Integrand size = 21, antiderivative size = 170 \[ \int F^{a+\frac {b}{(c+d x)^2}} (c+d x)^6 \, dx=\frac {F^{a+\frac {b}{(c+d x)^2}} (c+d x)^7}{7 d}+\frac {2 b F^{a+\frac {b}{(c+d x)^2}} (c+d x)^5 \log (F)}{35 d}+\frac {4 b^2 F^{a+\frac {b}{(c+d x)^2}} (c+d x)^3 \log ^2(F)}{105 d}+\frac {8 b^3 F^{a+\frac {b}{(c+d x)^2}} (c+d x) \log ^3(F)}{105 d}-\frac {8 b^{7/2} F^a \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (F)}}{c+d x}\right ) \log ^{\frac {7}{2}}(F)}{105 d} \]
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Time = 0.15 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2245, 2237, 2242, 2235} \[ \int F^{a+\frac {b}{(c+d x)^2}} (c+d x)^6 \, dx=-\frac {8 \sqrt {\pi } b^{7/2} F^a \log ^{\frac {7}{2}}(F) \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (F)}}{c+d x}\right )}{105 d}+\frac {8 b^3 \log ^3(F) (c+d x) F^{a+\frac {b}{(c+d x)^2}}}{105 d}+\frac {4 b^2 \log ^2(F) (c+d x)^3 F^{a+\frac {b}{(c+d x)^2}}}{105 d}+\frac {(c+d x)^7 F^{a+\frac {b}{(c+d x)^2}}}{7 d}+\frac {2 b \log (F) (c+d x)^5 F^{a+\frac {b}{(c+d x)^2}}}{35 d} \]
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Rule 2235
Rule 2237
Rule 2242
Rule 2245
Rubi steps \begin{align*} \text {integral}& = \frac {F^{a+\frac {b}{(c+d x)^2}} (c+d x)^7}{7 d}+\frac {1}{7} (2 b \log (F)) \int F^{a+\frac {b}{(c+d x)^2}} (c+d x)^4 \, dx \\ & = \frac {F^{a+\frac {b}{(c+d x)^2}} (c+d x)^7}{7 d}+\frac {2 b F^{a+\frac {b}{(c+d x)^2}} (c+d x)^5 \log (F)}{35 d}+\frac {1}{35} \left (4 b^2 \log ^2(F)\right ) \int F^{a+\frac {b}{(c+d x)^2}} (c+d x)^2 \, dx \\ & = \frac {F^{a+\frac {b}{(c+d x)^2}} (c+d x)^7}{7 d}+\frac {2 b F^{a+\frac {b}{(c+d x)^2}} (c+d x)^5 \log (F)}{35 d}+\frac {4 b^2 F^{a+\frac {b}{(c+d x)^2}} (c+d x)^3 \log ^2(F)}{105 d}+\frac {1}{105} \left (8 b^3 \log ^3(F)\right ) \int F^{a+\frac {b}{(c+d x)^2}} \, dx \\ & = \frac {F^{a+\frac {b}{(c+d x)^2}} (c+d x)^7}{7 d}+\frac {2 b F^{a+\frac {b}{(c+d x)^2}} (c+d x)^5 \log (F)}{35 d}+\frac {4 b^2 F^{a+\frac {b}{(c+d x)^2}} (c+d x)^3 \log ^2(F)}{105 d}+\frac {8 b^3 F^{a+\frac {b}{(c+d x)^2}} (c+d x) \log ^3(F)}{105 d}+\frac {1}{105} \left (16 b^4 \log ^4(F)\right ) \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^2} \, dx \\ & = \frac {F^{a+\frac {b}{(c+d x)^2}} (c+d x)^7}{7 d}+\frac {2 b F^{a+\frac {b}{(c+d x)^2}} (c+d x)^5 \log (F)}{35 d}+\frac {4 b^2 F^{a+\frac {b}{(c+d x)^2}} (c+d x)^3 \log ^2(F)}{105 d}+\frac {8 b^3 F^{a+\frac {b}{(c+d x)^2}} (c+d x) \log ^3(F)}{105 d}-\frac {\left (16 b^4 \log ^4(F)\right ) \text {Subst}\left (\int F^{a+b x^2} \, dx,x,\frac {1}{c+d x}\right )}{105 d} \\ & = \frac {F^{a+\frac {b}{(c+d x)^2}} (c+d x)^7}{7 d}+\frac {2 b F^{a+\frac {b}{(c+d x)^2}} (c+d x)^5 \log (F)}{35 d}+\frac {4 b^2 F^{a+\frac {b}{(c+d