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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]