Optimal. Leaf size=321 \[ \frac{1}{24} b^4 d^4 e^2 \log ^4(F) F^{a+b c} \text{Ei}(b d x \log (F))-\frac{b^2 d^2 e^2 \log ^2(F) F^{a+b c+b d x}}{24 x^2}-\frac{b^3 d^3 e^2 \log ^3(F) F^{a+b c+b d x}}{24 x}+\frac{1}{3} b^3 d^3 e f \log ^3(F) F^{a+b c} \text{Ei}(b d x \log (F))-\frac{b^2 d^2 e f \log ^2(F) F^{a+b c+b d x}}{3 x}+\frac{1}{2} b^2 d^2 f^2 \log ^2(F) F^{a+b c} \text{Ei}(b d x \log (F))-\frac{e^2 F^{a+b c+b d x}}{4 x^4}-\frac{b d e^2 \log (F) F^{a+b c+b d x}}{12 x^3}-\frac{2 e f F^{a+b c+b d x}}{3 x^3}-\frac{b d e f \log (F) F^{a+b c+b d x}}{3 x^2}-\frac{f^2 F^{a+b c+b d x}}{2 x^2}-\frac{b d f^2 \log (F) F^{a+b c+b d x}}{2 x} \]
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Rubi [A] time = 0.577845, antiderivative size = 321, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {2199, 2177, 2178} \[ \frac{1}{24} b^4 d^4 e^2 \log ^4(F) F^{a+b c} \text{Ei}(b d x \log (F))-\frac{b^2 d^2 e^2 \log ^2(F) F^{a+b c+b d x}}{24 x^2}-\frac{b^3 d^3 e^2 \log ^3(F) F^{a+b c+b d x}}{24 x}+\frac{1}{3} b^3 d^3 e f \log ^3(F) F^{a+b c} \text{Ei}(b d x \log (F))-\frac{b^2 d^2 e f \log ^2(F) F^{a+b c+b d x}}{3 x}+\frac{1}{2} b^2 d^2 f^2 \log ^2(F) F^{a+b c} \text{Ei}(b d x \log (F))-\frac{e^2 F^{a+b c+b d x}}{4 x^4}-\frac{b d e^2 \log (F) F^{a+b c+b d x}}{12 x^3}-\frac{2 e f F^{a+b c+b d x}}{3 x^3}-\frac{b d e f \log (F) F^{a+b c+b d x}}{3 x^2}-\frac{f^2 F^{a+b c+b d x}}{2 x^2}-\frac{b d f^2 \log (F) F^{a+b c+b d x}}{2 x} \]
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^5} \, dx &=\int \left (\frac{e^2 F^{a+b c+b d x}}{x^5}+\frac{2 e f F^{a+b c+b d x}}{x^4}+\frac{f^2 F^{a+b c+b d x}}{x^3}\right ) \, dx\\ &=e^2 \int \frac{F^{a+b c+b d x}}{x^5} \, dx+(2 e f) \int \frac{F^{a+b c+b d x}}{x^4} \, dx+f^2 \int \frac{F^{a+b c+b d x}}{x^3} \, dx\\ &=-\frac{e^2 F^{a+b c+b d x}}{4 x^4}-\frac{2 e f F^{a+b c+b d x}}{3 x^3}-\frac{f^2 F^{a+b c+b d x}}{2 x^2}+\frac{1}{4} \left (b d e^2 \log (F)\right ) \int \frac{F^{a+b c+b d x}}{x^4} \, dx+\frac{1}{3} (2 b d e f \log (F)) \int \frac{F^{a+b c+b d x}}{x^3} \, dx+\frac{1}{2} \left (b d f^2 \log (F)\right ) \int \frac{F^{a+b c+b d x}}{x^2} \, dx\\ &=-\frac{e^2 F^{a+b c+b d x}}{4 x^4}-\frac{2 e f F^{a+b c+b d x}}{3 x^3}-\frac{f^2 F^{a+b c+b d x}}{2 x^2}-\frac{b d e^2 F^{a+b c+b d x} \log (F)}{12 x^3}-\frac{b d e f F^{a+b c+b d x} \log (F)}{3 x^2}-\frac{b d f^2 F^{a+b c+b d x} \log (F)}{2 x}+\frac{1}{12} \left (b^2 d^2 e^2 \log ^2(F)\right ) \int \frac{F^{a+b c+b d x}}{x^3} \, dx+\frac{1}{3} \left (b^2 d^2 e f \log ^2(F)\right ) \int \frac{F^{a+b c+b d x}}{x^2} \, dx+\frac{1}{2} \left (b^2 d^2 f^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}}{4 x^4}-\frac{2 e f F^{a+b c+b d x}}{3 x^3}-\frac{f^2 F^{a+b c+b d x}}{2 x^2}-\frac{b d e^2 F^{a+b c+b d x} \log (F)}{12 x^3}-\frac{b d e f F^{a+b c+b d x} \log (F)}{3 x^2}-\frac{b d f^2 F^{a+b c+b d x} \log (F)}{2 x}-\frac{b^2 d^2 e^2 F^{a+b c+b d x} \log ^2(F)}{24 x^2}-\frac{b^2 d^2 e f F^{a+b c+b d x} \log ^2(F)}{3 x}+\frac{1}{2} b^2 d^2 f^2 F^{a+b c} \text{Ei}(b d x \log (F)) \log ^2(F)+\frac{1}{24} \left (b^3 d^3 e^2 \log ^3(F)\right ) \int \frac{F^{a+b c+b d x}}{x^2} \, dx+\frac{1}{3} \left (b^3 d^3 e f \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}}{4 x^4}-\frac{2 e f F^{a+b c+b d x}}{3 x^3}-\frac{f^2 F^{a+b c+b d x}}{2 x^2}-\frac{b d e^2 F^{a+b c+b d x} \log (F)}{12 x^3}-\frac{b d e f F^{a+b c+b d x} \log (F)}{3 x^2}-\frac{b d f^2 F^{a+b c+b d x} \log (F)}{2 x}-\frac{b^2 d^2 e^2 F^{a+b c+b d x} \log ^2(F)}{24 x^2}-\frac{b^2 d^2 e f F^{a+b c+b d x} \log ^2(F)}{3 x}+\frac{1}{2} b^2 d^2 f^2 F^{a+b c} \text{Ei}(b d x \log (F)) \log ^2(F)-\frac{b^3 d^3 e^2 F^{a+b c+b d x} \log ^3(F)}{24 x}+\frac{1}{3} b^3 d^3 e f F^{a+b c} \text{Ei}(b d x \log (F)) \log ^3(F)+\frac{1}{24} \left (b^4 d^4 e^2 \log ^4(F)\right ) \int \frac{F^{a+b c+b d x}}{x} \, dx\\ &=-\frac{e^2 F^{a+b c+b d x}}{4 x^4}-\frac{2 e f F^{a+b c+b d x}}{3 x^3}-\frac{f^2 F^{a+b c+b d x}}{2 x^2}-\frac{b d e^2 F^{a+b c+b d x} \log (F)}{12 x^3}-\frac{b d e f F^{a+b c+b d x} \log (F)}{3 x^2}-\frac{b d f^2 F^{a+b c+b d x} \log (F)}{2 x}-\frac{b^2 d^2 e^2 F^{a+b c+b d x} \log ^2(F)}{24 x^2}-\frac{b^2 d^2 e f F^{a+b c+b d x} \log ^2(F)}{3 x}+\frac{1}{2} b^2 d^2 f^2 F^{a+b c} \text{Ei}(b d x \log (F)) \log ^2(F)-\frac{b^3 d^3 e^2 F^{a+b c+b d x} \log ^3(F)}{24 x}+\frac{1}{3} b^3 d^3 e f F^{a+b c} \text{Ei}(b d x \log (F)) \log ^3(F)+\frac{1}{24} b^4 d^4 e^2 F^{a+b c} \text{Ei}(b d x \log (F)) \log ^4(F)\\ \end{align*}
Mathematica [A] time = 0.296526, size = 156, normalized size = 0.49 \[ \frac{F^{a+b c} \left (b^2 d^2 x^4 \log ^2(F) \left (b^2 d^2 e^2 \log ^2(F)+8 b d e f \log (F)+12 f^2\right ) \text{Ei}(b d x \log (F))-F^{b d x} \left (b^3 d^3 e^2 x^3 \log ^3(F)+b^2 d^2 e x^2 \log ^2(F) (e+8 f x)+2 b d x \log (F) \left (e^2+4 e f x+6 f^2 x^2\right )+2 \left (3 e^2+8 e f x+6 f^2 x^2\right )\right )\right )}{24 x^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.066, size = 382, normalized