3.2867 \(\int \frac{\sqrt{x}}{\sqrt{a+2 x} \sqrt{c+2 x}} \, dx\)

Optimal. Leaf size=86 \[ \frac{\sqrt{x} \sqrt{a-c} \sqrt{-\frac{c+2 x}{a-c}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+2 x}}{\sqrt{a-c}}\right )|1-\frac{c}{a}\right )}{\sqrt{2} \sqrt{-\frac{x}{a}} \sqrt{c+2 x}} \]

[Out]

(Sqrt[a - c]*Sqrt[x]*Sqrt[-((c + 2*x)/(a - c))]*EllipticE[ArcSin[Sqrt[a + 2*x]/S
qrt[a - c]], 1 - c/a])/(Sqrt[2]*Sqrt[-(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]

[Out]

(Sqrt[a - c]*Sqrt[x]*Sqrt[-((c + 2*x)/(a - c))]*EllipticE[ArcSin[Sqrt[a + 2*x]/S
qrt[a - c]], 1 - c/a])/(Sqrt[2]*Sqrt[-(x/a)]*Sqrt[c + 2*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)

[Out]

sqrt(2)*sqrt(x)*sqrt((-c - 2*x)/(a - c))*sqrt(a - c)*elliptic_e(asin(sqrt(a + 2*
x)/sqrt(a - c)), (a - c)/a)/(2*sqrt(-x/a)*sqrt(c + 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]

[Out]

((-I)*c*Sqrt[1 + (2*x)/a]*Sqrt[1 + (2*x)/c]*(EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[a^
(-1)]*Sqrt[x]], a/c] - EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[a^(-1)]*Sqrt[x]], a/c]))
/(Sqrt[2]*Sqrt[a^(-1)]*Sqrt[a + 2*x]*Sqrt[c + 2*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)

[Out]

-1/2*(c*EllipticF(((a+2*x)/a)^(1/2),(a/(a-c))^(1/2))+EllipticE(((a+2*x)/a)^(1/2)
,(a/(a-c))^(1/2))*a-EllipticE(((a+2*x)/a)^(1/2),(a/(a-c))^(1/2))*c)*2^(1/2)*(-x/
a)^(1/2)*(-(c+2*x)/(a-c))^(1/2)*((a+2*x)/a)^(1/2)*a*(c+2*x)^(1/2)*(a+2*x)^(1/2)/
x^(1/2)/(a*c+2*a*x+2*c*x+4*x^2)

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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")

[Out]

integrate(sqrt(x)/(sqrt(a + 2*x)*sqrt(c + 2*x)), x)

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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")

[Out]

integral(sqrt(x)/(sqrt(a + 2*x)*sqrt(c + 2*x)), x)

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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]

Integral(sqrt(x)/(sqrt(a + 2*x)*sqrt(c + 2*x)), x)

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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")

[Out]

integrate(sqrt(x)/(sqrt(a + 2*x)*sqrt(c + 2*x)), x)