Optimal. Leaf size=128 \[ \frac{b^3 c^3 \log ^3(F) F^{c \left (a-\frac{b d}{e}\right )} \text{ExpIntegralEi}\left (\frac{b c \log (F) (d+e x)}{e}\right )}{6 e^4}-\frac{b^2 c^2 \log ^2(F) F^{c (a+b x)}}{6 e^3 (d+e x)}-\frac{b c \log (F) F^{c (a+b x)}}{6 e^2 (d+e x)^2}-\frac{F^{c (a+b x)}}{3 e (d+e x)^3} \]
[Out]
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Rubi [A] time = 0.227158, 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 \[ \frac{b^3 c^3 \log ^3(F) F^{c \left (a-\frac{b d}{e}\right )} \text{ExpIntegralEi}\left (\frac{b c \log (F) (d+e x)}{e}\right )}{6 e^4}-\frac{b^2 c^2 \log ^2(F) F^{c (a+b x)}}{6 e^3 (d+e x)}-\frac{b c \log (F) F^{c (a+b x)}}{6 e^2 (d+e x)^2}-\frac{F^{c (a+b x)}}{3 e (d+e x)^3} \]
Antiderivative was successfully verified.
[In] Int[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),x]
[Out]
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Rubi in Sympy [A] time = 54.4037, size = 119, normalized size = 0.93 \[ - \frac{F^{c \left (a + b x\right )} b^{2} c^{2} \log{\left (F \right )}^{2}}{6 e^{3} \left (d + e x\right )} - \frac{F^{c \left (a + b x\right )} b c \log{\left (F \right )}}{6 e^{2} \left (d + e x\right )^{2}} - \frac{F^{c \left (a + b x\right )}}{3 e \left (d + e x\right )^{3}} + \frac{F^{\frac{c \left (a e - b d\right )}{e}} b^{3} c^{3} \log{\left (F \right )}^{3} \operatorname{Ei}{\left (\frac{b c \left (d + e x\right ) \log{\left (F \right )}}{e} \right )}}{6 e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(F**(c*(b*x+a))/(e**4*x**4+4*d*e**3*x**3+6*d**2*e**2*x**2+4*d**3*e*x+d**4),x)
[Out]
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Mathematica [A] time = 0.0271266, size = 99, normalized size = 0.77 \[ \frac{F^{a c} \left (b^3 c^3 \log ^3(F) F^{-\frac{b c d}{e}} \text{ExpIntegralEi}\left (\frac{b c \log (F) (d+e x)}{e}\right )-\frac{e F^{b c x} \left (b^2 c^2 \log ^2(F) (d+e x)^2+b c e \log (F) (d+e x)+2 e^2\right )}{(d+e x)^3}\right )}{6 e^4} \]
Antiderivative was successfully verified.
[In] Integrate[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),x]
[Out]
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Maple [A] time = 0.094, size = 193, normalized size = 1.5 \[ -{\frac{{b}^{3}{c}^{3} \left ( \ln \left ( F \right ) \right ) ^{3}{F}^{c \left ( bx+a \right ) }}{3\,{e}^{4}} \left ( bcx\ln \left ( F \right ) +{\frac{\ln \left ( F \right ) bcd}{e}} \right ) ^{-3}}-{\frac{{b}^{3}{c}^{3} \left ( \ln \left ( F \right ) \right ) ^{3}{F}^{c \left ( bx+a \right ) }}{6\,{e}^{4}} \left ( bcx\ln \left ( F \right ) +{\frac{\ln \left ( F \right ) bcd}{e}} \right ) ^{-2}}-{\frac{{b}^{3}{c}^{3} \left ( \ln \left ( F \right ) \right ) ^{3}{F}^{c \left ( bx+a \right ) }}{6\,{e}^{4}} \left ( bcx\ln \left ( F \right ) +{\frac{\ln \left ( F \right ) bcd}{e}} \right ) ^{-1}}-{\frac{{b}^{3}{c}^{3} \left ( \ln \left ( F \right ) \right ) ^{3}}{6\,{e}^{4}}{F}^{{\frac{c \left ( ea-bd \right ) }{e}}}{\it Ei} \left ( 1,-bcx\ln \left ( F \right ) -\ln \left ( F \right ) ac-{\frac{-eac\ln \left ( F \right ) +\ln \left ( F \right ) bcd}{e}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(F^(c*(b*x+a))/(e^4*x^4+4*d*e^3*x^3+6*d^2*e^2*x^2+4*d^3*e*x+d^4),x)
[Out]
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Maxima [A] time = 0.844392, size = 59, normalized size = 0.46 \[ -\frac{F^{a c} exp_integral_e\left (4, -\frac{{\left (e x + d\right )} b c \log \left (F\right )}{e}\right )}{{\left (e x + d\right )}^{3} F^{\frac{b c d}{e}} e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(F^((b*x + a)*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),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.239548, size = 282, normalized size = 2.2 \[ \frac{\frac{{\left (b^{3} c^{3} e^{3} x^{3} + 3 \, b^{3} c^{3} d e^{2} x^{2} + 3 \, b^{3} c^{3} d^{2} e x + b^{3} c^{3} d^{3}\right )}{\rm Ei}\left (\frac{{\left (b c e x + b c d\right )} \log \left (F\right )}{e}\right ) \log \left (F\right )^{3}}{F^{\frac{b c d - a c e}{e}}} -{\left (2 \, e^{3} +{\left (b^{2} c^{2} e^{3} x^{2} + 2 \, b^{2} c^{2} d e^{2} x + b^{2} c^{2} d^{2} e\right )} \log \left (F\right )^{2} +{\left (b c e^{3} x + b c d e^{2}\right )} \log \left (F\right )\right )} F^{b c x + a c}}{6 \,{\left (e^{7} x^{3} + 3 \, d e^{6} x^{2} + 3 \, d^{2} e^{5} x + d^{3} e^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(F^((b*x + a)*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),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(F**(c*(b*x+a))/(e**4*x**4+4*d*e**3*x**3+6*d**2*e**2*x**2+4*d**3*e*x+d**4),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{{\left (b x + a\right )} c}}{e^{4} x^{4} + 4 \, d e^{3} x^{3} + 6 \, d^{2} e^{2} x^{2} + 4 \, d^{3} e x + d^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(F^((b*x + a)*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),x, algorithm="giac")
[Out]