}\left (\frac {-6 x^{2}-25 x -24}{x^{2}+4 x +4}\right )^{4} \left (16 x^{4}+96 x^{3}+216 x^{2}+216 x +81\right ) \] Input:
int(((192*x^5+1760*x^4+6352*x^3+11304*x^2+9936*x+3456)*log((-6*x^2-25*x-24 )/(x^2+4*x+4))^4+(-32*x^4-80*x^3+72*x^2+324*x+216)*log((-6*x^2-25*x-24)/(x ^2+4*x+4))^3)/(3*x^2+14*x+16),x)
Output:
log(( - 6*x**2 - 25*x - 24)/(x**2 + 4*x + 4))**4*(16*x**4 + 96*x**3 + 216* x**2 + 216*x + 81)