\(\int \frac {a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {d f-e f x}}}}{d^2-e^2 x^2} \, dx\) [475]

Optimal result

Integrand size = 45, antiderivative size = 68 \[ \int \frac {a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {d f-e f x}}}}{d^2-e^2 x^2} \, dx=\frac {b \operatorname {ExpIntegralEi}\left (\frac {c \sqrt {d+e x} \log (F)}{\sqrt {d f-e f x}}\right )}{d e}+\frac {a \log \left (\frac {\sqrt {d+e x}}{\sqrt {d f-e f x}}\right )}{d e} \] Output:

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

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.93, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2729, 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.

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

\(\Big \downarrow \) 2729

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

\(\Big \downarrow \) 2010

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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 2729

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

Maple [F]

\[\int \frac {a +b \,F^{\frac {c \sqrt {e x +d}}{\sqrt {-e f x +d f}}}}{-e^{2} x^{2}+d^{2}}d x\]

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

-Integral(a/(-d**2 + e**2*x**2), x) - Integral(F**(c*sqrt(d + e*x)/sqrt(d* 
f - e*f*x))*b/(-d**2 + e**2*x**2), x)
 

Maxima [F]

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

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

Output:

1/2*a*(log(e*x + d)/(d*e) - log(e*x - d)/(d*e)) - b*integrate(F^(sqrt(e*x 
+ d)*c/(sqrt(-e*x + d)*sqrt(f)))/(e^2*x^2 - d^2), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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