3.17 \(\int \frac{F^{c (a+b x)}}{d^4+4 d^3 e x+6 d^2 e^2 x^2+4 d e^3 x^3+e^4 x^4} \, dx\)

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]

-F^(c*(a + b*x))/(3*e*(d + e*x)^3) - (b*c*F^(c*(a + b*x))*Log[F])/(6*e^2*(d + e*
x)^2) - (b^2*c^2*F^(c*(a + b*x))*Log[F]^2)/(6*e^3*(d + e*x)) + (b^3*c^3*F^(c*(a
- (b*d)/e))*ExpIntegralEi[(b*c*(d + e*x)*Log[F])/e]*Log[F]^3)/(6*e^4)

_______________________________________________________________________________________

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]

-F^(c*(a + b*x))/(3*e*(d + e*x)^3) - (b*c*F^(c*(a + b*x))*Log[F])/(6*e^2*(d + e*
x)^2) - (b^2*c^2*F^(c*(a + b*x))*Log[F]^2)/(6*e^3*(d + e*x)) + (b^3*c^3*F^(c*(a
- (b*d)/e))*ExpIntegralEi[(b*c*(d + e*x)*Log[F])/e]*Log[F]^3)/(6*e^4)

_______________________________________________________________________________________

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]

-F**(c*(a + b*x))*b**2*c**2*log(F)**2/(6*e**3*(d + e*x)) - F**(c*(a + b*x))*b*c*
log(F)/(6*e**2*(d + e*x)**2) - F**(c*(a + b*x))/(3*e*(d + e*x)**3) + F**(c*(a*e
- b*d)/e)*b**3*c**3*log(F)**3*Ei(b*c*(d + e*x)*log(F)/e)/(6*e**4)

_______________________________________________________________________________________

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]

(F^(a*c)*((b^3*c^3*ExpIntegralEi[(b*c*(d + e*x)*Log[F])/e]*Log[F]^3)/F^((b*c*d)/
e) - (e*F^(b*c*x)*(2*e^2 + b*c*e*(d + e*x)*Log[F] + b^2*c^2*(d + e*x)^2*Log[F]^2
))/(d + e*x)^3))/(6*e^4)

_______________________________________________________________________________________

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]

-1/3*b^3*c^3*ln(F)^3/e^4*F^(c*(b*x+a))/(b*c*x*ln(F)+1/e*ln(F)*b*c*d)^3-1/6*b^3*c
^3*ln(F)^3/e^4*F^(c*(b*x+a))/(b*c*x*ln(F)+1/e*ln(F)*b*c*d)^2-1/6*b^3*c^3*ln(F)^3
/e^4*F^(c*(b*x+a))/(b*c*x*ln(F)+1/e*ln(F)*b*c*d)-1/6*b^3*c^3*ln(F)^3/e^4*F^(c*(a
*e-b*d)/e)*Ei(1,-b*c*x*ln(F)-ln(F)*a*c-(-e*a*c*ln(F)+ln(F)*b*c*d)/e)

_______________________________________________________________________________________

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]

-F^(a*c)*exp_integral_e(4, -(e*x + d)*b*c*log(F)/e)/((e*x + d)^3*F^(b*c*d/e)*e)

_______________________________________________________________________________________

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]

1/6*((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)*E
i((b*c*e*x + b*c*d)*log(F)/e)*log(F)^3/F^((b*c*d - a*c*e)/e) - (2*e^3 + (b^2*c^2
*e^3*x^2 + 2*b^2*c^2*d*e^2*x + b^2*c^2*d^2*e)*log(F)^2 + (b*c*e^3*x + b*c*d*e^2)
*log(F))*F^(b*c*x + a*c))/(e^7*x^3 + 3*d*e^6*x^2 + 3*d^2*e^5*x + d^3*e^4)

_______________________________________________________________________________________

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]

Timed out

_______________________________________________________________________________________

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]

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)