Integrand size = 21, antiderivative size = 126 \[ \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^9} \, dx=\frac {3 F^{a+\frac {b}{(c+d x)^2}}}{b^4 d \log ^4(F)}-\frac {3 F^{a+\frac {b}{(c+d x)^2}}}{b^3 d (c+d x)^2 \log ^3(F)}+\frac {3 F^{a+\frac {b}{(c+d x)^2}}}{2 b^2 d (c+d x)^4 \log ^2(F)}-\frac {F^{a+\frac {b}{(c+d x)^2}}}{2 b d (c+d x)^6 \log (F)} \]
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Time = 0.12 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2243, 2240} \[ \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^9} \, dx=\frac {3 F^{a+\frac {b}{(c+d x)^2}}}{b^4 d \log ^4(F)}-\frac {3 F^{a+\frac {b}{(c+d x)^2}}}{b^3 d \log ^3(F) (c+d x)^2}+\frac {3 F^{a+\frac {b}{(c+d x)^2}}}{2 b^2 d \log ^2(F) (c+d x)^4}-\frac {F^{a+\frac {b}{(c+d x)^2}}}{2 b d \log (F) (c+d x)^6} \]
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Rule 2240
Rule 2243
Rubi steps \begin{align*} \text {integral}& = -\frac {F^{a+\frac {b}{(c+d x)^2}}}{2 b d (c+d x)^6 \log (F)}-\frac {3 \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^7} \, dx}{b \log (F)} \\ & = \frac {3 F^{a+\frac {b}{(c+d x)^2}}}{2 b^2 d (c+d x)^4 \log ^2(F)}-\frac {F^{a+\frac {b}{(c+d x)^2}}}{2 b d (c+d x)^6 \log (F)}+\frac {6 \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^5} \, dx}{b^2 \log ^2(F)} \\ & = -\frac {3 F^{a+\frac {b}{(c+d x)^2}}}{b^3 d (c+d x)^2 \log ^3(F)}+\frac {3 F^{a+\frac {b}{(c+d x)^2}}}{2 b^2 d (c+d x)^4 \log ^2(F)}-\frac {F^{a+\frac {b}{(c+d x)^2}}}{2 b d (c+d x)^6 \log (F)}-\frac {6 \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^3} \, dx}{b^3 \log ^3(F)} \\ & = \frac {3 F^{a+\frac {b}{(c+d x)^2}}}{b^4 d \log ^4(F)}-\frac {3 F^{a+\frac {b}{(c+d x)^2}}}{b^3 d (c+d x)^2 \log ^3(F)}+\frac {3 F^{a+\frac {b}{(c+d x)^2}}}{2 b^2 d (c+d x)^4 \log ^2(F)}-\frac {F^{a+\frac {b}{(c+d x)^2}}}{2 b d (c+d x)^6 \log (F)} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.64 \[ \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^9} \, dx=\frac {F^{a+\frac {b}{(c+d x)^2}} \left (6 (c+d x)^6-6 b (c+d x)^4 \log (F)+3 b^2 (c+d x)^2 \log ^2(F)-b^3 \log ^3(F)\right )}{2 b^4 d (c+d x)^6 \log ^4(F)} \]
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Time = 1.68 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.71
method | result | size |
risch | \(-\frac {\left (-6 d^{6} x^{6}-36 c \,d^{5} x^{5}+6 \ln \left (F \right ) b \,d^{4} x^{4}-90 c^{2} d^{4} x^{4}+24 \ln \left (F \right ) b c \,d^{3} x^{3}-120 c^{3} d^{3} x^{3}-3 b^{2} d^{2} x^{2} \ln \left (F \right )^{2}+36 \ln \left (F \right ) b \,c^{2} d^{2} x^{2}-90 c^{4} d^{2} x^{2}-6 \ln \left (F \right )^{2} b^{2} c d x +24 \ln \left (F \right ) b \,c^{3} d x -36 c^{5} d x +\ln \left (F \right )^{3} b^{3}-3 \ln \left (F \right )^{2} b^{2} c^{2}+6 \ln \left (F \right ) b \,c^{4}-6 c^{6}\right ) F^{\frac {a \,d^{2} x^{2}+2 a c d x +a \,c^{2}+b}{\left (d x +c \right )^{2}}}}{2 b^{4} \ln \left (F \right )^{4} d \left (d x +c \right )^{6}}\) | \(216\) |
