Integrand size = 67, antiderivative size = 29 \[ \int \frac {-900+2370 x-1836 x^2+541 x^3-165 x^4+97 x^5+4 x^7}{900 x-1440 x^2+516 x^3-72 x^4+97 x^5+4 x^6+4 x^7} \, dx=x-\frac {1}{x-\frac {6 (5-4 x)}{x (1+2 x)}}-\log (x) \]
Time = 0.02 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {-900+2370 x-1836 x^2+541 x^3-165 x^4+97 x^5+4 x^7}{900 x-1440 x^2+516 x^3-72 x^4+97 x^5+4 x^6+4 x^7} \, dx=x+\frac {-x-2 x^2}{-30+24 x+x^2+2 x^3}-\log (x) \]
Integrate[(-900 + 2370*x - 1836*x^2 + 541*x^3 - 165*x^4 + 97*x^5 + 4*x^7)/ (900*x - 1440*x^2 + 516*x^3 - 72*x^4 + 97*x^5 + 4*x^6 + 4*x^7),x]
Leaf count is larger than twice the leaf count of optimal. \(1356\) vs. \(2(29)=58\).
Time = 5.34 (sec) , antiderivative size = 1356, normalized size of antiderivative = 46.76, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {2026, 2462, 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.
\(\displaystyle \int \frac {4 x^7+97 x^5-165 x^4+541 x^3-1836 x^2+2370 x-900}{4 x^7+4 x^6+97 x^5-72 x^4+516 x^3-1440 x^2+900 x} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {4 x^7+97 x^5-165 x^4+541 x^3-1836 x^2+2370 x-900}{x \left (4 x^6+4 x^5+97 x^4-72 x^3+516 x^2-1440 x+900\right )}dx\) |
\(\Big \downarrow \) 2462 |
\(\displaystyle \int \left (\frac {2 x+1}{2 x^3+x^2+24 x-30}-\frac {12 \left (8 x^2-13 x-5\right )}{\left (2 x^3+x^2+24 x-30\right )^2}-\frac {1}{x}+1\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle x-\frac {150994368 \sqrt {\frac {3}{9658657+66060 \sqrt {19418}+20449 \left (1835+18 \sqrt {19418}\right )^{2/3}+286 \left (1835+18 \sqrt {19418}\right )^{4/3}}} \left (63233632048153+453497825892 \sqrt {19418}+\left (1835+18 \sqrt {19418}\right )^{2/3} \left (38627831641+253569696 \sqrt {19418}\right )\right ) \arctan \left (\frac {2 \left (1835+18 \sqrt {19418}\right )^{2/3} (6 x+1)-143 \sqrt [3]{1835+18 \sqrt {19418}}+18 \sqrt {19418}+1835}{\sqrt {3 \left (9658657+66060 \sqrt {19418}+20449 \left (1835+18 \sqrt {19418}\right )^{2/3}+286 \left (1835+18 \sqrt {19418}\right )^{4/3}\right )}}\right )}{\left (143+\left (1835+18 \sqrt {19418}\right )^{2/3}\right )^2 \left (20449-143 \left (1835+18 \sqrt {19418}\right )^{2/3}+\left (1835+18 \sqrt {19418}\right )^{4/3}\right )^3}+\frac {6 \left (3394489+23736 \sqrt {19418}+\left (5593-12 \sqrt {19418}\right ) \left (1835+18 \sqrt {19418}\right )^{2/3}\right ) \sqrt {\frac {3}{9658657+66060 \sqrt {19418}+20449 \left (1835+18 \sqrt {19418}\right )^{2/3}+286 \left (1835+18 \sqrt {19418}\right )^{4/3}}} \arctan \left (\frac {2 \left (1835+18 \sqrt {19418}\right )^{2/3} (6 x+1)-143 \sqrt [3]{1835+18 \sqrt {19418}}+18 \sqrt {19418}+1835}{\sqrt {3 \left (9658657+66060 \sqrt {19418}+20449 \left (1835+18 \sqrt {19418}\right )^{2/3}+286 \left (1835+18 \sqrt {19418}\right )^{4/3}\right )}}\right )}{20449-143 \left (1835+18 \sqrt {19418}\right )^{2/3}+\left (1835+18 \sqrt {19418}\right )^{4/3}}-\log (x)+\frac {25165728 \left (40813191035+295075926 \sqrt {19418}-143 \sqrt [3]{1835+18 \sqrt {19418}} \left (9658657+66060 \sqrt {19418}\right )+2 \left (1835+18 \sqrt {19418}\right )^{2/3} \left (9658657+66060 \sqrt {19418}\right )\right ) \log \left (6 \left (1835+18 \sqrt {19418}\right )^{2/3} x^2+2 \left (1835+18 \sqrt {19418}\right )^{2/3} x-143 \sqrt [3]{1835+18 \sqrt {19418}} x+18 \sqrt {19418} x+1835 x+3 \left (1238+\sqrt {19418}+\left (94+\sqrt {19418}\right ) \sqrt [3]{1835+18 \sqrt {19418}}+8 \left (1835+18 \sqrt {19418}\right )^{2/3}\right )\right )}{\left (143+\left (1835+18 \sqrt {19418}\right )^{2/3}\right )^2 \left (20449-143 \left (1835+18 \sqrt {19418}\right )^{2/3}+\left (1835+18 \sqrt {19418}\right )^{4/3}\right )^3}-\frac {\left (1835+18 \sqrt {19418}-143 \sqrt [3]{1835+18 \sqrt {19418}}+2 \left (1835+18 \sqrt {19418}\right )^{2/3}\right ) \log \left (6 \left (1835+18 \sqrt {19418}\right )^{2/3} x^2+2 \left (1835+18 \sqrt {19418}\right )^{2/3} x-143 \sqrt [3]{1835+18 \sqrt {19418}} x+18 \sqrt {19418} x+1835 x+3 \left (1238+\sqrt {19418}+\left (94+\sqrt {19418}\right ) \sqrt [3]{1835+18 \sqrt {19418}}+8 \left (1835+18 \sqrt {19418}\right )^{2/3}\right )\right )}{20449-143 \left (1835+18 \sqrt {19418}\right )^{2/3}+\left (1835+18 \sqrt {19418}\right )^{4/3}}-\frac {2 \sqrt [3]{1835+18 \sqrt {19418}} \left (11-\sqrt [3]{1835+18 \sqrt {19418}}\right ) \left (13+\sqrt [3]{1835+18 \sqrt {19418}}\right ) \log \left (\sqrt [3]{1835+18 \sqrt {19418}} (6 x+1)-\left (1835+18 \sqrt {19418}\right )^{2/3}+143\right )}{20449-143 \left (1835+18 \sqrt {19418}\right )^{2/3}+\left (1835+18 \sqrt {19418}\right )^{4/3}}+\frac {50331456 \left (1835+18 \sqrt {19418}\right )^{7/3} \left (143-2 \sqrt [3]{1835+18 \sqrt {19418}}-\left (1835+18 \sqrt {19418}\right )^{2/3}\right ) \log \left (\sqrt [3]{1835+18 \sqrt {19418}} (6 x+1)-\left (1835+18 \sqrt {19418}\right )^{2/3}+143\right )}{\left (143+\left (1835+18 \sqrt {19418}\right )^{2/3}\right )^2 \left (20449-143 \left (1835+18 \sqrt {19418}\right )^{2/3}+\left (1835+18 \sqrt {19418}\right )^{4/3}\right )^3}-\frac {16}{-2 x^3-x^2-24 x+30}-\frac {6 \sqrt [3]{1835+18 \sqrt {19418}}}{\sqrt [3]{1835+18 \sqrt {19418}} (6 x+1)-\left (1835+18 \sqrt {19418}\right )^{2/3}+143}+\frac {108 \sqrt [3]{2 \left (349524+1835 \sqrt {19418}\right )} \left (19418-\frac {\left (9709+619 \sqrt {19418}-\sqrt [3]{1835+18 \sqrt {19418}} \left (9709+47 \sqrt {19418}\right )\right ) (6 x+1)}{\left (1835+18 \sqrt {19418}\right )^{2/3}}\right )}{9709^{2/3} \left (143+\left (1835+18 \sqrt {19418}\right )^{2/3}\right ) \left (6 x-\sqrt [3]{1835+18 \sqrt {19418}}+\frac {143}{\sqrt [3]{1835+18 \sqrt {19418}}}+1\right ) \left ((6 x+1)^2-\frac {\left (143-\left (1835+18 \sqrt {19418}\right )^{2/3}\right ) (6 x+1)}{\sqrt [3]{1835+18 \sqrt {19418}}}+\left (1835+18 \sqrt {19418}\right )^{2/3}+\frac {20449}{\left (1835+18 \sqrt {19418}\right )^{2/3}}+143\right )}\) |
Int[(-900 + 2370*x - 1836*x^2 + 541*x^3 - 165*x^4 + 97*x^5 + 4*x^7)/(900*x - 1440*x^2 + 516*x^3 - 72*x^4 + 97*x^5 + 4*x^6 + 4*x^7),x]
x - 16/(30 - 24*x - x^2 - 2*x^3) - (6*(1835 + 18*Sqrt[19418])^(1/3))/(143 - (1835 + 18*Sqrt[19418])^(2/3) + (1835 + 18*Sqrt[19418])^(1/3)*(1 + 6*x)) + (108*(2*(349524 + 1835*Sqrt[19418]))^(1/3)*(19418 - ((9709 + 619*Sqrt[1 9418] - (1835 + 18*Sqrt[19418])^(1/3)*(9709 + 47*Sqrt[19418]))*(1 + 6*x))/ (1835 + 18*Sqrt[19418])^(2/3)))/(9709^(2/3)*(143 + (1835 + 18*Sqrt[19418]) ^(2/3))*(1 + 143/(1835 + 18*Sqrt[19418])^(1/3) - (1835 + 18*Sqrt[19418])^( 1/3) + 6*x)*(143 + 20449/(1835 + 18*Sqrt[19418])^(2/3) + (1835 + 18*Sqrt[1 9418])^(2/3) - ((143 - (1835 + 18*Sqrt[19418])^(2/3))*(1 + 6*x))/(1835 + 1 8*Sqrt[19418])^(1/3) + (1 + 6*x)^2)) + (6*(3394489 + 23736*Sqrt[19418] + ( 5593 - 12*Sqrt[19418])*(1835 + 18*Sqrt[19418])^(2/3))*Sqrt[3/(9658657 + 66 060*Sqrt[19418] + 20449*(1835 + 18*Sqrt[19418])^(2/3) + 286*(1835 + 18*Sqr t[19418])^(4/3))]*ArcTan[(1835 + 18*Sqrt[19418] - 143*(1835 + 18*Sqrt[1941 8])^(1/3) + 2*(1835 + 18*Sqrt[19418])^(2/3)*(1 + 6*x))/Sqrt[3*(9658657 + 6 6060*Sqrt[19418] + 20449*(1835 + 18*Sqrt[19418])^(2/3) + 286*(1835 + 18*Sq rt[19418])^(4/3))]])/(20449 - 143*(1835 + 18*Sqrt[19418])^(2/3) + (1835 + 18*Sqrt[19418])^(4/3)) - (150994368*Sqrt[3/(9658657 + 66060*Sqrt[19418] + 20449*(1835 + 18*Sqrt[19418])^(2/3) + 286*(1835 + 18*Sqrt[19418])^(4/3))]* (63233632048153 + 453497825892*Sqrt[19418] + (1835 + 18*Sqrt[19418])^(2/3) *(38627831641 + 253569696*Sqrt[19418]))*ArcTan[(1835 + 18*Sqrt[19418] - 14 3*(1835 + 18*Sqrt[19418])^(1/3) + 2*(1835 + 18*Sqrt[19418])^(2/3)*(1 + ...
