Optimal. Leaf size=42 \[ -\frac{2 E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{b x}}{\sqrt{-b}}\right )|-\frac{c}{d}\right )}{\sqrt{-b} \sqrt{d}} \]
[Out]
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Rubi [A] time = 0.0713888, antiderivative size = 42, 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 \[ -\frac{2 E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{b x}}{\sqrt{-b}}\right )|-\frac{c}{d}\right )}{\sqrt{-b} \sqrt{d}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[1 - c*x]/(Sqrt[b*x]*Sqrt[1 + d*x]),x]
[Out]
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Rubi in Sympy [A] time = 6.75966, size = 39, normalized size = 0.93 \[ - \frac{2 E\left (\operatorname{asin}{\left (\frac{\sqrt{d} \sqrt{b x}}{\sqrt{- b}} \right )}\middle | - \frac{c}{d}\right )}{\sqrt{d} \sqrt{- b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-c*x+1)**(1/2)/(b*x)**(1/2)/(d*x+1)**(1/2),x)
[Out]
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Mathematica [B] time = 0.766444, size = 112, normalized size = 2.67 \[ \frac{-2 x^{3/2} \sqrt{1-\frac{1}{c x}} \sqrt{\frac{1}{d x}+1} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{1}{c}}}{\sqrt{x}}\right )|-\frac{c}{d}\right )-\frac{2 \sqrt{\frac{1}{c}} (c x-1) (d x+1)}{d}}{\sqrt{\frac{1}{c}} \sqrt{b x} \sqrt{1-c x} \sqrt{d x+1}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[1 - c*x]/(Sqrt[b*x]*Sqrt[1 + d*x]),x]
[Out]
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Maple [A] time = 0.022, size = 67, normalized size = 1.6 \[ -2\,{\frac{ \left ( c+d \right ) \sqrt{-dx}\sqrt{-cx+1}}{ \left ( cx-1 \right ) \sqrt{bx}{d}^{2}}{\it EllipticE} \left ( \sqrt{dx+1},\sqrt{{\frac{c}{c+d}}} \right ) \sqrt{-{\frac{ \left ( cx-1 \right ) d}{c+d}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-c*x+1)^(1/2)/(b*x)^(1/2)/(d*x+1)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{-c x + 1}}{\sqrt{b x} \sqrt{d x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-c*x + 1)/(sqrt(b*x)*sqrt(d*x + 1)),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{-c x + 1}}{\sqrt{b x} \sqrt{d x + 1}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-c*x + 1)/(sqrt(b*x)*sqrt(d*x + 1)),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-c*x+1)**(1/2)/(b*x)**(1/2)/(d*x+1)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{-c x + 1}}{\sqrt{b x} \sqrt{d x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-c*x + 1)/(sqrt(b*x)*sqrt(d*x + 1)),x, algorithm="giac")
[Out]