Optimal. Leaf size=113 \[ \frac {F^{a+\frac {b}{(c+d x)^3}} \left (-b^5 \log ^5(F)+5 b^4 \log ^4(F) (c+d x)^3-20 b^3 \log ^3(F) (c+d x)^6+60 b^2 \log ^2(F) (c+d x)^9-120 b \log (F) (c+d x)^{12}+120 (c+d x)^{15}\right )}{3 b^6 d \log ^6(F) (c+d x)^{15}} \]
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Rubi [C] time = 0.04, antiderivative size = 31, normalized size of antiderivative = 0.27, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2218} \[ \frac {F^a \text {Gamma}\left (6,-\frac {b \log (F)}{(c+d x)^3}\right )}{3 b^6 d \log ^6(F)} \]
Antiderivative was successfully verified.
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Rule 2218
Rubi steps
\begin {align*} \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)}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 31, normalized size = 0.27 \[ \frac {F^a \Gamma \left (6,-\frac {b \log (F)}{(c+d x)^3}\right )}{3 b^6 d \log ^6(F)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.58, size = 863, normalized size = 7.64 \[ \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 \relax (F)^{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 \relax (F)^{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 \relax (F)^{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 \relax (F)^{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 \relax (F)\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 \relax (F)^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {F^{a + \frac {b}{{\left (d x + c\right )}^{3}}}}{{\left (d x + c\right )}^{19}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 733, normalized size = 6.49 \[ -\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}+120 b \,d^{12} x^{12} \ln \relax (F )-163800 c^{4} d^{11} x^{11}+1440 b c \,d^{11} x^{11} \ln \relax (F )-360360 c^{5} d^{10} x^{10}+7920 b \,c^{2} d^{10} x^{10} \ln \relax (F )-600600 c^{6} d^{9} x^{9}+26400 b \,c^{3} d^{9} x^{9} \ln \relax (F )-772200 c^{7} d^{8} x^{8}-60 b^{2} d^{9} x^{9} \ln \relax (F )^{2}+59400 b \,c^{4} d^{8} x^{8} \ln \relax (F )-772200 c^{8} d^{7} x^{7}-540 b^{2} c \,d^{8} x^{8} \ln \relax (F )^{2}+95040 b \,c^{5} d^{7} x^{7} \ln \relax (F )-600600 c^{9} d^{6} x^{6}-2160 b^{2} c^{2} d^{7} x^{7} \ln \relax (F )^{2}+110880 b \,c^{6} d^{6} x^{6} \ln \relax (F )-360360 c^{10} d^{5} x^{5}-5040 b^{2} c^{3} d^{6} x^{6} \ln \relax (F )^{2}+95040 b \,c^{7} d^{5} x^{5} \ln \relax (F )-163800 c^{11} d^{4} x^{4}+20 b^{3} d^{6} x^{6} \ln \relax (F )^{3}-7560 b^{2} c^{4} d^{5} x^{5} \ln \relax (F )^{2}+59400 b \,c^{8} d^{4} x^{4} \ln \relax (F )-54600 c^{12} d^{3} x^{3}+120 b^{3} c \,d^{5} x^{5} \ln \relax (F )^{3}-7560 b^{2} c^{5} d^{4} x^{4} \ln \relax (F )^{2}+26400 b \,c^{9} d^{3} x^{3} \ln \relax (F )-12600 c^{13} d^{2} x^{2}+300 b^{3} c^{2} d^{4} x^{4} \ln \relax (F )^{3}-5040 b^{2} c^{6} d^{3} x^{3} \ln \relax (F )^{2}+7920 b \,c^{10} d^{2} x^{2} \ln \relax (F )-1800 c^{14} d x +400 b^{3} c^{3} d^{3} x^{3} \ln \relax (F )^{3}-2160 b^{2} c^{7} d^{2} x^{2} \ln \relax (F )^{2}+1440 b \,c^{11} d x \ln \relax (F )-120 c^{15}-5 b^{4} d^{3} x^{3} \ln \relax (F )^{4}+300 