Integrand size = 21, antiderivative size = 105 \[ \int F^{a+b (c+d x)^3} (c+d x)^{17} \, dx=-\frac {F^{a+b (c+d x)^3} \left (120-120 b (c+d x)^3 \log (F)+60 b^2 (c+d x)^6 \log ^2(F)-20 b^3 (c+d x)^9 \log ^3(F)+5 b^4 (c+d x)^{12} \log ^4(F)-b^5 (c+d x)^{15} \log ^5(F)\right )}{3 b^6 d \log ^6(F)} \]
[Out]
Time = 0.07 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2249} \[ \int F^{a+b (c+d x)^3} (c+d x)^{17} \, dx=-\frac {F^{a+b (c+d x)^3} \left (-b^5 \log ^5(F) (c+d x)^{15}+5 b^4 \log ^4(F) (c+d x)^{12}-20 b^3 \log ^3(F) (c+d x)^9+60 b^2 \log ^2(F) (c+d x)^6-120 b \log (F) (c+d x)^3+120\right )}{3 b^6 d \log ^6(F)} \]
[In]
[Out]
Rule 2249
Rubi steps \begin{align*} \text {integral}& = -\frac {F^{a+b (c+d x)^3} \left (120-120 b (c+d x)^3 \log (F)+60 b^2 (c+d x)^6 \log ^2(F)-20 b^3 (c+d x)^9 \log ^3(F)+5 b^4 (c+d x)^{12} \log ^4(F)-b^5 (c+d x)^{15} \log ^5(F)\right )}{3 b^6 d \log ^6(F)} \\ \end{align*}
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.22 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.30 \[ \int F^{a+b (c+d x)^3} (c+d x)^{17} \, dx=-\frac {F^a \Gamma \left (6,-b (c+d x)^3 \log (F)\right )}{3 b^6 d \log ^6(F)} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(856\) vs. \(2(103)=206\).
Time = 4.91 (sec) , antiderivative size = 857, normalized size of antiderivative = 8.16
method | result | size |
gosper | \(\frac {\left (-120+120 \ln \left (F \right ) b \,c^{3}+120 \ln \left (F \right ) b \,d^{3} x^{3}+15 d^{14} c \,x^{14} \ln \left (F \right )^{5} b^{5}+105 d^{13} c^{2} x^{13} \ln \left (F \right )^{5} b^{5}+455 \ln \left (F \right )^{5} b^{5} c^{3} d^{12} x^{12}+1365 \ln \left (F \right )^{5} b^{5} c^{4} d^{11} x^{11}+3003 \ln \left (F \right )^{5} b^{5} c^{5} d^{10} x^{10}+5005 \ln \left (F \right )^{5} b^{5} c^{6} d^{9} x^{9}+6435 \ln \left (F \right )^{5} b^{5} c^{7} d^{8} x^{8}+6435 \ln \left (F \right )^{5} b^{5} c^{8} d^{7} x^{7}+5005 \ln \left (F \right )^{5} b^{5} c^{9} d^{6} x^{6}-60 c \,d^{11} x^{11} \ln \left (F \right )^{4} b^{4}+3003 \ln \left (F \right )^{5} b^{5} c^{10} d^{5} x^{5}-330 c^{2} d^{10} x^{10} \ln \left (F \right )^{4} b^{4}+1365 \ln \left (F \right )^{5} b^{5} c^{11} d^{4} x^{4}-1100 \ln \left (F \right )^{4} b^{4} c^{3} d^{9} x^{9}+455 \ln \left (F \right )^{5} b^{5} c^{12} d^{3} x^{3}-2475 \ln \left (F \right )^{4} b^{4} c^{4} d^{8} x^{8}+105 \ln \left (F \right )^{5} b^{5} c^{13} d^{2} x^{2}-3960 \ln \left (F \right )^{4} b^{4} c^{5} d^{7} x^{7}+15 \ln \left (F \right )^{5} b^{5} c^{14} d x -4620 \ln \left (F \right )^{4} b^{4} c^{6} d^{6} x^{6}-3960 \ln \left (F \right )^{4} b^{4} c^{7} d^{5} x^{5}-2475 \ln \left (F \right )^{4} b^{4} c^{8} d^{4} x^{4}-5 \ln \left (F \right )^{4} b^{4} c^{12}+20 \ln \left (F \right )^{3} b^{3} c^{9}-60 \ln \left (F \right )^{2} b^{2} c^{6}+\ln \left (F \right )^{5} b^{5} c^{15}+d^{15} x^{15} \ln \left (F \right )^{5} b^{5}-5 d^{12} x^{12} \ln \left (F \right )^{4} b^{4}+20 d^{9} x^{9} \ln \left (F \right )^{3} b^{3}-60 d^{6} x^{6} \ln \left (F \right )^{2} b^{2}+360 \ln \left (F \right ) b c \,d^{2} x^{2}+360 \ln \left (F \right ) b \,c^{2} d x -1100 \ln \left (F \right )^{4} b^{4} c^{9} d^{3} x^{3}+180 c \,d^{8} x^{8} \ln \left (F \right )^{3} b^{3}-330 \ln \left (F \right )^{4} b^{4} c^{10} d^{2} x^{2}+720 c^{2} d^{7} x^{7} \ln \left (F \right )^{3} b^{3}-60 \ln \left (F \right )^{4} b^{4} c^{11} d x +1680 \ln \left (F \right )^{3} b^{3} c^{3} d^{6} x^{6}+2520 \ln \left (F \right )^{3} b^{3} c^{4} d^{5} x^{5}+2520 \ln \left (F \right )^{3} b^{3} c^{5} d^{4} x^{4}+1680 \ln \left (F \right )^{3} b^{3} c^{6} d^{3} x^{3}+720 \ln \left (F \right )^{3} b^{3} c^{7} d^{2} x^{2}+180 \ln \left (F \right )^{3} b^{3} c^{8} d x -360 c \,d^{5} x^{5} \ln \left (F \right )^{2} b^{2}-900 c^{2} d^{4} x^{4} \ln \left (F \right )^{2} b^{2}-1200 \ln \left (F \right )^{2} b^{2} c^{3} d^{3} x^{3}-900 \ln \left (F \right )^{2} b^{2} c^{4} d^{2} x^{2}-360 \ln \left (F \right )^{2} b^{2} c^{5} d x \right ) F^{b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a}}{3 \ln \left (F \right )^{6} b^{6} d}\) | \(857\) |
risch | \(\frac {\left (-120+120 \ln \left (F \right ) b \,c^{3}+120 \ln \left (F \right ) b \,d^{3} x^{3}+15 d^{14} c \,x^{14} \ln \left (F \right )^{5} b^{5}+105 d^{13} c^{2} x^{13} \ln \left (F \right )^{5} b^{5}+455 \ln \left (F \right )^{5} b^{5} c^{3} d^{12} x^{12}+1365 \ln \left (F \right )^{5} b^{5} c^{4} d^{11} x^{11}+3003 \ln \left (F \right )^{5} b^{5} c^{5} d^{10} x^{10}+5005 \ln \left (F \right )^{5} b^{5} c^{6} d^{9} x^{9}+6435 \ln \left (F \right )^{5} b^{5} c^{7} d^{8} x^{8}+6435 \ln \left (F \right )^{5} b^{5} c^{8} d^{7} x^{7}+5005 \ln \left (F \right )^{5} b^{5} c^{9} d^{6} x^{6}-60 