parallelrisch | \(\frac {6 d^{17} F^{a +\frac {b}{\left (d x +c \right )^{2}}} x^{6}+36 d^{16} c \,F^{a +\frac {b}{\left (d x +c \right )^{2}}} x^{5}-6 \ln \left (F \right ) x^{4} F^{a +\frac {b}{\left (d x +c \right )^{2}}} b \,d^{15}+90 d^{15} c^{2} F^{a +\frac {b}{\left (d x +c \right )^{2}}} x^{4}-24 \ln \left (F \right ) x^{3} F^{a +\frac {b}{\left (d x +c \right )^{2}}} b c \,d^{14}+120 c^{3} d^{14} F^{a +\frac {b}{\left (d x +c \right )^{2}}} x^{3}+3 \ln \left (F \right )^{2} x^{2} F^{a +\frac {b}{\left (d x +c \right )^{2}}} b^{2} d^{13}-36 \ln \left (F \right ) x^{2} F^{a +\frac {b}{\left (d x +c \right )^{2}}} b \,c^{2} d^{13}+90 d^{13} c^{4} F^{a +\frac {b}{\left (d x +c \right )^{2}}} x^{2}+6 \ln \left (F \right )^{2} x \,F^{a +\frac {b}{\left (d x +c \right )^{2}}} b^{2} c \,d^{12}-24 \ln \left (F \right ) x \,F^{a +\frac {b}{\left (d x +c \right )^{2}}} b \,c^{3} d^{12}+36 c^{5} F^{a +\frac {b}{\left (d x +c \right )^{2}}} x \,d^{12}-\ln \left (F \right )^{3} F^{a +\frac {b}{\left (d x +c \right )^{2}}} b^{3} d^{11}+3 \ln \left (F \right )^{2} F^{a +\frac {b}{\left (d x +c \right )^{2}}} b^{2} c^{2} d^{11}-6 \ln \left (F \right ) F^{a +\frac {b}{\left (d x +c \right )^{2}}} b \,c^{4} d^{11}+6 c^{6} F^{a +\frac {b}{\left (d x +c \right )^{2}}} d^{11}}{2 \left (d x +c \right )^{6} \ln \left (F \right )^{4} b^{4} d^{12}}\) | \(412\) |
norman | \(\frac {\frac {3 d^{7} x^{8} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{4} b^{4}}-\frac {c \left (\ln \left (F \right )^{3} b^{3}-6 \ln \left (F \right )^{2} b^{2} c^{2}+18 \ln \left (F \right ) b \,c^{4}-24 c^{6}\right ) x \,{\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) \ln \left (F \right )}}{b^{4} \ln \left (F \right )^{4}}-\frac {d \left (\ln \left (F \right )^{3} b^{3}-18 \ln \left (F \right )^{2} b^{2} c^{2}+90 \ln \left (F \right ) b \,c^{4}-168 c^{6}\right ) x^{2} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) \ln \left (F \right )}}{2 \ln \left (F \right )^{4} b^{4}}+\frac {3 d^{3} \left (\ln \left (F \right )^{2} b^{2}-30 \ln \left (F \right ) b \,c^{2}+140 c^{4}\right ) x^{4} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) \ln \left (F \right )}}{2 \ln \left (F \right )^{4} b^{4}}-\frac {3 d^{5} \left (b \ln \left (F \right )-28 c^{2}\right ) x^{6} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{4} b^{4}}+\frac {24 d^{6} c \,x^{7} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{4} b^{4}}-\frac {\left (\ln \left (F \right )^{3} b^{3}-3 \ln \left (F \right )^{2} b^{2} c^{2}+6 \ln \left (F \right ) b \,c^{4}-6 c^{6}\right ) c^{2} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) \ln \left (F \right )}}{2 b^{4} \ln \left (F \right )^{4} d}+\frac {6 c \,d^{2} \left (\ln \left (F \right )^{2} b^{2}-10 \ln \left (F \right ) b \,c^{2}+28 c^{4}\right ) x^{3} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{4} b^{4}}-\frac {6 c \,d^{4} \left (3 b \ln \left (F \right )-28 c^{2}\right ) x^{5} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{4} b^{4}}}{\left (d x +c \right )^{8}}\) | \(444\) |
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Leaf count of result is larger than twice the leaf count of optimal. 287 vs. \(2 (122) = 244\).
