\(\int \frac {(a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {d f-e f x}}})^3}{d^2-e^2 x^2} \, dx\) [551]

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.064, Rules used = {2329, 2214, 2209} \[ \int \frac {\left (a+b F^{\frac {c \sqrt {d+e x}}{\sqrt {d f-e f x}}}\right )^3}{d^2-e^2 x^2} \, dx=\frac {a^3 \log \left (\frac {\sqrt {d+e x}}{\sqrt {d f-e f x}}\right )}{d e}+\frac {3 a^2 b \operatorname {ExpIntegralEi}\left (\frac {c \sqrt {d+e x} \log (F)}{\sqrt {d f-e f x}}\right )}{d e}+\frac {3 a b^2 \operatorname {ExpIntegralEi}\left (\frac {2 c \sqrt {d+e x} \log (F)}{\sqrt {d f-e f x}}\right )}{d e}+\frac {b^3 \operatorname {ExpIntegralEi}\left (\frac {3 c \sqrt {d+e x} \log (F)}{\sqrt {d f-e f x}}\right )}{d e} \]

[In]

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

[Out]

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

Rule 2209

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - c*(f/d)))/d)*ExpInteg
ralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]

Rule 2214

Int[((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.))^(p_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> In
t[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 2329

Int[((a_.) + (b_.)*(F_)^(((c_.)*Sqrt[(d_.) + (e_.)*(x_)])/Sqrt[(f_.) + (g_.)*(x_)]))^(n_.)/((A_) + (C_.)*(x_)^
2), x_Symbol] :> Dist[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
]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (a+b F^{c x}\right )^3}{x} \, dx,x,\frac {\sqrt {d+e x}}{\sqrt {d f-e f x}}\right )}{d e} \\ & = \frac {\text {Subst}\left (\int \left (\frac {a^3}{x}+\frac {3 a^2 b F^{c x}}{x}+\frac {3 a b^2 F^{2 c x}}{x}+\frac {b^3 F^{3 c x}}{x}\right ) \, dx,x,\frac {\sqrt {d+e x}}{\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}+\frac {\left (3 a^2 b\right ) \text {Subst}\left (\int \frac {F^{c x}}{x} \, dx,x,\frac {\sqrt {d+e x}}{\sqrt {d f-e f x}}\right )}{d e}+\frac {\left (3 a b^2\right ) \text {Subst}\left (\int \frac {F^{2 c x}}{x} \, dx,x,\frac {\sqrt {d+e x}}{\sqrt {d f-e f x}}\right )}{d e}+\frac {b^3 \text {Subst}\left (\int \frac {F^{3 c x}}{x} \, dx,x,\frac {\sqrt {d+e x}}{\sqrt {d f-e f x}}\right )}{d e} \\ & = \frac {3 a^2 b \text {Ei}\left (\frac {c \sqrt {d+e x} \log (F)}{\sqrt {d f-e f x}}\right )}{d e}+\frac {3 a b^2 \text {Ei}\left (\frac {2 c \sqrt {d+e x} \log (F)}{\sqrt {d f-e f x}}\right )}{d e}+\frac {b^3 \text {Ei}\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} \\ \end{align*}

Mathematica [F]

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

[In]

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

[Out]

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

Maple [F]

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

[In]

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)

[Out]

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]

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

[In]

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")

[Out]

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]

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

[In]

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)

[Out]

-Integral(a**3/(-d**2 + e**2*x**2), x) - Integral(F**(3*c*sqrt(d + e*x)/sqrt(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]

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

[In]

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")

[Out]

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

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

[In]

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")

[Out]

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

[In]

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)

[Out]

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)