x)^2}} (c+d x)^3 \log ^2(F)}{105 d}+\frac {8 b^3 F^{a+\frac {b}{(c+d x)^2}} (c+d x) \log ^3(F)}{105 d}-\frac {8 b^{7/2} F^a \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (F)}}{c+d x}\right ) \log ^{\frac {7}{2}}(F)}{105 d} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.66 \[ \int F^{a+\frac {b}{(c+d x)^2}} (c+d x)^6 \, dx=\frac {F^a \left (-8 b^{7/2} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (F)}}{c+d x}\right ) \log ^{\frac {7}{2}}(F)+F^{\frac {b}{(c+d x)^2}} (c+d x) \left (15 (c+d x)^6+6 b (c+d x)^4 \log (F)+4 b^2 (c+d x)^2 \log ^2(F)+8 b^3 \log ^3(F)\right )\right )}{105 d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(542\) vs. \(2(150)=300\).
Time = 1.52 (sec) , antiderivative size = 543, normalized size of antiderivative = 3.19
method | result | size |
risch | \(\frac {F^{a} d^{6} F^{\frac {b}{\left (d x +c \right )^{2}}} x^{7}}{7}+F^{a} d^{5} F^{\frac {b}{\left (d x +c \right )^{2}}} c \,x^{6}+3 F^{a} d^{4} F^{\frac {b}{\left (d x +c \right )^{2}}} c^{2} x^{5}+5 F^{a} d^{3} F^{\frac {b}{\left (d x +c \right )^{2}}} c^{3} x^{4}+5 F^{a} d^{2} F^{\frac {b}{\left (d x +c \right )^{2}}} c^{4} x^{3}+3 F^{a} d \,F^{\frac {b}{\left (d x +c \right )^{2}}} c^{5} x^{2}+F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} c^{6} x +\frac {F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} c^{7}}{7 d}+\frac {2 F^{a} d^{4} b \ln \left (F \right ) F^{\frac {b}{\left (d x +c \right )^{2}}} x^{5}}{35}+\frac {2 F^{a} d^{3} b \ln \left (F \right ) F^{\frac {b}{\left (d x +c \right )^{2}}} c \,x^{4}}{7}+\frac {4 F^{a} d^{2} b \ln \left (F \right ) F^{\frac {b}{\left (d x +c \right )^{2}}} c^{2} x^{3}}{7}+\frac {4 F^{a} d b \ln \left (F \right ) F^{\frac {b}{\left (d x +c \right )^{2}}} c^{3} x^{2}}{7}+\frac {2 F^{a} b \ln \left (F \right ) F^{\frac {b}{\left (d x +c \right )^{2}}} c^{4} x}{7}+\frac {2 F^{a} b \ln \left (F \right ) F^{\frac {b}{\left (d x +c \right )^{2}}} c^{5}}{35 d}+\frac {4 F^{a} d^{2} b^{2} \ln \left (F \right )^{2} F^{\frac {b}{\left (d x +c \right )^{2}}} x^{3}}{105}+\frac {4 F^{a} d \,b^{2} \ln \left (F \right )^{2} F^{\frac {b}{\left (d x +c \right )^{2}}} c \,x^{2}}{35}+\frac {4 F^{a} b^{2} \ln \left (F \right )^{2} F^{\frac {b}{\left (d x +c \right )^{2}}} c^{2} x}{35}+\frac {4 F^{a} b^{2} \ln \left (F \right )^{2} F^{\frac {b}{\left (d x +c \right )^{2}}} c^{3}}{105 d}+\frac {8 F^{a} b^{3} \ln \left (F \right )^{3} F^{\frac {b}{\left (d x +c \right )^{2}}} x}{105}+\frac {8 F^{a} b^{3} \ln \left (F \right )^{3} F^{\frac {b}{\left (d x +c \right )^{2}}} c}{105 d}-\frac {8 F^{a} b^{4} \ln \left (F \right )^{4} \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-b \ln \left (F \right )}}{d x +c}\right )}{105 d \sqrt {-b \ln \left (F \right )}}\) | \(543\) |
