\(\int \frac {f+g x}{\sqrt {d+e (F^{c (a+b x)})^n}} \, dx\) [48]

Optimal result

Integrand size = 25, antiderivative size = 246 \[ \int \frac {f+g x}{\sqrt {d+e \left (F^{c (a+b x)}\right )^n}} \, dx=\frac {2 g \text {arctanh}\left (\frac {\sqrt {d+e \left (F^{c (a+b x)}\right )^n}}{\sqrt {d}}\right )^2}{b^2 c^2 \sqrt {d} n^2 \log ^2(F)}-\frac {2 (f+g x) \text {arctanh}\left (\frac {\sqrt {d+e \left (F^{c (a+b x)}\right )^n}}{\sqrt {d}}\right )}{b c \sqrt {d} n \log (F)}-\frac {4 g \text {arctanh}\left (\frac {\sqrt {d+e \left (F^{c (a+b x)}\right )^n}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e \left (F^{c (a+b x)}\right )^n}}\right )}{b^2 c^2 \sqrt {d} n^2 \log ^2(F)}-\frac {2 g \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e \left (F^{c (a+b x)}\right )^n}}\right )}{b^2 c^2 \sqrt {d} n^2 \log ^2(F)} \] Output:

2*g*arctanh((d+e*(F^(c*(b*x+a)))^n)^(1/2)/d^(1/2))^2/b^2/c^2/d^(1/2)/n^2/l 
n(F)^2-2*(g*x+f)*arctanh((d+e*(F^(c*(b*x+a)))^n)^(1/2)/d^(1/2))/b/c/d^(1/2 
)/n/ln(F)-4*g*arctanh((d+e*(F^(c*(b*x+a)))^n)^(1/2)/d^(1/2))*ln(2*d^(1/2)/ 
(d^(1/2)-(d+e*(F^(c*(b*x+a)))^n)^(1/2)))/b^2/c^2/d^(1/2)/n^2/ln(F)^2-2*g*p 
olylog(2,1-2*d^(1/2)/(d^(1/2)-(d+e*(F^(c*(b*x+a)))^n)^(1/2)))/b^2/c^2/d^(1 
/2)/n^2/ln(F)^2
                                                                                    
                                                                                    
 

Mathematica [F]

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

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

Output:

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

Rubi [F]

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.

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

\(\Big \downarrow \) 2618

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

\(\Big \downarrow \) 2619

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

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 2618

Int[((a_.) + (b_.)*((F_)^((g_.)*(v_)))^(n_.))^(p_.)*((c_.) + (d_.)*(x_))^(m 
_.), x_Symbol] :> Int[(c + d*x)^m*(a + b*(F^(g*ExpandToSum[v, x]))^n)^p, x] 
 /; FreeQ[{F, a, b, c, d, g, n, p}, x] && LinearQ[v, x] &&  !LinearMatchQ[v 
, x] && IntegerQ[m]
 

rule 2619

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-2)]

Exception generated. \[ \int \frac {f+g x}{\sqrt {d+e \left (F^{c (a+b x)}\right )^n}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((g*x+f)/(d+e*(F^((b*x+a)*c))^n)^(1/2),x, algorithm="fricas")
 

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (constant residues)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((g*x+f)/(d+e*(F^((b*x+a)*c))^n)^(1/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((g*x+f)/(d+e*(F^((b*x+a)*c))^n)^(1/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {f+g x}{\sqrt {d+e \left (F^{c (a+b x)}\right )^n}} \, dx=\left (\int \frac {x}{\sqrt {f^{b c n x +a c n} e +d}}d x \right ) g +\left (\int \frac {1}{\sqrt {f^{b c n x +a c n} e +d}}d x \right ) f \] Input:

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

Output:

int(x/sqrt(f**(a*c*n + b*c*n*x)*e + d),x)*g + int(1/sqrt(f**(a*c*n + b*c*n 
*x)*e + d),x)*f