3.11 \(\int \frac{F^{c (a+b x)}}{(d+e x)^5} \, dx\)

Optimal. Leaf size=161 \[ \frac{b^4 c^4 \log ^4(F) F^{c \left (a-\frac{b d}{e}\right )} \text{ExpIntegralEi}\left (\frac{b c \log (F) (d+e x)}{e}\right )}{24 e^5}-\frac{b^3 c^3 \log ^3(F) F^{c (a+b x)}}{24 e^4 (d+e x)}-\frac{b^2 c^2 \log ^2(F) F^{c (a+b x)}}{24 e^3 (d+e x)^2}-\frac{b c \log (F) F^{c (a+b x)}}{12 e^2 (d+e x)^3}-\frac{F^{c (a+b x)}}{4 e (d+e x)^4} \]

[Out]

-F^(c*(a + b*x))/(4*e*(d + e*x)^4) - (b*c*F^(c*(a + b*x))*Log[F])/(12*e^2*(d + e
*x)^3) - (b^2*c^2*F^(c*(a + b*x))*Log[F]^2)/(24*e^3*(d + e*x)^2) - (b^3*c^3*F^(c
*(a + b*x))*Log[F]^3)/(24*e^4*(d + e*x)) + (b^4*c^4*F^(c*(a - (b*d)/e))*ExpInteg
ralEi[(b*c*(d + e*x)*Log[F])/e]*Log[F]^4)/(24*e^5)

_______________________________________________________________________________________

Rubi [A]  time = 0.233636, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{b^4 c^4 \log ^4(F) F^{c \left (a-\frac{b d}{e}\right )} \text{ExpIntegralEi}\left (\frac{b c \log (F) (d+e x)}{e}\right )}{24 e^5}-\frac{b^3 c^3 \log ^3(F) F^{c (a+b x)}}{24 e^4 (d+e x)}-\frac{b^2 c^2 \log ^2(F) F^{c (a+b x)}}{24 e^3 (d+e x)^2}-\frac{b c \log (F) F^{c (a+b x)}}{12 e^2 (d+e x)^3}-\frac{F^{c (a+b x)}}{4 e (d+e x)^4} \]

Antiderivative was successfully verified.

[In]  Int[F^(c*(a + b*x))/(d + e*x)^5,x]

[Out]

-F^(c*(a + b*x))/(4*e*(d + e*x)^4) - (b*c*F^(c*(a + b*x))*Log[F])/(12*e^2*(d + e
*x)^3) - (b^2*c^2*F^(c*(a + b*x))*Log[F]^2)/(24*e^3*(d + e*x)^2) - (b^3*c^3*F^(c
*(a + b*x))*Log[F]^3)/(24*e^4*(d + e*x)) + (b^4*c^4*F^(c*(a - (b*d)/e))*ExpInteg
ralEi[(b*c*(d + e*x)*Log[F])/e]*Log[F]^4)/(24*e^5)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 37.7373, size = 151, normalized size = 0.94 \[ - \frac{F^{c \left (a + b x\right )} b^{3} c^{3} \log{\left (F \right )}^{3}}{24 e^{4} \left (d + e x\right )} - \frac{F^{c \left (a + b x\right )} b^{2} c^{2} \log{\left (F \right )}^{2}}{24 e^{3} \left (d + e x\right )^{2}} - \frac{F^{c \left (a + b x\right )} b c \log{\left (F \right )}}{12 e^{2} \left (d + e x\right )^{3}} - \frac{F^{c \left (a + b x\right )}}{4 e \left (d + e x\right )^{4}} + \frac{F^{\frac{c \left (a e - b d\right )}{e}} b^{4} c^{4} \log{\left (F \right )}^{4} \operatorname{Ei}{\left (\frac{b c \left (d + e x\right ) \log{\left (F \right )}}{e} \right )}}{24 e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(F**(c*(b*x+a))/(e*x+d)**5,x)

[Out]

