3.2644 \(\int \frac{1}{\sqrt{-c+d x} \sqrt{c+d x} \sqrt{e+f x}} \, dx\)

Optimal. Leaf size=101 \[ \frac{2 \sqrt{c} \sqrt{\frac{c-d x}{c}} \sqrt{\frac{d (e+f x)}{d e-c f}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{2} \sqrt{c}}\right ),-\frac{2 c f}{d e-c f}\right )}{d \sqrt{d x-c} \sqrt{e+f x}} \]

[Out]

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

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Rubi [A]  time = 0.0695599, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {121, 119} \[ \frac{2 \sqrt{c} \sqrt{\frac{c-d x}{c}} \sqrt{\frac{d (e+f x)}{d e-c f}} F\left (\sin ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{2} \sqrt{c}}\right )|-\frac{2 c f}{d e-c f}\right )}{d \sqrt{d x-c} \sqrt{e+f x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[c]*Sqrt[(c - d*x)/c]*Sqrt[(d*(e + f*x))/(d*e - c*f)]*EllipticF[ArcSin[Sqrt[c + d*x]/(Sqrt[2]*Sqrt[c])]
, (-2*c*f)/(d*e - c*f)])/(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 119

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*c - a*d)/b, 0] && GtQ[(b*e - a*f)/b, 0] &
& PosQ[-(b/d)] &&  !(SimplerQ[c + d*x, a + b*x] && GtQ[(d*e - c*f)/d, 0] && GtQ[-(d/b), 0]) &&  !(SimplerQ[c +
d*x, a + b*x] && GtQ[(-(b*e) + a*f)/f, 0] && GtQ[-(f/b), 0]) &&  !(SimplerQ[e + f*x, a + b*x] && GtQ[(-(d*e)
+ c*f)/f, 0] && GtQ[(-(b*e) + a*f)/f, 0] && (PosQ[-(f/d)] || PosQ[-(f/b)]))

Rubi steps

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

Mathematica [A]  time = 0.299787, size = 123, normalized size = 1.22 \[ \frac{\sqrt{2} (c-d x) \sqrt{\frac{c+d x}{d x-c}} \sqrt{\frac{d (e+f x)}{f (d x-c)}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{-c}}{\sqrt{d x-c}}\right ),\frac{1}{2} \left (\frac{d e}{c f}+1\right )\right )}{\sqrt{-c} d \sqrt{c+d x} \sqrt{e+f x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[2]*(c - d*x)*Sqrt[(c + d*x)/(-c + d*x)]*Sqrt[(d*(e + f*x))/(f*(-c + d*x))]*EllipticF[ArcSin[(Sqrt[2]*Sqr
t[-c])/Sqrt[-c + d*x]], (1 + (d*e)/(c*f))/2])/(Sqrt[-c]*d*Sqrt[c + d*x]*Sqrt[e + f*x])

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Maple [A]  time = 0.079, size = 174, normalized size = 1.7 \begin{align*} -2\,{\frac{\sqrt{fx+e}\sqrt{dx+c}\sqrt{dx-c} \left ( cf-de \right ) }{df \left ({d}^{2}f{x}^{3}+{d}^{2}e{x}^{2}-{c}^{2}fx-{c}^{2}e \right ) }\sqrt{-{\frac{ \left ( fx+e \right ) d}{cf-de}}}\sqrt{-{\frac{ \left ( dx-c \right ) f}{cf+de}}}\sqrt{{\frac{ \left ( dx+c \right ) f}{cf-de}}}{\it EllipticF} \left ( \sqrt{-{\frac{ \left ( fx+e \right ) d}{cf-de}}},\sqrt{-{\frac{cf-de}{cf+de}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{d x + c} \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/(d*x-c)^(1/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(d*x + c)*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{d x + c} \sqrt{d x - c} \sqrt{f x + e}}{d^{2} f x^{3} + d^{2} e x^{2} - c^{2} f x - c^{2} e}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{- c + d 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/(d*x-c)**(1/2)/(d*x+c)**(1/2)/(f*x+e)**(1/2),x)

[Out]

Integral(1/(sqrt(-c + d*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{d x + c} \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/(d*x-c)^(1/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2),x, algorithm="giac")

[Out]

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