3.2627 \(\int \frac{1}{\sqrt{-c+d x} \sqrt{c+d x} \sqrt{e+f x}} \, dx\)

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

[Out]

(2*Sqrt[c]*Sqrt[(c - d*x)/c]*Sqrt[(d*(e + f*x))/(d*e - c*f)]*EllipticF[ArcSin[Sq
rt[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 = 0.394566, 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 \[ \frac{2 \sqrt{c} \sqrt{\frac{c-d x}{c}} \sqrt{\frac{d (e+f x)}{d e-c f}} F\left (\sin ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{2} \sqrt{c}}\right )|-\frac{2 c f}{d e-c f}\right )}{d \sqrt{d x-c} \sqrt{e+f x}} \]

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[Sq
rt[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 in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(d*x-c)**(1/2)/(d*x+c)**(1/2)/(f*x+e)**(1/2),x)

[Out]

Timed out

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Mathematica [A]  time = 0.422492, size = 123, normalized size = 1.22 \[ \frac{\sqrt{2} (c-d x) \sqrt{\frac{c+d x}{d x-c}} \sqrt{\frac{d (e+f x)}{f (d x-c)}} F\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{-c}}{\sqrt{d x-c}}\right )|\frac{1}{2} \left (\frac{d e}{c f}+1\right )\right )}{\sqrt{-c} d \sqrt{c+d x} \sqrt{e+f x}} \]

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

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Maple [A]  time = 0.119, size = 174, normalized size = 1.7 \[ -2\,{\frac{\sqrt{fx+e}\sqrt{dx+c}\sqrt{dx-c} \left ( cf-de \right ) }{df \left ({d}^{2}f{x}^{3}+{d}^{2}e{x}^{2}-{c}^{2}fx-{c}^{2}e \right ) }\sqrt{-{\frac{ \left ( fx+e \right ) d}{cf-de}}}\sqrt{-{\frac{ \left ( dx-c \right ) f}{cf+de}}}\sqrt{{\frac{ \left ( dx+c \right ) f}{cf-de}}}{\it EllipticF} \left ( \sqrt{-{\frac{ \left ( fx+e \right ) d}{cf-de}}},\sqrt{-{\frac{cf-de}{cf+de}}} \right ) } \]

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. \[ \int \frac{1}{\sqrt{d x + c} \sqrt{d x - c} \sqrt{f x + e}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(d*x + c)*sqrt(d*x - c)*sqrt(f*x + e)),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. \[{\rm integral}\left (\frac{1}{\sqrt{d x + c} \sqrt{d x - c} \sqrt{f x + e}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(d*x + c)*sqrt(d*x - c)*sqrt(f*x + e)),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{- c + d x} \sqrt{c + d x} \sqrt{e + f x}}\, dx \]

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/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{d x + c} \sqrt{d x - c} \sqrt{f x + e}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(d*x + c)*sqrt(d*x - c)*sqrt(f*x + e)),x, algorithm="giac")

[Out]

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