Optimal. Leaf size=128 \[ \frac{b^3 c^3 \log ^3(F) F^{c \left (a-\frac{b d}{e}\right )} \text{Ei}\left (\frac{b c (d+e x) \log (F)}{e}\right )}{6 e^4}-\frac{b^2 c^2 \log ^2(F) F^{c (a+b x)}}{6 e^3 (d+e x)}-\frac{b c \log (F) F^{c (a+b x)}}{6 e^2 (d+e x)^2}-\frac{F^{c (a+b x)}}{3 e (d+e x)^3} \]
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Rubi [A] time = 0.133982, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 50, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.06, Rules used = {2187, 2177, 2178} \[ \frac{b^3 c^3 \log ^3(F) F^{c \left (a-\frac{b d}{e}\right )} \text{Ei}\left (\frac{b c (d+e x) \log (F)}{e}\right )}{6 e^4}-\frac{b^2 c^2 \log ^2(F) F^{c (a+b x)}}{6 e^3 (d+e x)}-\frac{b c \log (F) F^{c (a+b x)}}{6 e^2 (d+e x)^2}-\frac{F^{c (a+b x)}}{3 e (d+e x)^3} \]
Antiderivative was successfully verified.
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Rule 2187
Rule 2177
Rule 2178
Rubi steps
\begin{align*} \int \frac{F^{c (a+b x)}}{d^4+4 d^3 e x+6 d^2 e^2 x^2+4 d e^3 x^3+e^4 x^4} \, dx &=\int \frac{F^{c (a+b x)}}{(d+e x)^4} \, dx\\ &=-\frac{F^{c (a+b x)}}{3 e (d+e x)^3}+\frac{(b c \log (F)) \int \frac{F^{c (a+b x)}}{(d+e x)^3} \, dx}{3 e}\\ &=-\frac{F^{c (a+b x)}}{3 e (d+e x)^3}-\frac{b c F^{c (a+b x)} \log (F)}{6 e^2 (d+e x)^2}+\frac{\left (b^2 c^2 \log ^2(F)\right ) \int \frac{F^{c (a+b x)}}{(d+e x)^2} \, dx}{6 e^2}\\ &=-\frac{F^{c (a+b x)}}{3 e (d+e x)^3}-\frac{b c F^{c (a+b x)} \log (F)}{6 e^2 (d+e x)^2}-\frac{b^2 c^2 F^{c (a+b x)} \log ^2(F)}{6 e^3 (d+e x)}+\frac{\left (b^3 c^3 \log ^3(F)\right ) \int \frac{F^{c (a+b x)}}{d+e x} \, dx}{6 e^3}\\ &=-\frac{F^{c (a+b x)}}{3 e (d+e x)^3}-\frac{b c F^{c (a+b x)} \log (F)}{6 e^2 (d+e x)^2}-\frac{b^2 c^2 F^{c (a+b x)} \log ^2(F)}{6 e^3 (d+e x)}+\frac{b^3 c^3 F^{c \left (a-\frac{b d}{e}\right )} \text{Ei}\left (\frac{b c (d+e x) \log (F)}{e}\right ) \log ^3(F)}{6 e^4}\\ \end{align*}
Mathematica [A] time = 0.0800948, size = 99, normalized size = 0.77 \[ \frac{F^{a c} \left (b^3 c^3 \log ^3(F) F^{-\frac{b c d}{e}} \text{Ei}\left (\frac{b c (d+e x) \log (F)}{e}\right )-\frac{e F^{b c x} \left (b^2 c^2 \log ^2(F) (d+e x)^2+b c e \log (F) (d+e x)+2 e^2\right )}{(d+e x)^3}\right )}{6 e^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.076, size = 199, normalized size = 1.6 \begin{align*} -{\frac{{b}^{3}{c}^{3} \left ( \ln \left ( F \right ) \right ) ^{3}{F}^{bcx}{F}^{ac}}{3\,{e}^{4}} \left ( bcx\ln \left ( F \right ) +{\frac{\ln \left ( F \right ) bcd}{e}} \right ) ^{-3}}-{\frac{{b}^{3}{c}^{3} \left ( \ln \left ( F \right ) \right ) ^{3}{F}^{bcx}{F}^{ac}}{6\,{e}^{4}} \left ( bcx\ln \left ( F \right ) +{\frac{\ln \left ( F \right ) bcd}{e}} \right ) ^{-2}}-{\frac{{b}^{3}{c}^{3} \left ( \ln \left ( F \right ) \right ) ^{3}{F}^{bcx}{F}^{ac}}{6\,{e}^{4}} \left ( bcx\ln \left ( F \right ) +{\frac{\ln \left ( F \right ) bcd}{e}} \right ) ^{-1}}-{\frac{{b}^{3}{c}^{3} \left ( \ln \left ( F \right ) \right ) ^{3}}{6\,{e}^{4}}{F}^{{\frac{c \left ( ae-bd \right ) }{e}}}{\it Ei} \left ( 1,-bcx\ln \left ( F \right ) -ac\ln \left ( F \right ) -{\frac{-\ln \left ( F \right ) ace+\ln \left ( F \right ) bcd}{e}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{F^{{\left (b x + a\right )} c}}{e^{4} x^{4} + 4 \, d e^{3} x^{3} + 6 \, d^{2} e^{2} x^{2} + 4 \, d^{3} e x + d^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57511, size = 425, normalized size = 3.32 \begin{align*} \frac{\frac{{\left (b^{3} c^{3} e^{3} x^{3} + 3 \, b^{3} c^{3} d e^{2} x^{2} + 3 \, b^{3} c^{3} d^{2} e x + b^{3} c^{3} d^{3}\right )}{\rm Ei}\left (\frac{{\left (b c e x + b c d\right )} \log \left (F\right )}{e}\right ) \log \left (F\right )^{3}}{F^{\frac{b c d - a c e}{e}}} -{\left (2 \, e^{3} +{\left (b^{2} c^{2} e^{3} x^{2} + 2 \, b^{2} c^{2} d e^{2} x + b^{2} c^{2} d^{2} e\right )} \log \left (F\right )^{2} +{\left (b c e^{3} x + b c d e^{2}\right )} \log \left (F\right )\right )} F^{b c x + a c}}{6 \,{\left (e^{7} x^{3} + 3 \, d e^{6} x^{2} + 3 \, d^{2} e^{5} x + d^{3} e^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{F^{{\left (b x + a\right )} c}}{e^{4} x^{4} + 4 \, d e^{3} x^{3} + 6 \, d^{2} e^{2} x^{2} + 4 \, d^{3} e x + d^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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