Integrand size = 83, antiderivative size = 24 \[ \int \frac {200-75 x-25 x^2-200 x^3-120 x^4+40 x^5+\left (50-80 x^3-60 x^4\right ) \log (x)}{100+100 x+25 x^2+160 x^3+160 x^4+40 x^5+64 x^6+64 x^7+16 x^8} \, dx=\frac {5 x (3-x+\log (x))}{(2+x) \left (5+4 x^3\right )} \]
Time = 0.59 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.12 \[ \int \frac {200-75 x-25 x^2-200 x^3-120 x^4+40 x^5+\left (50-80 x^3-60 x^4\right ) \log (x)}{100+100 x+25 x^2+160 x^3+160 x^4+40 x^5+64 x^6+64 x^7+16 x^8} \, dx=\frac {5 x (3-x+\log (x))}{10+5 x+8 x^3+4 x^4} \]
Integrate[(200 - 75*x - 25*x^2 - 200*x^3 - 120*x^4 + 40*x^5 + (50 - 80*x^3 - 60*x^4)*Log[x])/(100 + 100*x + 25*x^2 + 160*x^3 + 160*x^4 + 40*x^5 + 64 *x^6 + 64*x^7 + 16*x^8),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.
| \(\displaystyle \int \frac {40 x^5-120 x^4-200 x^3-25 x^2+\left (-60 x^4-80 x^3+50\right ) \log (x)-75 x+200}{16 x^8+64 x^7+64 x^6+40 x^5+160 x^4+160 x^3+25 x^2+100 x+100} \, dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {32 \left (40 x^5-120 x^4-200 x^3-25 x^2+\left (-60 x^4-80 x^3+50\right ) \log (x)-75 x+200\right )}{6561 (x+2)}-\frac {4 \left (32 x^2-55 x+92\right ) \left (40 x^5-120 x^4-200 x^3-25 x^2+\left (-60 x^4-80 x^3+50\right ) \log (x)-75 x+200\right )}{6561 \left (4 x^3+5\right )}+\frac {40 x^5-120 x^4-200 x^3-25 x^2+\left (-60 x^4-80 x^3+50\right ) \log (x)-75 x+200}{729 (x+2)^2}+\frac {4 \left (16 x^2-23 x+28\right ) \left (40 x^5-120 x^4-200 x^3-25 x^2+\left (-60 x^4-80 x^3+50\right ) \log (x)-75 x+200\right )}{243 \left (4 x^3+5\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {20 \log (x) x^3}{27 \left (4 x^3+5\right )}-\frac {5 \log (x) x}{27 (x+2)}-\frac {16 \sqrt [3]{-10} \log (x) x}{9 \left (1+\sqrt [3]{-1}\right )^4 \left (2 x+(-1)^{2/3} \sqrt [3]{10}\right )}+\frac {16 \sqrt [3]{10} \log (x) x}{81 \left (2 (-1)^{2/3} x+\sqrt [3]{10}\right )}+\frac {80 \log (x) x}{81 \left (10^{2/3} x+5\right )}+\frac {200 \left (16 x^2-23 x+28\right ) x}{729 \left (4 x^3+5\right )}-\frac {20 \left (23 x^2-28 x+20\right ) x}{243 \left (4 x^3+5\right )}-\frac {10 \left (112 x^2-80 x+115\right ) x}{729 \left (4 x^3+5\right )}+\frac {4\ 10^{2/3} \left (7+10^{2/3}\right ) \arctan \left (\frac {\sqrt [3]{5}-2\ 2^{2/3} x}{\sqrt {3} \sqrt [3]{5}}\right )}{243 \sqrt {3}}+\frac {20\ 10^{2/3} \left (23-2\ 10^{2/3}\right ) \arctan \left (\frac {\sqrt [3]{5}-2\ 2^{2/3} x}{\sqrt {3} \sqrt [3]{5}}\right )}{2187 \sqrt {3}}+\frac {16\ 10^{2/3} \left (7-2\ 10^{2/3}\right ) \arctan \left (\frac {\sqrt [3]{5}-2\ 2^{2/3} x}{\sqrt {3} \sqrt [3]{5}}\right )}{243 \sqrt {3}}-\frac {20 \sqrt [3]{10} \left (23-10 \sqrt [3]{10}\right ) \arctan \left (\frac {\sqrt [3]{5}-2\ 2^{2/3} x}{\sqrt {3} \sqrt [3]{5}}\right )}{729 \sqrt {3}}-\frac {25 \sqrt [3]{5} \left (55-16 \sqrt [3]{10}\right ) \arctan \left (\frac {\sqrt [3]{5}-2\ 2^{2/3} x}{\sqrt {3} \sqrt [3]{5}}\right )}{2187\ 2^{2/3} \sqrt {3}}-\frac {5 \sqrt [3]{5} \left (23-16 \sqrt [3]{10}\right ) \arctan \left (\frac {\sqrt [3]{5}-2\ 2^{2/3} x}{\sqrt {3} \sqrt [3]{5}}\right )}{729\ 2^{2/3} \sqrt {3}}+\frac {20 \sqrt [3]{10} \left (28-23 \sqrt [3]{10}\right ) \arctan \left (\frac {\sqrt [3]{5}-2\ 2^{2/3} x}{\sqrt {3} \sqrt [3]{5}}\right )}{729 \sqrt {3}}-\frac {8\ 2^{2/3} \sqrt [3]{5} \left (56\ 2^{2/3}-23 \sqrt [3]{5}\right ) \arctan \left (\frac {\sqrt [3]{5}-2\ 2^{2/3} x}{\sqrt {3} \sqrt [3]{5}}\right )}{729 \sqrt {3}}+\frac {10\ 2^{2/3} \sqrt [3]{5} \left (92\ 2^{2/3}-55 \sqrt [3]{5}\right ) \arctan \left (\frac {\sqrt [3]{5}-2\ 2^{2/3} x}{\sqrt {3} \sqrt [3]{5}}\right )}{729 \sqrt {3}}+\frac {8 \sqrt [3]{-10} \log \left (-2 x-(-1)^{2/3} \sqrt [3]{10}\right )}{9 \left (1+\sqrt [3]{-1}\right )^4}+\frac {5 \left (275+80 \sqrt [3]{10}+92\ 10^{2/3}\right ) \log \left (2 x+\sqrt [3]{10}\right )}{6561}+\frac {2}{729} \left (345+160 \sqrt [3]{10}+56\ 10^{2/3}\right ) \log \left (2 x+\sqrt [3]{10}\right )-\frac {10 \left (160+184 \sqrt [3]{10}+55\ 10^{2/3}\right ) \log \left (2 x+\sqrt [3]{10}\right )}{2187}-\frac {20 \left (120+28 \sqrt [3]{10}+23\ 10^{2/3}\right ) \log \left (2 x+\sqrt [3]{10}\right )}{2187}+\frac {25 \left (184+55 \sqrt [3]{10}+16\ 10^{2/3}\right ) \log \left (2 x+\sqrt [3]{10}\right )}{13122}+\frac {20 \left (42+23 \sqrt [3]{10}+10\ 10^{2/3}\right ) \log \left (2 x+\sqrt [3]{10}\right )}{2187}+\frac {8}{81} \sqrt [3]{-10} \log \left (2 (-1)^{2/3} x+\sqrt [3]{10}\right )+\frac {4}{729} 10^{2/3} \left (7-10^{2/3}\right ) \log \left (2^{2/3} x+\sqrt [3]{5}\right )+\frac {8 \sqrt [3]{10} \left (112+23 \sqrt [3]{10}\right ) \log \left (2^{2/3} x+\sqrt [3]{5}\right )}{2187}+\frac {5 \sqrt [3]{5} \left (23+16 \sqrt [3]{10}\right ) \log \left (2^{2/3} x+\sqrt [3]{5}\right )}{2187\ 2^{2/3}}-\frac {8}{81} \sqrt [3]{10} \log \left (2^{2/3} x+\sqrt [3]{5}\right )-\frac {5}{729} (-1)^{2/3} \left (2 (-10)^{2/3}-5 \sqrt [3]{-1}+32 \sqrt [3]{10}\right ) \log (x) \log \left (1-\sqrt [3]{-\frac {1}{5}} 2^{2/3} x\right )-\frac {1}{729} (-1)^{2/3} \left (8 (-10)^{2/3}+25 \sqrt [3]{-1}-16 \sqrt [3]{10}\right ) \log (x) \log \left (1-\sqrt [3]{-\frac {1}{5}} 2^{2/3} x\right )-\frac {16 i \sqrt [3]{10} \log (x) \log \left (1-\sqrt [3]{-\frac {1}{5}} 2^{2/3} x\right )}{3 \sqrt {3} \left (1+\sqrt [3]{-1}\right )^5}-\frac {5}{729} \left (5-32 \sqrt [3]{-10}+2 (-10)^{2/3}\right ) \log (x) \log \left (\frac {(-2)^{2/3} x}{\sqrt [3]{5}}+1\right )+\frac {1}{729} \left (25-16 \sqrt [3]{-10}-8 (-10)^{2/3}\right ) \log (x) \log \left (\frac {(-2)^{2/3} x}{\sqrt [3]{5}}+1\right )-\frac {5}{729} \left (5+32 \sqrt [3]{10}+2\ 10^{2/3}\right ) \log (x) \log \left (\frac {2^{2/3} x}{\sqrt [3]{5}}+1\right )+\frac {1}{729} \left (25+16 \sqrt [3]{10}-8\ 10^{2/3}\right ) \log (x) \log \left (\frac {2^{2/3} x}{\sqrt [3]{5}}+1\right )+\frac {16}{81} \sqrt [3]{10} \log (x) \log \left (\frac {2^{2/3} x}{\sqrt [3]{5}}+1\right )-\frac {32 \sqrt [3]{10} \log (x) \log \left (1-\frac {\left (1-i \sqrt {3}\right ) x}{\sqrt [3]{10}}\right )}{81 \left (1-i \sqrt {3}\right )}+\frac {10 \left (84-23 \sqrt [3]{10}-10\ 10^{2/3}\right ) \log \left (4 x^2-2 \sqrt [3]{10} x+10^{2/3}\right )}{2187}+\frac {25 \left (368-55 \sqrt [3]{10}-16\ 10^{2/3}\right ) \log \left (4 x^2-2 \sqrt [3]{10} x+10^{2/3}\right )}{26244}-\frac {10 \left (240-28 \sqrt [3]{10}-23\ 10^{2/3}\right ) \log \left (4 x^2-2 \sqrt [3]{10} x+10^{2/3}\right )}{2187}+\frac {2}{729} \left (345-80 \sqrt [3]{10}-28\ 10^{2/3}\right ) \log \left (4 x^2-2 \sqrt [3]{10} x+10^{2/3}\right )+\frac {5 \left (275-40 \sqrt [3]{10}-46\ 10^{2/3}\right ) \log \left (4 x^2-2 \sqrt [3]{10} x+10^{2/3}\right )}{6561}-\frac {5 \left (320-184 \sqrt [3]{10}-55\ 10^{2/3}\right ) \log \left (4 x^2-2 \sqrt [3]{10} x+10^{2/3}\right )}{2187}-\frac {2}{729} 10^{2/3} \left (7-10^{2/3}\right ) \log \left (2 \sqrt [3]{2} x^2-2^{2/3} \sqrt [3]{5} x+5^{2/3}\right )-\frac {4 \sqrt [3]{10} \left (112+23 \sqrt [3]{10}\right ) \log \left (2 \sqrt [3]{2} x^2-2^{2/3} \sqrt [3]{5} x+5^{2/3}\right )}{2187}-\frac {5 \sqrt [3]{5} \left (23+16 \sqrt [3]{10}\right ) \log \left (2 \sqrt [3]{2} x^2-2^{2/3} \sqrt [3]{5} x+5^{2/3}\right )}{4374\ 2^{2/3}}-\frac {5}{81} \log \left (4 x^3+5\right )-\frac {5}{729} (-1)^{2/3} \left (2 (-10)^{2/3}-5 \sqrt [3]{-1}+32 \sqrt [3]{10}\right ) \operatorname {PolyLog}\left (2,\sqrt [3]{-\frac {1}{5}} 2^{2/3} x\right )-\frac {1}{729} (-1)^{2/3} \left (8 (-10)^{2/3}+25 \sqrt [3]{-1}-16 \sqrt [3]{10}\right ) \operatorname {PolyLog}\left (2,\sqrt [3]{-\frac {1}{5}} 2^{2/3} x\right )-\frac {16 i \sqrt [3]{10} \operatorname {PolyLog}\left (2,\sqrt [3]{-\frac {1}{5}} 2^{2/3} x\right )}{3 \sqrt {3} \left (1+\sqrt [3]{-1}\right )^5}-\frac {5}{729} \left (5-32 \sqrt [3]{-10}+2 (-10)^{2/3}\right ) \operatorname {PolyLog}\left (2,-\frac {(-2)^{2/3} x}{\sqrt [3]{5}}\right )+\frac {1}{729} \left (25-16 \sqrt [3]{-10}-8 (-10)^{2/3}\right ) \operatorname {PolyLog}\left (2,-\frac {(-2)^{2/3} x}{\sqrt [3]{5}}\right )-\frac {5}{729} \left (5+32 \sqrt [3]{10}+2\ 10^{2/3}\right ) \operatorname {PolyLog}\left (2,-\frac {2^{2/3} x}{\sqrt [3]{5}}\right )+\frac {1}{729} \left (25+16 \sqrt [3]{10}-8\ 10^{2/3}\right ) \operatorname {PolyLog}\left (2,-\frac {2^{2/3} x}{\sqrt [3]{5}}\right )+\frac {16}{81} \sqrt [3]{10} \operatorname {PolyLog}\left (2,-\frac {2^{2/3} x}{\sqrt [3]{5}}\right )-\frac {32 \sqrt [3]{10} \operatorname {PolyLog}\left (2,\frac {\left (1-i \sqrt {3}\right ) x}{\sqrt [3]{10}}\right )}{81 \left (1-i \sqrt {3}\right )}-\frac {200}{9} \int \frac {x \log (x)}{\left (4 x^3+5\right )^2}dx+\frac {50}{27 (x+2)}-\frac {160 (20-(28-23 x) x)}{729 \left (4 x^3+5\right )}+\frac {25 \left (16 x^2-23 x+28\right )}{729 \left (4 x^3+5\right )}\) |
Int[(200 - 75*x - 25*x^2 - 200*x^3 - 120*x^4 + 40*x^5 + (50 - 80*x^3 - 60* x^4)*Log[x])/(100 + 100*x + 25*x^2 + 160*x^3 + 160*x^4 + 40*x^5 + 64*x^6 + 64*x^7 + 16*x^8),x]
3.30.8.3.1 Defintions of rubi rules used
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] && Gt Q[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
Time = 0.25 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.38
| method | result | size |
| norman | \(\frac {15 x -5 x^{2}+5 x \ln \left (x \right )}{4 x^{4}+8 x^{3}+5 x +10}\) | \(33\) |
| parallelrisch | \(\frac {-20 x^{2}+20 x \ln \left (x \right )+60 x}{16 x^{4}+32 x^{3}+20 x +40}\) | \(34\) |
| risch | \(\frac {5 x \ln \left (x \right )}{4 x^{4}+8 x^{3}+5 x +10}-\frac {5 x \left (-3+x \right )}{4 x^{4}+8 x^{3}+5 x +10}\) | \(47\) |
