Optimal. Leaf size=86 \[ \frac{\sqrt{x} \sqrt{a-c} \sqrt{-\frac{c+2 x}{a-c}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+2 x}}{\sqrt{a-c}}\right )|1-\frac{c}{a}\right )}{\sqrt{2} \sqrt{-\frac{x}{a}} \sqrt{c+2 x}} \]
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Rubi [A] time = 0.166935, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{\sqrt{x} \sqrt{a-c} \sqrt{-\frac{c+2 x}{a-c}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+2 x}}{\sqrt{a-c}}\right )|1-\frac{c}{a}\right )}{\sqrt{2} \sqrt{-\frac{x}{a}} \sqrt{c+2 x}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[x]/(Sqrt[a + 2*x]*Sqrt[c + 2*x]),x]
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Rubi in Sympy [A] time = 15.2992, size = 70, normalized size = 0.81 \[ \frac{\sqrt{2} \sqrt{x} \sqrt{\frac{- c - 2 x}{a - c}} \sqrt{a - c} E\left (\operatorname{asin}{\left (\frac{\sqrt{a + 2 x}}{\sqrt{a - c}} \right )}\middle | \frac{a - c}{a}\right )}{2 \sqrt{- \frac{x}{a}} \sqrt{c + 2 x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(1/2)/(a+2*x)**(1/2)/(c+2*x)**(1/2),x)
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Mathematica [C] time = 0.181684, size = 120, normalized size = 1.4 \[ -\frac{i c \sqrt{\frac{2 x}{a}+1} \sqrt{\frac{2 x}{c}+1} \left (E\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{1}{a}} \sqrt{x}\right )|\frac{a}{c}\right )-F\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{1}{a}} \sqrt{x}\right )|\frac{a}{c}\right )\right )}{\sqrt{2} \sqrt{\frac{1}{a}} \sqrt{a+2 x} \sqrt{c+2 x}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[x]/(Sqrt[a + 2*x]*Sqrt[c + 2*x]),x]
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Maple [B] time = 0.086, size = 155, normalized size = 1.8 \[ -{\frac{\sqrt{2}a}{2\,ac+4\,ax+4\,cx+8\,{x}^{2}} \left ( c{\it EllipticF} \left ( \sqrt{{\frac{a+2\,x}{a}}},\sqrt{{\frac{a}{a-c}}} \right ) +{\it EllipticE} \left ( \sqrt{{\frac{a+2\,x}{a}}},\sqrt{{\frac{a}{a-c}}} \right ) a-{\it EllipticE} \left ( \sqrt{{\frac{a+2\,x}{a}}},\sqrt{{\frac{a}{a-c}}} \right ) c \right ) \sqrt{-{\frac{x}{a}}}\sqrt{-{\frac{c+2\,x}{a-c}}}\sqrt{{\frac{a+2\,x}{a}}}\sqrt{c+2\,x}\sqrt{a+2\,x}{\frac{1}{\sqrt{x}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(1/2)/(a+2*x)^(1/2)/(c+2*x)^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x}}{\sqrt{a + 2 \, x} \sqrt{c + 2 \, x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x)/(sqrt(a + 2*x)*sqrt(c + 2*x)),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{x}}{\sqrt{a + 2 \, x} \sqrt{c + 2 \, x}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x)/(sqrt(a + 2*x)*sqrt(c + 2*x)),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x}}{\sqrt{a + 2 x} \sqrt{c + 2 x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(1/2)/(a+2*x)**(1/2)/(c+2*x)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x}}{\sqrt{a + 2 \, x} \sqrt{c + 2 \, x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x)/(sqrt(a + 2*x)*sqrt(c + 2*x)),x, algorithm="giac")
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