b^{3} c^{4} d^{2} x^{2} \ln \relax (F )^{3}-540 b^{2} c^{8} d x \ln \relax (F )^{2}+120 b \,c^{12} \ln \relax (F )-15 b^{4} c \,d^{2} x^{2} \ln \relax (F )^{4}+120 b^{3} c^{5} d x \ln \relax (F )^{3}-60 b^{2} c^{9} \ln \relax (F )^{2}-15 b^{4} c^{2} d x \ln \relax (F )^{4}+20 b^{3} c^{6} \ln \relax (F )^{3}-5 b^{4} c^{3} \ln \relax (F )^{4}+b^{5} \ln \relax (F )^{5}\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 \left (d x +c \right )^{15} b^{6} d \ln \relax (F )^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.34, size = 1085, normalized size = 9.60 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.77, size = 854, normalized size = 7.56 \[ \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 \relax (F)}^6}-\frac {b^5\,{\ln \relax (F)}^5-5\,b^4\,c^3\,{\ln \relax (F)}^4+20\,b^3\,c^6\,{\ln \relax (F)}^3-60\,b^2\,c^9\,{\ln \relax (F)}^2+120\,b\,c^{12}\,\ln \relax (F)-120\,c^{15}}{3\,b^6\,d^{16}\,{\ln \relax (F)}^6}+\frac {600\,c\,x^{14}}{b^6\,d^2\,{\ln \relax (F)}^6}+\frac {4200\,c^2\,x^{13}}{b^6\,d^3\,{\ln \relax (F)}^6}+\frac {5\,x^3\,\left (b^4\,{\ln \relax (F)}^4-80\,b^3\,c^3\,{\ln \relax (F)}^3+1008\,b^2\,c^6\,{\ln \relax (F)}^2-5280\,b\,c^9\,\ln \relax (F)+10920\,c^{12}\right )}{3\,b^6\,d^{13}\,{\ln \relax (F)}^6}-\frac {20\,x^6\,\left (b^3\,{\ln \relax (F)}^3-252\,b^2\,c^3\,{\ln \relax (F)}^2+5544\,b\,c^6\,\ln \relax (F)-30030\,c^9\right )}{3\,b^6\,d^{10}\,{\ln \relax (F)}^6}+\frac {20\,x^9\,\left (b^2\,{\ln \relax (F)}^2-440\,b\,c^3\,\ln \relax (F)+10010\,c^6\right )}{b^6\,d^7\,{\ln \relax (F)}^6}-\frac {40\,x^{12}\,\left (b\,\ln \relax (F)-455\,c^3\right )}{b^6\,d^4\,{\ln \relax (F)}^6}+\frac {5\,c^2\,x\,\left (b^4\,{\ln \relax (F)}^4-8\,b^3\,c^3\,{\ln \relax (F)}^3+36\,b^2\,c^6\,{\ln \relax (F)}^2-96\,b\,c^9\,\ln \relax (F)+120\,c^{12}\right )}{b^6\,d^{15}\,{\ln \relax (F)}^6}+\frac {5\,c\,x^2\,\left (b^4\,{\ln \relax (F)}^4-20\,b^3\,c^3\,{\ln \relax (F)}^3+144\,b^2\,c^6\,{\ln \relax (F)}^2-528\,b\,c^9\,\ln \relax (F)+840\,c^{12}\right )}{b^6\,d^{14}\,{\ln \relax (F)}^6}-\frac {40\,c\,x^5\,\left (b^3\,{\ln \relax (F)}^3-63\,b^2\,c^3\,{\ln \relax (F)}^2+792\,b\,c^6\,\ln \relax (F)-3003\,c^9\right )}{b^6\,d^{11}\,{\ln \relax (F)}^6}+\frac {180\,c\,x^8\,\left (b^2\,{\ln \relax (F)}^2-110\,b\,c^3\,\ln \relax (F)+1430\,c^6\right )}{b^6\,d^8\,{\ln \relax (F)}^6}-\frac {120\,c\,x^{11}\,\left (4\,b\,\ln \relax (F)-455\,c^3\right )}{b^6\,d^5\,{\ln \relax (F)}^6}-\frac {20\,c^2\,x^4\,\left (5\,b^3\,{\ln \relax (F)}^3-126\,b^2\,c^3\,{\ln \relax (F)}^2+990\,b\,c^6\,\ln \relax (F)-2730\,c^9\right )}{b^6\,d^{12}\,{\ln \relax (F)}^6}+\frac {360\,c^2\,x^7\,\left (2\,b^2\,{\ln \relax (F)}^2-88\,b\,c^3\,\ln \relax (F)+715\,c^6\right )}{b^6\,d^9\,{\ln \relax (F)}^6}-\frac {1320\,c^2\,x^{10}\,\left (2\,b\,\ln \relax (F)-91\,c^3\right )}{b^6\,d^6\,{\ln \relax (F)}^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}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 3.44, size = 1096, normalized size = 9.70 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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