| default | \(\frac {50}{27 \left (2+x \right )}+\frac {-\frac {50}{27} x^{2}+\frac {265}{108} x -\frac {125}{108}}{x^{3}+\frac {5}{4}}-\frac {5 \ln \left (x \right ) x}{27 \left (2+x \right )}+\frac {20 \ln \left (x \right ) x \left (x^{2}-2 x +4\right )}{27 \left (4 x^{3}+5\right )}\) | \(60\) |
| parts | \(\frac {50}{27 \left (2+x \right )}+\frac {-\frac {50}{27} x^{2}+\frac {265}{108} x -\frac {125}{108}}{x^{3}+\frac {5}{4}}-\frac {5 \ln \left (x \right ) x}{27 \left (2+x \right )}+\frac {20 \ln \left (x \right ) x \left (x^{2}-2 x +4\right )}{27 \left (4 x^{3}+5\right )}\) | \(60\) |
int(((-60*x^4-80*x^3+50)*ln(x)+40*x^5-120*x^4-200*x^3-25*x^2-75*x+200)/(16 *x^8+64*x^7+64*x^6+40*x^5+160*x^4+160*x^3+25*x^2+100*x+100),x,method=_RETU RNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.29 \[ \int \frac {200-75 x-25 x^2-200 x^3-120 x^4+40 x^5+\left (50-80 x^3-60 x^4\right ) \log (x)}{100+100 x+25 x^2+160 x^3+160 x^4+40 x^5+64 x^6+64 x^7+16 x^8} \, dx=-\frac {5 \, {\left (x^{2} - x \log \left (x\right ) - 3 \, x\right )}}{4 \, x^{4} + 8 \, x^{3} + 5 \, x + 10} \]
integrate(((-60*x^4-80*x^3+50)*log(x)+40*x^5-120*x^4-200*x^3-25*x^2-75*x+2 00)/(16*x^8+64*x^7+64*x^6+40*x^5+160*x^4+160*x^3+25*x^2+100*x+100),x, algo rithm=\
Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (20) = 40\).
Time = 0.14 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.83 \[ \int \frac {200-75 x-25 x^2-200 x^3-120 x^4+40 x^5+\left (50-80 x^3-60 x^4\right ) \log (x)}{100+100 x+25 x^2+160 x^3+160 x^4+40 x^5+64 x^6+64 x^7+16 x^8} \, dx=\frac {5 x \log {\left (x \right )}}{4 x^{4} + 8 x^{3} + 5 x + 10} + \frac {- 5 x^{2} + 15 x}{4 x^{4} + 8 x^{3} + 5 x + 10} \]
integrate(((-60*x**4-80*x**3+50)*ln(x)+40*x**5-120*x**4-200*x**3-25*x**2-7 5*x+200)/(16*x**8+64*x**7+64*x**6+40*x**5+160*x**4+160*x**3+25*x**2+100*x+ 100),x)
Leaf count of result is larger than twice the leaf count of optimal. 1153 vs. \(2 (24) = 48\).
Time = 2.85 (sec) , antiderivative size = 1153, normalized size of antiderivative = 48.04 \[ \int \frac {200-75 x-25 x^2-200 x^3-120 x^4+40 x^5+\left (50-80 x^3-60 x^4\right ) \log (x)}{100+100 x+25 x^2+160 x^3+160 x^4+40 x^5+64 x^6+64 x^7+16 x^8} \, dx=\text {Too large to display} \]
integrate(((-60*x^4-80*x^3+50)*log(x)+40*x^5-120*x^4-200*x^3-25*x^2-75*x+2 00)/(16*x^8+64*x^7+64*x^6+40*x^5+160*x^4+160*x^3+25*x^2+100*x+100),x, algo rithm=\
