Optimal. Leaf size=38 \[ \frac{2 E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{b x}}{\sqrt{b}}\right )|-\frac{c}{d}\right )}{\sqrt{b} \sqrt{d}} \]
[Out]
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Rubi [A] time = 0.0609315, 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 \[ \frac{2 E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{b x}}{\sqrt{b}}\right )|-\frac{c}{d}\right )}{\sqrt{b} \sqrt{d}} \]
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 = 5.70482, size = 34, normalized size = 0.89 \[ \frac{2 E\left (\operatorname{asin}{\left (\frac{\sqrt{d} \sqrt{b x}}{\sqrt{b}} \right )}\middle | - \frac{c}{d}\right )}{\sqrt{b} \sqrt{d}} \]
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 = 0.572853, size = 102, normalized size = 2.68 \[ \frac{2 \sqrt{1-d x} \left (\frac{\sqrt{x} \sqrt{\frac{1}{c x}+1} E\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{1}{c}}}{\sqrt{x}}\right )|-\frac{c}{d}\right )}{\sqrt{-\frac{1}{c}} \sqrt{1-\frac{1}{d x}}}-c x-1\right )}{d \sqrt{b x} \sqrt{c x+1}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[1 + c*x]/(Sqrt[b*x]*Sqrt[1 - d*x]),x]
[Out]
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Maple [B] time = 0.066, size = 129, normalized size = 3.4 \[ -2\,{\frac{\sqrt{-cx}\sqrt{-dx+1}}{ \left ( dx-1 \right ) \sqrt{bx}cd} \left ({\it EllipticF} \left ( \sqrt{cx+1},\sqrt{{\frac{d}{c+d}}} \right ) c+{\it EllipticF} \left ( \sqrt{cx+1},\sqrt{{\frac{d}{c+d}}} \right ) d-{\it EllipticE} \left ( \sqrt{cx+1},\sqrt{{\frac{d}{c+d}}} \right ) c-{\it EllipticE} \left ( \sqrt{cx+1},\sqrt{{\frac{d}{c+d}}} \right ) d \right ) \sqrt{-{\frac{ \left ( dx-1 \right ) c}{c+d}}}} \]
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]