Integrand size = 22, antiderivative size = 97 \[ \int \frac {1}{(1-2 x) (2+3 x)^7 (3+5 x)} \, dx=\frac {1}{14 (2+3 x)^6}+\frac {111}{245 (2+3 x)^5}+\frac {3897}{1372 (2+3 x)^4}+\frac {45473}{2401 (2+3 x)^3}+\frac {4774713}{33614 (2+3 x)^2}+\frac {167115051}{117649 (2+3 x)}-\frac {128 \log (1-2 x)}{9058973}-\frac {5849026977 \log (2+3 x)}{823543}+\frac {78125}{11} \log (3+5 x) \]
1/14/(2+3*x)^6+111/245/(2+3*x)^5+3897/1372/(2+3*x)^4+45473/2401/(2+3*x)^3+ 4774713/33614/(2+3*x)^2+167115051/117649/(2+3*x)-128/9058973*ln(1-2*x)-584 9026977/823543*ln(2+3*x)+78125/11*ln(3+5*x)
Time = 0.06 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(1-2 x) (2+3 x)^7 (3+5 x)} \, dx=\frac {1}{14 (2+3 x)^6}+\frac {111}{245 (2+3 x)^5}+\frac {3897}{1372 (2+3 x)^4}+\frac {45473}{2401 (2+3 x)^3}+\frac {4774713}{33614 (2+3 x)^2}+\frac {167115051}{117649 (2+3 x)}-\frac {128 \log (1-2 x)}{9058973}-\frac {5849026977 \log (4+6 x)}{823543}+\frac {78125}{11} \log (6+10 x) \]
1/(14*(2 + 3*x)^6) + 111/(245*(2 + 3*x)^5) + 3897/(1372*(2 + 3*x)^4) + 454 73/(2401*(2 + 3*x)^3) + 4774713/(33614*(2 + 3*x)^2) + 167115051/(117649*(2 + 3*x)) - (128*Log[1 - 2*x])/9058973 - (5849026977*Log[4 + 6*x])/823543 + (78125*Log[6 + 10*x])/11
Time = 0.22 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {93, 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.
\(\displaystyle \int \frac {1}{(1-2 x) (3 x+2)^7 (5 x+3)} \, dx\) |
\(\Big \downarrow \) 93 |
\(\displaystyle \int \left (-\frac {17547080931}{823543 (3 x+2)}+\frac {390625}{11 (5 x+3)}-\frac {501345153}{117649 (3 x+2)^2}-\frac {14324139}{16807 (3 x+2)^3}-\frac {409257}{2401 (3 x+2)^4}-\frac {11691}{343 (3 x+2)^5}-\frac {333}{49 (3 x+2)^6}-\frac {9}{7 (3 x+2)^7}-\frac {256}{9058973 (2 x-1)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {167115051}{117649 (3 x+2)}+\frac {4774713}{33614 (3 x+2)^2}+\frac {45473}{2401 (3 x+2)^3}+\frac {3897}{1372 (3 x+2)^4}+\frac {111}{245 (3 x+2)^5}+\frac {1}{14 (3 x+2)^6}-\frac {128 \log (1-2 x)}{9058973}-\frac {5849026977 \log (3 x+2)}{823543}+\frac {78125}{11} \log (5 x+3)\) |
1/(14*(2 + 3*x)^6) + 111/(245*(2 + 3*x)^5) + 3897/(1372*(2 + 3*x)^4) + 454 73/(2401*(2 + 3*x)^3) + 4774713/(33614*(2 + 3*x)^2) + 167115051/(117649*(2 + 3*x)) - (128*Log[1 - 2*x])/9058973 - (5849026977*Log[2 + 3*x])/823543 + (78125*Log[3 + 5*x])/11
3.15.100.3.1 Defintions of rubi rules used
Int[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_] :> Int[ExpandIntegrand[(e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; Fre eQ[{a, b, c, d, e, f}, x] && IntegerQ[p]
Time = 2.55 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.61
method | result | size |
