\(\int F^{a+\frac {b}{c+d x}} (c+d x)^4 \, dx\) [236]

Optimal result

Integrand size = 21, antiderivative size = 29 \[ \int F^{a+\frac {b}{c+d x}} (c+d x)^4 \, dx=-\frac {b^5 F^a \Gamma \left (-5,-\frac {b \log (F)}{c+d x}\right ) \log ^5(F)}{d} \] Output:

F^a*(d*x+c)^5*Ei(6,-b*ln(F)/(d*x+c))/d
 

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int F^{a+\frac {b}{c+d x}} (c+d x)^4 \, dx=-\frac {b^5 F^a \Gamma \left (-5,-\frac {b \log (F)}{c+d x}\right ) \log ^5(F)}{d} \] Input:

Integrate[F^(a + b/(c + d*x))*(c + d*x)^4,x]
 

Output:

-((b^5*F^a*Gamma[-5, -((b*Log[F])/(c + d*x))]*Log[F]^5)/d)
 

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2648}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int (c+d x)^4 F^{a+\frac {b}{c+d x}} \, dx\)

\(\Big \downarrow \) 2648

\(\displaystyle -\frac {b^5 F^a \log ^5(F) \Gamma \left (-5,-\frac {b \log (F)}{c+d x}\right )}{d}\)

Input:

Int[F^(a + b/(c + d*x))*(c + d*x)^4,x]
 

Output:

-((b^5*F^a*Gamma[-5, -((b*Log[F])/(c + d*x))]*Log[F]^5)/d)
 

Defintions of rubi rules used

rule 2648

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_ 
.), x_Symbol] :> Simp[(-F^a)*((e + f*x)^(m + 1)/(f*n*((-b)*(c + d*x)^n*Log[ 
F])^((m + 1)/n)))*Gamma[(m + 1)/n, (-b)*(c + d*x)^n*Log[F]], x] /; FreeQ[{F 
, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(533\) vs. \(2(28)=56\).

Time = 0.43 (sec) , antiderivative size = 534, normalized size of antiderivative = 18.41

method	result	size
risch	\(\frac {d^{4} F^{a} F^{\frac {b}{d x +c}} x^{5}}{5}+d^{3} F^{a} F^{\frac {b}{d x +c}} c \,x^{4}+2 d^{2} F^{a} F^{\frac {b}{d x +c}} c^{2} x^{3}+2 d \,F^{a} F^{\frac {b}{d x +c}} c^{3} x^{2}+F^{a} F^{\frac {b}{d x +c}} c^{4} x +\frac {F^{a} F^{\frac {b}{d x +c}} c^{5}}{5 d}+\frac {d^{3} b \ln \left (F \right ) F^{a} F^{\frac {b}{d x +c}} x^{4}}{20}+\frac {d^{2} b \ln \left (F \right ) F^{a} F^{\frac {b}{d x +c}} c \,x^{3}}{5}+\frac {3 d b \ln \left (F \right ) F^{a} F^{\frac {b}{d x +c}} c^{2} x^{2}}{10}+\frac {b \ln \left (F \right ) F^{a} F^{\frac {b}{d x +c}} c^{3} x}{5}+\frac {b \ln \left (F \right ) F^{a} F^{\frac {b}{d x +c}} c^{4}}{20 d}+\frac {d^{2} b^{2} \ln \left (F \right )^{2} F^{a} F^{\frac {b}{d x +c}} x^{3}}{60}+\frac {d \,b^{2} \ln \left (F \right )^{2} F^{a} F^{\frac {b}{d x +c}} c \,x^{2}}{20}+\frac {b^{2} \ln \left (F \right )^{2} F^{a} F^{\frac {b}{d x +c}} c^{2} x}{20}+\frac {b^{2} \ln \left (F \right )^{2} F^{a} F^{\frac {b}{d x +c}} c^{3}}{60 d}+\frac {d \,b^{3} \ln \left (F \right )^{3} F^{a} F^{\frac {b}{d x +c}} x^{2}}{120}+\frac {b^{3} \ln \left (F \right )^{3} F^{a} F^{\frac {b}{d x +c}} c x}{60}+\frac {b^{3} \ln \left (F \right )^{3} F^{a} F^{\frac {b}{d x +c}} c^{2}}{120 d}+\frac {b^{4} \ln \left (F \right )^{4} F^{a} F^{\frac {b}{d x +c}} x}{120}+\frac {b^{4} \ln \left (F \right )^{4} F^{a} F^{\frac {b}{d x +c}} c}{120 d}+\frac {b^{5} \ln \left (F \right )^{5} F^{a} \operatorname {expIntegral}_{1}\left (-\frac {b \ln \left (F \right )}{d x +c}\right )}{120 d}\)	\(534\)

