Integrand size = 21, antiderivative size = 113 \[ \int \frac {F^{a+\frac {b}{(c+d x)^3}}}{(c+d x)^{19}} \, dx=\frac {F^{a+\frac {b}{(c+d x)^3}} \left (120 (c+d x)^{15}-120 b (c+d x)^{12} \log (F)+60 b^2 (c+d x)^9 \log ^2(F)-20 b^3 (c+d x)^6 \log ^3(F)+5 b^4 (c+d x)^3 \log ^4(F)-b^5 \log ^5(F)\right )}{3 b^6 d (c+d x)^{15} \log ^6(F)} \]
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Time = 0.06 (sec) , antiderivative size = 113, 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} \[ \int \frac {F^{a+\frac {b}{(c+d x)^3}}}{(c+d x)^{19}} \, dx=\frac {F^{a+\frac {b}{(c+d x)^3}} \left (-b^5 \log ^5(F)+5 b^4 \log ^4(F) (c+d x)^3-20 b^3 \log ^3(F) (c+d x)^6+60 b^2 \log ^2(F) (c+d x)^9-120 b \log (F) (c+d x)^{12}+120 (c+d x)^{15}\right )}{3 b^6 d \log ^6(F) (c+d x)^{15}} \]
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Rule 2249
Rubi steps \begin{align*} \text {integral}& = \frac {F^{a+\frac {b}{(c+d x)^3}} \left (120 (c+d x)^{15}-120 b (c+d x)^{12} \log (F)+60 b^2 (c+d x)^9 \log ^2(F)-20 b^3 (c+d x)^6 \log ^3(F)+5 b^4 (c+d x)^3 \log ^4(F)-b^5 \log ^5(F)\right )}{3 b^6 d (c+d x)^{15} \log ^6(F)} \\ \end{align*}
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.27 \[ \int \frac {F^{a+\frac {b}{(c+d x)^3}}}{(c+d x)^{19}} \, dx=\frac {F^a \Gamma \left (6,-\frac {b \log (F)}{(c+d x)^3}\right )}{3 b^6 d \log ^6(F)} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(732\) vs. \(2(111)=222\).
Time = 17.86 (sec) , antiderivative size = 733, normalized size of antiderivative = 6.49
method | result | size |
risch | \(-\frac {\left (120 c \,d^{5} x^{5} b^{3} \ln \left (F \right )^{3}+300 \ln \left (F \right )^{3} b^{3} c^{2} d^{4} x^{4}+400 \ln \left (F \right )^{3} b^{3} c^{3} d^{3} x^{3}+300 \ln \left (F \right )^{3} b^{3} c^{4} d^{2} x^{2}+120 \ln \left (F \right )^{3} b^{3} c^{5} d x +120 \ln \left (F \right ) b \,c^{12}-60 \ln \left (F \right )^{2} b^{2} c^{9}-5 \ln \left (F \right )^{4} b^{4} c^{3}-120 c^{15}+b^{5} \ln \left (F \right )^{5}+120 \ln \left (F \right ) b \,d^{12} x^{12}-60 \ln \left (F \right )^{2} b^{2} d^{9} x^{9}+1440 \ln \left (F \right ) b c \,d^{11} x^{11}+7920 \ln \left (F \right ) b \,c^{2} d^{10} x^{10}+26400 \ln \left (F \right ) b \,c^{3} d^{9} x^{9}+59400 \ln \left (F \right ) b \,c^{4} d^{8} x^{8}-540 \ln \left (F \right )^{2} b^{2} c \,d^{8} x^{8}+95040 \ln \left (F \right ) b \,c^{5} d^{7} x^{7}-2160 \ln \left (F \right )^{2} b^{2} c^{2} d^{7} x^{7}+110880 \ln \left (F \right ) b \,c^{6} d^{6} x^{6}-5040 \ln \left (F \right )^{2} b^{2} c^{3} d^{6} x^{6}+95040 \ln \left (F \right ) b \,c^{7} d^{5} x^{5}-7560 \ln \left (F \right )^{2} b^{2} c^{4} d^{5} x^{5}+59400 \ln \left (F \right ) b \,c^{8} d^{4} x^{4}-7560 \ln \left (F \right )^{2} b^{2} c^{5} d^{4} x^{4}+26400 \ln \left (F \right ) b \,c^{9} d^{3} x^{3}-5040 \ln \left (F \right )^{2} b^{2} c^{6} d^{3} x^{3}+7920 \ln \left (F \right ) b \,c^{10} d^{2} x^{2}-2160 \ln \left (F \right )^{2} b^{2} c^{7} d^{2} x^{2}+1440 \ln \left (F \right ) b \,c^{11} d x -540 \ln \left (F \right )^{2} b^{2} c^{8} d x -15 \ln \left (F \right )^{4} b^{4} c \,d^{2} x^{2}-5 \ln \left (F \right )^{4} b^{4} d^{3} x^{3}+20 \ln \left (F \right )^{3} b^{3} c^{6}-120 d^{15} x^{15}+20 d^{6} x^{6} b^{3} \ln \left (F \right )^{3}-1800 c \,d^{14} x^{14}-12600 c^{2} d^{13} x^{13}-54600 c^{3} d^{12} x^{12}-163800 c^{4} d^{11} x^{11}-360360 c^{5} d^{10} x^{10}-600600 c^{6} d^{9} x^{9}-772200 c^{7} d^{8} x^{8}-772200 c^{8} d^{7} x^{7}-600600 c^{9} d^{6} x^{6}-360360 c^{10} d^{5} x^{5}-163800 c^{11} d^{4} x^{4}-54600 c^{12} d^{3} x^{3}-12600 c^{13} d^{2} x^{2}-1800 c^{14} d x -15 \ln \left (F \right )^{4} b^{4} c^{2} d x \right ) F^{\frac {a \,d^{3} x^{3}+3 a c \,d^{2} x^{2}+3 a \,c^{2} d x +a \,c^{3}+b}{\left (d x +c \right )^{3}}}}{3 \ln \left (F \right )^{6} b^{6} d \left (d x +c \right )^{15}}\) | \(733\) |
parallelrisch | \(\text {Expression too large to display}\) | \(1382\) |
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Leaf count of result is larger than twice the leaf count of optimal. 863 vs. \(2 (111) = 222\).
Time = 0.38 (sec) , antiderivative size = 863, normalized size of antiderivative = 7.64 \[ \int \frac {F^{a+\frac {b}{(c+d x)^3}}}{(c+d x)^{19}} \, dx=\frac {{\left (120 \, d^{15} x^{15} + 1800 \, c d^{14} x^{14} + 12600 \, c^{2} d^{13} x^{13} + 54600 \, c^{3} d^{12} x^{12} + 163800 \, c^{4} d^{11} x^{11} + 360360 \, c^{5} d^{10} x^{10} + 600600 \, c^{6} d^{9} x^{9} + 772200 \, c^{7} d^{8} x^{8} + 772200 \, c^{8} d^{7} x^{7} + 600600 \, c^{9} d^{6} x^{6} + 360360 \, c^{10} d^{5} x^{5} + 163800 \, c^{11} d^{4} x^{4} + 54600 \, c^{12} d^{3} x^{3} + 12600 \, c^{13} d^{2} x^{2} + 1800 \, c^{14} d x + 120 \, c^{15} - b^{5} \log \left (F\right )^{5} + 5 \, {\left (b^{4} d^{3} x^{3} + 3 \, b^{4} c d^{2} x^{2} + 3 \, b^{4} c^{2} d x + b^{4} c^{3}\right )} \log \left (F\right )^{4} - 20 \, {\left (b^{3} d^{6} x^{6} + 6 \, b^{3} c d^{5} x^{5} + 15 \, b^{3} c^{2} d^{4} x^{4} + 20 \, b^{3} c^{3} d^{3} x^{3} + 15 \, b^{3} c^{4} d^{2} x^{2} + 6 \, b^{3} c^{5} d x + b^{3} c^{6}\right )} \log \left (F\right )^{3} + 60 \, {\left (b^{2} d^{9} x^{9} + 9 \, b^{2} c d^{8} x^{8} + 36 \, b^{2} c^{2} d^{7} x^{7} + 84 \, b^{2} c^{3} d^{6} x^{6} + 126 \, b^{2} c^{4} d^{5} x^{5} + 