Integrand size = 65, antiderivative size = 29 \[ \int \frac {384 x^2+288 x^3+112 x^4-84 x^5+40 x^6+10 x^7}{128+96 x+24 x^2+2 x^3+\left (64+48 x+12 x^2+x^3\right ) \log (3)} \, dx=\frac {2 x^2 \left (x+\frac {x \left (-x+x^2\right )^2}{(4+x)^2}\right )}{2+\log (3)} \]
Time = 0.02 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.59 \[ \int \frac {384 x^2+288 x^3+112 x^4-84 x^5+40 x^6+10 x^7}{128+96 x+24 x^2+2 x^3+\left (64+48 x+12 x^2+x^3\right ) \log (3)} \, dx=\frac {2 \left (478208+239104 x+29888 x^2+16 x^3+8 x^4+2 x^5-2 x^6+x^7\right )}{(4+x)^2 (2+\log (3))} \]
Integrate[(384*x^2 + 288*x^3 + 112*x^4 - 84*x^5 + 40*x^6 + 10*x^7)/(128 + 96*x + 24*x^2 + 2*x^3 + (64 + 48*x + 12*x^2 + x^3)*Log[3]),x]
(2*(478208 + 239104*x + 29888*x^2 + 16*x^3 + 8*x^4 + 2*x^5 - 2*x^6 + x^7)) /((4 + x)^2*(2 + Log[3]))
Leaf count is larger than twice the leaf count of optimal. \(80\) vs. \(2(29)=58\).
Time = 0.33 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.76, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.046, Rules used = {2007, 2389, 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.
\(\displaystyle \int \frac {10 x^7+40 x^6-84 x^5+112 x^4+288 x^3+384 x^2}{2 x^3+24 x^2+\left (x^3+12 x^2+48 x+64\right ) \log (3)+96 x+128} \, dx\) |
\(\Big \downarrow \) 2007 |
\(\displaystyle \int \frac {10 x^7+40 x^6-84 x^5+112 x^4+288 x^3+384 x^2}{\left (x \sqrt [3]{2+\log (3)}+4 \sqrt [3]{2+\log (3)}\right )^3}dx\) |
\(\Big \downarrow \) 2389 |
\(\displaystyle \int \left (\frac {10 x^4}{2+\log (3)}-\frac {80 x^3}{2+\log (3)}+\frac {396 x^2}{2+\log (3)}-\frac {1440 x}{2+\log (3)}-\frac {84480}{(x+4)^2 (2+\log (3))}+\frac {102400}{(x+4)^3 (2+\log (3))}+\frac {3680}{2+\log (3)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2 x^5}{2+\log (3)}-\frac {20 x^4}{2+\log (3)}+\frac {132 x^3}{2+\log (3)}-\frac {720 x^2}{2+\log (3)}+\frac {3680 x}{2+\log (3)}+\frac {84480}{(x+4) (2+\log (3))}-\frac {51200}{(x+4)^2 (2+\log (3))}\) |
Int[(384*x^2 + 288*x^3 + 112*x^4 - 84*x^5 + 40*x^6 + 10*x^7)/(128 + 96*x + 24*x^2 + 2*x^3 + (64 + 48*x + 12*x^2 + x^3)*Log[3]),x]
(3680*x)/(2 + Log[3]) - (720*x^2)/(2 + Log[3]) + (132*x^3)/(2 + Log[3]) - (20*x^4)/(2 + Log[3]) + (2*x^5)/(2 + Log[3]) - 51200/((4 + x)^2*(2 + Log[3 ])) + 84480/((4 + x)*(2 + Log[3]))
3.2.69.3.1 Defintions of rubi rules used
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[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand [Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p , 0] || EqQ[n, 1])
Time = 0.07 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.52
method | result | size |
parallelrisch | \(\frac {2 x^{7}-4 x^{6}+4 x^{5}+16 x^{4}+32 x^{3}}{\left (2+\ln \left (3\right )\right ) \left (x^{2}+8 x +16\right )}\) | \(44\) |
