Integrand size = 123, antiderivative size = 28 \begin {dmath*} \int \frac {953694 x+1262196 x^2+694960 x^3+196000 x^4+28000 x^5+1600 x^6+\left (-33614 x-48020 x^2-27440 x^3-7840 x^4-1120 x^5-64 x^6\right ) \log (x)+\left (-33614 x-48020 x^2-27440 x^3-7840 x^4-1120 x^5-64 x^6\right ) \log ^2(x)}{16807+24010 x+13720 x^2+3920 x^3+560 x^4+32 x^5} \, dx=e^4+x^2 \left (\left (5+\frac {16}{(7+2 x)^2}\right )^2-\log ^2(x)\right ) \end {dmath*}
Time = 0.11 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.39 \begin {dmath*} \int \frac {953694 x+1262196 x^2+694960 x^3+196000 x^4+28000 x^5+1600 x^6+\left (-33614 x-48020 x^2-27440 x^3-7840 x^4-1120 x^5-64 x^6\right ) \log (x)+\left (-33614 x-48020 x^2-27440 x^3-7840 x^4-1120 x^5-64 x^6\right ) \log ^2(x)}{16807+24010 x+13720 x^2+3920 x^3+560 x^4+32 x^5} \, dx=25 x^2-\frac {8 \left (12005+13720 x+4868 x^2+560 x^3\right )}{(7+2 x)^4}-x^2 \log ^2(x) \end {dmath*}
Integrate[(953694*x + 1262196*x^2 + 694960*x^3 + 196000*x^4 + 28000*x^5 + 1600*x^6 + (-33614*x - 48020*x^2 - 27440*x^3 - 7840*x^4 - 1120*x^5 - 64*x^ 6)*Log[x] + (-33614*x - 48020*x^2 - 27440*x^3 - 7840*x^4 - 1120*x^5 - 64*x ^6)*Log[x]^2)/(16807 + 24010*x + 13720*x^2 + 3920*x^3 + 560*x^4 + 32*x^5), x]
Leaf count is larger than twice the leaf count of optimal. \(77\) vs. \(2(28)=56\).
Time = 0.74 (sec) , antiderivative size = 77, normalized size of antiderivative = 2.75, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.041, Rules used = {2007, 7239, 27, 7293, 2009}
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.
\(\displaystyle \int \frac {1600 x^6+28000 x^5+196000 x^4+694960 x^3+1262196 x^2+\left (-64 x^6-1120 x^5-7840 x^4-27440 x^3-48020 x^2-33614 x\right ) \log ^2(x)+\left (-64 x^6-1120 x^5-7840 x^4-27440 x^3-48020 x^2-33614 x\right ) \log (x)+953694 x}{32 x^5+560 x^4+3920 x^3+13720 x^2+24010 x+16807} \, dx\) |
\(\Big \downarrow \) 2007 |
\(\displaystyle \int \frac {1600 x^6+28000 x^5+196000 x^4+694960 x^3+1262196 x^2+\left (-64 x^6-1120 x^5-7840 x^4-27440 x^3-48020 x^2-33614 x\right ) \log ^2(x)+\left (-64 x^6-1120 x^5-7840 x^4-27440 x^3-48020 x^2-33614 x\right ) \log (x)+953694 x}{(2 x+7)^5}dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {2 x \left (800 x^5+14000 x^4+98000 x^3+347480 x^2+631098 x-(2 x+7)^5 \log ^2(x)-(2 x+7)^5 \log (x)+476847\right )}{(2 x+7)^5}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 2 \int \frac {x \left (800 x^5+14000 x^4+98000 x^3+347480 x^2+631098 x-(2 x+7)^5 \log ^2(x)-(2 x+7)^5 \log (x)+476847\right )}{(2 x+7)^5}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle 2 \int \left (\frac {800 x^6}{(2 x+7)^5}+\frac {14000 x^5}{(2 x+7)^5}+\frac {98000 x^4}{(2 x+7)^5}+\frac {347480 x^3}{(2 x+7)^5}+\frac {631098 x^2}{(2 x+7)^5}-\log ^2(x) x-\log (x) x+\frac {476847 x}{(2 x+7)^5}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \left (\frac {12410 x^4}{(2 x+7)^4}+\frac {25 x^2}{2}-\frac {1}{2} x^2 \log ^2(x)+\frac {42875}{2 (2 x+7)}-\frac {908087}{4 (2 x+7)^2}+\frac {2127419}{2 (2 x+7)^3}-\frac {14885661}{8 (2 x+7)^4}\right )\) |