size = 1.2 \begin{align*} -{\frac{ \left ( \ln \left ( F \right ) \right ) ^{4}{b}^{4}{d}^{4}{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 ) }{24}}-{\frac{2\,fe{F}^{bdx}{F}^{bc+a}}{3\,{x}^{3}}}-{\frac{fe\ln \left ( F \right ) bd{F}^{bdx}{F}^{bc+a}}{3\,{x}^{2}}}-{\frac{{b}^{2}{d}^{2} \left ( \ln \left ( F \right ) \right ) ^{2}fe{F}^{bdx}{F}^{bc+a}}{3\,x}}-{\frac{ \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}{d}^{3}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 ) }{3}}-{\frac{bd\ln \left ( F \right ){f}^{2}{F}^{bdx}{F}^{bc+a}}{2\,x}}-{\frac{ \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{d}^{2}{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 ) }{2}}-{\frac{{f}^{2}{F}^{bdx}{F}^{bc+a}}{2\,{x}^{2}}}-{\frac{{e}^{2}{F}^{bdx}{F}^{bc+a}}{4\,{x}^{4}}}-{\frac{\ln \left ( F \right ) bd{e}^{2}{F}^{bdx}{F}^{bc+a}}{12\,{x}^{3}}}-{\frac{ \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{d}^{2}{e}^{2}{F}^{bdx}{F}^{bc+a}}{24\,{x}^{2}}}-{\frac{ \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}{d}^{3}{e}^{2}{F}^{bdx}{F}^{bc+a}}{24\,x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.24327, size = 126, normalized size = 0.39 \begin{align*} -F^{b c + a} b^{4} d^{4} e^{2} \Gamma \left (-4, -b d x \log \left (F\right )\right ) \log \left (F\right )^{4} + 2 \, F^{b c + a} b^{3} d^{3} e f \Gamma \left (-3, -b d x \log \left (F\right )\right ) \log \left (F\right )^{3} - F^{b c + a} b^{2} d^{2} f^{2} \Gamma \left (-2, -b d x \log \left (F\right )\right ) \log \left (F\right )^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.50121, size = 421, normalized size = 1.31 \begin{align*} \frac{{\left (b^{4} d^{4} e^{2} x^{4} \log \left (F\right )^{4} + 8 \, b^{3} d^{3} e f x^{4} \log \left (F\right )^{3} + 12 \, b^{2} d^{2} f^{2} x^{4} \log \left (F\right )^{2}\right )} F^{b c + a}{\rm Ei}\left (b d x \log \left (F\right )\right ) -{\left (b^{3} d^{3} e^{2} x^{3} \log \left (F\right )^{3} + 12 \, f^{2} x^{2} + 16 \, e f x +{\left (8 \, b^{2} d^{2} e f x^{3} + b^{2} d^{2} e^{2} x^{2}\right )} \log \left (F\right )^{2} + 6 \, e^{2} + 2 \,{\left (6 \, b d f^{2} x^{3} + 4 \, b d e f x^{2} + b d e^{2} x\right )} \log \left (F\right )\right )} F^{b d x + b c + a}}{24 \, x^{4}} \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^{5}}\, 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^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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