5/39366*4^(2/3)*sqrt(3)*(689*5^(2/3)*4^(2/3) - 2848*5^(1/3))*arctan(1/60*5 ^(2/3)*4^(2/3)*sqrt(3)*(2*4^(2/3)*x - 5^(1/3)*4^(1/3))) - 2/6561*4^(2/3)*s qrt(3)*(241*5^(2/3)*4^(2/3) - 920*5^(1/3))*arctan(1/60*5^(2/3)*4^(2/3)*sqr t(3)*(2*4^(2/3)*x - 5^(1/3)*4^(1/3))) - 5/39366*4^(2/3)*sqrt(3)*(220*5^(2/ 3)*4^(2/3) - 1103*5^(1/3))*arctan(1/60*5^(2/3)*4^(2/3)*sqrt(3)*(2*4^(2/3)* x - 5^(1/3)*4^(1/3))) - 5/39366*4^(2/3)*sqrt(3)*(56*5^(2/3)*4^(2/3) - 241* 5^(1/3))*arctan(1/60*5^(2/3)*4^(2/3)*sqrt(3)*(2*4^(2/3)*x - 5^(1/3)*4^(1/3 ))) + 8/19683*4^(2/3)*sqrt(3)*(17*5^(2/3)*4^(2/3) + 44*5^(1/3))*arctan(1/6 0*5^(2/3)*4^(2/3)*sqrt(3)*(2*4^(2/3)*x - 5^(1/3)*4^(1/3))) + 4/6561*4^(2/3 )*sqrt(3)*(13*5^(2/3)*4^(2/3) - 95*5^(1/3))*arctan(1/60*5^(2/3)*4^(2/3)*sq rt(3)*(2*4^(2/3)*x - 5^(1/3)*4^(1/3))) + 1/162*4^(2/3)*sqrt(3)*(5^(2/3)*4^ (2/3) - 8*5^(1/3))*arctan(1/60*5^(2/3)*4^(2/3)*sqrt(3)*(2*4^(2/3)*x - 5^(1 /3)*4^(1/3))) + 1/78732*5^(1/3)*(9728*5^(2/3) - 4400*5^(1/3)*4^(1/3) - 551 5*4^(2/3))*log(4^(2/3)*x^2 - 5^(1/3)*4^(1/3)*x + 5^(2/3)) - 5/19683*5^(1/3 )*(1184*5^(2/3) - 689*5^(1/3)*4^(1/3) - 712*4^(2/3))*log(4^(2/3)*x^2 - 5^( 1/3)*4^(1/3)*x + 5^(2/3)) + 4/6561*5^(1/3)*(448*5^(2/3) - 241*5^(1/3)*4^(1 /3) - 230*4^(2/3))*log(4^(2/3)*x^2 - 5^(1/3)*4^(1/3)*x + 5^(2/3)) + 5/7873 2*5^(1/3)*(368*5^(2/3) - 224*5^(1/3)*4^(1/3) - 241*4^(2/3))*log(4^(2/3)*x^ 2 - 5^(1/3)*4^(1/3)*x + 5^(2/3)) - 1/6561*5^(1/3)*(275*5^(2/3) - 104*5^(1/ 3)*4^(1/3) - 190*4^(2/3))*log(4^(2/3)*x^2 - 5^(1/3)*4^(1/3)*x + 5^(2/3)...
Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (24) = 48\).
Time = 0.28 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.04 \[ \int \frac {200-75 x-25 x^2-200 x^3-120 x^4+40 x^5+\left (50-80 x^3-60 x^4\right ) \log (x)}{100+100 x+25 x^2+160 x^3+160 x^4+40 x^5+64 x^6+64 x^7+16 x^8} \, dx=\frac {5 \, x \log \left (x\right )}{4 \, x^{4} + 8 \, x^{3} + 5 \, x + 10} - \frac {5 \, {\left (x^{2} - 3 \, x\right )}}{4 \, x^{4} + 8 \, x^{3} + 5 \, x + 10} \]
integrate(((-60*x^4-80*x^3+50)*log(x)+40*x^5-120*x^4-200*x^3-25*x^2-75*x+2 00)/(16*x^8+64*x^7+64*x^6+40*x^5+160*x^4+160*x^3+25*x^2+100*x+100),x, algo rithm=\
Time = 9.11 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.12 \[ \int \frac {200-75 x-25 x^2-200 x^3-120 x^4+40 x^5+\left (50-80 x^3-60 x^4\right ) \log (x)}{100+100 x+25 x^2+160 x^3+160 x^4+40 x^5+64 x^6+64 x^7+16 x^8} \, dx=\frac {5\,x\,\left (\ln \left (x\right )-x+3\right )}{4\,x^4+8\,x^3+5\,x+10} \]