\(\int F^{a+\frac {b}{c+d x}} (c+d x)^3 \, dx\) [237]

Optimal result

Integrand size = 21, antiderivative size = 28 \[ \int F^{a+\frac {b}{c+d x}} (c+d x)^3 \, dx=\frac {b^4 F^a \Gamma \left (-4,-\frac {b \log (F)}{c+d x}\right ) \log ^4(F)}{d} \] Output:

F^a*(d*x+c)^4*Ei(5,-b*ln(F)/(d*x+c))/d
 

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int F^{a+\frac {b}{c+d x}} (c+d x)^3 \, dx=\frac {b^4 F^a \Gamma \left (-4,-\frac {b \log (F)}{c+d x}\right ) \log ^4(F)}{d} \] Input:

Integrate[F^(a + b/(c + d*x))*(c + d*x)^3,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 28, 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.

\(\displaystyle \int (c+d x)^3 F^{a+\frac {b}{c+d x}} \, dx\)

\(\Big \downarrow \) 2648

\(\displaystyle \frac {b^4 F^a \log ^4(F) \Gamma \left (-4,-\frac {b \log (F)}{c+d x}\right )}{d}\)

Input:

Int[F^(a + b/(c + d*x))*(c + d*x)^3,x]
 

Output:

(b^4*F^a*Gamma[-4, -((b*Log[F])/(c + d*x))]*Log[F]^4)/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. \(367\) vs. \(2(28)=56\).

Time = 0.30 (sec) , antiderivative size = 368, normalized size of antiderivative = 13.14

method	result	size
risch	\(\frac {d^{3} F^{a} F^{\frac {b}{d x +c}} x^{4}}{4}+d^{2} F^{a} F^{\frac {b}{d x +c}} c \,x^{3}+\frac {3 d \,F^{a} F^{\frac {b}{d x +c}} c^{2} x^{2}}{2}+F^{a} F^{\frac {b}{d x +c}} c^{3} x +\frac {F^{a} F^{\frac {b}{d x +c}} c^{4}}{4 d}+\frac {d^{2} b \ln \left (F \right ) F^{a} F^{\frac {b}{d x +c}} x^{3}}{12}+\frac {d b \ln \left (F \right ) F^{a} F^{\frac {b}{d x +c}} c \,x^{2}}{4}+\frac {b \ln \left (F \right ) F^{a} F^{\frac {b}{d x +c}} c^{2} x}{4}+\frac {b \ln \left (F \right ) F^{a} F^{\frac {b}{d x +c}} c^{3}}{12 d}+\frac {d \,b^{2} \ln \left (F \right )^{2} F^{a} F^{\frac {b}{d x +c}} x^{2}}{24}+\frac {b^{2} \ln \left (F \right )^{2} F^{a} F^{\frac {b}{d x +c}} c x}{12}+\frac {b^{2} \ln \left (F \right )^{2} F^{a} F^{\frac {b}{d x +c}} c^{2}}{24 d}+\frac {b^{3} \ln \left (F \right )^{3} F^{a} F^{\frac {b}{d x +c}} x}{24}+\frac {b^{3} \ln \left (F \right )^{3} F^{a} F^{\frac {b}{d x +c}} c}{24 d}+\frac {b^{4} \ln \left (F \right )^{4} F^{a} \operatorname {expIntegral}_{1}\left (-\frac {b \ln \left (F \right )}{d x +c}\right )}{24 d}\)	\(368\)

Input:

int(F^(a+b/(d*x+c))*(d*x+c)^3,x,method=_RETURNVERBOSE)
 

Output:

1/4*d^3*F^a*F^(b/(d*x+c))*x^4+d^2*F^a*F^(b/(d*x+c))*c*x^3+3/2*d*F^a*F^(b/( 
d*x+c))*c^2*x^2+F^a*F^(b/(d*x+c))*c^3*x+1/4/d*F^a*F^(b/(d*x+c))*c^4+1/12*d 
^2*b*ln(F)*F^a*F^(b/(d*x+c))*x^3+1/4*d*b*ln(F)*F^a*F^(b/(d*x+c))*c*x^2+1/4 
*b*ln(F)*F^a*F^(b/(d*x+c))*c^2*x+1/12/d*b*ln(F)*F^a*F^(b/(d*x+c))*c^3+1/24 
*d*b^2*ln(F)^2*F^a*F^(b/(d*x+c))*x^2+1/12*b^2*ln(F)^2*F^a*F^(b/(d*x+c))*c* 
x+1/24/d*b^2*ln(F)^2*F^a*F^(b/(d*x+c))*c^2+1/24*b^3*ln(F)^3*F^a*F^(b/(d*x+ 
c))*x+1/24/d*b^3*ln(F)^3*F^a*F^(b/(d*x+c))*c+1/24/d*b^4*ln(F)^4*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. 175 vs. \(2 (28) = 56\).