Time = 0.32 (sec) , antiderivative size = 287, normalized size of antiderivative = 2.28 \[ \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^9} \, dx=\frac {{\left (6 \, d^{6} x^{6} + 36 \, c d^{5} x^{5} + 90 \, c^{2} d^{4} x^{4} + 120 \, c^{3} d^{3} x^{3} + 90 \, c^{4} d^{2} x^{2} + 36 \, c^{5} d x + 6 \, c^{6} - b^{3} \log \left (F\right )^{3} + 3 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \log \left (F\right )^{2} - 6 \, {\left (b d^{4} x^{4} + 4 \, b c d^{3} x^{3} + 6 \, b c^{2} d^{2} x^{2} + 4 \, b c^{3} d x + b c^{4}\right )} \log \left (F\right )\right )} F^{\frac {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{2 \, {\left (b^{4} d^{7} x^{6} + 6 \, b^{4} c d^{6} x^{5} + 15 \, b^{4} c^{2} d^{5} x^{4} + 20 \, b^{4} c^{3} d^{4} x^{3} + 15 \, b^{4} c^{4} d^{3} x^{2} + 6 \, b^{4} c^{5} d^{2} x + b^{4} c^{6} d\right )} \log \left (F\right )^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 333 vs. \(2 (112) = 224\).
Time = 0.22 (sec) , antiderivative size = 333, normalized size of antiderivative = 2.64 \[ \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^9} \, dx=\frac {F^{a + \frac {b}{\left (c + d x\right )^{2}}} \left (- b^{3} \log {\left (F \right )}^{3} + 3 b^{2} c^{2} \log {\left (F \right )}^{2} + 6 b^{2} c d x \log {\left (F \right )}^{2} + 3 b^{2} d^{2} x^{2} \log {\left (F \right )}^{2} - 6 b c^{4} \log {\left (F \right )} - 24 b c^{3} d x \log {\left (F \right )} - 36 b c^{2} d^{2} x^{2} \log {\left (F \right )} - 24 b c d^{3} x^{3} \log {\left (F \right )} - 6 b d^{4} x^{4} \log {\left (F \right )} + 6 c^{6} + 36 c^{5} d x + 90 c^{4} d^{2} x^{2} + 120 c^{3} d^{3} x^{3} + 90 c^{2} d^{4} x^{4} + 36 c d^{5} x^{5} + 6 d^{6} x^{6}\right )}{2 b^{4} c^{6} d \log {\left (F \right )}^{4} + 12 b^{4} c^{5} d^{2} x \log {\left (F \right )}^{4} + 30 b^{4} c^{4} d^{3} x^{2} \log {\left (F \right )}^{4} + 40 b^{4} c^{3} d^{4} x^{3} \log {\left (F \right )}^{4} + 30 b^{4} c^{2} d^{5} x^{4} \log {\left (F \right )}^{4} + 12 b^{4} c d^{6} x^{5} \log {\left (F \right )}^{4} + 2 b^{4} d^{7} x^{6} \log {\left (F \right )}^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 349 vs. \(2 (122) = 244\).