default | \(\frac {2 x^{5}-20 x^{4}+132 x^{3}-720 x^{2}+3680 x +\frac {84480}{4+x}-\frac {51200}{\left (4+x \right )^{2}}}{2+\ln \left (3\right )}\) | \(45\) |
gosper | \(\frac {2 x^{3} \left (x^{4}-2 x^{3}+2 x^{2}+8 x +16\right )}{x^{2} \ln \left (3\right )+8 x \ln \left (3\right )+2 x^{2}+16 \ln \left (3\right )+16 x +32}\) | \(51\) |
norman | \(\frac {\frac {32 x^{3}}{2+\ln \left (3\right )}+\frac {16 x^{4}}{2+\ln \left (3\right )}+\frac {4 x^{5}}{2+\ln \left (3\right )}-\frac {4 x^{6}}{2+\ln \left (3\right )}+\frac {2 x^{7}}{2+\ln \left (3\right )}}{\left (4+x \right )^{2}}\) | \(63\) |
risch | \(\frac {2 x^{5}}{2+\ln \left (3\right )}-\frac {20 x^{4}}{2+\ln \left (3\right )}+\frac {132 x^{3}}{2+\ln \left (3\right )}-\frac {720 x^{2}}{2+\ln \left (3\right )}+\frac {3680 x}{2+\ln \left (3\right )}+\frac {\left (84480 \ln \left (3\right )+168960\right ) x +286720 \ln \left (3\right )+573440}{\left (2+\ln \left (3\right )\right ) \left (x^{2} \ln \left (3\right )+8 x \ln \left (3\right )+2 x^{2}+16 \ln \left (3\right )+16 x +32\right )}\) | \(103\) |
int((10*x^7+40*x^6-84*x^5+112*x^4+288*x^3+384*x^2)/((x^3+12*x^2+48*x+64)*l n(3)+2*x^3+24*x^2+96*x+128),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.00 \[ \int \frac {384 x^2+288 x^3+112 x^4-84 x^5+40 x^6+10 x^7}{128+96 x+24 x^2+2 x^3+\left (64+48 x+12 x^2+x^3\right ) \log (3)} \, dx=\frac {2 \, {\left (x^{7} - 2 \, x^{6} + 2 \, x^{5} + 8 \, x^{4} + 16 \, x^{3} + 8960 \, x^{2} + 71680 \, x + 143360\right )}}{2 \, x^{2} + {\left (x^{2} + 8 \, x + 16\right )} \log \left (3\right ) + 16 \, x + 32} \]
integrate((10*x^7+40*x^6-84*x^5+112*x^4+288*x^3+384*x^2)/((x^3+12*x^2+48*x +64)*log(3)+2*x^3+24*x^2+96*x+128),x, algorithm=\
2*(x^7 - 2*x^6 + 2*x^5 + 8*x^4 + 16*x^3 + 8960*x^2 + 71680*x + 143360)/(2* x^2 + (x^2 + 8*x + 16)*log(3) + 16*x + 32)
Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (24) = 48\).
Time = 0.41 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.62 \[ \int \frac {384 x^2+288 x^3+112 x^4-84 x^5+40 x^6+10 x^7}{128+96 x+24 x^2+2 x^3+\left (64+48 x+12 x^2+x^3\right ) \log (3)} \, dx=\frac {2 x^{5}}{\log {\left (3 \right )} + 2} - \frac {20 x^{4}}{\log {\left (3 \right )} + 2} + \frac {132 x^{3}}{\log {\left (3 \right )} + 2} - \frac {720 x^{2}}{\log {\left (3 \right )} + 2} + \frac {3680 x}{\log {\left (3 \right )} + 2} + \frac {84480 x + 286720}{x^{2} \left (\log {\left (3 \right )} + 2\right ) + x \left (8 \log {\left (3 \right )} + 16\right ) + 16 \log {\left (3 \right )} + 32} \]
integrate((10*x**7+40*x**6-84*x**5+112*x**4+288*x**3+384*x**2)/((x**3+12*x **2+48*x+64)*ln(3)+2*x**3+24*x**2+96*x+128),x)
2*x**5/(log(3) + 2) - 20*x**4/(log(3) + 2) + 132*x**3/(log(3) + 2) - 720*x **2/(log(3) + 2) + 3680*x/(log(3) + 2) + (84480*x + 286720)/(x**2*(log(3) + 2) + x*(8*log(3) + 16) + 16*log(3) + 32)
Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (29) = 58\).
