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]

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

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Rubi [A]  time = 0.04, antiderivative size = 86, normalized size of antiderivative = 1.00, 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 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])

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]

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*}

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Mathematica [C]  time = 0.17, size = 120, normalized size = 1.40 \[ -\frac {i c \sqrt {\frac {2 x}{a}+1} \sqrt {\frac {2 x}{c}+1} \left (E\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {1}{a}} \sqrt {x}\right )|\frac {a}{c}\right )-\operatorname {EllipticF}\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {1}{a}} \sqrt {x}\right ),\frac {a}{c}\right )\right )}{\sqrt {2} \sqrt {\frac {1}{a}} \sqrt {a+2 x} \sqrt {c+2 x}} \]

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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fricas [F]  time = 1.02, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a + 2 \, x} \sqrt {c + 2 \, x} \sqrt {x}}{a c + 2 \, {\left (a + c\right )} x + 4 \, x^{2}}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {x}}{\sqrt {a + 2 \, x} \sqrt {c + 2 \, x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(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)

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maple [B]  time = 0.06, size = 155, normalized size = 1.80 \[ -\frac {\left (a \EllipticE \left (\sqrt {\frac {a +2 x}{a}}, \sqrt {\frac {a}{a -c}}\right )-c \EllipticE \left (\sqrt {\frac {a +2 x}{a}}, \sqrt {\frac {a}{a -c}}\right )+c \EllipticF \left (\sqrt {\frac {a +2 x}{a}}, \sqrt {\frac {a}{a -c}}\right )\right ) \sqrt {2}\, \sqrt {-\frac {x}{a}}\, \sqrt {-\frac {c +2 x}{a -c}}\, \sqrt {\frac {a +2 x}{a}}\, \sqrt {c +2 x}\, \sqrt {a +2 x}\, a}{2 \left (a c +2 a x +2 c x +4 x^{2}\right ) \sqrt {x}} \]

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)*(-1/a*x)^(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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {x}}{\sqrt {a + 2 \, x} \sqrt {c + 2 \, x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {x}}{\sqrt {a+2\,x}\,\sqrt {c+2\,x}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1/2)/((a + 2*x)^(1/2)*(c + 2*x)^(1/2)),x)

[Out]

int(x^(1/2)/((a + 2*x)^(1/2)*(c + 2*x)^(1/2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {x}}{\sqrt {a + 2 x} \sqrt {c + 2 x}}\, dx \]

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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