Int[(953694*x + 1262196*x^2 + 694960*x^3 + 196000*x^4 + 28000*x^5 + 1600*x ^6 + (-33614*x - 48020*x^2 - 27440*x^3 - 7840*x^4 - 1120*x^5 - 64*x^6)*Log [x] + (-33614*x - 48020*x^2 - 27440*x^3 - 7840*x^4 - 1120*x^5 - 64*x^6)*Lo g[x]^2)/(16807 + 24010*x + 13720*x^2 + 3920*x^3 + 560*x^4 + 32*x^5),x]
2*((25*x^2)/2 - 14885661/(8*(7 + 2*x)^4) + (12410*x^4)/(7 + 2*x)^4 + 21274 19/(2*(7 + 2*x)^3) - 908087/(4*(7 + 2*x)^2) + 42875/(2*(7 + 2*x)) - (x^2*L og[x]^2)/2)
3.6.38.3.1 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]]
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^(Ex pon[Px, x]*p), x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; IntegerQ[p] && Pol yQ[Px, x] && GtQ[Expon[Px, x], 1] && NeQ[Coeff[Px, x, 0], 0]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 0.13 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.86
method | result | size |
default | \(-x^{2} \ln \left (x \right )^{2}+25 x^{2}+\frac {2024}{\left (2 x +7\right )^{2}}-\frac {896}{\left (2 x +7\right )^{3}}-\frac {560}{2 x +7}+\frac {3136}{\left (2 x +7\right )^{4}}\) | \(52\) |
parts | \(-x^{2} \ln \left (x \right )^{2}+25 x^{2}+\frac {2024}{\left (2 x +7\right )^{2}}-\frac {896}{\left (2 x +7\right )^{3}}-\frac {560}{2 x +7}+\frac {3136}{\left (2 x +7\right )^{4}}\) | \(52\) |
risch | \(-x^{2} \ln \left (x \right )^{2}+\frac {400 x^{6}+5600 x^{5}+29400 x^{4}+64120 x^{3}+21081 x^{2}-109760 x -96040}{16 x^{4}+224 x^{3}+1176 x^{2}+2744 x +2401}\) | \(64\) |
norman | \(\frac {-5151860 x -2139819 x^{2}-347480 x^{3}+5600 x^{5}+400 x^{6}-2401 x^{2} \ln \left (x \right )^{2}-2744 x^{3} \ln \left (x \right )^{2}-1176 x^{4} \ln \left (x \right )^{2}-224 x^{5} \ln \left (x \right )^{2}-16 x^{6} \ln \left (x \right )^{2}-\frac {9015755}{2}}{\left (2 x +7\right )^{4}}\) | \(79\) |
parallelrisch | \(\frac {-72126040-82429760 x -256 x^{6} \ln \left (x \right )^{2}-18816 x^{4} \ln \left (x \right )^{2}-3584 x^{5} \ln \left (x \right )^{2}-43904 x^{3} \ln \left (x \right )^{2}-38416 x^{2} \ln \left (x \right )^{2}+6400 x^{6}+89600 x^{5}-5559680 x^{3}-34237104 x^{2}}{256 x^{4}+3584 x^{3}+18816 x^{2}+43904 x +38416}\) | \(95\) |
int(((-64*x^6-1120*x^5-7840*x^4-27440*x^3-48020*x^2-33614*x)*ln(x)^2+(-64* x^6-1120*x^5-7840*x^4-27440*x^3-48020*x^2-33614*x)*ln(x)+1600*x^6+28000*x^ 5+196000*x^4+694960*x^3+1262196*x^2+953694*x)/(32*x^5+560*x^4+3920*x^3+137 20*x^2+24010*x+16807),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (27) = 54\).
