Optimal. Leaf size=217 \[ \frac{1}{6} b^3 d^3 e^2 \log ^3(F) F^{a+b c} \text{Ei}(b d x \log (F))-\frac{b^2 d^2 e^2 \log ^2(F) F^{a+b c+b d x}}{6 x}+b^2 d^2 e f \log ^2(F) F^{a+b c} \text{Ei}(b d x \log (F))-\frac{e^2 F^{a+b c+b d x}}{3 x^3}-\frac{b d e^2 \log (F) F^{a+b c+b d x}}{6 x^2}-\frac{e f F^{a+b c+b d x}}{x^2}-\frac{b d e f \log (F) F^{a+b c+b d x}}{x}+b d f^2 \log (F) F^{a+b c} \text{Ei}(b d x \log (F))-\frac{f^2 F^{a+b c+b d x}}{x} \]
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Rubi [A] time = 0.458988, antiderivative size = 217, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {2199, 2177, 2178} \[ \frac{1}{6} b^3 d^3 e^2 \log ^3(F) F^{a+b c} \text{Ei}(b d x \log (F))-\frac{b^2 d^2 e^2 \log ^2(F) F^{a+b c+b d x}}{6 x}+b^2 d^2 e f \log ^2(F) F^{a+b c} \text{Ei}(b d x \log (F))-\frac{e^2 F^{a+b c+b d x}}{3 x^3}-\frac{b d e^2 \log (F) F^{a+b c+b d x}}{6 x^2}-\frac{e f F^{a+b c+b d x}}{x^2}-\frac{b d e f \log (F) F^{a+b c+b d x}}{x}+b d f^2 \log (F) F^{a+b c} \text{Ei}(b d x \log (F))-\frac{f^2 F^{a+b c+b d x}}{x} \]
Antiderivative was successfully verified.
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Rule 2199
Rule 2177
Rule 2178
Rubi steps
\begin{align*} \int \frac{F^{a+b (c+d x)} (e+f x)^2}{x^4} \, dx &=\int \left (\frac{e^2 F^{a+b c+b d x}}{x^4}+\frac{2 e f F^{a+b c+b d x}}{x^3}+\frac{f^2 F^{a+b c+b d x}}{x^2}\right ) \, dx\\ &=e^2 \int \frac{F^{a+b c+b d x}}{x^4} \, dx+(2 e f) \int \frac{F^{a+b c+b d x}}{x^3} \, dx+f^2 \int \frac{F^{a+b c+b d x}}{x^2} \, dx\\ &=-\frac{e^2 F^{a+b c+b d x}}{3 x^3}-\frac{e f F^{a+b c+b d x}}{x^2}-\frac{f^2 F^{a+b c+b d x}}{x}+\frac{1}{3} \left (b d e^2 \log (F)\right ) \int \frac{F^{a+b c+b d x}}{x^3} \, dx+(b d e f \log (F)) \int \frac{F^{a+b c+b d x}}{x^2} \, dx+\left (b d f^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}}{3 x^3}-\frac{e f F^{a+b c+b d x}}{x^2}-\frac{f^2 F^{a+b c+b d x}}{x}-\frac{b d e^2 F^{a+b c+b d x} \log (F)}{6 x^2}-\frac{b d e f F^{a+b c+b d x} \log (F)}{x}+b d f^2 F^{a+b c} \text{Ei}(b d x \log (F)) \log (F)+\frac{1}{6} \left (b^2 d^2 e^2 \log ^2(F)\right ) \int \frac{F^{a+b c+b d x}}{x^2} \, dx+\left (b^2 d^2 e f \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}}{3 x^3}-\frac{e f F^{a+b c+b d x}}{x^2}-\frac{f^2 F^{a+b c+b d x}}{x}-\frac{b d e^2 F^{a+b c+b d x} \log (F)}{6 x^2}-\frac{b d e f F^{a+b c+b d x} \log (F)}{x}+b d f^2 F^{a+b c} \text{Ei}(b d x \log (F)) \log (F)-\frac{b^2 d^2 e^2 F^{a+b c+b d x} \log ^2(F)}{6 x}+b^2 d^2 e f F^{a+b c} \text{Ei}(b d x \log (F)) \log ^2(F)+\frac{1}{6} \left (b^3 d^3 e^2 \log ^3(F)\right ) \int \frac{F^{a+b c+b d x}}{x} \, dx\\ &=-\frac{e^2 F^{a+b c+b d x}}{3 x^3}-\frac{e f F^{a+b c+b d x}}{x^2}-\frac{f^2 F^{a+b c+b d x}}{x}-\frac{b d e^2 F^{a+b c+b d x} \log (F)}{6 x^2}-\frac{b d e f F^{a+b c+b d x} \log (F)}{x}+b d f^2 F^{a+b c} \text{Ei}(b d x \log (F)) \log (F)-\frac{b^2 d^2 e^2 F^{a+b c+b d x} \log ^2(F)}{6 x}+b^2 d^2 e f F^{a+b c} \text{Ei}(b d x \log (F)) \log ^2(F)+\frac{1}{6} b^3 d^3 e^2 F^{a+b c} \text{Ei}(b d x \log (F)) \log ^3(F)\\ \end{align*}
