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

Optimal. Leaf size=152 \[ \frac{3 a^2 b \text{Ei}\left (\frac{c \sqrt{d+e x} \log (F)}{\sqrt{d f-e f x}}\right )}{d e}+\frac{a^3 \log \left (\frac{\sqrt{d+e x}}{\sqrt{d f-e f x}}\right )}{d e}+\frac{3 a b^2 \text{Ei}\left (\frac{2 c \sqrt{d+e x} \log (F)}{\sqrt{d f-e f x}}\right )}{d e}+\frac{b^3 \text{Ei}\left (\frac{3 c \sqrt{d+e x} \log (F)}{\sqrt{d f-e f x}}\right )}{d e} \]

[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)

________________________________________________________________________________________

Rubi [A]  time = 0.326986, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 47, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.064, Rules used = {2291, 2183, 2178} \[ \frac{3 a^2 b \text{Ei}\left (\frac{c \sqrt{d+e x} \log (F)}{\sqrt{d f-e f x}}\right )}{d e}+\frac{a^3 \log \left (\frac{\sqrt{d+e x}}{\sqrt{d f-e f x}}\right )}{d e}+\frac{3 a b^2 \text{Ei}\left (\frac{2 c \sqrt{d+e x} \log (F)}{\sqrt{d f-e f x}}\right )}{d e}+\frac{b^3 \text{Ei}\left (\frac{3 c \sqrt{d+e x} \log (F)}{\sqrt{d f-e f x}}\right )}{d e} \]

Antiderivative was successfully verified.

[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 2291

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
]

Rule 2183

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 2178

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

Rubi steps

\begin{align*} \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{\operatorname{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{\operatorname{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 ) \operatorname{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 ) \operatorname{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 \operatorname{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]  time = 1.31904, size = 0, normalized size = 0. \[ \int \frac{\left (a+b F^{\frac{c \sqrt{d+e x}}{\sqrt{d f-e f x}}}\right )^3}{d^2-e^2 x^2} \, dx \]

Verification is Not applicable to the result.

[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]  time = 0.023, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{-{e}^{2}{x}^{2}+{d}^{2}} \left ( a+b{F}^{{c\sqrt{ex+d}{\frac{1}{\sqrt{-efx+df}}}}} \right ) ^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \, a^{3}{\left (\frac{\log \left (e x + d\right )}{d e} - \frac{\log \left (e x - d\right )}{d e}\right )} - b^{3} \int \frac{F^{\frac{3 \, \sqrt{e x + d} c}{\sqrt{-e x + d} \sqrt{f}}}}{e^{2} x^{2} - d^{2}}\,{d x} - 3 \, a b^{2} \int \frac{F^{\frac{2 \, \sqrt{e x + d} c}{\sqrt{-e x + d} \sqrt{f}}}}{e^{2} x^{2} - d^{2}}\,{d x} - 3 \, a^{2} b \int \frac{F^{\frac{\sqrt{e x + d} c}{\sqrt{-e x + d} \sqrt{f}}}}{e^{2} x^{2} - d^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{a^{3} + \frac{3 \, a^{2} b}{F^{\frac{\sqrt{-e f x + d f} \sqrt{e x + d} c}{e f x - d f}}} + \frac{3 \, a b^{2}}{F^{\frac{2 \, \sqrt{-e f x + d f} \sqrt{e x + d} c}{e f x - d f}}} + \frac{b^{3}}{F^{\frac{3 \, \sqrt{-e f x + d f} \sqrt{e x + d} c}{e f x - d f}}}}{e^{2} x^{2} - d^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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]  time = 0., size = 0, normalized size = 0. \begin{align*} - \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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)