Input:

int(F^(a+b/(d*x+c))*(d*x+c)^4,x,method=_RETURNVERBOSE)
 

Output:

1/5*d^4*F^a*F^(b/(d*x+c))*x^5+d^3*F^a*F^(b/(d*x+c))*c*x^4+2*d^2*F^a*F^(b/( 
d*x+c))*c^2*x^3+2*d*F^a*F^(b/(d*x+c))*c^3*x^2+F^a*F^(b/(d*x+c))*c^4*x+1/5/ 
d*F^a*F^(b/(d*x+c))*c^5+1/20*d^3*b*ln(F)*F^a*F^(b/(d*x+c))*x^4+1/5*d^2*b*l 
n(F)*F^a*F^(b/(d*x+c))*c*x^3+3/10*d*b*ln(F)*F^a*F^(b/(d*x+c))*c^2*x^2+1/5* 
b*ln(F)*F^a*F^(b/(d*x+c))*c^3*x+1/20/d*b*ln(F)*F^a*F^(b/(d*x+c))*c^4+1/60* 
d^2*b^2*ln(F)^2*F^a*F^(b/(d*x+c))*x^3+1/20*d*b^2*ln(F)^2*F^a*F^(b/(d*x+c)) 
*c*x^2+1/20*b^2*ln(F)^2*F^a*F^(b/(d*x+c))*c^2*x+1/60/d*b^2*ln(F)^2*F^a*F^( 
b/(d*x+c))*c^3+1/120*d*b^3*ln(F)^3*F^a*F^(b/(d*x+c))*x^2+1/60*b^3*ln(F)^3* 
F^a*F^(b/(d*x+c))*c*x+1/120/d*b^3*ln(F)^3*F^a*F^(b/(d*x+c))*c^2+1/120*b^4* 
ln(F)^4*F^a*F^(b/(d*x+c))*x+1/120/d*b^4*ln(F)^4*F^a*F^(b/(d*x+c))*c+1/120/ 
d*b^5*ln(F)^5*F^a*Ei(1,-b*ln(F)/(d*x+c))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 244 vs. \(2 (29) = 58\).

Time = 0.08 (sec) , antiderivative size = 244, normalized size of antiderivative = 8.41 \[ \int F^{a+\frac {b}{c+d x}} (c+d x)^4 \, dx=-\frac {F^{a} b^{5} {\rm Ei}\left (\frac {b \log \left (F\right )}{d x + c}\right ) \log \left (F\right )^{5} - {\left (24 \, d^{5} x^{5} + 120 \, c d^{4} x^{4} + 240 \, c^{2} d^{3} x^{3} + 240 \, c^{3} d^{2} x^{2} + 120 \, c^{4} d x + 24 \, c^{5} + {\left (b^{4} d x + b^{4} c\right )} \log \left (F\right )^{4} + {\left (b^{3} d^{2} x^{2} + 2 \, b^{3} c d x + b^{3} c^{2}\right )} \log \left (F\right )^{3} + 2 \, {\left (b^{2} d^{3} x^{3} + 3 \, b^{2} c d^{2} x^{2} + 3 \, b^{2} c^{2} d x + b^{2} c^{3}\right )} \log \left (F\right )^{2} + 6 \, {\left (b d^{4} x^{4} + 4 \, b c d^{3} x^{3} + 6 \, b c^{2} d^{2} x^{2} + 4 \, b c^{3} d x + b c^{4}\right )} \log \left (F\right )\right )} F^{\frac {a d x + a c + b}{d x + c}}}{120 \, d} \] Input:

integrate(F^(a+b/(d*x+c))*(d*x+c)^4,x, algorithm="fricas")
 

