Integrand size = 21, antiderivative size = 122 \[ \int \frac {F^{a+\frac {b}{c+d x}}}{(c+d x)^5} \, dx=\frac {6 F^{a+\frac {b}{c+d x}}}{b^4 d \log ^4(F)}-\frac {6 F^{a+\frac {b}{c+d x}}}{b^3 d (c+d x) \log ^3(F)}+\frac {3 F^{a+\frac {b}{c+d x}}}{b^2 d (c+d x)^2 \log ^2(F)}-\frac {F^{a+\frac {b}{c+d x}}}{b d (c+d x)^3 \log (F)} \]
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Time = 0.12 (sec) , antiderivative size = 122, 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} \[ \int \frac {F^{a+\frac {b}{c+d x}}}{(c+d x)^5} \, dx=\frac {6 F^{a+\frac {b}{c+d x}}}{b^4 d \log ^4(F)}-\frac {6 F^{a+\frac {b}{c+d x}}}{b^3 d \log ^3(F) (c+d x)}+\frac {3 F^{a+\frac {b}{c+d x}}}{b^2 d \log ^2(F) (c+d x)^2}-\frac {F^{a+\frac {b}{c+d x}}}{b d \log (F) (c+d x)^3} \]
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Rule 2240
Rule 2243
Rubi steps \begin{align*} \text {integral}& = -\frac {F^{a+\frac {b}{c+d x}}}{b d (c+d x)^3 \log (F)}-\frac {3 \int \frac {F^{a+\frac {b}{c+d x}}}{(c+d x)^4} \, dx}{b \log (F)} \\ & = \frac {3 F^{a+\frac {b}{c+d x}}}{b^2 d (c+d x)^2 \log ^2(F)}-\frac {F^{a+\frac {b}{c+d x}}}{b d (c+d x)^3 \log (F)}+\frac {6 \int \frac {F^{a+\frac {b}{c+d x}}}{(c+d x)^3} \, dx}{b^2 \log ^2(F)} \\ & = -\frac {6 F^{a+\frac {b}{c+d x}}}{b^3 d (c+d x) \log ^3(F)}+\frac {3 F^{a+\frac {b}{c+d x}}}{b^2 d (c+d x)^2 \log ^2(F)}-\frac {F^{a+\frac {b}{c+d x}}}{b d (c+d x)^3 \log (F)}-\frac {6 \int \frac {F^{a+\frac {b}{c+d x}}}{(c+d x)^2} \, dx}{b^3 \log ^3(F)} \\ & = \frac {6 F^{a+\frac {b}{c+d x}}}{b^4 d \log ^4(F)}-\frac {6 F^{a+\frac {b}{c+d x}}}{b^3 d (c+d x) \log ^3(F)}+\frac {3 F^{a+\frac {b}{c+d x}}}{b^2 d (c+d x)^2 \log ^2(F)}-\frac {F^{a+\frac {b}{c+d x}}}{b d (c+d x)^3 \log (F)} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.62 \[ \int \frac {F^{a+\frac {b}{c+d x}}}{(c+d x)^5} \, dx=\frac {F^{a+\frac {b}{c+d x}} \left (6 (c+d x)^3-6 b (c+d x)^2 \log (F)+3 b^2 (c+d x) \log ^2(F)-b^3 \log ^3(F)\right )}{b^4 d (c+d x)^3 \log ^4(F)} \]
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Time = 0.54 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.02
method | result | size |
risch | \(-\frac {\left (\ln \left (F \right )^{3} b^{3}-3 \ln \left (F \right )^{2} b^{2} d x +6 \ln \left (F \right ) b \,d^{2} x^{2}-6 d^{3} x^{3}-3 \ln \left (F \right )^{2} b^{2} c +12 \ln \left (F \right ) b c d x -18 c \,d^{2} x^{2}+6 \ln \left (F \right ) b \,c^{2}-18 c^{2} d x -6 c^{3}\right ) F^{\frac {x a d +c a +b}{d x +c}}}{b^{4} \ln \left (F \right )^{4} d \left (d x +c \right )^{3}}\) | \(125\) |
norman | \(\frac {-\frac {\left (\ln \left (F \right )^{3} b^{3}-6 \ln \left (F \right )^{2} b^{2} c +18 \ln \left (F \right ) b \,c^{2}-24 c^{3}\right ) x \,{\mathrm e}^{\left (a +\frac {b}{d x +c}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{4} b^{4}}+\frac {6 d^{3} x^{4} {\mathrm e}^{\left (a +\frac {b}{d x +c}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{4} b^{4}}+\frac {3 d \left (\ln \left (F \right )^{2} b^{2}-6 c b \ln \left (F \right )+12 c^{2}\right ) x^{2} {\mathrm e}^{\left (a +\frac {b}{d x +c}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{4} b^{4}}-\frac {6 d^{2} \left (b \ln \left (F \right )-4 c \right ) x^{3} {\mathrm e}^{\left (a +\frac {b}{d x +c}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{4} b^{4}}-\frac {\left (\ln \left (F \right )^{3} b^{3}-3 \ln \left (F \right )^{2} b^{2} c +6 \ln \left (F \right ) b \,c^{2}-6 c^{3}\right ) c \,{\mathrm e}^{\left (a +\frac {b}{d x +c}\right ) \ln \left (F \right )}}{b^{4} \ln \left (F \right )^{4} d}}{\left (d x +c \right )^{4}}\) | \(243\) |
