Integrand size = 71, antiderivative size = 22 \[ \int \frac {-512000000-2048000000 x^2-3840000 x^3-2048000000 x^4-10240000 x^5-5120000 x^7+\left (2048000000 x^2+4096000000 x^4\right ) \log (x)}{x^{10}+1200 x^7 \log (x)+480000 x^4 \log ^2(x)+64000000 x \log ^3(x)} \, dx=\frac {\left (2+4 x^2\right )^2}{\left (\frac {x^3}{400}+\log (x)\right )^2} \]
Time = 0.33 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {-512000000-2048000000 x^2-3840000 x^3-2048000000 x^4-10240000 x^5-5120000 x^7+\left (2048000000 x^2+4096000000 x^4\right ) \log (x)}{x^{10}+1200 x^7 \log (x)+480000 x^4 \log ^2(x)+64000000 x \log ^3(x)} \, dx=\frac {640000 \left (1+2 x^2\right )^2}{\left (x^3+400 \log (x)\right )^2} \]
Integrate[(-512000000 - 2048000000*x^2 - 3840000*x^3 - 2048000000*x^4 - 10 240000*x^5 - 5120000*x^7 + (2048000000*x^2 + 4096000000*x^4)*Log[x])/(x^10 + 1200*x^7*Log[x] + 480000*x^4*Log[x]^2 + 64000000*x*Log[x]^3),x]
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.
\(\displaystyle \int \frac {-5120000 x^7-10240000 x^5-2048000000 x^4-3840000 x^3-2048000000 x^2+\left (4096000000 x^4+2048000000 x^2\right ) \log (x)-512000000}{x^{10}+1200 x^7 \log (x)+480000 x^4 \log ^2(x)+64000000 x \log ^3(x)} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {1280000 \left (2 x^2+1\right ) \left (-2 x^5-3 x^3-800 x^2+1600 x^2 \log (x)-400\right )}{x \left (x^3+400 \log (x)\right )^3}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 1280000 \int -\frac {\left (2 x^2+1\right ) \left (2 x^5+3 x^3-1600 \log (x) x^2+800 x^2+400\right )}{x \left (x^3+400 \log (x)\right )^3}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -1280000 \int \frac {\left (2 x^2+1\right ) \left (2 x^5+3 x^3-1600 \log (x) x^2+800 x^2+400\right )}{x \left (x^3+400 \log (x)\right )^3}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -1280000 \int \left (\frac {\left (2 x^2+1\right )^2 \left (3 x^3+400\right )}{x \left (x^3+400 \log (x)\right )^3}-\frac {4 x \left (2 x^2+1\right )}{\left (x^3+400 \log (x)\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -1280000 \left (400 \int \frac {1}{x \left (x^3+400 \log (x)\right )^3}dx+1600 \int \frac {x}{\left (x^3+400 \log (x)\right )^3}dx+1600 \int \frac {x^3}{\left (x^3+400 \log (x)\right )^3}dx-4 \int \frac {x}{\left (x^3+400 \log (x)\right )^2}dx-8 \int \frac {x^3}{\left (x^3+400 \log (x)\right )^2}dx+12 \int \frac {x^6}{\left (x^3+400 \log (x)\right )^3}dx+12 \int \frac {x^4}{\left (x^3+400 \log (x)\right )^3}dx+3 \int \frac {x^2}{\left (x^3+400 \log (x)\right )^3}dx\right )\) |
Int[(-512000000 - 2048000000*x^2 - 3840000*x^3 - 2048000000*x^4 - 10240000 *x^5 - 5120000*x^7 + (2048000000*x^2 + 4096000000*x^4)*Log[x])/(x^10 + 120 0*x^7*Log[x] + 480000*x^4*Log[x]^2 + 64000000*x*Log[x]^3),x]
3.12.12.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Time = 0.69 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14
method | result | size |
default | \(\frac {2560000 x^{4}+2560000 x^{2}+640000}{\left (x^{3}+400 \ln \left (x \right )\right )^{2}}\) | \(25\) |
risch | \(\frac {2560000 x^{4}+2560000 x^{2}+640000}{\left (x^{3}+400 \ln \left (x \right )\right )^{2}}\) | \(25\) |
parallelrisch | \(-\frac {-2048000000 x^{4}-2048000000 x^{2}-512000000}{800 \left (x^{6}+800 x^{3} \ln \left (x \right )+160000 \ln \left (x \right )^{2}\right )}\) | \(34\) |