c \,d^{11} x^{11} \ln \left (F \right )^{4} b^{4}+3003 \ln \left (F \right )^{5} b^{5} c^{10} d^{5} x^{5}-330 c^{2} d^{10} x^{10} \ln \left (F \right )^{4} b^{4}+1365 \ln \left (F \right )^{5} b^{5} c^{11} d^{4} x^{4}-1100 \ln \left (F \right )^{4} b^{4} c^{3} d^{9} x^{9}+455 \ln \left (F \right )^{5} b^{5} c^{12} d^{3} x^{3}-2475 \ln \left (F \right )^{4} b^{4} c^{4} d^{8} x^{8}+105 \ln \left (F \right )^{5} b^{5} c^{13} d^{2} x^{2}-3960 \ln \left (F \right )^{4} b^{4} c^{5} d^{7} x^{7}+15 \ln \left (F \right )^{5} b^{5} c^{14} d x -4620 \ln \left (F \right )^{4} b^{4} c^{6} d^{6} x^{6}-3960 \ln \left (F \right )^{4} b^{4} c^{7} d^{5} x^{5}-2475 \ln \left (F \right )^{4} b^{4} c^{8} d^{4} x^{4}-5 \ln \left (F \right )^{4} b^{4} c^{12}+20 \ln \left (F \right )^{3} b^{3} c^{9}-60 \ln \left (F \right )^{2} b^{2} c^{6}+\ln \left (F \right )^{5} b^{5} c^{15}+d^{15} x^{15} \ln \left (F \right )^{5} b^{5}-5 d^{12} x^{12} \ln \left (F \right )^{4} b^{4}+20 d^{9} x^{9} \ln \left (F \right )^{3} b^{3}-60 d^{6} x^{6} \ln \left (F \right )^{2} b^{2}+360 \ln \left (F \right ) b c \,d^{2} x^{2}+360 \ln \left (F \right ) b \,c^{2} d x -1100 \ln \left (F \right )^{4} b^{4} c^{9} d^{3} x^{3}+180 c \,d^{8} x^{8} \ln \left (F \right )^{3} b^{3}-330 \ln \left (F \right )^{4} b^{4} c^{10} d^{2} x^{2}+720 c^{2} d^{7} x^{7} \ln \left (F \right )^{3} b^{3}-60 \ln \left (F \right )^{4} b^{4} c^{11} d x +1680 \ln \left (F \right )^{3} b^{3} c^{3} d^{6} x^{6}+2520 \ln \left (F \right )^{3} b^{3} c^{4} d^{5} x^{5}+2520 \ln \left (F \right )^{3} b^{3} c^{5} d^{4} x^{4}+1680 \ln \left (F \right )^{3} b^{3} c^{6} d^{3} x^{3}+720 \ln \left (F \right )^{3} b^{3} c^{7} d^{2} x^{2}+180 \ln \left (F \right )^{3} b^{3} c^{8} d x -360 c \,d^{5} x^{5} \ln \left (F \right )^{2} b^{2}-900 c^{2} d^{4} x^{4} \ln \left (F \right )^{2} b^{2}-1200 \ln \left (F \right )^{2} b^{2} c^{3} d^{3} x^{3}-900 \ln \left (F \right )^{2} b^{2} c^{4} d^{2} x^{2}-360 \ln \left (F \right )^{2} b^{2} c^{5} d x \right ) F^{b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a}}{3 \ln \left (F \right )^{6} b^{6} d}\) | \(857\) |
norman | \(\text {Expression too large to display}\) | \(883\) |
parallelrisch | \(\text {Expression too large to display}\) | \(1486\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 688 vs. \(2 (102) = 204\).
