3.2884 \(\int \frac{\sqrt{x}}{\sqrt{a+2 x} \sqrt{c+2 x}} \, dx\)

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]/Sqrt[a - c]], 1 - c/a])/(Sqrt[2
]*Sqrt[-(x/a)]*Sqrt[c + 2*x])

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Rubi [A]  time = 0.0377975, 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, Rules used = {114, 12, 113} \[ \frac{\sqrt{x} \sqrt{a-c} \sqrt{-\frac{c+2 x}{a-c}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+2 x}}{\sqrt{a-c}}\right )|1-\frac{c}{a}\right )}{\sqrt{2} \sqrt{-\frac{x}{a}} \sqrt{c+2 x}} \]

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]/Sqrt[a - c]], 1 - c/a])/(Sqrt[2
]*Sqrt[-(x/a)]*Sqrt[c + 2*x])

Rule 114

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Dist[(Sqrt[e + f*
x]*Sqrt[(b*(c + d*x))/(b*c - a*d)])/(Sqrt[c + d*x]*Sqrt[(b*(e + f*x))/(b*e - a*f)]), Int[Sqrt[(b*e)/(b*e - a*f
) + (b*f*x)/(b*e - a*f)]/(Sqrt[a + b*x]*Sqrt[(b*c)/(b*c - a*d) + (b*d*x)/(b*c - a*d)]), x], x] /; FreeQ[{a, b,
c, d, e, f}, x] &&  !(GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0]) &&  !LtQ[-((b*c - a*d)/d), 0]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 113

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[(2*Rt[-((b*e
- a*f)/d), 2]*EllipticE[ArcSin[Sqrt[a + b*x]/Rt[-((b*c - a*d)/d), 2]], (f*(b*c - a*d))/(d*(b*e - a*f))])/b, x
] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] &&  !LtQ[-((b*c - a*d)/d),
0] &&  !(SimplerQ[c + d*x, a + b*x] && GtQ[-(d/(b*c - a*d)), 0] && GtQ[d/(d*e - c*f), 0] &&  !LtQ[(b*c - a*d)
/b, 0])

Rubi steps

\begin{align*} \int \frac{\sqrt{x}}{\sqrt{a+2 x} \sqrt{c+2 x}} \, dx &=\frac{\left (\sqrt{x} \sqrt{\frac{c+2 x}{-2 a+2 c}}\right ) \int \frac{\sqrt{2} \sqrt{-\frac{x}{a}}}{\sqrt{a+2 x} \sqrt{\frac{2 c}{-2 a+2 c}+\frac{4 x}{-2 a+2 c}}} \, dx}{\sqrt{-\frac{x}{a}} \sqrt{c+2 x}}\\ &=\frac{\left (\sqrt{2} \sqrt{x} \sqrt{\frac{c+2 x}{-2 a+2 c}}\right ) \int \frac{\sqrt{-\frac{x}{a}}}{\sqrt{a+2 x} \sqrt{\frac{2 c}{-2 a+2 c}+\frac{4 x}{-2 a+2 c}}} \, dx}{\sqrt{-\frac{x}{a}} \sqrt{c+2 x}}\\ &=\frac{\sqrt{a-c} \sqrt{x} \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}}\\ \end{align*}

Mathematica [C]  time = 0.125349, size = 120, normalized size = 1.4 \[ -\frac{i c \sqrt{\frac{2 x}{a}+1} \sqrt{\frac{2 x}{c}+1} \left (E\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{1}{a}} \sqrt{x}\right )|\frac{a}{c}\right )-\text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{1}{a}} \sqrt{x}\right ),\frac{a}{c}\right )\right )}{\sqrt{2} \sqrt{\frac{1}{a}} \sqrt{a+2 x} \sqrt{c+2 x}} \]

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] - Ellipti
cF[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 = 0.048, size = 155, normalized size = 1.8 \begin{align*} -{\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}}}} \end{align*}

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. \begin{align*} \int \frac{\sqrt{x}}{\sqrt{a + 2 \, x} \sqrt{c + 2 \, x}}\,{d x} \end{align*}

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, 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. \begin{align*}{\rm integral}\left (\frac{\sqrt{a + 2 \, x} \sqrt{c + 2 \, x} \sqrt{x}}{a c + 2 \,{\left (a + c\right )} x + 4 \, x^{2}}, x\right ) \end{align*}

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, algorithm="fricas")

[Out]

integral(sqrt(a + 2*x)*sqrt(c + 2*x)*sqrt(x)/(a*c + 2*(a + c)*x + 4*x^2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{x}}{\sqrt{a + 2 x} \sqrt{c + 2 x}}\, dx \end{align*}

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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{x}}{\sqrt{a + 2 \, x} \sqrt{c + 2 \, x}}\,{d x} \end{align*}

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, algorithm="giac")

[Out]

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