int(((4096000000*x^4+2048000000*x^2)*ln(x)-5120000*x^7-10240000*x^5-204800 0000*x^4-3840000*x^3-2048000000*x^2-512000000)/(64000000*x*ln(x)^3+480000* x^4*ln(x)^2+1200*x^7*ln(x)+x^10),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.50 \[ \int \frac {-512000000-2048000000 x^2-3840000 x^3-2048000000 x^4-10240000 x^5-5120000 x^7+\left (2048000000 x^2+4096000000 x^4\right ) \log (x)}{x^{10}+1200 x^7 \log (x)+480000 x^4 \log ^2(x)+64000000 x \log ^3(x)} \, dx=\frac {640000 \, {\left (4 \, x^{4} + 4 \, x^{2} + 1\right )}}{x^{6} + 800 \, x^{3} \log \left (x\right ) + 160000 \, \log \left (x\right )^{2}} \]
integrate(((4096000000*x^4+2048000000*x^2)*log(x)-5120000*x^7-10240000*x^5 -2048000000*x^4-3840000*x^3-2048000000*x^2-512000000)/(64000000*x*log(x)^3 +480000*x^4*log(x)^2+1200*x^7*log(x)+x^10),x, algorithm=\
Time = 0.06 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.32 \[ \int \frac {-512000000-2048000000 x^2-3840000 x^3-2048000000 x^4-10240000 x^5-5120000 x^7+\left (2048000000 x^2+4096000000 x^4\right ) \log (x)}{x^{10}+1200 x^7 \log (x)+480000 x^4 \log ^2(x)+64000000 x \log ^3(x)} \, dx=\frac {2560000 x^{4} + 2560000 x^{2} + 640000}{x^{6} + 800 x^{3} \log {\left (x \right )} + 160000 \log {\left (x \right )}^{2}} \]
integrate(((4096000000*x**4+2048000000*x**2)*ln(x)-5120000*x**7-10240000*x **5-2048000000*x**4-3840000*x**3-2048000000*x**2-512000000)/(64000000*x*ln (x)**3+480000*x**4*ln(x)**2+1200*x**7*ln(x)+x**10),x)
Time = 0.24 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.50 \[ \int \frac {-512000000-2048000000 x^2-3840000 x^3-2048000000 x^4-10240000 x^5-5120000 x^7+\left (2048000000 x^2+4096000000 x^4\right ) \log (x)}{x^{10}+1200 x^7 \log (x)+480000 x^4 \log ^2(x)+64000000 x \log ^3(x)} \, dx=\frac {640000 \, {\left (4 \, x^{4} + 4 \, x^{2} + 1\right )}}{x^{6} + 800 \, x^{3} \log \left (x\right ) + 160000 \, \log \left (x\right )^{2}} \]
integrate(((4096000000*x^4+2048000000*x^2)*log(x)-5120000*x^7-10240000*x^5 -2048000000*x^4-3840000*x^3-2048000000*x^2-512000000)/(64000000*x*log(x)^3 +480000*x^4*log(x)^2+1200*x^7*log(x)+x^10),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (21) = 42\).
Time = 0.28 (sec) , antiderivative size = 71, normalized size of antiderivative = 3.23 \[ \int \frac {-512000000-2048000000 x^2-3840000 x^3-2048000000 x^4-10240000 x^5-5120000 x^7+\left (2048000000 x^2+4096000000 x^4\right ) \log (x)}{x^{10}+1200 x^7 \log (x)+480000 x^4 \log ^2(x)+64000000 x \log ^3(x)} \, dx=\frac {640000 \, {\left (12 \, x^{7} + 12 \, x^{5} + 1600 \, x^{4} + 3 \, x^{3} + 1600 \, x^{2} + 400\right )}}{3 \, x^{9} + 2400 \, x^{6} \log \left (x\right ) + 400 \, x^{6} + 480000 \, x^{3} \log \left (x\right )^{2} + 320000 \, x^{3} \log \left (x\right ) + 64000000 \, \log \left (x\right )^{2}} \]
integrate(((4096000000*x^4+2048000000*x^2)*log(x)-5120000*x^7-10240000*x^5 -2048000000*x^4-3840000*x^3-2048000000*x^2-512000000)/(64000000*x*log(x)^3 +480000*x^4*log(x)^2+1200*x^7*log(x)+x^10),x, algorithm=\
640000*(12*x^7 + 12*x^5 + 1600*x^4 + 3*x^3 + 1600*x^2 + 400)/(3*x^9 + 2400 *x^6*log(x) + 400*x^6 + 480000*x^3*log(x)^2 + 320000*x^3*log(x) + 64000000 *log(x)^2)
Time = 12.71 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {-512000000-2048000000 x^2-3840000 x^3-2048000000 x^4-10240000 x^5-5120000 x^7+\left (2048000000 x^2+4096000000 x^4\right ) \log (x)}{x^{10}+1200 x^7 \log (x)+480000 x^4 \log ^2(x)+64000000 x \log ^3(x)} \, dx=\frac {640000\,{\left (2\,x^2+1\right )}^2}{{\left (400\,\ln \left (x\right )+x^3\right )}^2} \]