Integrand size = 21, antiderivative size = 49 \[ \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^{12}} \, dx=\frac {F^a \Gamma \left (\frac {11}{2},-\frac {b \log (F)}{(c+d x)^2}\right )}{2 d (c+d x)^{11} \left (-\frac {b \log (F)}{(c+d x)^2}\right )^{11/2}} \]
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Time = 0.03 (sec) , antiderivative size = 49, 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 = {2250} \[ \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^{12}} \, dx=\frac {F^a \Gamma \left (\frac {11}{2},-\frac {b \log (F)}{(c+d x)^2}\right )}{2 d (c+d x)^{11} \left (-\frac {b \log (F)}{(c+d x)^2}\right )^{11/2}} \]
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Rule 2250
Rubi steps \begin{align*} \text {integral}& = \frac {F^a \Gamma \left (\frac {11}{2},-\frac {b \log (F)}{(c+d x)^2}\right )}{2 d (c+d x)^{11} \left (-\frac {b \log (F)}{(c+d x)^2}\right )^{11/2}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00 \[ \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^{12}} \, dx=\frac {F^a \Gamma \left (\frac {11}{2},-\frac {b \log (F)}{(c+d x)^2}\right )}{2 d (c+d x)^{11} \left (-\frac {b \log (F)}{(c+d x)^2}\right )^{11/2}} \]
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Time = 7.00 (sec) , antiderivative size = 208, normalized size of antiderivative = 4.24
method | result | size |
risch | \(-\frac {F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}}{2 d \left (d x +c \right )^{9} b \ln \left (F \right )}+\frac {9 F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}}{4 d \,b^{2} \ln \left (F \right )^{2} \left (d x +c \right )^{7}}-\frac {63 F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}}{8 d \,b^{3} \ln \left (F \right )^{3} \left (d x +c \right )^{5}}+\frac {315 F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}}{16 d \,b^{4} \ln \left (F \right )^{4} \left (d x +c \right )^{3}}-\frac {945 F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}}{32 d \,b^{5} \ln \left (F \right )^{5} \left (d x +c \right )}+\frac {945 F^{a} \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-b \ln \left (F \right )}}{d x +c}\right )}{64 d \,b^{5} \ln \left (F \right )^{5} \sqrt {-b \ln \left (F \right )}}\) | \(208\) |
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none
Time = 0.13 (sec) , antiderivative size = 601, normalized size of antiderivative = 12.27 \[ \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^{12}} \, dx=-\frac {945 \, \sqrt {\pi } {\left (d^{10} x^{9} + 9 \, c d^{9} x^{8} + 36 \, c^{2} d^{8} x^{7} + 84 \, c^{3} d^{7} x^{6} + 126 \, c^{4} d^{6} x^{5} + 126 \, c^{5} d^{5} x^{4} + 84 \, c^{6} d^{4} x^{3} + 36 \, c^{7} d^{3} x^{2} + 9 \, c^{8} d^{2} x + c^{9} d\right )} F^{a} \sqrt {-\frac {b \log \left (F\right )}{d^{2}}} \operatorname {erf}\left (\frac {d \sqrt {-\frac {b \log \left (F\right )}{d^{2}}}}{d x + c}\right ) + 2 \, {\left (16 \, b^{5} \log \left (F\right )^{5} - 72 \, {\left (b^{4} d^{2} x^{2} + 2 \, b^{4} c d x + b^{4} c^{2}\right )} \log \left (F\right )^{4} + 252 \, {\left (b^{3} d^{4} x^{4} + 4 \, b^{3} c d^{3} x^{3} + 6 \, b^{3} c^{2} d^{2} x^{2} + 4 \, b^{3} c^{3} d x + b^{3} c^{4}\right )} \log \left (F\right )^{3} - 630 \, {\left (b^{2} d^{6} x^{6} + 6 \, b^{2} c d^{5} x^{5} + 15 \, b^{2} c^{2} d^{4} x^{4} + 20 \, b^{2} c^{3} d^{3} x^{3} + 15 \, b^{2} c^{4} d^{2} x^{2} + 6 \, b^{2} c^{5} d x + b^{2} c^{6}\right )} \log \left (F\right )^{2} + 945 \, {\left (b d^{8} x^{8} + 8 \, b c d^{7} x^{7} + 28 \, b c^{2} d^{6} x^{6} + 56 \, b c^{3} d^{5} x^{5} + 70 \, b c^{4} d^{4} x^{4} + 56 \, b c^{5} d^{3} x^{3} + 28 \, b c^{6} d^{2} x^{2} + 8 \, b c^{7} d x + b c^{8}\right )} \log \left (F\right )\right )} F^{\frac {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{64 \, {\left (b^{6} d^{10} x^{9} + 9 \, b^{6} c d^{9} x^{8} + 36 \, b^{6} c^{2} d^{8} x^{7} + 84 \, b^{6} c^{3} d^{7} x^{6} + 126 \, b^{6} c^{4} d^{6} x^{5} + 126 \, b^{6} c^{5} d^{5} x^{4} + 84 \, b^{6} c^{6} d^{4} x^{3} + 36 \, b^{6} c^{7} d^{3} x^{2} + 9 \, b^{6} c^{8} d^{2} x + b^{6} c^{9} d\right )} \log \left (F\right )^{6}} \]
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Timed out. \[ \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^{12}} \, dx=\text {Timed out} \]
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\[ \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^{12}} \, dx=\int { \frac {F^{a + \frac {b}{{\left (d x + c\right )}^{2}}}}{{\left (d x + c\right )}^{12}} \,d x } \]
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\[ \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^{12}} \, dx=\int { \frac {F^{a + \frac {b}{{\left (d x + c\right )}^{2}}}}{{\left (d x + c\right )}^{12}} \,d x } \]
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Time = 1.50 (sec) , antiderivative size = 189, normalized size of antiderivative = 3.86 \[ \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^{12}} \, dx=\frac {\frac {F^a\,\left (945\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {b\,\ln \left (F\right )}{\sqrt {b\,\ln \left (F\right )}\,\left (c+d\,x\right )}\right )-\frac {1890\,F^{\frac {b}{{\left (c+d\,x\right )}^2}}\,\sqrt {b\,\ln \left (F\right )}}{c+d\,x}\right )}{64\,\sqrt {b\,\ln \left (F\right )}}-\frac {63\,F^a\,F^{\frac {b}{{\left (c+d\,x\right )}^2}}\,b^2\,{\ln \left (F\right )}^2}{8\,{\left (c+d\,x\right )}^5}+\frac {9\,F^a\,F^{\frac {b}{{\left (c+d\,x\right )}^2}}\,b^3\,{\ln \left (F\right )}^3}{4\,{\left (c+d\,x\right )}^7}-\frac {F^a\,F^{\frac {b}{{\left (c+d\,x\right )}^2}}\,b^4\,{\ln \left (F\right )}^4}{2\,{\left (c+d\,x\right )}^9}+\frac {315\,F^a\,F^{\frac {b}{{\left (c+d\,x\right )}^2}}\,b\,\ln \left (F\right )}{16\,{\left (c+d\,x\right )}^3}}{b^5\,d\,{\ln \left (F\right )}^5} \]
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