Time = 0.29 (sec) , antiderivative size = 349, normalized size of antiderivative = 2.77 \[ \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^9} \, dx=\frac {{\left (6 \, F^{a} d^{6} x^{6} + 36 \, F^{a} c d^{5} x^{5} + 6 \, F^{a} c^{6} - 6 \, F^{a} b c^{4} \log \left (F\right ) + 3 \, F^{a} b^{2} c^{2} \log \left (F\right )^{2} - F^{a} b^{3} \log \left (F\right )^{3} + 6 \, {\left (15 \, F^{a} c^{2} d^{4} - F^{a} b d^{4} \log \left (F\right )\right )} x^{4} + 24 \, {\left (5 \, F^{a} c^{3} d^{3} - F^{a} b c d^{3} \log \left (F\right )\right )} x^{3} + 3 \, {\left (30 \, F^{a} c^{4} d^{2} - 12 \, F^{a} b c^{2} d^{2} \log \left (F\right ) + F^{a} b^{2} d^{2} \log \left (F\right )^{2}\right )} x^{2} + 6 \, {\left (6 \, F^{a} c^{5} d - 4 \, F^{a} b c^{3} d \log \left (F\right ) + F^{a} b^{2} c d \log \left (F\right )^{2}\right )} x\right )} F^{\frac {b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{2 \, {\left (b^{4} d^{7} x^{6} \log \left (F\right )^{4} + 6 \, b^{4} c d^{6} x^{5} \log \left (F\right )^{4} + 15 \, b^{4} c^{2} d^{5} x^{4} \log \left (F\right )^{4} + 20 \, b^{4} c^{3} d^{4} x^{3} \log \left (F\right )^{4} + 15 \, b^{4} c^{4} d^{3} x^{2} \log \left (F\right )^{4} + 6 \, b^{4} c^{5} d^{2} x \log \left (F\right )^{4} + b^{4} c^{6} d \log \left (F\right )^{4}\right )}} \]
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\[ \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^9} \, dx=\int { \frac {F^{a + \frac {b}{{\left (d x + c\right )}^{2}}}}{{\left (d x + c\right )}^{9}} \,d x } \]
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Time = 0.87 (sec) , antiderivative size = 292, normalized size of antiderivative = 2.32 \[ \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^9} \, dx=\frac {F^a\,F^{\frac {b}{c^2+2\,c\,d\,x+d^2\,x^2}}\,\left (\frac {3\,x^6}{b^4\,d\,{\ln \left (F\right )}^4}-\frac {b^3\,{\ln \left (F\right )}^3-3\,b^2\,c^2\,{\ln \left (F\right )}^2+6\,b\,c^4\,\ln \left (F\right )-6\,c^6}{2\,b^4\,d^7\,{\ln \left (F\right )}^4}+\frac {18\,c\,x^5}{b^4\,d^2\,{\ln \left (F\right )}^4}+\frac {3\,x^2\,\left (b^2\,{\ln \left (F\right )}^2-12\,b\,c^2\,\ln \left (F\right )+30\,c^4\right )}{2\,b^4\,d^5\,{\ln \left (F\right )}^4}-\frac {3\,x^4\,\left (b\,\ln \left (F\right )-15\,c^2\right )}{b^4\,d^3\,{\ln \left (F\right )}^4}-\frac {12\,c\,x^3\,\left (b\,\ln \left (F\right )-5\,c^2\right )}{b^4\,d^4\,{\ln \left (F\right )}^4}+\frac {3\,c\,x\,\left (b^2\,{\ln \left (F\right )}^2-4\,b\,c^2\,\ln \left (F\right )+6\,c^4\right )}{b^4\,d^6\,{\ln \left (F\right )}^4}\right )}{x^6+\frac {c^6}{d^6}+\frac {6\,c\,x^5}{d}+\frac {6\,c^5\,x}{d^5}+\frac {15\,c^2\,x^4}{d^2}+\frac {20\,c^3\,x^3}{d^3}+\frac {15\,c^4\,x^2}{d^4}} \]
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