norman | \(\frac {\frac {40608957393}{117649} x^{5}+\frac {184154098887}{117649} x^{3}+\frac {208981500498}{588245} x +\frac {273433644891}{235298} x^{4}+\frac {496223395263}{470596} x^{2}+\frac {56343426549}{1176490}}{\left (2+3 x \right )^{6}}-\frac {128 \ln \left (-1+2 x \right )}{9058973}-\frac {5849026977 \ln \left (2+3 x \right )}{823543}+\frac {78125 \ln \left (3+5 x \right )}{11}\) | \(59\) |
risch | \(\frac {\frac {40608957393}{117649} x^{5}+\frac {184154098887}{117649} x^{3}+\frac {208981500498}{588245} x +\frac {273433644891}{235298} x^{4}+\frac {496223395263}{470596} x^{2}+\frac {56343426549}{1176490}}{\left (2+3 x \right )^{6}}-\frac {128 \ln \left (-1+2 x \right )}{9058973}-\frac {5849026977 \ln \left (2+3 x \right )}{823543}+\frac {78125 \ln \left (3+5 x \right )}{11}\) | \(60\) |
default | \(\frac {78125 \ln \left (3+5 x \right )}{11}-\frac {128 \ln \left (-1+2 x \right )}{9058973}+\frac {1}{14 \left (2+3 x \right )^{6}}+\frac {111}{245 \left (2+3 x \right )^{5}}+\frac {3897}{1372 \left (2+3 x \right )^{4}}+\frac {45473}{2401 \left (2+3 x \right )^{3}}+\frac {4774713}{33614 \left (2+3 x \right )^{2}}+\frac {167115051}{117649 \left (2+3 x \right )}-\frac {5849026977 \ln \left (2+3 x \right )}{823543}\) | \(80\) |
parallelrisch | \(-\frac {439221985392960 x -88942644000000000 \ln \left (x +\frac {3}{5}\right ) x^{2}+177885287646105600 \ln \left (\frac {2}{3}+x \right ) x^{3}-23718038400000000 \ln \left (x +\frac {3}{5}\right ) x +88942643823052800 \ln \left (\frac {2}{3}+x \right ) x^{2}+23718038352814080 \ln \left (\frac {2}{3}+x \right ) x +10649692829573028 x^{5}+3162725562475017 x^{6}+9666963414108000 x^{3}+14347432073052540 x^{4}+3257566473989520 x^{2}+398131200 \ln \left (x -\frac {1}{2}\right ) x^{4}+200120948601868800 \ln \left (\frac {2}{3}+x \right ) x^{4}+2635337594757120 \ln \left (\frac {2}{3}+x \right )+353894400 \ln \left (x -\frac {1}{2}\right ) x^{3}+176947200 \ln \left (x -\frac {1}{2}\right ) x^{2}+47185920 \ln \left (x -\frac {1}{2}\right ) x -2635337600000000 \ln \left (x +\frac {3}{5}\right )+120072569161121280 \ln \left (\frac {2}{3}+x \right ) x^{5}-177885288000000000 \ln \left (x +\frac {3}{5}\right ) x^{3}-120072569400000000 \ln \left (x +\frac {3}{5}\right ) x^{5}-200120949000000000 \ln \left (x +\frac {3}{5}\right ) x^{4}+30018142290280320 \ln \left (\frac {2}{3}+x \right ) x^{6}-30018142350000000 \ln \left (x +\frac {3}{5}\right ) x^{6}+5242880 \ln \left (x -\frac {1}{2}\right )+59719680 \ln \left (x -\frac {1}{2}\right ) x^{6}+238878720 \ln \left (x -\frac {1}{2}\right ) x^{5}}{5797742720 \left (2+3 x \right )^{6}}\) | \(213\) |
(40608957393/117649*x^5+184154098887/117649*x^3+208981500498/588245*x+2734 33644891/235298*x^4+496223395263/470596*x^2+56343426549/1176490)/(2+3*x)^6 -128/9058973*ln(-1+2*x)-5849026977/823543*ln(2+3*x)+78125/11*ln(3+5*x)
Leaf count of result is larger than twice the leaf count of optimal. 173 vs. \(2 (79) = 158\).
