\(\int \frac {(a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {d f-e f x}}})^3}{d^2-e^2 x^2} \, dx\) [473]

Optimal result

Integrand size = 47, antiderivative size = 152 \[ \int \frac {\left (a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {d f-e f x}}}\right )^3}{d^2-e^2 x^2} \, dx=\frac {3 a^2 b \operatorname {ExpIntegralEi}\left (\frac {c \sqrt {d+e x} \log (F)}{\sqrt {d f-e f x}}\right )}{d e}+\frac {3 a b^2 \operatorname {ExpIntegralEi}\left (\frac {2 c \sqrt {d+e x} \log (F)}{\sqrt {d f-e f x}}\right )}{d e}+\frac {b^3 \operatorname {ExpIntegralEi}\left (\frac {3 c \sqrt {d+e x} \log (F)}{\sqrt {d f-e f x}}\right )}{d e}+\frac {a^3 \log \left (\frac {\sqrt {d+e x}}{\sqrt {d f-e f x}}\right )}{d e} \] Output:

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

Mathematica [F]

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

Integrate[(a + b*F^((c*Sqrt[d + e*x])/Sqrt[d*f - e*f*x]))^3/(d^2 - e^2*x^2 
),x]
 

Output:

Integrate[(a + b*F^((c*Sqrt[d + e*x])/Sqrt[d*f - e*f*x]))^3/(d^2 - e^2*x^2 
), x]
 

Rubi [A] (verified)

Time = 0.74 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.89, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.064, Rules used = {2729, 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.

\(\displaystyle \int \frac {\left (a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {d f-e f x}}}\right )^3}{d^2-e^2 x^2} \, dx\)

\(\Big \downarrow \) 2729

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

\(\Big \downarrow \) 2614

\(\displaystyle \frac {\int \left (\frac {3 a^2 b \sqrt {d f-e f x} F^{\frac {c \sqrt {d+e x}}{\sqrt {d f-e f x}}}}{\sqrt {d+e x}}+\frac {3 a b^2 \sqrt {d f-e f x} F^{\frac {2 c \sqrt {d+e x}}{\sqrt {d f-e f x}}}}{\sqrt {d+e x}}+\frac {b^3 \sqrt {d f-e f x} F^{\frac {3 c \sqrt {d+e x}}{\sqrt {d f-e f x}}}}{\sqrt {d+e x}}+\frac {a^3 \sqrt {d f-e f x}}{\sqrt {d+e x}}\right )d\frac {\sqrt {d+e x}}{\sqrt {d f-e f x}}}{d e}\)

\(\Big \downarrow \) 2009

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

Input:

Int[(a + b*F^((c*Sqrt[d + e*x])/Sqrt[d*f - e*f*x]))^3/(d^2 - e^2*x^2),x]
 

Output:

(3*a^2*b*ExpIntegralEi[(c*Sqrt[d + e*x]*Log[F])/Sqrt[d*f - e*f*x]] + 3*a*b 
^2*ExpIntegralEi[(2*c*Sqrt[d + e*x]*Log[F])/Sqrt[d*f - e*f*x]] + b^3*ExpIn 
tegralEi[(3*c*Sqrt[d + e*x]*Log[F])/Sqrt[d*f - e*f*x]] + a^3*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 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]
 

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]

\[\int \frac {\left (a +b \,F^{\frac {c \sqrt {e x +d}}{\sqrt {-e f x +d f}}}\right )^{3}}{-e^{2} x^{2}+d^{2}}d x\]

Input:

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

Fricas [F]

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

integrate((a+b*F^(c*(e*x+d)^(1/2)/(-e*f*x+d*f)^(1/2)))^3/(-e^2*x^2+d^2),x, 
 algorithm="fricas")
 

Output:

integral(-(a^3 + 3*a^2*b/F^(sqrt(-e*f*x + d*f)*sqrt(e*x + d)*c/(e*f*x - d* 
f)) + 3*a*b^2/F^(2*sqrt(-e*f*x + d*f)*sqrt(e*x + d)*c/(e*f*x - d*f)) + b^3 
/F^(3*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]

