Optimal. Leaf size=113 \[ \frac {F^{a+\frac {b}{(c+d x)^2}} \left (-b^5 \log ^5(F)+5 b^4 \log ^4(F) (c+d x)^2-20 b^3 \log ^3(F) (c+d x)^4+60 b^2 \log ^2(F) (c+d x)^6-120 b \log (F) (c+d x)^8+120 (c+d x)^{10}\right )}{2 b^6 d \log ^6(F) (c+d x)^{10}} \]
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Rubi [C] time = 0.05, 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} \[ \frac {F^a \text {Gamma}\left (6,-\frac {b \log (F)}{(c+d x)^2}\right )}{2 b^6 d \log ^6(F)} \]
Antiderivative was successfully verified.
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Rule 2218
Rubi steps
\begin {align*} \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^{13}} \, dx &=\frac {F^a \Gamma \left (6,-\frac {b \log (F)}{(c+d x)^2}\right )}{2 b^6 d \log ^6(F)}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 31, normalized size = 0.27 \[ \frac {F^a \Gamma \left (6,-\frac {b \log (F)}{(c+d x)^2}\right )}{2 b^6 d \log ^6(F)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 583, normalized size = 5.16 \[ \frac {{\left (120 \, d^{10} x^{10} + 1200 \, c d^{9} x^{9} + 5400 \, c^{2} d^{8} x^{8} + 14400 \, c^{3} d^{7} x^{7} + 25200 \, c^{4} d^{6} x^{6} + 30240 \, c^{5} d^{5} x^{5} + 25200 \, c^{6} d^{4} x^{4} + 14400 \, c^{7} d^{3} x^{3} + 5400 \, c^{8} d^{2} x^{2} + 1200 \, c^{9} d x + 120 \, c^{10} - b^{5} \log \relax (F)^{5} + 5 \, {\left (b^{4} d^{2} x^{2} + 2 \, b^{4} c d x + b^{4} c^{2}\right )} \log \relax (F)^{4} - 20 \, {\left (b^{3} d^{4} x^{4} + 4 \, b^{3} c d^{3} x^{3} + 6 \, b^{3} c^{2} d^{2} x^{2} + 4 \, b^{3} c^{3} d x + b^{3} c^{4}\right )} \log \relax (F)^{3} + 60 \, {\left (b^{2} d^{6} x^{6} + 6 \, b^{2} c d^{5} x^{5} + 15 \, b^{2} c^{2} d^{4} x^{4} + 20 \, b^{2} c^{3} d^{3} x^{3} + 15 \, b^{2} c^{4} d^{2} x^{2} + 6 \, b^{2} c^{5} d x + b^{2} c^{6}\right )} \log \relax (F)^{2} - 120 \, {\left (b d^{8} x^{8} + 8 \, b c d^{7} x^{7} + 28 \, b c^{2} d^{6} x^{6} + 56 \, b c^{3} d^{5} x^{5} + 70 \, b c^{4} d^{4} x^{4} + 56 \, b c^{5} d^{3} x^{3} + 28 \, b c^{6} d^{2} x^{2} + 8 \, b c^{7} d x + b c^{8}\right )} \log \relax (F)\right )} F^{\frac {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{2 \, {\left (b^{6} d^{11} x^{10} + 10 \, b^{6} c d^{10} x^{9} + 45 \, b^{6} c^{2} d^{9} x^{8} + 120 \, b^{6} c^{3} d^{8} x^{7} + 210 \, b^{6} c^{4} d^{7} x^{6} + 252 \, b^{6} c^{5} d^{6} x^{5} + 210 \, b^{6} c^{6} d^{5} x^{4} + 120 \, b^{6} c^{7} d^{4} x^{3} + 45 \, b^{6} c^{8} d^{3} x^{2} + 10 \, b^{6} c^{9} d^{2} x + b^{6} c^{10} d\right )} \log \relax (F)^{6}} \]
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 \[ \int \frac {F^{a + \frac {b}{{\left (d x + c\right )}^{2}}}}{{\left (d x + c\right )}^{13}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.21, size = 797, normalized size = 7.05 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.19, size = 740, normalized size = 6.55 \[ \frac {{\left (120 \, F^{a} d^{10} x^{10} + 1200 \, F^{a} c d^{9} x^{9} + 120 \, F^{a} c^{10} - 120 \, F^{a} b c^{8} \log \relax (F) + 60 \, F^{a} b^{2} c^{6} \log \relax (F)^{2} - 20 \, F^{a} b^{3} c^{4} \log \relax (F)^{3} + 5 \, F^{a} b^{4} c^{2} \log \relax (F)^{4} - F^{a} b^{5} \log \relax (F)^{5} + 120 \, {\left (45 \, F^{a} c^{2} d^{8} - F^{a} b d^{8} \log \relax (F)\right )} x^{8} + 960 \, {\left (15 \, F^{a} c^{3} d^{7} - F^{a} b c d^{7} \log \relax (F)\right )} x^{7} + 60 \, {\left (420 \, F^{a} c^{4} d^{6} - 56 \, F^{a} b c^{2} d^{6} \log \relax (F) + F^{a} b^{2} d^{6} \log \relax (F)^{2}\right )} x^{6} + 120 \, {\left (252 \, F^{a} c^{5} d^{5} - 56 \, F^{a} b c^{3} d^{5} \log \relax (F) + 3 \, F^{a} b^{2} c d^{5} \log \relax (F)^{2}\right )} x^{5} + 20 \, {\left (1260 \, F^{a} c^{6} d^{4} - 420 \, F^{a} b c^{4} d^{4} \log \relax (F) + 45 \, F^{a} b^{2} c^{2} d^{4} \log \relax (F)^{2} - F^{a} b^{3} d^{4} \log \relax (F)^{3}\right )} x^{4} + 80 \, {\left (180 \, F^{a} c^{7} d^{3} - 84 \, F^{a} b c^{5} d^{3} \log \relax (F) + 15 \, F^{a} b^{2} c^{3} d^{3} \log \relax (F)^{2} - F^{a} b^{3} c d^{3} \log \relax (F)^{3}\right )} x^{3} + 5 \, {\left (1080 \, F^{a} c^{8} d^{2} - 672 \, F^{a} b c^{6} d^{2} \log \relax (F) + 180 \, F^{a} b^{2} c^{4} d^{2} \log \relax (F)^{2} - 24 \, F^{a} b^{3} c^{2} d^{2} \log \relax (F)^{3} + F^{a} b^{4} d^{2} \log \relax (F)^{4}\right )} x^{2} + 10 \, {\left (120 \, F^{a} c^{9} d - 96 \, F^{a} b c^{7} d \log \relax (F) + 36 \, F^{a} b^{2} c^{5} d \log \relax (F)^{2} - 8 \, F^{a} b^{3} c^{3} d \log \relax (F)^{3} + F^{a} b^{4} c d \log \relax (F)^{4}\right )} x\right )} F^{\frac {b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{2 \, {\left (b^{6} d^{11} x^{10} \log \relax (F)^{6} + 10 \, b^{6} c d^{10} x^{9} \log \relax (F)^{6} + 45 \, b^{6} c^{2} d^{9} x^{8} \log \relax (F)^{6} + 120 \, b^{6} c^{3} d^{8} x^{7} \log \relax (F)^{6} + 210 \, b^{6} c^{4} d^{7} x^{6} \log \relax (F)^{6} + 252 \, b^{6} c^{5} d^{6} x^{5} \log \relax (F)^{6} + 210 \, b^{6} c^{6} d^{5} x^{4} \log \relax (F)^{6} + 120 \, b^{6} c^{7} d^{4} x^{3} \log \relax (F)^{6} + 45 \, b^{6} c^{8} d^{3} x^{2} \log \relax (F)^{6} + 10 \, b^{6} c^{9} d^{2} x \log \relax (F)^{6} + b^{6} c^{10} d \log \relax (F)^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.99, size = 583, normalized size = 5.16 \[ \frac {F^a\,F^{\frac {b}{c^2+2\,c\,d\,x+d^2\,x^2}}\,\left (\frac {60\,x^{10}}{b^6\,d\,{\ln \relax (F)}^6}-\frac {b^5\,{\ln \relax (F)}^5-5\,b^4\,c^2\,{\ln \relax (F)}^4+20\,b^3\,c^4\,{\ln \relax (F)}^3-60\,b^2\,c^6\,{\ln \relax (F)}^2+120\,b\,c^8\,\ln \relax (F)-120\,c^{10}}{2\,b^6\,d^{11}\,{\ln \relax (F)}^6}+\frac {600\,c\,x^9}{b^6\,d^2\,{\ln \relax (F)}^6}+\frac {5\,x^2\,\left (b^4\,{\ln \relax (F)}^4-24\,b^3\,c^2\,{\ln \relax (F)}^3+180\,b^2\,c^4\,{\ln \relax (F)}^2-672\,b\,c^6\,\ln \relax (F)+1080\,c^8\right )}{2\,b^6\,d^9\,{\ln \relax (F)}^6}-\frac {10\,x^4\,\left (b^3\,{\ln \relax (F)}^3-45\,b^2\,c^2\,{\ln \relax (F)}^2+420\,b\,c^4\,\ln \relax (F)-1260\,c^6\right )}{b^6\,d^7\,{\ln \relax (F)}^6}+\frac {30\,x^6\,\left (b^2\,{\ln \relax (F)}^2-56\,b\,c^2\,\ln \relax (F)+420\,c^4\right )}{b^6\,d^5\,{\ln \relax (F)}^6}-\frac {60\,x^8\,\left (b\,\ln \relax (F)-45\,c^2\right )}{b^6\,d^3\,{\ln \relax (F)}^6}-\frac {40\,c\,x^3\,\left (b^3\,{\ln \relax (F)}^3-15\,b^2\,c^2\,{\ln \relax (F)}^2+84\,b\,c^4\,\ln \relax (F)-180\,c^6\right )}{b^6\,d^8\,{\ln \relax (F)}^6}+\frac {60\,c\,x^5\,\left (3\,b^2\,{\ln \relax (F)}^2-56\,b\,c^2\,\ln \relax (F)+252\,c^4\right )}{b^6\,d^6\,{\ln \relax (F)}^6}-\frac {480\,c\,x^7\,\left (b\,\ln \relax (F)-15\,c^2\right )}{b^6\,d^4\,{\ln \relax (F)}^6}+\frac {5\,c\,x\,\left (b^4\,{\ln \relax (F)}^4-8\,b^3\,c^2\,{\ln \relax (F)}^3+36\,b^2\,c^4\,{\ln \relax (F)}^2-96\,b\,c^6\,\ln \relax (F)+120\,c^8\right )}{b^6\,d^{10}\,{\ln \relax (F)}^6}\right )}{x^{10}+\frac {c^{10}}{d^{10}}+\frac {10\,c\,x^9}{d}+\frac {10\,c^9\,x}{d^9}+\frac {45\,c^2\,x^8}{d^2}+\frac {120\,c^3\,x^7}{d^3}+\frac {210\,c^4\,x^6}{d^4}+\frac {252\,c^5\,x^5}{d^5}+\frac {210\,c^6\,x^4}{d^6}+\frac {120\,c^7\,x^3}{d^7}+\frac {45\,c^8\,x^2}{d^8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.83, size = 745, normalized size = 6.59 \[ \frac {F^{a + \frac {b}{\left (c + d x\right )^{2}}} \left (- b^{5} \log {\relax (F )}^{5} + 5 b^{4} c^{2} \log {\relax (F )}^{4} + 10 b^{4} c d x \log {\relax (F )}^{4} + 5 b^{4} d^{2} x^{2} \log {\relax (F )}^{4} - 20 b^{3} c^{4} \log {\relax (F )}^{3} - 80 b^{3} c^{3} d x \log {\relax (F )}^{3} - 120 b^{3} c^{2} d^{2} x^{2} \log {\relax (F )}^{3} - 80 b^{3} c d^{3} x^{3} \log {\relax (F )}^{3} - 20 b^{3} d^{4} x^{4} \log {\relax (F )}^{3} + 60 b^{2} c^{6} \log {\relax (F )}^{2} + 360 b^{2} c^{5} d x \log {\relax (F )}^{2} + 900 b^{2} c^{4} d^{2} x^{2} \log {\relax (F )}^{2} + 1200 b^{2} c^{3} d^{3} x^{3} \log {\relax (F )}^{2} + 900 b^{2} c^{2} d^{4} x^{4} \log {\relax (F )}^{2} + 360 b^{2} c d^{5} x^{5} \log {\relax (F )}^{2} + 60 b^{2} d^{6} x^{6} \log {\relax (F )}^{2} - 120 b c^{8} \log {\relax (F )} - 960 b c^{7} d x \log {\relax (F )} - 3360 b c^{6} d^{2} x^{2} \log {\relax (F )} - 6720 b c^{5} d^{3} x^{3} \log {\relax (F )} - 8400 b c^{4} d^{4} x^{4} \log {\relax (F )} - 6720 b c^{3} d^{5} x^{5} \log {\relax (F )} - 3360 b c^{2} d^{6} x^{6} \log {\relax (F )} - 960 b c d^{7} x^{7} \log {\relax (F )} - 120 b d^{8} x^{8} \log {\relax (F )} + 120 c^{10} + 1200 c^{9} d x + 5400 c^{8} d^{2} x^{2} + 14400 c^{7} d^{3} x^{3} + 25200 c^{6} d^{4} x^{4} + 30240 c^{5} d^{5} x^{5} + 25200 c^{4} d^{6} x^{6} + 14400 c^{3} d^{7} x^{7} + 5400 c^{2} d^{8} x^{8} + 1200 c d^{9} x^{9} + 120 d^{10} x^{10}\right )}{2 b^{6} c^{10} d \log {\relax (F )}^{6} + 20 b^{6} c^{9} d^{2} x \log {\relax (F )}^{6} + 90 b^{6} c^{8} d^{3} x^{2} \log {\relax (F )}^{6} + 240 b^{6} c^{7} d^{4} x^{3} \log {\relax (F )}^{6} + 420 b^{6} c^{6} d^{5} x^{4} \log {\relax (F )}^{6} + 504 b^{6} c^{5} d^{6} x^{5} \log {\relax (F )}^{6} + 420 b^{6} c^{4} d^{7} x^{6} \log {\relax (F )}^{6} + 240 b^{6} c^{3} d^{8} x^{7} \log {\relax (F )}^{6} + 90 b^{6} c^{2} d^{9} x^{8} \log {\relax (F )}^{6} + 20 b^{6} c d^{10} x^{9} \log {\relax (F )}^{6} + 2 b^{6} d^{11} x^{10} \log {\relax (F )}^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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