Output:

-1/120*(F^a*b^5*Ei(b*log(F)/(d*x + c))*log(F)^5 - (24*d^5*x^5 + 120*c*d^4* 
x^4 + 240*c^2*d^3*x^3 + 240*c^3*d^2*x^2 + 120*c^4*d*x + 24*c^5 + (b^4*d*x 
+ b^4*c)*log(F)^4 + (b^3*d^2*x^2 + 2*b^3*c*d*x + b^3*c^2)*log(F)^3 + 2*(b^ 
2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*log(F)^2 + 6*(b*d^4 
*x^4 + 4*b*c*d^3*x^3 + 6*b*c^2*d^2*x^2 + 4*b*c^3*d*x + b*c^4)*log(F))*F^(( 
a*d*x + a*c + b)/(d*x + c)))/d
 

Sympy [F]

\[ \int F^{a+\frac {b}{c+d x}} (c+d x)^4 \, dx=\int F^{a + \frac {b}{c + d x}} \left (c + d x\right )^{4}\, dx \] Input:

integrate(F**(a+b/(d*x+c))*(d*x+c)**4,x)
 

Output:

Integral(F**(a + b/(c + d*x))*(c + d*x)**4, x)
 

Maxima [F]

\[ \int F^{a+\frac {b}{c+d x}} (c+d x)^4 \, dx=\int { {\left (d x + c\right )}^{4} F^{a + \frac {b}{d x + c}} \,d x } \] Input:

integrate(F^(a+b/(d*x+c))*(d*x+c)^4,x, algorithm="maxima")
 

Output:

1/120*(24*F^a*d^4*x^5 + 6*(F^a*b*d^3*log(F) + 20*F^a*c*d^3)*x^4 + 2*(F^a*b 
^2*d^2*log(F)^2 + 12*F^a*b*c*d^2*log(F) + 120*F^a*c^2*d^2)*x^3 + (F^a*b^3* 
d*log(F)^3 + 6*F^a*b^2*c*d*log(F)^2 + 36*F^a*b*c^2*d*log(F) + 240*F^a*c^3* 
d)*x^2 + (F^a*b^4*log(F)^4 + 2*F^a*b^3*c*log(F)^3 + 6*F^a*b^2*c^2*log(F)^2 
 + 24*F^a*b*c^3*log(F) + 120*F^a*c^4)*x)*F^(b/(d*x + c)) + integrate(1/120 
*(F^a*b^5*d*x*log(F)^5 - F^a*b^4*c^2*log(F)^4 - 2*F^a*b^3*c^3*log(F)^3 - 6 
*F^a*b^2*c^4*log(F)^2 - 24*F^a*b*c^5*log(F))*F^(b/(d*x + c))/(d^2*x^2 + 2* 
c*d*x + c^2), x)
 

Giac [F]

\[ \int F^{a+\frac {b}{c+d x}} (c+d x)^4 \, dx=\int { {\left (d x + c\right )}^{4} F^{a + \frac {b}{d x + c}} \,d x } \] Input:

integrate(F^(a+b/(d*x+c))*(d*x+c)^4,x, algorithm="giac")
 

Output:

integrate((d*x + c)^4*F^(a + b/(d*x + c)), x)
 