126 \, b^{2} c^{5} d^{4} x^{4} + 84 \, b^{2} c^{6} d^{3} x^{3} + 36 \, b^{2} c^{7} d^{2} x^{2} + 9 \, b^{2} c^{8} d x + b^{2} c^{9}\right )} \log \left (F\right )^{2} - 120 \, {\left (b d^{12} x^{12} + 12 \, b c d^{11} x^{11} + 66 \, b c^{2} d^{10} x^{10} + 220 \, b c^{3} d^{9} x^{9} + 495 \, b c^{4} d^{8} x^{8} + 792 \, b c^{5} d^{7} x^{7} + 924 \, b c^{6} d^{6} x^{6} + 792 \, b c^{7} d^{5} x^{5} + 495 \, b c^{8} d^{4} x^{4} + 220 \, b c^{9} d^{3} x^{3} + 66 \, b c^{10} d^{2} x^{2} + 12 \, b c^{11} d x + b c^{12}\right )} \log \left (F\right )\right )} F^{\frac {a d^{3} x^{3} + 3 \, a c d^{2} x^{2} + 3 \, a c^{2} d x + a c^{3} + b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}}}{3 \, {\left (b^{6} d^{16} x^{15} + 15 \, b^{6} c d^{15} x^{14} + 105 \, b^{6} c^{2} d^{14} x^{13} + 455 \, b^{6} c^{3} d^{13} x^{12} + 1365 \, b^{6} c^{4} d^{12} x^{11} + 3003 \, b^{6} c^{5} d^{11} x^{10} + 5005 \, b^{6} c^{6} d^{10} x^{9} + 6435 \, b^{6} c^{7} d^{9} x^{8} + 6435 \, b^{6} c^{8} d^{8} x^{7} + 5005 \, b^{6} c^{9} d^{7} x^{6} + 3003 \, b^{6} c^{10} d^{6} x^{5} + 1365 \, b^{6} c^{11} d^{5} x^{4} + 455 \, b^{6} c^{12} d^{4} x^{3} + 105 \, b^{6} c^{13} d^{3} x^{2} + 15 \, b^{6} c^{14} d^{2} x + b^{6} c^{15} d\right )} \log \left (F\right )^{6}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1096 vs. \(2 (110) = 220\).
Time = 52.11 (sec) , antiderivative size = 1096, normalized size of antiderivative = 9.70 \[ \int \frac {F^{a+\frac {b}{(c+d x)^3}}}{(c+d x)^{19}} \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1085 vs. \(2 (111) = 222\).
Time = 0.29 (sec) , antiderivative size = 1085, normalized size of antiderivative = 9.60 \[ \int \frac {F^{a+\frac {b}{(c+d x)^3}}}{(c+d x)^{19}} \, dx=\text {Too large to display} \]
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\[ \int \frac {F^{a+\frac {b}{(c+d x)^3}}}{(c+d x)^{19}} \, dx=\int { \frac {F^{a + \frac {b}{{\left (d x + c\right )}^{3}}}}{{\left (d x + c\right )}^{19}} \,d x } \]
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Time = 2.54 (sec) , antiderivative size = 854, normalized size of antiderivative = 7.56 \[ \int \frac {F^{a+\frac {b}{(c+d x)^3}}}{(c+d x)^{19}} \, dx=\frac {F^a\,F^{\frac {b}{c^3+3\,c^2\,d\,x+3\,c\,d^2\,x^2+d^3\,x^3}}\,\left (\frac {40\,x^{15}}{b^6\,d\,{\ln \left (F\right )}^6}-\frac {b^5\,{\ln \left (F\right )}^5-5\,b^4\,c^3\,{\ln \left (F\right )}^4+20\,b^3\,c^6\,{\ln \left (F\right )}^3-60\,b^2\,c^9\,{\ln \left (F\right )}^2+120\,b\,c^{12}\,\ln \left (F\right )-120\,c^{15}}{3\,b^6\,d^{16}\,{\ln \left (F\right )}^6}+\frac {600\,c\,x^{14}}{b^6\,d^2\,{\ln \left (F\right )}^6}+\frac {4200\,c^2\,x^{13}}{b^6\,d^3\,{\ln \left (F\right )}^6}+\frac {5\,x^3\,\left (b^4\,{\ln \left (F\right )}^4-80\,b^3\,c^3\,{\ln \left (F\right )}^3+1008\,b^2\,c^6\,{\ln \left (F\right )}^2-5280\,b\,c^9\,\ln \left (F\right )+10920\,c^{12}\right )}{3\,b^6\,d^{13}\,{\ln \left (F\right )}^6}-\frac {20\,x^6\,\left (b^3\,{\ln \left (F\right )}^3-252\,b^2\,c^3\,{\ln \left (F\right )}^2+5544\,b\,c^6\,\ln \left (F\right )-30030\,c^9\right )}{3\,b^6\,d^{10}\,{\ln \left (F\right )}^6}+\frac {20\,x^9\,\left (b^2\,{\ln \left (F\right )}^2-440\,b\,c^3\,\ln \left (F\right )+10010\,c^6\right )}{b^6\,d^7\,{\ln \left (F\right )}^6}-\frac {40\,x^{12}\,\left (b\,\ln \left (F\right )-455\,c^3\right )}{b^6\,d^4\,{\ln \left (F\right )}^6}+\frac {5\,c^2\,x\,\left (b^4\,{\ln \left (F\right )}^4-8\,b^3\,c^3\,{\ln \left (F\right )}^3+36\,b^2\,c^6\,{\ln \left (F\right )}^2-96\,b\,c^9\,\ln \left (F\right )+120\,c^{12}\right )}{b^6\,d^{15}\,{\ln \left (F\right )}^6}+\frac {5\,c\,x^2\,\left (b^4\,{\ln \left (F\right )}^4-20\,b^3\,c^3\,{\ln \left (F\right )}^3+144\,b^2\,c^6\,{\ln \left (F\right )}^2-528\,b\,c^9\,\ln \left (F\right )+840\,c^{12}\right )}{b^6\,d^{14}\,{\ln \left (F\right )}^6}-\frac {40\,c\,x^5\,\left (b^3\,{\ln \left (F\right )}^3-63\,b^2\,c^3\,{\ln \left (F\right )}^2+792\,b\,c^6\,\ln \left (F\right )-3003\,c^9\right )}{b^6\,d^{11}\,{\ln \left (F\right )}^6}+\frac {180\,c\,x^8\,\left (b^2\,{\ln \left (F\right )}^2-110\,b\,c^3\,\ln \left (F\right )+1430\,c^6\right )}{b^6\,d^8\,{\ln \left (F\right )}^6}-\frac {120\,c\,x^{11}\,\left (4\,b\,\ln \left (F\right )-455\,c^3\right )}{b^6\,d^5\,{\ln \left (F\right )}^6}-\frac {20\,c^2\,x^4\,\left (5\,b^3\,{\ln \left (F\right )}^3-126\,b^2\,c^3\,{\ln \left (F\right )}^2+990\,b\,c^6\,\ln \left (F\right )-2730\,c^9\right )}{b^6\,d^{12}\,{\ln \left (F\right )}^6}+\frac {360\,c^2\,x^7\,\left (2\,b^2\,{\ln \left (F\right )}^2-88\,b\,c^3\,\ln \left (F\right )+715\,c^6\right )}{b^6\,d^9\,{\ln \left (F\right )}^6}-\frac {1320\,c^2\,x^{10}\,\left (2\,b\,\ln \left (F\right )-91\,c^3\right )}{b^6\,d^6\,{\ln \left (F\right )}^6}\right )}{x^{15}+\frac {c^{15}}{d^{15}}+\frac {15\,c\,x^{14}}{d}+\frac {15\,c^{14}\,x}{d^{14}}+\frac {105\,c^2\,x^{13}}{d^2}+\frac {455\,c^3\,x^{12}}{d^3}+\frac {1365\,c^4\,x^{11}}{d^4}+\frac {3003\,c^5\,x^{10}}{d^5}+\frac {5005\,c^6\,x^9}{d^6}+\frac {6435\,c^7\,x^8}{d^7}+\frac {6435\,c^8\,x^7}{d^8}+\frac {5005\,c^9\,x^6}{d^9}+\frac {3003\,c^{10}\,x^5}{d^{10}}+\frac {1365\,c^{11}\,x^4}{d^{11}}+\frac {455\,c^{12}\,x^3}{d^{12}}+\frac {105\,c^{13}\,x^2}{d^{13}}} \]
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