Time = 0.08 (sec) , antiderivative size = 175, normalized size of antiderivative = 6.25 \[ \int F^{a+\frac {b}{c+d x}} (c+d x)^3 \, dx=-\frac {F^{a} b^{4} {\rm Ei}\left (\frac {b \log \left (F\right )}{d x + c}\right ) \log \left (F\right )^{4} - {\left (6 \, d^{4} x^{4} + 24 \, c d^{3} x^{3} + 36 \, c^{2} d^{2} x^{2} + 24 \, c^{3} d x + 6 \, c^{4} + {\left (b^{3} d x + b^{3} c\right )} \log \left (F\right )^{3} + {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \log \left (F\right )^{2} + 2 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \log \left (F\right )\right )} F^{\frac {a d x + a c + b}{d x + c}}}{24 \, d} \] Input:

integrate(F^(a+b/(d*x+c))*(d*x+c)^3,x, algorithm="fricas")
 

Output:

-1/24*(F^a*b^4*Ei(b*log(F)/(d*x + c))*log(F)^4 - (6*d^4*x^4 + 24*c*d^3*x^3 
 + 36*c^2*d^2*x^2 + 24*c^3*d*x + 6*c^4 + (b^3*d*x + b^3*c)*log(F)^3 + (b^2 
*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*log(F)^2 + 2*(b*d^3*x^3 + 3*b*c*d^2*x^2 
+ 3*b*c^2*d*x + b*c^3)*log(F))*F^((a*d*x + a*c + b)/(d*x + c)))/d
 

Sympy [F]

\[ \int F^{a+\frac {b}{c+d x}} (c+d x)^3 \, dx=\int F^{a + \frac {b}{c + d x}} \left (c + d x\right )^{3}\, dx \] Input:

integrate(F**(a+b/(d*x+c))*(d*x+c)**3,x)
 

Output:

Integral(F**(a + b/(c + d*x))*(c + d*x)**3, x)
 

Maxima [F]

\[ \int F^{a+\frac {b}{c+d x}} (c+d x)^3 \, dx=\int { {\left (d x + c\right )}^{3} F^{a + \frac {b}{d x + c}} \,d x } \] Input:

integrate(F^(a+b/(d*x+c))*(d*x+c)^3,x, algorithm="maxima")
 

Output:

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

Giac [F]

\[ \int F^{a+\frac {b}{c+d x}} (c+d x)^3 \, dx=\int { {\left (d x + c\right )}^{3} F^{a + \frac {b}{d x + c}} \,d x } \] Input:

integrate(F^(a+b/(d*x+c))*(d*x+c)^3,x, algorithm="giac")
 

Output:

integrate((d*x + c)^3*F^(a + b/(d*x + c)), x)
 

Mupad [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 148, normalized size of antiderivative = 5.29 \[ \int F^{a+\frac {b}{c+d x}} (c+d x)^3 \, dx=\frac {F^a\,F^{\frac {b}{c+d\,x}}\,{\left (c+d\,x\right )}^4}{4\,d}+\frac {F^a\,b^4\,{\ln \left (F\right )}^4\,\mathrm {expint}\left (-\frac {b\,\ln \left (F\right )}{c+d\,x}\right )}{24\,d}+\frac {F^a\,F^{\frac {b}{c+d\,x}}\,b^2\,{\ln \left (F\right )}^2\,{\left (c+d\,x\right )}^2}{24\,d}+\frac {F^a\,F^{\frac {b}{c+d\,x}}\,b\,\ln \left (F\right )\,{\left (c+d\,x\right )}^3}{12\,d}+\frac {F^a\,F^{\frac {b}{c+d\,x}}\,b^3\,{\ln \left (F\right )}^3\,\left (c+d\,x\right )}{24\,d} \] Input:

int(F^(a + b/(c + d*x))*(c + d*x)^3,x)
 

Output:

(F^a*F^(b/(c + d*x))*(c + d*x)^4)/(4*d) + (F^a*b^4*log(F)^4*expint(-(b*log 
(F))/(c + d*x)))/(24*d) + (F^a*F^(b/(c + d*x))*b^2*log(F)^2*(c + d*x)^2)/( 
24*d) + (F^a*F^(b/(c + d*x))*b*log(F)*(c + d*x)^3)/(12*d) + (F^a*F^(b/(c + 
 d*x))*b^3*log(F)^3*(c + d*x))/(24*d)
 

Reduce [F]

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

int(F^(a+b/(d*x+c))*(d*x+c)^3,x)
 

Output:

(f**((a*c + a*d*x + b)/(c + d*x))*log(f)**3*b**3*d**2*x**2 + 2*f**((a*c + 
a*d*x + b)/(c + d*x))*log(f)**2*b**2*c**2*d*x + 3*f**((a*c + a*d*x + b)/(c 
 + d*x))*log(f)**2*b**2*c*d**2*x**2 + f**((a*c + a*d*x + b)/(c + d*x))*log 
(f)**2*b**2*d**3*x**3 + 2*f**((a*c + a*d*x + b)/(c + d*x))*log(f)*b*c**4 + 
 8*f**((a*c + a*d*x + b)/(c + d*x))*log(f)*b*c**3*d*x + 12*f**((a*c + a*d* 
x + b)/(c + d*x))*log(f)*b*c**2*d**2*x**2 + 8*f**((a*c + a*d*x + b)/(c + d 
*x))*log(f)*b*c*d**3*x**3 + 2*f**((a*c + a*d*x + b)/(c + d*x))*log(f)*b*d* 
*4*x**4 + 6*f**((a*c + a*d*x + b)/(c + d*x))*c**5 + 30*f**((a*c + a*d*x + 
b)/(c + d*x))*c**4*d*x + 60*f**((a*c + a*d*x + b)/(c + d*x))*c**3*d**2*x** 
2 + 60*f**((a*c + a*d*x + b)/(c + d*x))*c**2*d**3*x**3 + 30*f**((a*c + a*d 
*x + b)/(c + d*x))*c*d**4*x**4 + 6*f**((a*c + a*d*x + b)/(c + d*x))*d**5*x 
**5 + 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)**4*b**4*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)*lo 
g(f)**4*b**4*d**4*x)/(24*d*(c + d*x))