Integrand size = 22, antiderivative size = 136 \[ \int \frac {F^{a+b (c+d x)} (e+f x)^2}{x^3} \, dx=-\frac {e^2 F^{a+b c+b d x}}{2 x^2}-\frac {2 e f F^{a+b c+b d x}}{x}+f^2 F^{a+b c} \operatorname {ExpIntegralEi}(b d x \log (F))-\frac {b d e^2 F^{a+b c+b d x} \log (F)}{2 x}+2 b d e f F^{a+b c} \operatorname {ExpIntegralEi}(b d x \log (F)) \log (F)+\frac {1}{2} b^2 d^2 e^2 F^{a+b c} \operatorname {ExpIntegralEi}(b d x \log (F)) \log ^2(F) \]
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Time = 0.23 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {2230, 2208, 2209} \[ \int \frac {F^{a+b (c+d x)} (e+f x)^2}{x^3} \, dx=\frac {1}{2} b^2 d^2 e^2 \log ^2(F) F^{a+b c} \operatorname {ExpIntegralEi}(b d x \log (F))-\frac {e^2 F^{a+b c+b d x}}{2 x^2}-\frac {b d e^2 \log (F) F^{a+b c+b d x}}{2 x}+2 b d e f \log (F) F^{a+b c} \operatorname {ExpIntegralEi}(b d x \log (F))-\frac {2 e f F^{a+b c+b d x}}{x}+f^2 F^{a+b c} \operatorname {ExpIntegralEi}(b d x \log (F)) \]
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Rule 2208
Rule 2209
Rule 2230
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {e^2 F^{a+b c+b d x}}{x^3}+\frac {2 e f F^{a+b c+b d x}}{x^2}+\frac {f^2 F^{a+b c+b d x}}{x}\right ) \, dx \\ & = e^2 \int \frac {F^{a+b c+b d x}}{x^3} \, dx+(2 e f) \int \frac {F^{a+b c+b d x}}{x^2} \, dx+f^2 \int \frac {F^{a+b c+b d x}}{x} \, dx \\ & = -\frac {e^2 F^{a+b c+b d x}}{2 x^2}-\frac {2 e f F^{a+b c+b d x}}{x}+f^2 F^{a+b c} \text {Ei}(b d x \log (F))+\frac {1}{2} \left (b d e^2 \log (F)\right ) \int \frac {F^{a+b c+b d x}}{x^2} \, dx+(2 b d e f \log (F)) \int \frac {F^{a+b c+b d x}}{x} \, dx \\ & = -\frac {e^2 F^{a+b c+b d x}}{2 x^2}-\frac {2 e f F^{a+b c+b d x}}{x}+f^2 F^{a+b c} \text {Ei}(b d x \log (F))-\frac {b d e^2 F^{a+b c+b d x} \log (F)}{2 x}+2 b d e f F^{a+b c} \text {Ei}(b d x \log (F)) \log (F)+\frac {1}{2} \left (b^2 d^2 e^2 \log ^2(F)\right ) \int \frac {F^{a+b c+b d x}}{x} \, dx \\ & = -\frac {e^2 F^{a+b c+b d x}}{2 x^2}-\frac {2 e f F^{a+b c+b d x}}{x}+f^2 F^{a+b c} \text {Ei}(b d x \log (F))-\frac {b d e^2 F^{a+b c+b d x} \log (F)}{2 x}+2 b d e f F^{a+b c} \text {Ei}(b d x \log (F)) \log (F)+\frac {1}{2} b^2 d^2 e^2 F^{a+b c} \text {Ei}(b d x \log (F)) \log ^2(F) \\ \end{align*}
Time = 0.33 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.56 \[ \int \frac {F^{a+b (c+d x)} (e+f x)^2}{x^3} \, dx=\frac {F^{a+b c} \left (-e F^{b d x} (e+4 f x+b d e x \log (F))+x^2 \operatorname {ExpIntegralEi}(b d x \log (F)) \left (2 f^2+4 b d e f \log (F)+b^2 d^2 e^2 \log ^2(F)\right )\right )}{2 x^2} \]
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Time = 0.23 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.53
method | result | size |
risch | \(-\frac {\ln \left (F \right )^{2} F^{c b} F^{a} \operatorname {Ei}_{1}\left (c b \ln \left (F \right )+a \ln \left (F \right )-b d x \ln \left (F \right )-\left (c b +a \right ) \ln \left (F \right )\right ) b^{2} d^{2} e^{2} x^{2}+4 \ln \left (F \right ) F^{c b} F^{a} \operatorname {Ei}_{1}\left (c b \ln \left (F \right )+a \ln \left (F \right )-b d x \ln \left (F \right )-\left (c b +a \right ) \ln \left (F \right )\right ) b d e f \,x^{2}+\ln \left (F \right ) F^{b d x} F^{c b +a} b d \,e^{2} x +2 F^{c b} F^{a} \operatorname {Ei}_{1}\left (c b \ln \left (F \right )+a \ln \left (F \right )-b d x \ln \left (F \right )-\left (c b +a \right ) \ln \left (F \right )\right ) f^{2} x^{2}+4 F^{b d x} F^{c b +a} e f x +F^{b d x} F^{c b +a} e^{2}}{2 x^{2}}\) | \(208\) |
