Integrand size = 21, antiderivative size = 121 \[ \int F^{a+\frac {b}{(c+d x)^2}} (c+d x)^5 \, dx=\frac {F^{a+\frac {b}{(c+d x)^2}} (c+d x)^6}{6 d}+\frac {b F^{a+\frac {b}{(c+d x)^2}} (c+d x)^4 \log (F)}{12 d}+\frac {b^2 F^{a+\frac {b}{(c+d x)^2}} (c+d x)^2 \log ^2(F)}{12 d}-\frac {b^3 F^a \operatorname {ExpIntegralEi}\left (\frac {b \log (F)}{(c+d x)^2}\right ) \log ^3(F)}{12 d} \]
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Time = 0.11 (sec) , antiderivative size = 121, 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 = {2245, 2241} \[ \int F^{a+\frac {b}{(c+d x)^2}} (c+d x)^5 \, dx=-\frac {b^3 F^a \log ^3(F) \operatorname {ExpIntegralEi}\left (\frac {b \log (F)}{(c+d x)^2}\right )}{12 d}+\frac {b^2 \log ^2(F) (c+d x)^2 F^{a+\frac {b}{(c+d x)^2}}}{12 d}+\frac {(c+d x)^6 F^{a+\frac {b}{(c+d x)^2}}}{6 d}+\frac {b \log (F) (c+d x)^4 F^{a+\frac {b}{(c+d x)^2}}}{12 d} \]
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Rule 2241
Rule 2245
Rubi steps \begin{align*} \text {integral}& = \frac {F^{a+\frac {b}{(c+d x)^2}} (c+d x)^6}{6 d}+\frac {1}{3} (b \log (F)) \int F^{a+\frac {b}{(c+d x)^2}} (c+d x)^3 \, dx \\ & = \frac {F^{a+\frac {b}{(c+d x)^2}} (c+d x)^6}{6 d}+\frac {b F^{a+\frac {b}{(c+d x)^2}} (c+d x)^4 \log (F)}{12 d}+\frac {1}{6} \left (b^2 \log ^2(F)\right ) \int F^{a+\frac {b}{(c+d x)^2}} (c+d x) \, dx \\ & = \frac {F^{a+\frac {b}{(c+d x)^2}} (c+d x)^6}{6 d}+\frac {b F^{a+\frac {b}{(c+d x)^2}} (c+d x)^4 \log (F)}{12 d}+\frac {b^2 F^{a+\frac {b}{(c+d x)^2}} (c+d x)^2 \log ^2(F)}{12 d}+\frac {1}{6} \left (b^3 \log ^3(F)\right ) \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{c+d x} \, dx \\ & = \frac {F^{a+\frac {b}{(c+d x)^2}} (c+d x)^6}{6 d}+\frac {b F^{a+\frac {b}{(c+d x)^2}} (c+d x)^4 \log (F)}{12 d}+\frac {b^2 F^{a+\frac {b}{(c+d x)^2}} (c+d x)^2 \log ^2(F)}{12 d}-\frac {b^3 F^a \text {Ei}\left (\frac {b \log (F)}{(c+d x)^2}\right ) \log ^3(F)}{12 d} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.79 \[ \int F^{a+\frac {b}{(c+d x)^2}} (c+d x)^5 \, dx=\frac {F^a \left (2 F^{\frac {b}{(c+d x)^2}} (c+d x)^6+b \log (F) \left (F^{\frac {b}{(c+d x)^2}} (c+d x)^4+b \log (F) \left (F^{\frac {b}{(c+d x)^2}} (c+d x)^2-b \operatorname {ExpIntegralEi}\left (\frac {b \log (F)}{(c+d x)^2}\right ) \log (F)\right )\right )\right )}{12 d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(394\) vs. \(2(113)=226\).