Time = 0.28 (sec) , antiderivative size = 688, normalized size of antiderivative = 6.55 \[ \int F^{a+b (c+d x)^3} (c+d x)^{17} \, dx=\frac {{\left ({\left (b^{5} d^{15} x^{15} + 15 \, b^{5} c d^{14} x^{14} + 105 \, b^{5} c^{2} d^{13} x^{13} + 455 \, b^{5} c^{3} d^{12} x^{12} + 1365 \, b^{5} c^{4} d^{11} x^{11} + 3003 \, b^{5} c^{5} d^{10} x^{10} + 5005 \, b^{5} c^{6} d^{9} x^{9} + 6435 \, b^{5} c^{7} d^{8} x^{8} + 6435 \, b^{5} c^{8} d^{7} x^{7} + 5005 \, b^{5} c^{9} d^{6} x^{6} + 3003 \, b^{5} c^{10} d^{5} x^{5} + 1365 \, b^{5} c^{11} d^{4} x^{4} + 455 \, b^{5} c^{12} d^{3} x^{3} + 105 \, b^{5} c^{13} d^{2} x^{2} + 15 \, b^{5} c^{14} d x + b^{5} c^{15}\right )} \log \left (F\right )^{5} - 5 \, {\left (b^{4} d^{12} x^{12} + 12 \, b^{4} c d^{11} x^{11} + 66 \, b^{4} c^{2} d^{10} x^{10} + 220 \, b^{4} c^{3} d^{9} x^{9} + 495 \, b^{4} c^{4} d^{8} x^{8} + 792 \, b^{4} c^{5} d^{7} x^{7} + 924 \, b^{4} c^{6} d^{6} x^{6} + 792 \, b^{4} c^{7} d^{5} x^{5} + 495 \, b^{4} c^{8} d^{4} x^{4} + 220 \, b^{4} c^{9} d^{3} x^{3} + 66 \, b^{4} c^{10} d^{2} x^{2} + 12 \, b^{4} c^{11} d x + b^{4} c^{12}\right )} \log \left (F\right )^{4} + 20 \, {\left (b^{3} d^{9} x^{9} + 9 \, b^{3} c d^{8} x^{8} + 36 \, b^{3} c^{2} d^{7} x^{7} + 84 \, b^{3} c^{3} d^{6} x^{6} + 126 \, b^{3} c^{4} d^{5} x^{5} + 126 \, b^{3} c^{5} d^{4} x^{4} + 84 \, b^{3} c^{6} d^{3} x^{3} + 36 \, b^{3} c^{7} d^{2} x^{2} + 9 \, b^{3} c^{8} d x + b^{3} c^{9}\right )} \log \left (F\right )^{3} - 60 \, {\left (b^{2} d^{6} x^{6} + 6 \, b^{2} c d^{5} x^{5} + 15 \, b^{2} c^{2} d^{4} x^{4} + 20 \, b^{2} c^{3} d^{3} x^{3} + 15 \, b^{2} c^{4} d^{2} x^{2} + 6 \, b^{2} c^{5} d x + b^{2} c^{6}\right )} \log \left (F\right )^{2} + 120 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \log \left (F\right ) - 120\right )} F^{b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a}}{3 \, b^{6} d \log \left (F\right )^{6}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 1170 vs. \(2 (105) = 210\).
Time = 0.36 (sec) , antiderivative size = 1170, normalized size of antiderivative = 11.14 \[ \int F^{a+b (c+d x)^3} (c+d x)^{17} \, dx=\text {Too large to display} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 1268 vs. \(2 (102) = 204\).