Mathematica [A] time = 0.223677, size = 116, normalized size = 0.53 \[ \frac{F^{a+b c} \left (b d x^3 \log (F) \left (b^2 d^2 e^2 \log ^2(F)+6 b d e f \log (F)+6 f^2\right ) \text{Ei}(b d x \log (F))-F^{b d x} \left (b^2 d^2 e^2 x^2 \log ^2(F)+b d e x \log (F) (e+6 f x)+2 \left (e^2+3 e f x+3 f^2 x^2\right )\right )\right )}{6 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.065, size = 290, normalized size = 1.3 \begin{align*} -{\frac{ \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}{d}^{3}{e}^{2}{F}^{bc}{F}^{a}{\it Ei} \left ( 1,bc\ln \left ( F \right ) +\ln \left ( F \right ) a-bdx\ln \left ( F \right ) - \left ( bc+a \right ) \ln \left ( F \right ) \right ) }{6}}-{\frac{fe{F}^{bdx}{F}^{bc+a}}{{x}^{2}}}-{\frac{fe\ln \left ( F \right ) bd{F}^{bdx}{F}^{bc+a}}{x}}- \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{d}^{2}fe{F}^{bc}{F}^{a}{\it Ei} \left ( 1,bc\ln \left ( F \right ) +\ln \left ( F \right ) a-bdx\ln \left ( F \right ) - \left ( bc+a \right ) \ln \left ( F \right ) \right ) -{\frac{{e}^{2}{F}^{bdx}{F}^{bc+a}}{3\,{x}^{3}}}-{\frac{\ln \left ( F \right ) bd{e}^{2}{F}^{bdx}{F}^{bc+a}}{6\,{x}^{2}}}-{\frac{ \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{d}^{2}{e}^{2}{F}^{bdx}{F}^{bc+a}}{6\,x}}-{\frac{{f}^{2}{F}^{bdx}{F}^{bc+a}}{x}}-\ln \left ( F \right ) bd{f}^{2}{F}^{bc}{F}^{a}{\it Ei} \left ( 1,bc\ln \left ( F \right ) +\ln \left ( F \right ) a-bdx\ln \left ( F \right ) - \left ( bc+a \right ) \ln \left ( F \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.21015, size = 115, normalized size = 0.53 \begin{align*} F^{b c + a} b^{3} d^{3} e^{2} \Gamma \left (-3, -b d x \log \left (F\right )\right ) \log \left (F\right )^{3} - 2 \, F^{b c + a} b^{2} d^{2} e f \Gamma \left (-2, -b d x \log \left (F\right )\right ) \log \left (F\right )^{2} + F^{b c + a} b d f^{2} \Gamma \left (-1, -b d x \log \left (F\right )\right ) \log \left (F\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53015, size = 317, normalized size = 1.46 \begin{align*} \frac{{\left (b^{3} d^{3} e^{2} x^{3} \log \left (F\right )^{3} + 6 \, b^{2} d^{2} e f x^{3} \log \left (F\right )^{2} + 6 \, b d f^{2} x^{3} \log \left (F\right )\right )} F^{b c + a}{\rm Ei}\left (b d x \log \left (F\right )\right ) -{\left (b^{2} d^{2} e^{2} x^{2} \log \left (F\right )^{2} + 6 \, f^{2} x^{2} + 6 \, e f x + 2 \, e^{2} +{\left (6 \, b d e f x^{2} + b d e^{2} x\right )} \log \left (F\right )\right )} F^{b d x + b c + a}}{6 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{F^{a + b \left (c + d x\right )} \left (e + f x\right )^{2}}{x^{4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (f x + e\right )}^{2} F^{{\left (d x + c\right )} b + a}}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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