Optimal. Leaf size=95 \[ \frac{b^2 c^2 \log ^2(F) F^{c \left (a-\frac{b d}{e}\right )} \text{ExpIntegralEi}\left (\frac{b c \log (F) (d+e x)}{e}\right )}{2 e^3}-\frac{b c \log (F) F^{c (a+b x)}}{2 e^2 (d+e x)}-\frac{F^{c (a+b x)}}{2 e (d+e x)^2} \]
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Rubi [A] time = 0.159986, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{b^2 c^2 \log ^2(F) F^{c \left (a-\frac{b d}{e}\right )} \text{ExpIntegralEi}\left (\frac{b c \log (F) (d+e x)}{e}\right )}{2 e^3}-\frac{b c \log (F) F^{c (a+b x)}}{2 e^2 (d+e x)}-\frac{F^{c (a+b x)}}{2 e (d+e x)^2} \]
Antiderivative was successfully verified.
[In] Int[F^(c*(a + b*x))/(d^3 + 3*d^2*e*x + 3*d*e^2*x^2 + e^3*x^3),x]
[Out]
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Rubi in Sympy [A] time = 34.8704, size = 87, normalized size = 0.92 \[ - \frac{F^{c \left (a + b x\right )} b c \log{\left (F \right )}}{2 e^{2} \left (d + e x\right )} - \frac{F^{c \left (a + b x\right )}}{2 e \left (d + e x\right )^{2}} + \frac{F^{\frac{c \left (a e - b d\right )}{e}} b^{2} c^{2} \log{\left (F \right )}^{2} \operatorname{Ei}{\left (\frac{b c \left (d + e x\right ) \log{\left (F \right )}}{e} \right )}}{2 e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(F**(c*(b*x+a))/(e**3*x**3+3*d*e**2*x**2+3*d**2*e*x+d**3),x)
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Mathematica [A] time = 0.0240768, size = 88, normalized size = 0.93 \[ \frac{F^{c \left (a-\frac{b d}{e}\right )} \left (b^2 c^2 \log ^2(F) (d+e x)^2 \text{ExpIntegralEi}\left (\frac{b c \log (F) (d+e x)}{e}\right )-e F^{\frac{b c (d+e x)}{e}} (b c \log (F) (d+e x)+e)\right )}{2 e^3 (d+e x)^2} \]
Antiderivative was successfully verified.
[In] Integrate[F^(c*(a + b*x))/(d^3 + 3*d^2*e*x + 3*d*e^2*x^2 + e^3*x^3),x]
[Out]
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Maple [A] time = 0.068, size = 151, normalized size = 1.6 \[ -{\frac{{b}^{2}{c}^{2} \left ( \ln \left ( F \right ) \right ) ^{2}{F}^{c \left ( bx+a \right ) }}{2\,{e}^{3}} \left ( bcx\ln \left ( F \right ) +{\frac{\ln \left ( F \right ) bcd}{e}} \right ) ^{-2}}-{\frac{{b}^{2}{c}^{2} \left ( \ln \left ( F \right ) \right ) ^{2}{F}^{c \left ( bx+a \right ) }}{2\,{e}^{3}} \left ( bcx\ln \left ( F \right ) +{\frac{\ln \left ( F \right ) bcd}{e}} \right ) ^{-1}}-{\frac{{b}^{2}{c}^{2} \left ( \ln \left ( F \right ) \right ) ^{2}}{2\,{e}^{3}}{F}^{{\frac{ \left ( ea-bd \right ) c}{e}}}{\it Ei} \left ( 1,-bcx\ln \left ( F \right ) -\ln \left ( F \right ) ac-{\frac{-eac\ln \left ( F \right ) +\ln \left ( F \right ) bcd}{e}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(F^(c*(b*x+a))/(e^3*x^3+3*d*e^2*x^2+3*d^2*e*x+d^3),x)
[Out]
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Maxima [A] time = 0.811642, size = 59, normalized size = 0.62 \[ -\frac{F^{a c} exp_integral_e\left (3, -\frac{{\left (e x + d\right )} b c \log \left (F\right )}{e}\right )}{{\left (e x + d\right )}^{2} F^{\frac{b c d}{e}} e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(F^((b*x + a)*c)/(e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x + d^3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.243352, size = 181, normalized size = 1.91 \[ \frac{\frac{{\left (b^{2} c^{2} e^{2} x^{2} + 2 \, b^{2} c^{2} d e x + b^{2} c^{2} d^{2}\right )}{\rm Ei}\left (\frac{{\left (b c e x + b c d\right )} \log \left (F\right )}{e}\right ) \log \left (F\right )^{2}}{F^{\frac{b c d - a c e}{e}}} -{\left (e^{2} +{\left (b c e^{2} x + b c d e\right )} \log \left (F\right )\right )} F^{b c x + a c}}{2 \,{\left (e^{5} x^{2} + 2 \, d e^{4} x + d^{2} e^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(F^((b*x + a)*c)/(e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x + d^3),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{c \left (a + b x\right )}}{\left (d + e x\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(F**(c*(b*x+a))/(e**3*x**3+3*d*e**2*x**2+3*d**2*e*x+d**3),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{{\left (b x + a\right )} c}}{e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(F^((b*x + a)*c)/(e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x + d^3),x, algorithm="giac")
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