Time = 0.18 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.10 \[ \int \frac {384 x^2+288 x^3+112 x^4-84 x^5+40 x^6+10 x^7}{128+96 x+24 x^2+2 x^3+\left (64+48 x+12 x^2+x^3\right ) \log (3)} \, dx=\frac {2560 \, {\left (33 \, x + 112\right )}}{x^{2} {\left (\log \left (3\right ) + 2\right )} + 8 \, x {\left (\log \left (3\right ) + 2\right )} + 16 \, \log \left (3\right ) + 32} + \frac {2 \, {\left (x^{5} - 10 \, x^{4} + 66 \, x^{3} - 360 \, x^{2} + 1840 \, x\right )}}{\log \left (3\right ) + 2} \]
integrate((10*x^7+40*x^6-84*x^5+112*x^4+288*x^3+384*x^2)/((x^3+12*x^2+48*x +64)*log(3)+2*x^3+24*x^2+96*x+128),x, algorithm=\
2560*(33*x + 112)/(x^2*(log(3) + 2) + 8*x*(log(3) + 2) + 16*log(3) + 32) + 2*(x^5 - 10*x^4 + 66*x^3 - 360*x^2 + 1840*x)/(log(3) + 2)
Leaf count of result is larger than twice the leaf count of optimal. 236 vs. \(2 (29) = 58\).
Time = 0.26 (sec) , antiderivative size = 236, normalized size of antiderivative = 8.14 \[ \int \frac {384 x^2+288 x^3+112 x^4-84 x^5+40 x^6+10 x^7}{128+96 x+24 x^2+2 x^3+\left (64+48 x+12 x^2+x^3\right ) \log (3)} \, dx=\frac {2 \, {\left (x^{5} \log \left (3\right )^{4} + 8 \, x^{5} \log \left (3\right )^{3} - 10 \, x^{4} \log \left (3\right )^{4} + 24 \, x^{5} \log \left (3\right )^{2} - 80 \, x^{4} \log \left (3\right )^{3} + 66 \, x^{3} \log \left (3\right )^{4} + 32 \, x^{5} \log \left (3\right ) - 240 \, x^{4} \log \left (3\right )^{2} + 528 \, x^{3} \log \left (3\right )^{3} - 360 \, x^{2} \log \left (3\right )^{4} + 16 \, x^{5} - 320 \, x^{4} \log \left (3\right ) + 1584 \, x^{3} \log \left (3\right )^{2} - 2880 \, x^{2} \log \left (3\right )^{3} + 1840 \, x \log \left (3\right )^{4} - 160 \, x^{4} + 2112 \, x^{3} \log \left (3\right ) - 8640 \, x^{2} \log \left (3\right )^{2} + 14720 \, x \log \left (3\right )^{3} + 1056 \, x^{3} - 11520 \, x^{2} \log \left (3\right ) + 44160 \, x \log \left (3\right )^{2} - 5760 \, x^{2} + 58880 \, x \log \left (3\right ) + 29440 \, x\right )}}{\log \left (3\right )^{5} + 10 \, \log \left (3\right )^{4} + 40 \, \log \left (3\right )^{3} + 80 \, \log \left (3\right )^{2} + 80 \, \log \left (3\right ) + 32} + \frac {2560 \, {\left (33 \, x + 112\right )}}{{\left (x + 4\right )}^{2} {\left (\log \left (3\right ) + 2\right )}} \]
integrate((10*x^7+40*x^6-84*x^5+112*x^4+288*x^3+384*x^2)/((x^3+12*x^2+48*x +64)*log(3)+2*x^3+24*x^2+96*x+128),x, algorithm=\
