Integrand size = 21, antiderivative size = 115 \[ \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^6} \, dx=-\frac {3 F^a \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (F)}}{c+d x}\right )}{8 b^{5/2} d \log ^{\frac {5}{2}}(F)}+\frac {3 F^{a+\frac {b}{(c+d x)^2}}}{4 b^2 d (c+d x) \log ^2(F)}-\frac {F^{a+\frac {b}{(c+d x)^2}}}{2 b d (c+d x)^3 \log (F)} \]
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Time = 0.10 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2243, 2242, 2235} \[ \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^6} \, dx=-\frac {3 \sqrt {\pi } F^a \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (F)}}{c+d x}\right )}{8 b^{5/2} d \log ^{\frac {5}{2}}(F)}+\frac {3 F^{a+\frac {b}{(c+d x)^2}}}{4 b^2 d \log ^2(F) (c+d x)}-\frac {F^{a+\frac {b}{(c+d x)^2}}}{2 b d \log (F) (c+d x)^3} \]
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Rule 2235
Rule 2242
Rule 2243
Rubi steps \begin{align*} \text {integral}& = -\frac {F^{a+\frac {b}{(c+d x)^2}}}{2 b d (c+d x)^3 \log (F)}-\frac {3 \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^4} \, dx}{2 b \log (F)} \\ & = \frac {3 F^{a+\frac {b}{(c+d x)^2}}}{4 b^2 d (c+d x) \log ^2(F)}-\frac {F^{a+\frac {b}{(c+d x)^2}}}{2 b d (c+d x)^3 \log (F)}+\frac {3 \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^2} \, dx}{4 b^2 \log ^2(F)} \\ & = \frac {3 F^{a+\frac {b}{(c+d x)^2}}}{4 b^2 d (c+d x) \log ^2(F)}-\frac {F^{a+\frac {b}{(c+d x)^2}}}{2 b d (c+d x)^3 \log (F)}-\frac {3 \text {Subst}\left (\int F^{a+b x^2} \, dx,x,\frac {1}{c+d x}\right )}{4 b^2 d \log ^2(F)} \\ & = -\frac {3 F^a \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (F)}}{c+d x}\right )}{8 b^{5/2} d \log ^{\frac {5}{2}}(F)}+\frac {3 F^{a+\frac {b}{(c+d x)^2}}}{4 b^2 d (c+d x) \log ^2(F)}-\frac {F^{a+\frac {b}{(c+d x)^2}}}{2 b d (c+d x)^3 \log (F)} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.83 \[ \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^6} \, dx=\frac {F^a \left (-3 \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (F)}}{c+d x}\right )-\frac {2 \sqrt {b} F^{\frac {b}{(c+d x)^2}} \sqrt {\log (F)} \left (-3 (c+d x)^2+2 b \log (F)\right )}{(c+d x)^3}\right )}{8 b^{5/2} d \log ^{\frac {5}{2}}(F)} \]
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Time = 1.40 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.95
method | result | size |
risch | \(-\frac {F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}}{2 d \left (d x +c \right )^{3} b \ln \left (F \right )}+\frac {3 F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}}}{4 d \,b^{2} \ln \left (F \right )^{2} \left (d x +c \right )}-\frac {3 F^{a} \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-b \ln \left (F \right )}}{d x +c}\right )}{8 d \,b^{2} \ln \left (F \right )^{2} \sqrt {-b \ln \left (F \right )}}\) | \(109\) |
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Leaf count of result is larger than twice the leaf count of optimal. 199 vs. \(2 (99) = 198\).
Time = 0.29 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.73 \[ \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^6} \, dx=\frac {3 \, \sqrt {\pi } {\left (d^{4} x^{3} + 3 \, c d^{3} x^{2} + 3 \, c^{2} d^{2} x + c^{3} d\right )} F^{a} \sqrt {-\frac {b \log \left (F\right )}{d^{2}}} \operatorname {erf}\left (\frac {d \sqrt {-\frac {b \log \left (F\right )}{d^{2}}}}{d x + c}\right ) - 2 \, {\left (2 \, b^{2} \log \left (F\right )^{2} - 3 \, {\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \log \left (F\right )\right )} F^{\frac {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{8 \, {\left (b^{3} d^{4} x^{3} + 3 \, b^{3} c d^{3} x^{2} + 3 \, b^{3} c^{2} d^{2} x + b^{3} c^{3} d\right )} \log \left (F\right )^{3}} \]
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Timed out. \[ \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^6} \, dx=\text {Timed out} \]
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\[ \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^6} \, dx=\int { \frac {F^{a + \frac {b}{{\left (d x + c\right )}^{2}}}}{{\left (d x + c\right )}^{6}} \,d x } \]
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\[ \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^6} \, dx=\int { \frac {F^{a + \frac {b}{{\left (d x + c\right )}^{2}}}}{{\left (d x + c\right )}^{6}} \,d x } \]
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Time = 0.77 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.91 \[ \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^6} \, dx=-\frac {F^a\,F^{\frac {b}{{\left (c+d\,x\right )}^2}}}{2\,b\,d\,\ln \left (F\right )\,{\left (c+d\,x\right )}^3}-\frac {F^a\,\left (3\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {b\,\ln \left (F\right )}{\sqrt {b\,\ln \left (F\right )}\,\left (c+d\,x\right )}\right )-\frac {6\,F^{\frac {b}{{\left (c+d\,x\right )}^2}}\,\sqrt {b\,\ln \left (F\right )}}{c+d\,x}\right )}{8\,b^2\,d\,{\ln \left (F\right )}^2\,\sqrt {b\,\ln \left (F\right )}} \]
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