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

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

Output:

-Integral(a**3/(-d**2 + e**2*x**2), x) - Integral(F**(3*c*sqrt(d + e*x)/sq 
rt(d*f - e*f*x))*b**3/(-d**2 + e**2*x**2), x) - Integral(3*F**(c*sqrt(d + 
e*x)/sqrt(d*f - e*f*x))*a**2*b/(-d**2 + e**2*x**2), x) - Integral(3*F**(2* 
c*sqrt(d + e*x)/sqrt(d*f - e*f*x))*a*b**2/(-d**2 + e**2*x**2), x)
 

Maxima [F]

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

integrate((a+b*F^(c*(e*x+d)^(1/2)/(-e*f*x+d*f)^(1/2)))^3/(-e^2*x^2+d^2),x, 
 algorithm="maxima")
 

Output:

1/2*a^3*(log(e*x + d)/(d*e) - log(e*x - d)/(d*e)) - b^3*integrate(F^(3*sqr 
t(e*x + d)*c/(sqrt(-e*x + d)*sqrt(f)))/(e^2*x^2 - d^2), x) - 3*a*b^2*integ 
rate(F^(2*sqrt(e*x + d)*c/(sqrt(-e*x + d)*sqrt(f)))/(e^2*x^2 - d^2), x) - 
3*a^2*b*integrate(F^(sqrt(e*x + d)*c/(sqrt(-e*x + d)*sqrt(f)))/(e^2*x^2 - 
d^2), x)
 

Giac [F]

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

integrate((a+b*F^(c*(e*x+d)^(1/2)/(-e*f*x+d*f)^(1/2)))^3/(-e^2*x^2+d^2),x, 
 algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {d f-e f x}}}\right )^3}{d^2-e^2 x^2} \, dx=\int \frac {{\left (a+b\,{\mathrm {e}}^{\frac {c\,\ln \left (F\right )\,\sqrt {d+e\,x}}{\sqrt {d\,f-e\,f\,x}}}\right )}^3}{d^2-e^2\,x^2} \,d x \] Input:

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

Reduce [F]

\[ \int \frac {\left (a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {d f-e f x}}}\right )^3}{d^2-e^2 x^2} \, dx=\frac {6 \left (\int \frac {f^{\frac {\sqrt {e x +d}\, c}{\sqrt {f}\, \sqrt {-e x +d}}}}{-e^{2} x^{2}+d^{2}}d x \right ) a^{2} b d e +2 \left (\int \frac {f^{\frac {3 \sqrt {e x +d}\, c}{\sqrt {f}\, \sqrt {-e x +d}}}}{-e^{2} x^{2}+d^{2}}d x \right ) b^{3} d e +6 \left (\int \frac {f^{\frac {2 \sqrt {e x +d}\, c}{\sqrt {f}\, \sqrt {-e x +d}}}}{-e^{2} x^{2}+d^{2}}d x \right ) a \,b^{2} d e +\mathrm {log}\left (-e x -d \right ) a^{3}-\mathrm {log}\left (-e x +d \right ) a^{3}}{2 d e} \] Input:

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

Output:

(6*int(f**((sqrt(d + e*x)*c)/(sqrt(f)*sqrt(d - e*x)))/(d**2 - e**2*x**2),x 
)*a**2*b*d*e + 2*int(f**((3*sqrt(d + e*x)*c)/(sqrt(f)*sqrt(d - e*x)))/(d** 
2 - e**2*x**2),x)*b**3*d*e + 6*int(f**((2*sqrt(d + e*x)*c)/(sqrt(f)*sqrt(d 
 - e*x)))/(d**2 - e**2*x**2),x)*a*b**2*d*e + log( - d - e*x)*a**3 - log(d 
- e*x)*a**3)/(2*d*e)