Optimal. Leaf size=38 \[ \frac{2 F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{b x}}{\sqrt{b}}\right )|-\frac{d}{c}\right )}{\sqrt{b} \sqrt{c}} \]
[Out]
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Rubi [A] time = 0.0620521, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.037 \[ \frac{2 F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{b x}}{\sqrt{b}}\right )|-\frac{d}{c}\right )}{\sqrt{b} \sqrt{c}} \]
Antiderivative was successfully verified.
[In] Int[1/(Sqrt[b*x]*Sqrt[1 - c*x]*Sqrt[1 + d*x]),x]
[Out]
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Rubi in Sympy [A] time = 5.97137, size = 34, normalized size = 0.89 \[ \frac{2 F\left (\operatorname{asin}{\left (\frac{\sqrt{c} \sqrt{b x}}{\sqrt{b}} \right )}\middle | - \frac{d}{c}\right )}{\sqrt{b} \sqrt{c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x)**(1/2)/(-c*x+1)**(1/2)/(d*x+1)**(1/2),x)
[Out]
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Mathematica [B] time = 0.197368, size = 89, normalized size = 2.34 \[ -\frac{2 x^{3/2} \sqrt{\frac{c-\frac{1}{x}}{c}} \sqrt{\frac{d+\frac{1}{x}}{d}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{1}{c}}}{\sqrt{x}}\right )|-\frac{c}{d}\right )}{\sqrt{\frac{1}{c}} \sqrt{b x} \sqrt{1-c x} \sqrt{d x+1}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(Sqrt[b*x]*Sqrt[1 - c*x]*Sqrt[1 + d*x]),x]
[Out]
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Maple [B] time = 0.088, size = 64, normalized size = 1.7 \[ -2\,{\frac{\sqrt{-cx+1}\sqrt{-dx}}{\sqrt{bx} \left ( cx-1 \right ) d}\sqrt{-{\frac{ \left ( cx-1 \right ) d}{c+d}}}{\it EllipticF} \left ( \sqrt{dx+1},\sqrt{{\frac{c}{c+d}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b*x)^(1/2)/(-c*x+1)^(1/2)/(d*x+1)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b x} \sqrt{-c x + 1} \sqrt{d x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(b*x)*sqrt(-c*x + 1)*sqrt(d*x + 1)),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{b x} \sqrt{-c x + 1} \sqrt{d x + 1}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(b*x)*sqrt(-c*x + 1)*sqrt(d*x + 1)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b x} \sqrt{- c x + 1} \sqrt{d x + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*x)**(1/2)/(-c*x+1)**(1/2)/(d*x+1)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b x} \sqrt{-c x + 1} \sqrt{d x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(b*x)*sqrt(-c*x + 1)*sqrt(d*x + 1)),x, algorithm="giac")
[Out]