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Time = 0.29 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.72 \[ \int F^{a+\frac {b}{(c+d x)^2}} (c+d x)^6 \, dx=\frac {8 \, \sqrt {\pi } F^{a} b^{3} d \sqrt {-\frac {b \log \left (F\right )}{d^{2}}} \operatorname {erf}\left (\frac {d \sqrt {-\frac {b \log \left (F\right )}{d^{2}}}}{d x + c}\right ) \log \left (F\right )^{3} + {\left (15 \, d^{7} x^{7} + 105 \, c d^{6} x^{6} + 315 \, c^{2} d^{5} x^{5} + 525 \, c^{3} d^{4} x^{4} + 525 \, c^{4} d^{3} x^{3} + 315 \, c^{5} d^{2} x^{2} + 105 \, c^{6} d x + 15 \, c^{7} + 8 \, {\left (b^{3} d x + b^{3} c\right )} \log \left (F\right )^{3} + 4 \, {\left (b^{2} d^{3} x^{3} + 3 \, b^{2} c d^{2} x^{2} + 3 \, b^{2} c^{2} d x + b^{2} c^{3}\right )} \log \left (F\right )^{2} + 6 \, {\left (b d^{5} x^{5} + 5 \, b c d^{4} x^{4} + 10 \, b c^{2} d^{3} x^{3} + 10 \, b c^{3} d^{2} x^{2} + 5 \, b c^{4} d x + b c^{5}\right )} \log \left (F\right )\right )} F^{\frac {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{105 \, d} \]
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\[ \int F^{a+\frac {b}{(c+d x)^2}} (c+d x)^6 \, dx=\int F^{a + \frac {b}{\left (c + d x\right )^{2}}} \left (c + d x\right )^{6}\, dx \]
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\[ \int F^{a+\frac {b}{(c+d x)^2}} (c+d x)^6 \, dx=\int { {\left (d x + c\right )}^{6} F^{a + \frac {b}{{\left (d x + c\right )}^{2}}} \,d x } \]
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\[ \int F^{a+\frac {b}{(c+d x)^2}} (c+d x)^6 \, dx=\int { {\left (d x + c\right )}^{6} F^{a + \frac {b}{{\left (d x + c\right )}^{2}}} \,d x } \]
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Time = 1.10 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.17 \[ \int F^{a+\frac {b}{(c+d x)^2}} (c+d x)^6 \, dx=\frac {F^a\,F^{\frac {b}{{\left (c+d\,x\right )}^2}}\,{\left (c+d\,x\right )}^7}{7\,d}-\frac {8\,F^a\,\sqrt {\pi }\,{\left (c+d\,x\right )}^7\,{\left (-\frac {b\,\ln \left (F\right )}{{\left (c+d\,x\right )}^2}\right )}^{7/2}}{105\,d}+\frac {4\,F^a\,F^{\frac {b}{{\left (c+d\,x\right )}^2}}\,b^2\,{\ln \left (F\right )}^2\,{\left (c+d\,x\right )}^3}{105\,d}+\frac {2\,F^a\,F^{\frac {b}{{\left (c+d\,x\right )}^2}}\,b\,\ln \left (F\right )\,{\left (c+d\,x\right )}^5}{35\,d}+\frac {8\,F^a\,F^{\frac {b}{{\left (c+d\,x\right )}^2}}\,b^3\,{\ln \left (F\right )}^3\,\left (c+d\,x\right )}{105\,d}+\frac {8\,F^a\,\sqrt {\pi }\,\mathrm {erfc}\left (\sqrt {-\frac {b\,\ln \left (F\right )}{{\left (c+d\,x\right )}^2}}\right )\,{\left (c+d\,x\right )}^7\,{\left (-\frac {b\,\ln \left (F\right )}{{\left (c+d\,x\right )}^2}\right )}^{7/2}}{105\,d} \]
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