-F**(c*(a + b*x))*b**3*c**3*log(F)**3/(24*e**4*(d + e*x)) - F**(c*(a + b*x))*b**
2*c**2*log(F)**2/(24*e**3*(d + e*x)**2) - F**(c*(a + b*x))*b*c*log(F)/(12*e**2*(
d + e*x)**3) - F**(c*(a + b*x))/(4*e*(d + e*x)**4) + F**(c*(a*e - b*d)/e)*b**4*c
**4*log(F)**4*Ei(b*c*(d + e*x)*log(F)/e)/(24*e**5)

_______________________________________________________________________________________

Mathematica [A]  time = 0.14671, size = 121, normalized size = 0.75 \[ \frac{F^{a c} \left (b^4 c^4 \log ^4(F) F^{-\frac{b c d}{e}} \text{ExpIntegralEi}\left (\frac{b c \log (F) (d+e x)}{e}\right )-\frac{e F^{b c x} \left (b^3 c^3 \log ^3(F) (d+e x)^3+b^2 c^2 e \log ^2(F) (d+e x)^2+2 b c e^2 \log (F) (d+e x)+6 e^3\right )}{(d+e x)^4}\right )}{24 e^5} \]

Antiderivative was successfully verified.

[In]  Integrate[F^(c*(a + b*x))/(d + e*x)^5,x]

[Out]

(F^(a*c)*((b^4*c^4*ExpIntegralEi[(b*c*(d + e*x)*Log[F])/e]*Log[F]^4)/F^((b*c*d)/
e) - (e*F^(b*c*x)*(6*e^3 + 2*b*c*e^2*(d + e*x)*Log[F] + b^2*c^2*e*(d + e*x)^2*Lo
g[F]^2 + b^3*c^3*(d + e*x)^3*Log[F]^3))/(d + e*x)^4))/(24*e^5)

_______________________________________________________________________________________

Maple [A]  time = 0.066, size = 235, normalized size = 1.5 \[ -{\frac{{b}^{4}{c}^{4} \left ( \ln \left ( F \right ) \right ) ^{4}{F}^{c \left ( bx+a \right ) }}{4\,{e}^{5}} \left ( bcx\ln \left ( F \right ) +{\frac{\ln \left ( F \right ) bcd}{e}} \right ) ^{-4}}-{\frac{{b}^{4}{c}^{4} \left ( \ln \left ( F \right ) \right ) ^{4}{F}^{c \left ( bx+a \right ) }}{12\,{e}^{5}} \left ( bcx\ln \left ( F \right ) +{\frac{\ln \left ( F \right ) bcd}{e}} \right ) ^{-3}}-{\frac{{b}^{4}{c}^{4} \left ( \ln \left ( F \right ) \right ) ^{4}{F}^{c \left ( bx+a \right ) }}{24\,{e}^{5}} \left ( bcx\ln \left ( F \right ) +{\frac{\ln \left ( F \right ) bcd}{e}} \right ) ^{-2}}-{\frac{{b}^{4}{c}^{4} \left ( \ln \left ( F \right ) \right ) ^{4}{F}^{c \left ( bx+a \right ) }}{24\,{e}^{5}} \left ( bcx\ln \left ( F \right ) +{\frac{\ln \left ( F \right ) bcd}{e}} \right ) ^{-1}}-{\frac{{b}^{4}{c}^{4} \left ( \ln \left ( F \right ) \right ) ^{4}}{24\,{e}^{5}}{F}^{{\frac{c \left ( ea-bd \right ) }{e}}}{\it Ei} \left ( 1,-bcx\ln \left ( F \right ) -\ln \left ( F \right ) ac-{\frac{-eac\ln \left ( F \right ) +\ln \left ( F \right ) bcd}{e}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(F^(c*(b*x+a))/(e*x+d)^5,x)

[Out]