meijerg | \(F^{c b +a} f^{2} \left (\ln \left (x \right )+\ln \left (-b d \right )+\ln \left (\ln \left (F \right )\right )-\ln \left (-b d x \ln \left (F \right )\right )-\operatorname {Ei}_{1}\left (-b d x \ln \left (F \right )\right )\right )-2 b d \ln \left (F \right ) F^{c b +a} f e \left (\frac {1}{b d x \ln \left (F \right )}+1-\ln \left (x \right )-\ln \left (-b d \right )-\ln \left (\ln \left (F \right )\right )-\frac {2+2 b d x \ln \left (F \right )}{2 b d x \ln \left (F \right )}+\frac {{\mathrm e}^{b d x \ln \left (F \right )}}{b d x \ln \left (F \right )}+\ln \left (-b d x \ln \left (F \right )\right )+\operatorname {Ei}_{1}\left (-b d x \ln \left (F \right )\right )\right )+F^{c b +a} e^{2} \ln \left (F \right )^{2} b^{2} d^{2} \left (-\frac {1}{2 b^{2} d^{2} x^{2} \ln \left (F \right )^{2}}-\frac {1}{b d x \ln \left (F \right )}-\frac {3}{4}+\frac {\ln \left (x \right )}{2}+\frac {\ln \left (-b d \right )}{2}+\frac {\ln \left (\ln \left (F \right )\right )}{2}+\frac {9 b^{2} d^{2} x^{2} \ln \left (F \right )^{2}+12 b d x \ln \left (F \right )+6}{12 b^{2} d^{2} x^{2} \ln \left (F \right )^{2}}-\frac {\left (3 b d x \ln \left (F \right )+3\right ) {\mathrm e}^{b d x \ln \left (F \right )}}{6 b^{2} d^{2} x^{2} \ln \left (F \right )^{2}}-\frac {\ln \left (-b d x \ln \left (F \right )\right )}{2}-\frac {\operatorname {Ei}_{1}\left (-b d x \ln \left (F \right )\right )}{2}\right )\) | \(314\) |
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Time = 0.23 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.65 \[ \int \frac {F^{a+b (c+d x)} (e+f x)^2}{x^3} \, dx=\frac {{\left (b^{2} d^{2} e^{2} x^{2} \log \left (F\right )^{2} + 4 \, b d e f x^{2} \log \left (F\right ) + 2 \, f^{2} x^{2}\right )} F^{b c + a} {\rm Ei}\left (b d x \log \left (F\right )\right ) - {\left (b d e^{2} x \log \left (F\right ) + 4 \, e f x + e^{2}\right )} F^{b d x + b c + a}}{2 \, x^{2}} \]
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\[ \int \frac {F^{a+b (c+d x)} (e+f x)^2}{x^3} \, dx=\int \frac {F^{a + b \left (c + d x\right )} \left (e + f x\right )^{2}}{x^{3}}\, dx \]
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Time = 0.27 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.54 \[ \int \frac {F^{a+b (c+d x)} (e+f x)^2}{x^3} \, dx=-F^{b c + a} b^{2} d^{2} e^{2} \Gamma \left (-2, -b d x \log \left (F\right )\right ) \log \left (F\right )^{2} + 2 \, F^{b c + a} b d e f \Gamma \left (-1, -b d x \log \left (F\right )\right ) \log \left (F\right ) + F^{b c + a} f^{2} {\rm Ei}\left (b d x \log \left (F\right )\right ) \]
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\[ \int \frac {F^{a+b (c+d x)} (e+f x)^2}{x^3} \, dx=\int { \frac {{\left (f x + e\right )}^{2} F^{{\left (d x + c\right )} b + a}}{x^{3}} \,d x } \]
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Time = 0.28 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.98 \[ \int \frac {F^{a+b (c+d x)} (e+f x)^2}{x^3} \, dx=F^{a+b\,c}\,f^2\,\mathrm {ei}\left (b\,d\,x\,\ln \left (F\right )\right )-\frac {2\,F^{b\,d\,x}\,F^{a+b\,c}\,e\,f}{x}-F^{a+b\,c}\,b^2\,d^2\,e^2\,{\ln \left (F\right )}^2\,\left (\frac {\mathrm {expint}\left (-b\,d\,x\,\ln \left (F\right )\right )}{2}+F^{b\,d\,x}\,\left (\frac {1}{2\,b\,d\,x\,\ln \left (F\right )}+\frac {1}{2\,b^2\,d^2\,x^2\,{\ln \left (F\right )}^2}\right )\right )-2\,F^{a+b\,c}\,b\,d\,e\,f\,\ln \left (F\right )\,\mathrm {expint}\left (-b\,d\,x\,\ln \left (F\right )\right ) \]
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