3.10.40.3.1 Defintions of rubi rules used
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u*Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && GtQ [Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0 ] && RationalFunctionQ[u, x]
Time = 0.11 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.10
method | result | size |
default | \(x -\ln \left (x \right )+\frac {-x^{2}-\frac {1}{2} x}{x^{3}+\frac {1}{2} x^{2}+12 x -15}\) | \(32\) |
risch | \(x -\ln \left (x \right )+\frac {-x^{2}-\frac {1}{2} x}{x^{3}+\frac {1}{2} x^{2}+12 x -15}\) | \(32\) |
norman | \(\frac {-43 x +\frac {43}{2} x^{2}+2 x^{4}+15}{2 x^{3}+x^{2}+24 x -30}-\ln \left (x \right )\) | \(37\) |
parallelrisch | \(-\frac {4 x^{3} \ln \left (x \right )-4 x^{4}-30+2 x^{2} \ln \left (x \right )+48 x \ln \left (x \right )-43 x^{2}-60 \ln \left (x \right )+86 x}{2 \left (2 x^{3}+x^{2}+24 x -30\right )}\) | \(56\) |
int((4*x^7+97*x^5-165*x^4+541*x^3-1836*x^2+2370*x-900)/(4*x^7+4*x^6+97*x^5 -72*x^4+516*x^3-1440*x^2+900*x),x,method=_RETURNVERBOSE)
Time = 0.24 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.72 \[ \int \frac {-900+2370 x-1836 x^2+541 x^3-165 x^4+97 x^5+4 x^7}{900 x-1440 x^2+516 x^3-72 x^4+97 x^5+4 x^6+4 x^7} \, dx=\frac {2 \, x^{4} + x^{3} + 22 \, x^{2} - {\left (2 \, x^{3} + x^{2} + 24 \, x - 30\right )} \log \left (x\right ) - 31 \, x}{2 \, x^{3} + x^{2} + 24 \, x - 30} \]
integrate((4*x^7+97*x^5-165*x^4+541*x^3-1836*x^2+2370*x-900)/(4*x^7+4*x^6+ 97*x^5-72*x^4+516*x^3-1440*x^2+900*x),x, algorithm=\
Time = 0.07 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90 \[ \int \frac {-900+2370 x-1836 x^2+541 x^3-165 x^4+97 x^5+4 x^7}{900 x-1440 x^2+516 x^3-72 x^4+97 x^5+4 x^6+4 x^7} \, dx=x + \frac {- 2 x^{2} - x}{2 x^{3} + x^{2} + 24 x - 30} - \log {\left (x \right )} \]
integrate((4*x**7+97*x**5-165*x**4+541*x**3-1836*x**2+2370*x-900)/(4*x**7+ 4*x**6+97*x**5-72*x**4+516*x**3-1440*x**2+900*x),x)
Time = 0.20 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.03 \[ \int \frac {-900+2370 x-1836 x^2+541 x^3-165 x^4+97 x^5+4 x^7}{900 x-1440 x^2+516 x^3-72 x^4+97 x^5+4 x^6+4 x^7} \, dx=x - \frac {2 \, x^{2} + x}{2 \, x^{3} + x^{2} + 24 \, x - 30} - \log \left (x\right ) \]
integrate((4*x^7+97*x^5-165*x^4+541*x^3-1836*x^2+2370*x-900)/(4*x^7+4*x^6+ 97*x^5-72*x^4+516*x^3-1440*x^2+900*x),x, algorithm=\
Time = 0.27 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {-900+2370 x-1836 x^2+541 x^3-165 x^4+97 x^5+4 x^7}{900 x-1440 x^2+516 x^3-72 x^4+97 x^5+4 x^6+4 x^7} \, dx=x - \frac {2 \, x^{2} + x}{2 \, x^{3} + x^{2} + 24 \, x - 30} - \log \left ({\left | x \right |}\right ) \]
integrate((4*x^7+97*x^5-165*x^4+541*x^3-1836*x^2+2370*x-900)/(4*x^7+4*x^6+ 97*x^5-72*x^4+516*x^3-1440*x^2+900*x),x, algorithm=\
Time = 0.08 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.03 \[ \int \frac {-900+2370 x-1836 x^2+541 x^3-165 x^4+97 x^5+4 x^7}{900 x-1440 x^2+516 x^3-72 x^4+97 x^5+4 x^6+4 x^7} \, dx=x-\ln \left (x\right )-\frac {x^2+\frac {x}{2}}{x^3+\frac {x^2}{2}+12\,x-15} \]