-1/4*b^4*c^4*ln(F)^4/e^5*F^(c*(b*x+a))/(b*c*x*ln(F)+1/e*ln(F)*b*c*d)^4-1/12*b^4*
c^4*ln(F)^4/e^5*F^(c*(b*x+a))/(b*c*x*ln(F)+1/e*ln(F)*b*c*d)^3-1/24*b^4*c^4*ln(F)
^4/e^5*F^(c*(b*x+a))/(b*c*x*ln(F)+1/e*ln(F)*b*c*d)^2-1/24*b^4*c^4*ln(F)^4/e^5*F^
(c*(b*x+a))/(b*c*x*ln(F)+1/e*ln(F)*b*c*d)-1/24*b^4*c^4*ln(F)^4/e^5*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 = 0.78876, size = 59, normalized size = 0.37 \[ -\frac{F^{a c} exp_integral_e\left (5, -\frac{{\left (e x + d\right )} b c \log \left (F\right )}{e}\right )}{{\left (e x + d\right )}^{4} F^{\frac{b c d}{e}} e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^((b*x + a)*c)/(e*x + d)^5,x, algorithm="maxima")

[Out]

-F^(a*c)*exp_integral_e(5, -(e*x + d)*b*c*log(F)/e)/((e*x + d)^4*F^(b*c*d/e)*e)

_______________________________________________________________________________________

Fricas [A]  time = 0.24805, size = 405, normalized size = 2.52 \[ \frac{\frac{{\left (b^{4} c^{4} e^{4} x^{4} + 4 \, b^{4} c^{4} d e^{3} x^{3} + 6 \, b^{4} c^{4} d^{2} e^{2} x^{2} + 4 \, b^{4} c^{4} d^{3} e x + b^{4} c^{4} d^{4}\right )}{\rm Ei}\left (\frac{{\left (b c e x + b c d\right )} \log \left (F\right )}{e}\right ) \log \left (F\right )^{4}}{F^{\frac{b c d - a c e}{e}}} -{\left (6 \, e^{4} +{\left (b^{3} c^{3} e^{4} x^{3} + 3 \, b^{3} c^{3} d e^{3} x^{2} + 3 \, b^{3} c^{3} d^{2} e^{2} x + b^{3} c^{3} d^{3} e\right )} \log \left (F\right )^{3} +{\left (b^{2} c^{2} e^{4} x^{2} + 2 \, b^{2} c^{2} d e^{3} x + b^{2} c^{2} d^{2} e^{2}\right )} \log \left (F\right )^{2} + 2 \,{\left (b c e^{4} x + b c d e^{3}\right )} \log \left (F\right )\right )} F^{b c x + a c}}{24 \,{\left (e^{9} x^{4} + 4 \, d e^{8} x^{3} + 6 \, d^{2} e^{7} x^{2} + 4 \, d^{3} e^{6} x + d^{4} e^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^((b*x + a)*c)/(e*x + d)^5,x, algorithm="fricas")

[Out]

1/24*((b^4*c^4*e^4*x^4 + 4*b^4*c^4*d*e^3*x^3 + 6*b^4*c^4*d^2*e^2*x^2 + 4*b^4*c^4
*d^3*e*x + b^4*c^4*d^4)*Ei((b*c*e*x + b*c*d)*log(F)/e)*log(F)^4/F^((b*c*d - a*c*
e)/e) - (6*e^4 + (b^3*c^3*e^4*x^3 + 3*b^3*c^3*d*e^3*x^2 + 3*b^3*c^3*d^2*e^2*x +
b^3*c^3*d^3*e)*log(F)^3 + (b^2*c^2*e^4*x^2 + 2*b^2*c^2*d*e^3*x + b^2*c^2*d^2*e^2
)*log(F)^2 + 2*(b*c*e^4*x + b*c*d*e^3)*log(F))*F^(b*c*x + a*c))/(e^9*x^4 + 4*d*e
^8*x^3 + 6*d^2*e^7*x^2 + 4*d^3*e^6*x + d^4*e^5)

_______________________________________________________________________________________

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*x+d)**5,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{{\left (b x + a\right )} c}}{{\left (e x + d\right )}^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^((b*x + a)*c)/(e*x + d)^5,x, algorithm="giac")

[Out]

integrate(F^((b*x + a)*c)/(e*x + d)^5, x)