Time = 1.03 (sec) , antiderivative size = 395, normalized size of antiderivative = 3.26
method | result | size |
risch | \(\frac {F^{a} d^{5} F^{\frac {b}{\left (d x +c \right )^{2}}} x^{6}}{6}+F^{a} d^{4} F^{\frac {b}{\left (d x +c \right )^{2}}} c \,x^{5}+\frac {5 F^{a} d^{3} F^{\frac {b}{\left (d x +c \right )^{2}}} c^{2} x^{4}}{2}+\frac {10 F^{a} d^{2} F^{\frac {b}{\left (d x +c \right )^{2}}} c^{3} x^{3}}{3}+\frac {5 F^{a} d \,F^{\frac {b}{\left (d x +c \right )^{2}}} c^{4} x^{2}}{2}+F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} c^{5} x +\frac {F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} c^{6}}{6 d}+\frac {F^{a} d^{3} b \ln \left (F \right ) F^{\frac {b}{\left (d x +c \right )^{2}}} x^{4}}{12}+\frac {F^{a} d^{2} b \ln \left (F \right ) F^{\frac {b}{\left (d x +c \right )^{2}}} c \,x^{3}}{3}+\frac {F^{a} d b \ln \left (F \right ) F^{\frac {b}{\left (d x +c \right )^{2}}} c^{2} x^{2}}{2}+\frac {F^{a} b \ln \left (F \right ) F^{\frac {b}{\left (d x +c \right )^{2}}} c^{3} x}{3}+\frac {F^{a} b \ln \left (F \right ) F^{\frac {b}{\left (d x +c \right )^{2}}} c^{4}}{12 d}+\frac {F^{a} d \,b^{2} \ln \left (F \right )^{2} F^{\frac {b}{\left (d x +c \right )^{2}}} x^{2}}{12}+\frac {F^{a} b^{2} \ln \left (F \right )^{2} F^{\frac {b}{\left (d x +c \right )^{2}}} c x}{6}+\frac {F^{a} b^{2} \ln \left (F \right )^{2} F^{\frac {b}{\left (d x +c \right )^{2}}} c^{2}}{12 d}+\frac {F^{a} b^{3} \ln \left (F \right )^{3} \operatorname {Ei}_{1}\left (-\frac {b \ln \left (F \right )}{\left (d x +c \right )^{2}}\right )}{12 d}\) | \(395\) |
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Time = 0.27 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.86 \[ \int F^{a+\frac {b}{(c+d x)^2}} (c+d x)^5 \, dx=-\frac {F^{a} b^{3} {\rm Ei}\left (\frac {b \log \left (F\right )}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) \log \left (F\right )^{3} - {\left (2 \, d^{6} x^{6} + 12 \, c d^{5} x^{5} + 30 \, c^{2} d^{4} x^{4} + 40 \, c^{3} d^{3} x^{3} + 30 \, c^{4} d^{2} x^{2} + 12 \, c^{5} d x + 2 \, c^{6} + {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \log \left (F\right )^{2} + {\left (b d^{4} x^{4} + 4 \, b c d^{3} x^{3} + 6 \, b c^{2} d^{2} x^{2} + 4 \, b c^{3} d x + b c^{4}\right )} \log \left (F\right )\right )} F^{\frac {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{12 \, d} \]
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\[ \int F^{a+\frac {b}{(c+d x)^2}} (c+d x)^5 \, dx=\int F^{a + \frac {b}{\left (c + d x\right )^{2}}} \left (c + d x\right )^{5}\, dx \]
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\[ \int F^{a+\frac {b}{(c+d x)^2}} (c+d x)^5 \, dx=\int { {\left (d x + c\right )}^{5} F^{a + \frac {b}{{\left (d x + c\right )}^{2}}} \,d x } \]
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\[ \int F^{a+\frac {b}{(c+d x)^2}} (c+d x)^5 \, dx=\int { {\left (d x + c\right )}^{5} F^{a + \frac {b}{{\left (d x + c\right )}^{2}}} \,d x } \]
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Time = 0.45 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.76 \[ \int F^{a+\frac {b}{(c+d x)^2}} (c+d x)^5 \, dx=\frac {F^a\,b^3\,{\ln \left (F\right )}^3\,\left (\frac {\mathrm {expint}\left (-\frac {b\,\ln \left (F\right )}{{\left (c+d\,x\right )}^2}\right )}{6}+F^{\frac {b}{{\left (c+d\,x\right )}^2}}\,\left (\frac {{\left (c+d\,x\right )}^2}{6\,b\,\ln \left (F\right )}+\frac {{\left (c+d\,x\right )}^4}{6\,b^2\,{\ln \left (F\right )}^2}+\frac {{\left (c+d\,x\right )}^6}{3\,b^3\,{\ln \left (F\right )}^3}\right )\right )}{2\,d} \]
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