Time = 0.24 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.78 \[ \int \frac {1}{(1-2 x) (2+3 x)^7 (3+5 x)} \, dx=\frac {62537794385220 \, x^{5} + 210543906566070 \, x^{4} + 283597312285980 \, x^{3} + 191046007176255 \, x^{2} + 1286785937500 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \log \left (5 \, x + 3\right ) - 1286785934940 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \log \left (3 \, x + 2\right ) - 2560 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \log \left (2 \, x - 1\right ) + 64366302153384 \, x + 8676887688546}{181179460 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]
1/181179460*(62537794385220*x^5 + 210543906566070*x^4 + 283597312285980*x^ 3 + 191046007176255*x^2 + 1286785937500*(729*x^6 + 2916*x^5 + 4860*x^4 + 4 320*x^3 + 2160*x^2 + 576*x + 64)*log(5*x + 3) - 1286785934940*(729*x^6 + 2 916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)*log(3*x + 2) - 2560 *(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)*log(2* x - 1) + 64366302153384*x + 8676887688546)/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)
Time = 0.12 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.90 \[ \int \frac {1}{(1-2 x) (2+3 x)^7 (3+5 x)} \, dx=- \frac {- 812179147860 x^{5} - 2734336448910 x^{4} - 3683081977740 x^{3} - 2481116976315 x^{2} - 835926001992 x - 112686853098}{1715322420 x^{6} + 6861289680 x^{5} + 11435482800 x^{4} + 10164873600 x^{3} + 5082436800 x^{2} + 1355316480 x + 150590720} - \frac {128 \log {\left (x - \frac {1}{2} \right )}}{9058973} + \frac {78125 \log {\left (x + \frac {3}{5} \right )}}{11} - \frac {5849026977 \log {\left (x + \frac {2}{3} \right )}}{823543} \]
-(-812179147860*x**5 - 2734336448910*x**4 - 3683081977740*x**3 - 248111697 6315*x**2 - 835926001992*x - 112686853098)/(1715322420*x**6 + 6861289680*x **5 + 11435482800*x**4 + 10164873600*x**3 + 5082436800*x**2 + 1355316480*x + 150590720) - 128*log(x - 1/2)/9058973 + 78125*log(x + 3/5)/11 - 5849026 977*log(x + 2/3)/823543
Time = 0.20 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.87 \[ \int \frac {1}{(1-2 x) (2+3 x)^7 (3+5 x)} \, dx=\frac {3 \, {\left (270726382620 \, x^{5} + 911445482970 \, x^{4} + 1227693992580 \, x^{3} + 827038992105 \, x^{2} + 278642000664 \, x + 37562284366\right )}}{2352980 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + \frac {78125}{11} \, \log \left (5 \, x + 3\right ) - \frac {5849026977}{823543} \, \log \left (3 \, x + 2\right ) - \frac {128}{9058973} \, \log \left (2 \, x - 1\right ) \]
3/2352980*(270726382620*x^5 + 911445482970*x^4 + 1227693992580*x^3 + 82703 8992105*x^2 + 278642000664*x + 37562284366)/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64) + 78125/11*log(5*x + 3) - 5849026977/ 823543*log(3*x + 2) - 128/9058973*log(2*x - 1)
Time = 0.27 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.64 \[ \int \frac {1}{(1-2 x) (2+3 x)^7 (3+5 x)} \, dx=\frac {3 \, {\left (270726382620 \, x^{5} + 911445482970 \, x^{4} + 1227693992580 \, x^{3} + 827038992105 \, x^{2} + 278642000664 \, x + 37562284366\right )}}{2352980 \, {\left (3 \, x + 2\right )}^{6}} + \frac {78125}{11} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac {5849026977}{823543} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac {128}{9058973} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \]
3/2352980*(270726382620*x^5 + 911445482970*x^4 + 1227693992580*x^3 + 82703 8992105*x^2 + 278642000664*x + 37562284366)/(3*x + 2)^6 + 78125/11*log(abs (5*x + 3)) - 5849026977/823543*log(abs(3*x + 2)) - 128/9058973*log(abs(2*x - 1))
Time = 1.42 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.77 \[ \int \frac {1}{(1-2 x) (2+3 x)^7 (3+5 x)} \, dx=\frac {78125\,\ln \left (x+\frac {3}{5}\right )}{11}-\frac {5849026977\,\ln \left (x+\frac {2}{3}\right )}{823543}-\frac {128\,\ln \left (x-\frac {1}{2}\right )}{9058973}+\frac {\frac {55705017\,x^5}{117649}+\frac {1125241337\,x^4}{705894}+\frac {6820522181\,x^3}{3176523}+\frac {18378644269\,x^2}{12706092}+\frac {7740055574\,x}{15882615}+\frac {18781142183}{285887070}}{x^6+4\,x^5+\frac {20\,x^4}{3}+\frac {160\,x^3}{27}+\frac {80\,x^2}{27}+\frac {64\,x}{81}+\frac {64}{729}} \]
(78125*log(x + 3/5))/11 - (5849026977*log(x + 2/3))/823543 - (128*log(x - 1/2))/9058973 + ((7740055574*x)/15882615 + (18378644269*x^2)/12706092 + (6 820522181*x^3)/3176523 + (1125241337*x^4)/705894 + (55705017*x^5)/117649 + 18781142183/285887070)/((64*x)/81 + (80*x^2)/27 + (160*x^3)/27 + (20*x^4) /3 + 4*x^5 + x^6 + 64/729)