\(\int \frac {a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {f+g x}}}}{d f+(e f+d g) x+e g x^2} \, dx\) [469]

Optimal result

Integrand size = 48, antiderivative size = 70 \[ \int \frac {a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {f+g x}}}}{d f+(e f+d g) x+e g x^2} \, dx=\frac {2 b \operatorname {ExpIntegralEi}\left (\frac {c \sqrt {d+e x} \log (F)}{\sqrt {f+g x}}\right )}{e f-d g}+\frac {2 a \log \left (\frac {\sqrt {d+e x}}{\sqrt {f+g x}}\right )}{e f-d g} \] Output:

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

Mathematica [F]

\[ \int \frac {a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {f+g x}}}}{d f+(e f+d g) x+e g x^2} \, dx=\int \frac {a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {f+g x}}}}{d f+(e f+d g) x+e g x^2} \, dx \] Input:

Integrate[(a + b*F^((c*Sqrt[d + e*x])/Sqrt[f + g*x]))/(d*f + (e*f + d*g)*x 
 + e*g*x^2),x]
 

Output:

Integrate[(a + b*F^((c*Sqrt[d + e*x])/Sqrt[f + g*x]))/(d*f + (e*f + d*g)*x 
 + e*g*x^2), x]
 

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.86, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {2728, 2010, 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.

\(\displaystyle \int \frac {a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {f+g x}}}}{x (d g+e f)+d f+e g x^2} \, dx\)

\(\Big \downarrow \) 2728

\(\displaystyle \frac {2 \int \frac {\left (b F^{\frac {c \sqrt {d+e x}}{\sqrt {f+g x}}}+a\right ) \sqrt {f+g x}}{\sqrt {d+e x}}d\frac {\sqrt {d+e x}}{\sqrt {f+g x}}}{e f-d g}\)

\(\Big \downarrow \) 2010

\(\displaystyle \frac {2 \int \left (\frac {b \sqrt {f+g x} F^{\frac {c \sqrt {d+e x}}{\sqrt {f+g x}}}}{\sqrt {d+e x}}+\frac {a \sqrt {f+g x}}{\sqrt {d+e x}}\right )d\frac {\sqrt {d+e x}}{\sqrt {f+g x}}}{e f-d g}\)

\(\Big \downarrow \) 2009

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

Input:

Int[(a + b*F^((c*Sqrt[d + e*x])/Sqrt[f + g*x]))/(d*f + (e*f + d*g)*x + e*g 
*x^2),x]
 

Output:

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

Defintions of rubi rules used

rule 2009

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

rule 2010

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] 
, x] /; FreeQ[{c, m}, x] && SumQ[u] &&  !LinearQ[u, x] &&  !MatchQ[u, (a_) 
+ (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
 

rule 2728

Int[((a_.) + (b_.)*(F_)^(((c_.)*Sqrt[(d_.) + (e_.)*(x_)])/Sqrt[(f_.) + (g_. 
)*(x_)]))^(n_.)/((A_.) + (B_.)*(x_) + (C_.)*(x_)^2), x_Symbol] :> Simp[2*e* 
(g/(C*(e*f - d*g)))   Subst[Int[(a + b*F^(c*x))^n/x, x], x, Sqrt[d + e*x]/S 
qrt[f + g*x]], x] /; FreeQ[{a, b, c, d, e, f, g, A, B, C, F}, x] && EqQ[C*d 
*f - A*e*g, 0] && EqQ[B*e*g - C*(e*f + d*g), 0] && IGtQ[n, 0]
 

Maple [F]

\[\int \frac {a +b \,F^{\frac {c \sqrt {e x +d}}{\sqrt {g x +f}}}}{d f +\left (d g +e f \right ) x +e g \,x^{2}}d x\]

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((F^(sqrt(e*x + d)*c/sqrt(g*x + f))*b + a)/(e*g*x^2 + d*f + (e*f + 
 d*g)*x), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

a*(log(e*x + d)/(e*f - d*g) - log(g*x + f)/(e*f - d*g)) + b*integrate(F^(s 
qrt(e*x + d)*c/sqrt(g*x + f))/(e*g*x^2 + d*f + (e*f + d*g)*x), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {f+g x}}}}{d f+(e f+d g) x+e g x^2} \, dx=\int \frac {a+F^{\frac {c\,\sqrt {d+e\,x}}{\sqrt {f+g\,x}}}\,b}{e\,g\,x^2+\left (d\,g+e\,f\right )\,x+d\,f} \,d x \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

(int(f**((sqrt(d + e*x)*c)/sqrt(f + g*x))/(d*f + d*g*x + e*f*x + e*g*x**2) 
,x)*b*d*g - int(f**((sqrt(d + e*x)*c)/sqrt(f + g*x))/(d*f + d*g*x + e*f*x 
+ e*g*x**2),x)*b*e*f - log(d + e*x)*a + log(f + g*x)*a)/(d*g - e*f)