Time = 0.37 (sec) , antiderivative size = 1268, normalized size of antiderivative = 12.08 \[ \int F^{a+b (c+d x)^3} (c+d x)^{17} \, dx=\text {Too large to display} \]
[In]
[Out]
Exception generated. \[ \int F^{a+b (c+d x)^3} (c+d x)^{17} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
Time = 1.01 (sec) , antiderivative size = 685, normalized size of antiderivative = 6.52 \[ \int F^{a+b (c+d x)^3} (c+d x)^{17} \, dx=F^{b\,d^3\,x^3}\,F^{3\,b\,c^2\,d\,x}\,F^a\,F^{b\,c^3}\,F^{3\,b\,c\,d^2\,x^2}\,\left (\frac {b^5\,c^{15}\,{\ln \left (F\right )}^5-5\,b^4\,c^{12}\,{\ln \left (F\right )}^4+20\,b^3\,c^9\,{\ln \left (F\right )}^3-60\,b^2\,c^6\,{\ln \left (F\right )}^2+120\,b\,c^3\,\ln \left (F\right )-120}{3\,b^6\,d\,{\ln \left (F\right )}^6}+\frac {d^{14}\,x^{15}}{3\,b\,\ln \left (F\right )}+\frac {5\,c\,d^{13}\,x^{14}}{b\,\ln \left (F\right )}+\frac {5\,d^2\,x^3\,\left (91\,b^4\,c^{12}\,{\ln \left (F\right )}^4-220\,b^3\,c^9\,{\ln \left (F\right )}^3+336\,b^2\,c^6\,{\ln \left (F\right )}^2-240\,b\,c^3\,\ln \left (F\right )+24\right )}{3\,b^5\,{\ln \left (F\right )}^5}+\frac {5\,d^5\,x^6\,\left (1001\,b^3\,c^9\,{\ln \left (F\right )}^3-924\,b^2\,c^6\,{\ln \left (F\right )}^2+336\,b\,c^3\,\ln \left (F\right )-12\right )}{3\,b^4\,{\ln \left (F\right )}^4}+\frac {5\,d^8\,x^9\,\left (1001\,b^2\,c^6\,{\ln \left (F\right )}^2-220\,b\,c^3\,\ln \left (F\right )+4\right )}{3\,b^3\,{\ln \left (F\right )}^3}+\frac {5\,d^{11}\,x^{12}\,\left (91\,b\,c^3\,\ln \left (F\right )-1\right )}{3\,b^2\,{\ln \left (F\right )}^2}+\frac {35\,c^2\,d^{12}\,x^{13}}{b\,\ln \left (F\right )}+\frac {5\,c^2\,x\,\left (b^4\,c^{12}\,{\ln \left (F\right )}^4-4\,b^3\,c^9\,{\ln \left (F\right )}^3+12\,b^2\,c^6\,{\ln \left (F\right )}^2-24\,b\,c^3\,\ln \left (F\right )+24\right )}{b^5\,{\ln \left (F\right )}^5}+\frac {5\,c^2\,d^3\,x^4\,\left (91\,b^3\,c^9\,{\ln \left (F\right )}^3-165\,b^2\,c^6\,{\ln \left (F\right )}^2+168\,b\,c^3\,\ln \left (F\right )-60\right )}{b^4\,{\ln \left (F\right )}^4}+\frac {15\,c^2\,d^6\,x^7\,\left (143\,b^2\,c^6\,{\ln \left (F\right )}^2-88\,b\,c^3\,\ln \left (F\right )+16\right )}{b^3\,{\ln \left (F\right )}^3}+\frac {11\,c^2\,d^9\,x^{10}\,\left (91\,b\,c^3\,\ln \left (F\right )-10\right )}{b^2\,{\ln \left (F\right )}^2}+\frac {5\,c\,d\,x^2\,\left (7\,b^4\,c^{12}\,{\ln \left (F\right )}^4-22\,b^3\,c^9\,{\ln \left (F\right )}^3+48\,b^2\,c^6\,{\ln \left (F\right )}^2-60\,b\,c^3\,\ln \left (F\right )+24\right )}{b^5\,{\ln \left (F\right )}^5}+\frac {c\,d^4\,x^5\,\left (1001\,b^3\,c^9\,{\ln \left (F\right )}^3-1320\,b^2\,c^6\,{\ln \left (F\right )}^2+840\,b\,c^3\,\ln \left (F\right )-120\right )}{b^4\,{\ln \left (F\right )}^4}+\frac {15\,c\,d^7\,x^8\,\left (143\,b^2\,c^6\,{\ln \left (F\right )}^2-55\,b\,c^3\,\ln \left (F\right )+4\right )}{b^3\,{\ln \left (F\right )}^3}+\frac {5\,c\,d^{10}\,x^{11}\,\left (91\,b\,c^3\,\ln \left (F\right )-4\right )}{b^2\,{\ln \left (F\right )}^2}\right ) \]
[In]
[Out]