Integrand size = 22, antiderivative size = 85 \[ \int \frac {F^{a+b (c+d x)} (e+f x)^2}{x^2} \, dx=-\frac {e^2 F^{a+b c+b d x}}{x}+2 e f F^{a+b c} \operatorname {ExpIntegralEi}(b d x \log (F))+\frac {f^2 F^{a+b c+b d x}}{b d \log (F)}+b d e^2 F^{a+b c} \operatorname {ExpIntegralEi}(b d x \log (F)) \log (F) \]
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Time = 0.18 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2230, 2225, 2208, 2209} \[ \int \frac {F^{a+b (c+d x)} (e+f x)^2}{x^2} \, dx=b d e^2 \log (F) F^{a+b c} \operatorname {ExpIntegralEi}(b d x \log (F))-\frac {e^2 F^{a+b c+b d x}}{x}+2 e f F^{a+b c} \operatorname {ExpIntegralEi}(b d x \log (F))+\frac {f^2 F^{a+b c+b d x}}{b d \log (F)} \]
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Rule 2208
Rule 2209
Rule 2225
Rule 2230
Rubi steps \begin{align*} \text {integral}& = \int \left (f^2 F^{a+b c+b d x}+\frac {e^2 F^{a+b c+b d x}}{x^2}+\frac {2 e f F^{a+b c+b d x}}{x}\right ) \, dx \\ & = e^2 \int \frac {F^{a+b c+b d x}}{x^2} \, dx+(2 e f) \int \frac {F^{a+b c+b d x}}{x} \, dx+f^2 \int F^{a+b c+b d x} \, dx \\ & = -\frac {e^2 F^{a+b c+b d x}}{x}+2 e f F^{a+b c} \text {Ei}(b d x \log (F))+\frac {f^2 F^{a+b c+b d x}}{b d \log (F)}+\left (b d e^2 \log (F)\right ) \int \frac {F^{a+b c+b d x}}{x} \, dx \\ & = -\frac {e^2 F^{a+b c+b d x}}{x}+2 e f F^{a+b c} \text {Ei}(b d x \log (F))+\frac {f^2 F^{a+b c+b d x}}{b d \log (F)}+b d e^2 F^{a+b c} \text {Ei}(b d x \log (F)) \log (F) \\ \end{align*}
Time = 0.25 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.68 \[ \int \frac {F^{a+b (c+d x)} (e+f x)^2}{x^2} \, dx=F^{a+b c} \left (F^{b d x} \left (-\frac {e^2}{x}+\frac {f^2}{b d \log (F)}\right )+e \operatorname {ExpIntegralEi}(b d x \log (F)) (2 f+b d e \log (F))\right ) \]
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Time = 0.23 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.80
method | result | size |
risch | \(-\frac {\ln \left (F \right )^{2} F^{c b} F^{a} \operatorname {Ei}_{1}\left (c b \ln \left (F \right )+a \ln \left (F \right )-b d x \ln \left (F \right )-\left (c b +a \right ) \ln \left (F \right )\right ) b^{2} d^{2} e^{2} x +2 \ln \left (F \right ) F^{c b} F^{a} \operatorname {Ei}_{1}\left (c b \ln \left (F \right )+a \ln \left (F \right )-b d x \ln \left (F \right )-\left (c b +a \right ) \ln \left (F \right )\right ) b d e f x +\ln \left (F \right ) F^{b d x} F^{c b +a} b d \,e^{2}-F^{b d x} F^{c b +a} f^{2} x}{\ln \left (F \right ) b d x}\) | \(153\) |
meijerg | \(-\frac {F^{c b +a} f^{2} \left (1-{\mathrm e}^{b d x \ln \left (F \right )}\right )}{b d \ln \left (F \right )}+2 F^{c b +a} f e \left (\ln \left (x \right )+\ln \left (-b d \right )+\ln \left (\ln \left (F \right )\right )-\ln \left (-b d x \ln \left (F \right )\right )-\operatorname {Ei}_{1}\left (-b d x \ln \left (F \right )\right )\right )-F^{c b +a} e^{2} b d \ln \left (F \right ) \left (\frac {1}{b d x \ln \left (F \right )}+1-\ln \left (x \right )-\ln \left (-b d \right )-\ln \left (\ln \left (F \right )\right )-\frac {2+2 b d x \ln \left (F \right )}{2 b d x \ln \left (F \right )}+\frac {{\mathrm e}^{b d x \ln \left (F \right )}}{b d x \ln \left (F \right )}+\ln \left (-b d x \ln \left (F \right )\right )+\operatorname {Ei}_{1}\left (-b d x \ln \left (F \right )\right )\right )\) | \(188\) |
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Time = 0.24 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.98 \[ \int \frac {F^{a+b (c+d x)} (e+f x)^2}{x^2} \, dx=\frac {{\left (b^{2} d^{2} e^{2} x \log \left (F\right )^{2} + 2 \, b d e f x \log \left (F\right )\right )} F^{b c + a} {\rm Ei}\left (b d x \log \left (F\right )\right ) - {\left (b d e^{2} \log \left (F\right ) - f^{2} x\right )} F^{b d x + b c + a}}{b d x \log \left (F\right )} \]
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\[ \int \frac {F^{a+b (c+d x)} (e+f x)^2}{x^2} \, dx=\int \frac {F^{a + b \left (c + d x\right )} \left (e + f x\right )^{2}}{x^{2}}\, dx \]
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Time = 0.26 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.80 \[ \int \frac {F^{a+b (c+d x)} (e+f x)^2}{x^2} \, dx=F^{b c + a} b d e^{2} \Gamma \left (-1, -b d x \log \left (F\right )\right ) \log \left (F\right ) + 2 \, F^{b c + a} e f {\rm Ei}\left (b d x \log \left (F\right )\right ) + \frac {F^{b d x + b c + a} f^{2}}{b d \log \left (F\right )} \]
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\[ \int \frac {F^{a+b (c+d x)} (e+f x)^2}{x^2} \, dx=\int { \frac {{\left (f x + e\right )}^{2} F^{{\left (d x + c\right )} b + a}}{x^{2}} \,d x } \]
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Time = 0.25 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.05 \[ \int \frac {F^{a+b (c+d x)} (e+f x)^2}{x^2} \, dx=2\,F^{a+b\,c}\,e\,f\,\mathrm {ei}\left (b\,d\,x\,\ln \left (F\right )\right )-\frac {F^{b\,d\,x}\,F^{a+b\,c}\,e^2}{x}+\frac {F^{a+b\,c+b\,d\,x}\,f^2}{b\,d\,\ln \left (F\right )}-F^{a+b\,c}\,b\,d\,e^2\,\ln \left (F\right )\,\mathrm {expint}\left (-b\,d\,x\,\ln \left (F\right )\right ) \]
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