Time = 0.24 (sec) , antiderivative size = 85, normalized size of antiderivative = 3.04 \begin {dmath*} \int \frac {953694 x+1262196 x^2+694960 x^3+196000 x^4+28000 x^5+1600 x^6+\left (-33614 x-48020 x^2-27440 x^3-7840 x^4-1120 x^5-64 x^6\right ) \log (x)+\left (-33614 x-48020 x^2-27440 x^3-7840 x^4-1120 x^5-64 x^6\right ) \log ^2(x)}{16807+24010 x+13720 x^2+3920 x^3+560 x^4+32 x^5} \, dx=\frac {400 \, x^{6} + 5600 \, x^{5} + 29400 \, x^{4} + 64120 \, x^{3} - {\left (16 \, x^{6} + 224 \, x^{5} + 1176 \, x^{4} + 2744 \, x^{3} + 2401 \, x^{2}\right )} \log \left (x\right )^{2} + 21081 \, x^{2} - 109760 \, x - 96040}{16 \, x^{4} + 224 \, x^{3} + 1176 \, x^{2} + 2744 \, x + 2401} \end {dmath*}
integrate(((-64*x^6-1120*x^5-7840*x^4-27440*x^3-48020*x^2-33614*x)*log(x)^ 2+(-64*x^6-1120*x^5-7840*x^4-27440*x^3-48020*x^2-33614*x)*log(x)+1600*x^6+ 28000*x^5+196000*x^4+694960*x^3+1262196*x^2+953694*x)/(32*x^5+560*x^4+3920 *x^3+13720*x^2+24010*x+16807),x, algorithm=\
(400*x^6 + 5600*x^5 + 29400*x^4 + 64120*x^3 - (16*x^6 + 224*x^5 + 1176*x^4 + 2744*x^3 + 2401*x^2)*log(x)^2 + 21081*x^2 - 109760*x - 96040)/(16*x^4 + 224*x^3 + 1176*x^2 + 2744*x + 2401)
Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (22) = 44\).
Time = 0.15 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.75 \begin {dmath*} \int \frac {953694 x+1262196 x^2+694960 x^3+196000 x^4+28000 x^5+1600 x^6+\left (-33614 x-48020 x^2-27440 x^3-7840 x^4-1120 x^5-64 x^6\right ) \log (x)+\left (-33614 x-48020 x^2-27440 x^3-7840 x^4-1120 x^5-64 x^6\right ) \log ^2(x)}{16807+24010 x+13720 x^2+3920 x^3+560 x^4+32 x^5} \, dx=- x^{2} \log {\left (x \right )}^{2} + 25 x^{2} + \frac {- 4480 x^{3} - 38944 x^{2} - 109760 x - 96040}{16 x^{4} + 224 x^{3} + 1176 x^{2} + 2744 x + 2401} \end {dmath*}
integrate(((-64*x**6-1120*x**5-7840*x**4-27440*x**3-48020*x**2-33614*x)*ln (x)**2+(-64*x**6-1120*x**5-7840*x**4-27440*x**3-48020*x**2-33614*x)*ln(x)+ 1600*x**6+28000*x**5+196000*x**4+694960*x**3+1262196*x**2+953694*x)/(32*x* *5+560*x**4+3920*x**3+13720*x**2+24010*x+16807),x)
-x**2*log(x)**2 + 25*x**2 + (-4480*x**3 - 38944*x**2 - 109760*x - 96040)/( 16*x**4 + 224*x**3 + 1176*x**2 + 2744*x + 2401)
Leaf count of result is larger than twice the leaf count of optimal. 515 vs. \(2 (27) = 54\).
Time = 0.27 (sec) , antiderivative size = 515, normalized size of antiderivative = 18.39 \begin {dmath*} \int \frac {953694 x+1262196 x^2+694960 x^3+196000 x^4+28000 x^5+1600 x^6+\left (-33614 x-48020 x^2-27440 x^3-7840 x^4-1120 x^5-64 x^6\right ) \log (x)+\left (-33614 x-48020 x^2-27440 x^3-7840 x^4-1120 x^5-64 x^6\right ) \log ^2(x)}{16807+24010 x+13720 x^2+3920 x^3+560 x^4+32 x^5} \, dx =\text {Too large to display} \end {dmath*}
integrate(((-64*x^6-1120*x^5-7840*x^4-27440*x^3-48020*x^2-33614*x)*log(x)^ 2+(-64*x^6-1120*x^5-7840*x^4-27440*x^3-48020*x^2-33614*x)*log(x)+1600*x^6+ 28000*x^5+196000*x^4+694960*x^3+1262196*x^2+953694*x)/(32*x^5+560*x^4+3920 *x^3+13720*x^2+24010*x+16807),x, algorithm=\
