\(\int \frac {d+e (F^{c (a+b x)})^n}{(f+g x)^3} \, dx\) [25]

Optimal result

Integrand size = 23, antiderivative size = 147 \[ \int \frac {d+e \left (F^{c (a+b x)}\right )^n}{(f+g x)^3} \, dx=-\frac {d}{2 g (f+g x)^2}-\frac {e \left (F^{a c+b c x}\right )^n}{2 g (f+g x)^2}-\frac {b c e \left (F^{a c+b c x}\right )^n n \log (F)}{2 g^2 (f+g x)}+\frac {b^2 c^2 e F^{c \left (a-\frac {b f}{g}\right ) n-c n (a+b x)} \left (F^{a c+b c x}\right )^n n^2 \operatorname {ExpIntegralEi}\left (\frac {b c n (f+g x) \log (F)}{g}\right ) \log ^2(F)}{2 g^3} \] Output:

-1/2*d/g/(g*x+f)^2-1/2*e*(F^(b*c*x+a*c))^n/g/(g*x+f)^2-1/2*b*c*e*(F^(b*c*x 
+a*c))^n*n*ln(F)/g^2/(g*x+f)+1/2*b^2*c^2*e*F^(c*(a-b*f/g)*n-c*n*(b*x+a))*( 
F^(b*c*x+a*c))^n*n^2*Ei(b*c*n*(g*x+f)*ln(F)/g)*ln(F)^2/g^3
 

Mathematica [A] (verified)

Time = 0.64 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.76 \[ \int \frac {d+e \left (F^{c (a+b x)}\right )^n}{(f+g x)^3} \, dx=-\frac {d g^2-b^2 c^2 e F^{-\frac {b c n (f+g x)}{g}} \left (F^{c (a+b x)}\right )^n n^2 (f+g x)^2 \operatorname {ExpIntegralEi}\left (\frac {b c n (f+g x) \log (F)}{g}\right ) \log ^2(F)+e \left (F^{c (a+b x)}\right )^n g (g+b c n (f+g x) \log (F))}{2 g^3 (f+g x)^2} \] Input:

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

Output:

-1/2*(d*g^2 - (b^2*c^2*e*(F^(c*(a + b*x)))^n*n^2*(f + g*x)^2*ExpIntegralEi 
[(b*c*n*(f + g*x)*Log[F])/g]*Log[F]^2)/F^((b*c*n*(f + g*x))/g) + e*(F^(c*( 
a + b*x)))^n*g*(g + b*c*n*(f + g*x)*Log[F]))/(g^3*(f + g*x)^2)
 

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2614, 2009}

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.

\(\displaystyle \int \frac {e \left (F^{c (a+b x)}\right )^n+d}{(f+g x)^3} \, dx\)

\(\Big \downarrow \) 2614

\(\displaystyle \int \left (\frac {e \left (F^{a c+b c x}\right )^n}{(f+g x)^3}+\frac {d}{(f+g x)^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {b^2 c^2 e n^2 \log ^2(F) \left (F^{a c+b c x}\right )^n F^{c n \left (a-\frac {b f}{g}\right )-c n (a+b x)} \operatorname {ExpIntegralEi}\left (\frac {b c n (f+g x) \log (F)}{g}\right )}{2 g^3}-\frac {b c e n \log (F) \left (F^{a c+b c x}\right )^n}{2 g^2 (f+g x)}-\frac {e \left (F^{a c+b c x}\right )^n}{2 g (f+g x)^2}-\frac {d}{2 g (f+g x)^2}\)

Input:

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

Output:

-1/2*d/(g*(f + g*x)^2) - (e*(F^(a*c + b*c*x))^n)/(2*g*(f + g*x)^2) - (b*c* 
e*(F^(a*c + b*c*x))^n*n*Log[F])/(2*g^2*(f + g*x)) + (b^2*c^2*e*F^(c*(a - ( 
b*f)/g)*n - c*n*(a + b*x))*(F^(a*c + b*c*x))^n*n^2*ExpIntegralEi[(b*c*n*(f 
 + g*x)*Log[F])/g]*Log[F]^2)/(2*g^3)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2614

Int[((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.))^(p_.)*((c_.) + 
 (d_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c + d*x)^m, (a + b*(F 
^(g*(e + f*x)))^n)^p, x], x] /; FreeQ[{F, a, b, c, d, e, f, g, m, n}, x] && 
 IGtQ[p, 0]
 

Maple [F]

\[\int \frac {d +e \left (F^{c \left (b x +a \right )}\right )^{n}}{\left (g x +f \right )^{3}}d x\]