Mupad [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 181, normalized size of antiderivative = 6.24 \[ \int F^{a+\frac {b}{c+d x}} (c+d x)^4 \, dx=\frac {F^a\,F^{\frac {b}{c+d\,x}}\,{\left (c+d\,x\right )}^5}{5\,d}+\frac {F^a\,b^5\,{\ln \left (F\right )}^5\,\mathrm {expint}\left (-\frac {b\,\ln \left (F\right )}{c+d\,x}\right )}{120\,d}+\frac {F^a\,F^{\frac {b}{c+d\,x}}\,b^2\,{\ln \left (F\right )}^2\,{\left (c+d\,x\right )}^3}{60\,d}+\frac {F^a\,F^{\frac {b}{c+d\,x}}\,b^3\,{\ln \left (F\right )}^3\,{\left (c+d\,x\right )}^2}{120\,d}+\frac {F^a\,F^{\frac {b}{c+d\,x}}\,b\,\ln \left (F\right )\,{\left (c+d\,x\right )}^4}{20\,d}+\frac {F^a\,F^{\frac {b}{c+d\,x}}\,b^4\,{\ln \left (F\right )}^4\,\left (c+d\,x\right )}{120\,d} \] Input:

int(F^(a + b/(c + d*x))*(c + d*x)^4,x)
 

Output:

(F^a*F^(b/(c + d*x))*(c + d*x)^5)/(5*d) + (F^a*b^5*log(F)^5*expint(-(b*log 
(F))/(c + d*x)))/(120*d) + (F^a*F^(b/(c + d*x))*b^2*log(F)^2*(c + d*x)^3)/ 
(60*d) + (F^a*F^(b/(c + d*x))*b^3*log(F)^3*(c + d*x)^2)/(120*d) + (F^a*F^( 
b/(c + d*x))*b*log(F)*(c + d*x)^4)/(20*d) + (F^a*F^(b/(c + d*x))*b^4*log(F 
)^4*(c + d*x))/(120*d)
 

Reduce [F]

\[ \int F^{a+\frac {b}{c+d x}} (c+d x)^4 \, dx=\frac {f^{\frac {a d x +a c +b}{d x +c}} \mathrm {log}\left (f \right )^{4} b^{4} d^{2} x^{2}+2 f^{\frac {a d x +a c +b}{d x +c}} \mathrm {log}\left (f \right )^{3} b^{3} c^{2} d x +3 f^{\frac {a d x +a c +b}{d x +c}} \mathrm {log}\left (f \right )^{3} b^{3} c \,d^{2} x^{2}+f^{\frac {a d x +a c +b}{d x +c}} \mathrm {log}\left (f \right )^{3} b^{3} d^{3} x^{3}+2 f^{\frac {a d x +a c +b}{d x +c}} \mathrm {log}\left (f \right )^{2} b^{2} c^{4}+8 f^{\frac {a d x +a c +b}{d x +c}} \mathrm {log}\left (f \right )^{2} b^{2} c^{3} d x +12 f^{\frac {a d x +a c +b}{d x +c}} \mathrm {log}\left (f \right )^{2} b^{2} c^{2} d^{2} x^{2}+8 f^{\frac {a d x +a c +b}{d x +c}} \mathrm {log}\left (f \right )^{2} b^{2} c \,d^{3} x^{3}+2 f^{\frac {a d x +a c +b}{d x +c}} \mathrm {log}\left (f \right )^{2} b^{2} d^{4} x^{4}+6 f^{\frac {a d x +a c +b}{d x +c}} \mathrm {log}\left (f \right ) b \,c^{5}+30 f^{\frac {a d x +a c +b}{d x +c}} \mathrm {log}\left (f \right ) b \,c^{4} d x +60 f^{\frac {a d x +a c +b}{d x +c}} \mathrm {log}\left (f \right ) b \,c^{3} d^{2} x^{2}+60 f^{\frac {a d x +a c +b}{d x +c}} \mathrm {log}\left (f \right ) b \,c^{2} d^{3} x^{3}+30 f^{\frac {a d x +a c +b}{d x +c}} \mathrm {log}\left (f \right ) b c \,d^{4} x^{4}+6 f^{\frac {a d x +a c +b}{d x +c}} \mathrm {log}\left (f \right ) b \,d^{5} x^{5}+24 f^{\frac {a d x +a c +b}{d x +c}} c^{6}+144 f^{\frac {a d x +a c +b}{d x +c}} c^{5} d x +360 f^{\frac {a d x +a c +b}{d x +c}} c^{4} d^{2} x^{2}+480 f^{\frac {a d x +a c +b}{d x +c}} c^{3} d^{3} x^{3}+360 f^{\frac {a d x +a c +b}{d x +c}} c^{2} d^{4} x^{4}+144 f^{\frac {a d x +a c +b}{d x +c}} c \,d^{5} x^{5}+24 f^{\frac {a d x +a c +b}{d x +c}} d^{6} x^{6}+\left (\int \frac {f^{\frac {a d x +a c +b}{d x +c}} x^{2}}{d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}}d x \right ) \mathrm {log}\left (f \right )^{5} b^{5} c \,d^{3}+\left (\int \frac {f^{\frac {a d x +a c +b}{d x +c}} x^{2}}{d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}}d x \right ) \mathrm {log}\left (f \right )^{5} b^{5} d^{4} x}{120 d \left (d x +c \right )} \] Input:

