3.29.84 \(\int \frac {\sqrt {x}}{\sqrt {a+2 x} \sqrt {c+2 x}} \, dx\) [2884]

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

[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 = 0.03, 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 = {115, 12, 114} \begin {gather*} \frac {\sqrt {x} \sqrt {a-c} \sqrt {-\frac {c+2 x}{a-c}} E\left (\text {ArcSin}\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 {gather*}

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 114

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[(2/b)*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)))], 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 115

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] Result contains complex when optimal does not.
time = 1.37, size = 120, normalized size = 1.40 \begin {gather*} -\frac {i c \sqrt {1+\frac {2 x}{a}} \sqrt {1+\frac {2 x}{c}} \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}} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. \(156\) vs. \(2(73)=146\).
time = 0.10, size = 157, normalized size = 1.83

method	result	size
default	\(-\frac {\left (a \EllipticF \left (\sqrt {\frac {c +2 x}{c}}, \sqrt {-\frac {c}{a -c}}\right )-\EllipticE \left (\sqrt {\frac {c +2 x}{c}}, \sqrt {-\frac {c}{a -c}}\right ) a +\EllipticE \left (\sqrt {\frac {c +2 x}{c}}, \sqrt {-\frac {c}{a -c}}\right ) c \right ) \sqrt {2}\, \sqrt {-\frac {x}{c}}\, \sqrt {\frac {a +2 x}{a -c}}\, \sqrt {\frac {c +2 x}{c}}\, c \sqrt {a +2 x}\, \sqrt {c +2 x}}{2 \sqrt {x}\, \left (a c +2 a x +2 c x +4 x^{2}\right )}\)	\(157\)
elliptic	\(\frac {\sqrt {\left (a +2 x \right ) \left (c +2 x \right ) x}\, c \sqrt {2}\, \sqrt {\frac {x +\frac {c}{2}}{c}}\, \sqrt {\frac {x +\frac {a}{2}}{-\frac {c}{2}+\frac {a}{2}}}\, \sqrt {-\frac {2 x}{c}}\, \left (\left (-\frac {c}{2}+\frac {a}{2}\right ) \EllipticE \left (\sqrt {2}\, \sqrt {\frac {x +\frac {c}{2}}{c}}, \frac {\sqrt {-\frac {2 c}{-\frac {c}{2}+\frac {a}{2}}}}{2}\right )-\frac {a \EllipticF \left (\sqrt {2}\, \sqrt {\frac {x +\frac {c}{2}}{c}}, \frac {\sqrt {-\frac {2 c}{-\frac {c}{2}+\frac {a}{2}}}}{2}\right )}{2}\right )}{\sqrt {a +2 x}\, \sqrt {c +2 x}\, \sqrt {x}\, \sqrt {a c x +2 a \,x^{2}+2 c \,x^{2}+4 x^{3}}}\)	\(173\)

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,method=_RETURNVERBOSE)

[Out]

-1/2*(a*EllipticF(((c+2*x)/c)^(1/2),(-c/(a-c))^(1/2))-EllipticE(((c+2*x)/c)^(1/2),(-c/(a-c))^(1/2))*a+Elliptic
E(((c+2*x)/c)^(1/2),(-c/(a-c))^(1/2))*c)*2^(1/2)*(-x/c)^(1/2)*((a+2*x)/(a-c))^(1/2)*((c+2*x)/c)^(1/2)*c/x^(1/2
)*(a+2*x)^(1/2)*(c+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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.22, size = 141, normalized size = 1.64 \begin {gather*} -\frac {1}{6} \, {\left (a + c\right )} {\rm weierstrassPInverse}\left (\frac {1}{3} \, a^{2} - \frac {1}{3} \, a c + \frac {1}{3} \, c^{2}, -\frac {1}{27} \, a^{3} + \frac {1}{18} \, a^{2} c + \frac {1}{18} \, a c^{2} - \frac {1}{27} \, c^{3}, \frac {1}{6} \, a + \frac {1}{6} \, c + x\right ) - {\rm weierstrassZeta}\left (\frac {1}{3} \, a^{2} - \frac {1}{3} \, a c + \frac {1}{3} \, c^{2}, -\frac {1}{27} \, a^{3} + \frac {1}{18} \, a^{2} c + \frac {1}{18} \, a c^{2} - \frac {1}{27} \, c^{3}, {\rm weierstrassPInverse}\left (\frac {1}{3} \, a^{2} - \frac {1}{3} \, a c + \frac {1}{3} \, c^{2}, -\frac {1}{27} \, a^{3} + \frac {1}{18} \, a^{2} c + \frac {1}{18} \, a c^{2} - \frac {1}{27} \, c^{3}, \frac {1}{6} \, a + \frac {1}{6} \, c + x\right )\right ) \end {gather*}

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]

-1/6*(a + c)*weierstrassPInverse(1/3*a^2 - 1/3*a*c + 1/3*c^2, -1/27*a^3 + 1/18*a^2*c + 1/18*a*c^2 - 1/27*c^3,
1/6*a + 1/6*c + x) - weierstrassZeta(1/3*a^2 - 1/3*a*c + 1/3*c^2, -1/27*a^3 + 1/18*a^2*c + 1/18*a*c^2 - 1/27*c
^3, weierstrassPInverse(1/3*a^2 - 1/3*a*c + 1/3*c^2, -1/27*a^3 + 1/18*a^2*c + 1/18*a*c^2 - 1/27*c^3, 1/6*a + 1
/6*c + x))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x}}{\sqrt {a + 2 x} \sqrt {c + 2 x}}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {x}}{\sqrt {a+2\,x}\,\sqrt {c+2\,x}} \,d x \end {gather*}

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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