25*x^2 + 1715/4*(32*x^3 + 168*x^2 + 392*x + 343)*log(x)/(16*x^4 + 224*x^3 + 1176*x^2 + 2744*x + 2401) + 12005/24*(24*x^2 + 56*x + 49)*log(x)/(16*x^4 + 224*x^3 + 1176*x^2 + 2744*x + 2401) + 16807/24*(8*x + 7)*log(x)/(16*x^4 + 224*x^3 + 1176*x^2 + 2744*x + 2401) - 1/8*(76832*x^3 + 8*(16*x^6 + 224* x^5 + 1176*x^4 + 2744*x^3 + 2401*x^2)*log(x)^2 + 701092*x^2 - 392*(24*x^4 + 56*x^3 + 49*x^2)*log(x) + 2151296*x + 2235331)/(16*x^4 + 224*x^3 + 1176* x^2 + 2744*x + 2401) - 42875/24*(960*x^3 + 8400*x^2 + 25480*x + 26411)/(16 *x^4 + 224*x^3 + 1176*x^2 + 2744*x + 2401) + 8575/8*(640*x^3 + 5880*x^2 + 18424*x + 19551)/(16*x^4 + 224*x^3 + 1176*x^2 + 2744*x + 2401) + 42875/12* (384*x^3 + 3024*x^2 + 8624*x + 8575)/(16*x^4 + 224*x^3 + 1176*x^2 + 2744*x + 2401) - 43435/4*(32*x^3 + 168*x^2 + 392*x + 343)/(16*x^4 + 224*x^3 + 11 76*x^2 + 2744*x + 2401) + 1715/24*(72*x^2 + 378*x + 539)/(8*x^3 + 84*x^2 + 294*x + 343) - 105183/8*(24*x^2 + 56*x + 49)/(16*x^4 + 224*x^3 + 1176*x^2 + 2744*x + 2401) - 343/48*(8*x^2 + 70*x + 49)/(8*x^3 + 84*x^2 + 294*x + 3 43) - 1715/48*(8*x^2 - 14*x - 49)/(8*x^3 + 84*x^2 + 294*x + 343) - 158949/ 8*(8*x + 7)/(16*x^4 + 224*x^3 + 1176*x^2 + 2744*x + 2401) - 147/2*log(x)
Time = 0.28 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.93 \begin {dmath*} \int \frac {953694 x+1262196 x^2+694960 x^3+196000 x^4+28000 x^5+1600 x^6+\left (-33614 x-48020 x^2-27440 x^3-7840 x^4-1120 x^5-64 x^6\right ) \log (x)+\left (-33614 x-48020 x^2-27440 x^3-7840 x^4-1120 x^5-64 x^6\right ) \log ^2(x)}{16807+24010 x+13720 x^2+3920 x^3+560 x^4+32 x^5} \, dx=-x^{2} \log \left (x\right )^{2} + 25 \, x^{2} - \frac {8 \, {\left (560 \, x^{3} + 4868 \, x^{2} + 13720 \, x + 12005\right )}}{16 \, x^{4} + 224 \, x^{3} + 1176 \, x^{2} + 2744 \, x + 2401} \end {dmath*}
integrate(((-64*x^6-1120*x^5-7840*x^4-27440*x^3-48020*x^2-33614*x)*log(x)^ 2+(-64*x^6-1120*x^5-7840*x^4-27440*x^3-48020*x^2-33614*x)*log(x)+1600*x^6+ 28000*x^5+196000*x^4+694960*x^3+1262196*x^2+953694*x)/(32*x^5+560*x^4+3920 *x^3+13720*x^2+24010*x+16807),x, algorithm=\
-x^2*log(x)^2 + 25*x^2 - 8*(560*x^3 + 4868*x^2 + 13720*x + 12005)/(16*x^4 + 224*x^3 + 1176*x^2 + 2744*x + 2401)
Time = 15.84 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.86 \begin {dmath*} \int \frac {953694 x+1262196 x^2+694960 x^3+196000 x^4+28000 x^5+1600 x^6+\left (-33614 x-48020 x^2-27440 x^3-7840 x^4-1120 x^5-64 x^6\right ) \log (x)+\left (-33614 x-48020 x^2-27440 x^3-7840 x^4-1120 x^5-64 x^6\right ) \log ^2(x)}{16807+24010 x+13720 x^2+3920 x^3+560 x^4+32 x^5} \, dx=25\,x^2-x^2\,{\ln \left (x\right )}^2-\frac {280\,x^3+2434\,x^2+6860\,x+\frac {12005}{2}}{x^4+14\,x^3+\frac {147\,x^2}{2}+\frac {343\,x}{2}+\frac {2401}{16}} \end {dmath*}
int((953694*x - log(x)*(33614*x + 48020*x^2 + 27440*x^3 + 7840*x^4 + 1120* x^5 + 64*x^6) + 1262196*x^2 + 694960*x^3 + 196000*x^4 + 28000*x^5 + 1600*x ^6 - log(x)^2*(33614*x + 48020*x^2 + 27440*x^3 + 7840*x^4 + 1120*x^5 + 64* x^6))/(24010*x + 13720*x^2 + 3920*x^3 + 560*x^4 + 32*x^5 + 16807),x)