3.855 \(\int \frac{\sqrt{1+c x}}{\sqrt{b x} \sqrt{1-c x}} \, dx\)

Optimal. Leaf size=33 \[ \frac{2 E\left (\left .\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{b x}}{\sqrt{b}}\right )\right |-1\right )}{\sqrt{b} \sqrt{c}} \]

[Out]

(2*EllipticE[ArcSin[(Sqrt[c]*Sqrt[b*x])/Sqrt[b]], -1])/(Sqrt[b]*Sqrt[c])

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Rubi [A]  time = 0.0578785, antiderivative size = 33, 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 \[ \frac{2 E\left (\left .\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{b x}}{\sqrt{b}}\right )\right |-1\right )}{\sqrt{b} \sqrt{c}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[1 + c*x]/(Sqrt[b*x]*Sqrt[1 - c*x]),x]

[Out]

(2*EllipticE[ArcSin[(Sqrt[c]*Sqrt[b*x])/Sqrt[b]], -1])/(Sqrt[b]*Sqrt[c])

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Rubi in Sympy [A]  time = 5.63096, size = 32, normalized size = 0.97 \[ \frac{2 E\left (\operatorname{asin}{\left (\frac{\sqrt{c} \sqrt{b x}}{\sqrt{b}} \right )}\middle | -1\right )}{\sqrt{b} \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x+1)**(1/2)/(b*x)**(1/2)/(-c*x+1)**(1/2),x)

[Out]

2*elliptic_e(asin(sqrt(c)*sqrt(b*x)/sqrt(b)), -1)/(sqrt(b)*sqrt(c))

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Mathematica [B]  time = 0.40957, size = 119, normalized size = 3.61 \[ -\frac{2 \sqrt{-\frac{1}{c}} (c x-1) \left (\sqrt{-\frac{1}{c}} \sqrt{1-\frac{1}{c x}} (c x+1)-\sqrt{x} \sqrt{\frac{1}{c x}+1} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{-\frac{1}{c}}}{\sqrt{x}}\right )\right |-1\right )\right )}{\sqrt{b x} \sqrt{1-\frac{1}{c x}} \sqrt{1-c^2 x^2}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[Sqrt[1 + c*x]/(Sqrt[b*x]*Sqrt[1 - c*x]),x]

[Out]

(-2*Sqrt[-c^(-1)]*(-1 + c*x)*(Sqrt[-c^(-1)]*Sqrt[1 - 1/(c*x)]*(1 + c*x) - Sqrt[1
+ 1/(c*x)]*Sqrt[x]*EllipticE[ArcSin[Sqrt[-c^(-1)]/Sqrt[x]], -1]))/(Sqrt[1 - 1/(
c*x)]*Sqrt[b*x]*Sqrt[1 - c^2*x^2])

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Maple [B]  time = 0.023, size = 49, normalized size = 1.5 \[ 2\,{\frac{\sqrt{2}\sqrt{-cx} \left ({\it EllipticF} \left ( \sqrt{cx+1},1/2\,\sqrt{2} \right ) -{\it EllipticE} \left ( \sqrt{cx+1},1/2\,\sqrt{2} \right ) \right ) }{c\sqrt{bx}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x+1)^(1/2)/(b*x)^(1/2)/(-c*x+1)^(1/2),x)

[Out]

2*2^(1/2)*(-c*x)^(1/2)*(EllipticF((c*x+1)^(1/2),1/2*2^(1/2))-EllipticE((c*x+1)^(
1/2),1/2*2^(1/2)))/c/(b*x)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x + 1}}{\sqrt{b x} \sqrt{-c x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x + 1)/(sqrt(b*x)*sqrt(-c*x + 1)),x, algorithm="maxima")

[Out]

integrate(sqrt(c*x + 1)/(sqrt(b*x)*sqrt(-c*x + 1)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{c x + 1}}{\sqrt{b x} \sqrt{-c x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(sqrt(c*x + 1)/(sqrt(b*x)*sqrt(-c*x + 1)), x)

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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)/(-c*x+1)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(c*x + 1)/(sqrt(b*x)*sqrt(-c*x + 1)), x)