parallelrisch | \(\frac {-\ln \left (F \right )^{3} F^{a +\frac {b}{d x +c}} b^{3} d^{6}+3 \ln \left (F \right )^{2} x \,F^{a +\frac {b}{d x +c}} b^{2} d^{7}-6 \ln \left (F \right ) x^{2} F^{a +\frac {b}{d x +c}} b \,d^{8}+6 x^{3} F^{a +\frac {b}{d x +c}} d^{9}+3 \ln \left (F \right )^{2} F^{a +\frac {b}{d x +c}} b^{2} c \,d^{6}-12 \ln \left (F \right ) x \,F^{a +\frac {b}{d x +c}} b c \,d^{7}+18 x^{2} F^{a +\frac {b}{d x +c}} c \,d^{8}-6 \ln \left (F \right ) F^{a +\frac {b}{d x +c}} b \,c^{2} d^{6}+18 x \,F^{a +\frac {b}{d x +c}} c^{2} d^{7}+6 F^{a +\frac {b}{d x +c}} c^{3} d^{6}}{\left (d x +c \right )^{3} d^{7} \ln \left (F \right )^{4} b^{4}}\) | \(254\) |
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Time = 0.27 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.23 \[ \int \frac {F^{a+\frac {b}{c+d x}}}{(c+d x)^5} \, dx=\frac {{\left (6 \, d^{3} x^{3} - b^{3} \log \left (F\right )^{3} + 18 \, c d^{2} x^{2} + 18 \, c^{2} d x + 6 \, c^{3} + 3 \, {\left (b^{2} d x + b^{2} c\right )} \log \left (F\right )^{2} - 6 \, {\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \log \left (F\right )\right )} F^{\frac {a d x + a c + b}{d x + c}}}{{\left (b^{4} d^{4} x^{3} + 3 \, b^{4} c d^{3} x^{2} + 3 \, b^{4} c^{2} d^{2} x + b^{4} c^{3} d\right )} \log \left (F\right )^{4}} \]
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Time = 0.14 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.45 \[ \int \frac {F^{a+\frac {b}{c+d x}}}{(c+d x)^5} \, dx=\frac {F^{a + \frac {b}{c + d x}} \left (- b^{3} \log {\left (F \right )}^{3} + 3 b^{2} c \log {\left (F \right )}^{2} + 3 b^{2} d x \log {\left (F \right )}^{2} - 6 b c^{2} \log {\left (F \right )} - 12 b c d x \log {\left (F \right )} - 6 b d^{2} x^{2} \log {\left (F \right )} + 6 c^{3} + 18 c^{2} d x + 18 c d^{2} x^{2} + 6 d^{3} x^{3}\right )}{b^{4} c^{3} d \log {\left (F \right )}^{4} + 3 b^{4} c^{2} d^{2} x \log {\left (F \right )}^{4} + 3 b^{4} c d^{3} x^{2} \log {\left (F \right )}^{4} + b^{4} d^{4} x^{3} \log {\left (F \right )}^{4}} \]
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\[ \int \frac {F^{a+\frac {b}{c+d x}}}{(c+d x)^5} \, dx=\int { \frac {F^{a + \frac {b}{d x + c}}}{{\left (d x + c\right )}^{5}} \,d x } \]
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\[ \int \frac {F^{a+\frac {b}{c+d x}}}{(c+d x)^5} \, dx=\int { \frac {F^{a + \frac {b}{d x + c}}}{{\left (d x + c\right )}^{5}} \,d x } \]
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Time = 0.47 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.32 \[ \int \frac {F^{a+\frac {b}{c+d x}}}{(c+d x)^5} \, dx=\frac {F^{a+\frac {b}{c+d\,x}}\,\left (\frac {6\,x^3}{b^4\,d\,{\ln \left (F\right )}^4}-\frac {b^3\,{\ln \left (F\right )}^3-3\,b^2\,c\,{\ln \left (F\right )}^2+6\,b\,c^2\,\ln \left (F\right )-6\,c^3}{b^4\,d^4\,{\ln \left (F\right )}^4}+\frac {x^2\,\left (18\,c-6\,b\,\ln \left (F\right )\right )}{b^4\,d^2\,{\ln \left (F\right )}^4}+\frac {3\,x\,\left (b^2\,{\ln \left (F\right )}^2-4\,b\,c\,\ln \left (F\right )+6\,c^2\right )}{b^4\,d^3\,{\ln \left (F\right )}^4}\right )}{x^3+\frac {c^3}{d^3}+\frac {3\,c\,x^2}{d}+\frac {3\,c^2\,x}{d^2}} \]
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