2*(x^5*log(3)^4 + 8*x^5*log(3)^3 - 10*x^4*log(3)^4 + 24*x^5*log(3)^2 - 80* x^4*log(3)^3 + 66*x^3*log(3)^4 + 32*x^5*log(3) - 240*x^4*log(3)^2 + 528*x^ 3*log(3)^3 - 360*x^2*log(3)^4 + 16*x^5 - 320*x^4*log(3) + 1584*x^3*log(3)^ 2 - 2880*x^2*log(3)^3 + 1840*x*log(3)^4 - 160*x^4 + 2112*x^3*log(3) - 8640 *x^2*log(3)^2 + 14720*x*log(3)^3 + 1056*x^3 - 11520*x^2*log(3) + 44160*x*l og(3)^2 - 5760*x^2 + 58880*x*log(3) + 29440*x)/(log(3)^5 + 10*log(3)^4 + 4 0*log(3)^3 + 80*log(3)^2 + 80*log(3) + 32) + 2560*(33*x + 112)/((x + 4)^2* (log(3) + 2))
Time = 8.44 (sec) , antiderivative size = 541, normalized size of antiderivative = 18.66 \[ \int \frac {384 x^2+288 x^3+112 x^4-84 x^5+40 x^6+10 x^7}{128+96 x+24 x^2+2 x^3+\left (64+48 x+12 x^2+x^3\right ) \log (3)} \, dx =\text {Too large to display} \]
int((384*x^2 + 288*x^3 + 112*x^4 - 84*x^5 + 40*x^6 + 10*x^7)/(96*x + log(3 )*(48*x + 12*x^2 + x^3 + 64) + 24*x^2 + 2*x^3 + 128),x)
x*(288/(log(3) + 2) - ((64*log(3) + 128)*(40/(log(3) + 2) - (10*(12*log(3) + 24))/(log(3) + 2)^2))/(log(3) + 2) - ((12*log(3) + 24)*(112/(log(3) + 2 ) - (10*(64*log(3) + 128))/(log(3) + 2)^2 - ((48*log(3) + 96)*(40/(log(3) + 2) - (10*(12*log(3) + 24))/(log(3) + 2)^2))/(log(3) + 2) + ((12*log(3) + 24)*(84/(log(3) + 2) + (10*(48*log(3) + 96))/(log(3) + 2)^2 + ((12*log(3) + 24)*(40/(log(3) + 2) - (10*(12*log(3) + 24))/(log(3) + 2)^2))/(log(3) + 2)))/(log(3) + 2)))/(log(3) + 2) + ((48*log(3) + 96)*(84/(log(3) + 2) + ( 10*(48*log(3) + 96))/(log(3) + 2)^2 + ((12*log(3) + 24)*(40/(log(3) + 2) - (10*(12*log(3) + 24))/(log(3) + 2)^2))/(log(3) + 2)))/(log(3) + 2)) + (84 480*x + 286720)/(16*log(3) + x*(8*log(3) + 16) + x^2*(log(3) + 2) + 32) - x^3*(28/(log(3) + 2) + (10*(48*log(3) + 96))/(3*(log(3) + 2)^2) + ((12*log (3) + 24)*(40/(log(3) + 2) - (10*(12*log(3) + 24))/(log(3) + 2)^2))/(3*(lo g(3) + 2))) + (2*x^5)/(log(3) + 2) + x^4*(10/(log(3) + 2) - (5*(12*log(3) + 24))/(2*(log(3) + 2)^2)) + x^2*(56/(log(3) + 2) - (5*(64*log(3) + 128))/ (log(3) + 2)^2 - ((48*log(3) + 96)*(40/(log(3) + 2) - (10*(12*log(3) + 24) )/(log(3) + 2)^2))/(2*(log(3) + 2)) + ((12*log(3) + 24)*(84/(log(3) + 2) + (10*(48*log(3) + 96))/(log(3) + 2)^2 + ((12*log(3) + 24)*(40/(log(3) + 2) - (10*(12*log(3) + 24))/(log(3) + 2)^2))/(log(3) + 2)))/(2*(log(3) + 2)))