Input:

int((d+e*(F^(c*(b*x+a)))^n)/(g*x+f)^3,x)
 

Output:

int((d+e*(F^(c*(b*x+a)))^n)/(g*x+f)^3,x)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.10 \[ \int \frac {d+e \left (F^{c (a+b x)}\right )^n}{(f+g x)^3} \, dx=-\frac {d g^{2} - \frac {{\left (b^{2} c^{2} e g^{2} n^{2} x^{2} + 2 \, b^{2} c^{2} e f g n^{2} x + b^{2} c^{2} e f^{2} n^{2}\right )} {\rm Ei}\left (\frac {{\left (b c g n x + b c f n\right )} \log \left (F\right )}{g}\right ) \log \left (F\right )^{2}}{F^{\frac {{\left (b c f - a c g\right )} n}{g}}} + {\left (e g^{2} + {\left (b c e g^{2} n x + b c e f g n\right )} \log \left (F\right )\right )} F^{b c n x + a c n}}{2 \, {\left (g^{5} x^{2} + 2 \, f g^{4} x + f^{2} g^{3}\right )}} \] Input:

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

Output:

-1/2*(d*g^2 - (b^2*c^2*e*g^2*n^2*x^2 + 2*b^2*c^2*e*f*g*n^2*x + b^2*c^2*e*f 
^2*n^2)*Ei((b*c*g*n*x + b*c*f*n)*log(F)/g)*log(F)^2/F^((b*c*f - a*c*g)*n/g 
) + (e*g^2 + (b*c*e*g^2*n*x + b*c*e*f*g*n)*log(F))*F^(b*c*n*x + a*c*n))/(g 
^5*x^2 + 2*f*g^4*x + f^2*g^3)
 

Sympy [F]

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

integrate((d+e*(F**((b*x+a)*c))**n)/(g*x+f)**3,x)
 

Output:

Integral((d + e*(F**(a*c + b*c*x))**n)/(f + g*x)**3, x)
 

Maxima [F]

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

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

Output:

F^(a*c*n)*e*integrate(F^(b*c*n*x)/(g^3*x^3 + 3*f*g^2*x^2 + 3*f^2*g*x + f^3 
), x) - 1/2*d/(g^3*x^2 + 2*f*g^2*x + f^2*g)
 

Giac [F]

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

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

Output:

integrate(((F^((b*x + a)*c))^n*e + d)/(g*x + f)^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {d+e \left (F^{c (a+b x)}\right )^n}{(f+g x)^3} \, dx=\int \frac {d+e\,{\left (F^{c\,\left (a+b\,x\right )}\right )}^n}{{\left (f+g\,x\right )}^3} \,d x \] Input:

int((d + e*(F^(c*(a + b*x)))^n)/(f + g*x)^3,x)
 

Output:

int((d + e*(F^(c*(a + b*x)))^n)/(f + g*x)^3, x)
 

Reduce [F]

\[ \int \frac {d+e \left (F^{c (a+b x)}\right )^n}{(f+g x)^3} \, dx=\frac {2 f^{a c n} \left (\int \frac {f^{b c n x}}{g^{3} x^{3}+3 f \,g^{2} x^{2}+3 f^{2} g x +f^{3}}d x \right ) e \,f^{2} g +4 f^{a c n} \left (\int \frac {f^{b c n x}}{g^{3} x^{3}+3 f \,g^{2} x^{2}+3 f^{2} g x +f^{3}}d x \right ) e f \,g^{2} x +2 f^{a c n} \left (\int \frac {f^{b c n x}}{g^{3} x^{3}+3 f \,g^{2} x^{2}+3 f^{2} g x +f^{3}}d x \right ) e \,g^{3} x^{2}-d}{2 g \left (g^{2} x^{2}+2 f g x +f^{2}\right )} \] Input:

int((d+e*(F^((b*x+a)*c))^n)/(g*x+f)^3,x)
 

Output:

(2*f**(a*c*n)*int(f**(b*c*n*x)/(f**3 + 3*f**2*g*x + 3*f*g**2*x**2 + g**3*x 
**3),x)*e*f**2*g + 4*f**(a*c*n)*int(f**(b*c*n*x)/(f**3 + 3*f**2*g*x + 3*f* 
g**2*x**2 + g**3*x**3),x)*e*f*g**2*x + 2*f**(a*c*n)*int(f**(b*c*n*x)/(f**3 
 + 3*f**2*g*x + 3*f*g**2*x**2 + g**3*x**3),x)*e*g**3*x**2 - d)/(2*g*(f**2 
+ 2*f*g*x + g**2*x**2))