3.2639 \(\int \frac{1}{\sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x}} \, dx\)

Optimal. Leaf size=134 \[ \frac{2 \sqrt{a d-b c} \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right ),\frac{f (b c-a d)}{d (b e-a f)}\right )}{b \sqrt{d} \sqrt{c+d x} \sqrt{e+f x}} \]

[Out]

(2*Sqrt[-(b*c) + a*d]*Sqrt[(b*(c + d*x))/(b*c - a*d)]*Sqrt[(b*(e + f*x))/(b*e - a*f)]*EllipticF[ArcSin[(Sqrt[d
]*Sqrt[a + b*x])/Sqrt[-(b*c) + a*d]], ((b*c - a*d)*f)/(d*(b*e - a*f))])/(b*Sqrt[d]*Sqrt[c + d*x]*Sqrt[e + f*x]
)

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Rubi [A]  time = 0.124484, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {121, 120} \[ \frac{2 \sqrt{a d-b c} \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}} F\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt{d} \sqrt{c+d x} \sqrt{e+f x}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]),x]

[Out]

(2*Sqrt[-(b*c) + a*d]*Sqrt[(b*(c + d*x))/(b*c - a*d)]*Sqrt[(b*(e + f*x))/(b*e - a*f)]*EllipticF[ArcSin[(Sqrt[d
]*Sqrt[a + b*x])/Sqrt[-(b*c) + a*d]], ((b*c - a*d)*f)/(d*(b*e - a*f))])/(b*Sqrt[d]*Sqrt[c + d*x]*Sqrt[e + f*x]
)

Rule 121

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Dist[Sqrt[(b*(c
+ d*x))/(b*c - a*d)]/Sqrt[c + d*x], Int[1/(Sqrt[a + b*x]*Sqrt[(b*c)/(b*c - a*d) + (b*d*x)/(b*c - a*d)]*Sqrt[e
+ f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] &&  !GtQ[(b*c - a*d)/b, 0] && SimplerQ[a + b*x, c + d*x] && Si
mplerQ[a + b*x, e + f*x]

Rule 120

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[(2*Rt[-(b/d
), 2]*EllipticF[ArcSin[Sqrt[a + b*x]/(Rt[-(b/d), 2]*Sqrt[(b*c - a*d)/b])], (f*(b*c - a*d))/(d*(b*e - a*f))])/(
b*Sqrt[(b*e - a*f)/b]), x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] &
& SimplerQ[a + b*x, c + d*x] && SimplerQ[a + b*x, e + f*x] && (PosQ[-((b*c - a*d)/d)] || NegQ[-((b*e - a*f)/f)
])

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x}} \, dx &=\frac{\sqrt{\frac{b (c+d x)}{b c-a d}} \int \frac{1}{\sqrt{a+b x} \sqrt{\frac{b c}{b c-a d}+\frac{b d x}{b c-a d}} \sqrt{e+f x}} \, dx}{\sqrt{c+d x}}\\ &=\frac{\left (\sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}}\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{\frac{b c}{b c-a d}+\frac{b d x}{b c-a d}} \sqrt{\frac{b e}{b e-a f}+\frac{b f x}{b e-a f}}} \, dx}{\sqrt{c+d x} \sqrt{e+f x}}\\ &=\frac{2 \sqrt{-b c+a d} \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}} F\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{-b c+a d}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt{d} \sqrt{c+d x} \sqrt{e+f x}}\\ \end{align*}

Mathematica [A]  time = 0.438752, size = 126, normalized size = 0.94 \[ -\frac{2 \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{f (a+b x)}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{a-\frac{b c}{d}}}{\sqrt{a+b x}}\right ),\frac{b d e-a d f}{b c f-a d f}\right )}{d \sqrt{e+f x} \sqrt{a-\frac{b c}{d}} \sqrt{\frac{b (c+d x)}{d (a+b x)}}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]),x]

[Out]

(-2*Sqrt[c + d*x]*Sqrt[(b*(e + f*x))/(f*(a + b*x))]*EllipticF[ArcSin[Sqrt[a - (b*c)/d]/Sqrt[a + b*x]], (b*d*e
- a*d*f)/(b*c*f - a*d*f)])/(Sqrt[a - (b*c)/d]*d*Sqrt[(b*(c + d*x))/(d*(a + b*x))]*Sqrt[e + f*x])

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Maple [A]  time = 0.075, size = 192, normalized size = 1.4 \begin{align*} 2\,{\frac{ \left ( ad-bc \right ) \sqrt{bx+a}\sqrt{dx+c}\sqrt{fx+e}}{bd \left ( bdf{x}^{3}+adf{x}^{2}+bcf{x}^{2}+bde{x}^{2}+acfx+adex+bcex+ace \right ) }{\it EllipticF} \left ( \sqrt{{\frac{d \left ( bx+a \right ) }{ad-bc}}},\sqrt{{\frac{ \left ( ad-bc \right ) f}{d \left ( af-be \right ) }}} \right ) \sqrt{-{\frac{ \left ( dx+c \right ) b}{ad-bc}}}\sqrt{-{\frac{ \left ( fx+e \right ) b}{af-be}}}\sqrt{{\frac{d \left ( bx+a \right ) }{ad-bc}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b x + a} \sqrt{d x + c} \sqrt{f x + e}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(b*x + a)*sqrt(d*x + c)*sqrt(f*x + e)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a + b x} \sqrt{c + d x} \sqrt{e + f x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(sqrt(a + b*x)*sqrt(c + d*x)*sqrt(e + f*x)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b x + a} \sqrt{d x + c} \sqrt{f x + e}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(b*x + a)*sqrt(d*x + c)*sqrt(f*x + e)), x)