int(F^(a+b/(d*x+c))*(d*x+c)^4,x)
 

Output:

(f**((a*c + a*d*x + b)/(c + d*x))*log(f)**4*b**4*d**2*x**2 + 2*f**((a*c + 
a*d*x + b)/(c + d*x))*log(f)**3*b**3*c**2*d*x + 3*f**((a*c + a*d*x + b)/(c 
 + d*x))*log(f)**3*b**3*c*d**2*x**2 + f**((a*c + a*d*x + b)/(c + d*x))*log 
(f)**3*b**3*d**3*x**3 + 2*f**((a*c + a*d*x + b)/(c + d*x))*log(f)**2*b**2* 
c**4 + 8*f**((a*c + a*d*x + b)/(c + d*x))*log(f)**2*b**2*c**3*d*x + 12*f** 
((a*c + a*d*x + b)/(c + d*x))*log(f)**2*b**2*c**2*d**2*x**2 + 8*f**((a*c + 
 a*d*x + b)/(c + d*x))*log(f)**2*b**2*c*d**3*x**3 + 2*f**((a*c + a*d*x + b 
)/(c + d*x))*log(f)**2*b**2*d**4*x**4 + 6*f**((a*c + a*d*x + b)/(c + d*x)) 
*log(f)*b*c**5 + 30*f**((a*c + a*d*x + b)/(c + d*x))*log(f)*b*c**4*d*x + 6 
0*f**((a*c + a*d*x + b)/(c + d*x))*log(f)*b*c**3*d**2*x**2 + 60*f**((a*c + 
 a*d*x + b)/(c + d*x))*log(f)*b*c**2*d**3*x**3 + 30*f**((a*c + a*d*x + b)/ 
(c + d*x))*log(f)*b*c*d**4*x**4 + 6*f**((a*c + a*d*x + b)/(c + d*x))*log(f 
)*b*d**5*x**5 + 24*f**((a*c + a*d*x + b)/(c + d*x))*c**6 + 144*f**((a*c + 
a*d*x + b)/(c + d*x))*c**5*d*x + 360*f**((a*c + a*d*x + b)/(c + d*x))*c**4 
*d**2*x**2 + 480*f**((a*c + a*d*x + b)/(c + d*x))*c**3*d**3*x**3 + 360*f** 
((a*c + a*d*x + b)/(c + d*x))*c**2*d**4*x**4 + 144*f**((a*c + a*d*x + b)/( 
c + d*x))*c*d**5*x**5 + 24*f**((a*c + a*d*x + b)/(c + d*x))*d**6*x**6 + in 
t((f**((a*c + a*d*x + b)/(c + d*x))*x**2)/(c**3 + 3*c**2*d*x + 3*c*d**2*x* 
*2 + d**3*x**3),x)*log(f)**5*b**5*c*d**3 + int((f**((a*c + a*d*x + b)/(c + 
 d*x))*x**2)/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3),x)*log(f)*...