3.13.51 \(\int \frac {-\sqrt {a b}+x}{\sqrt {x (a+x) (b+x)} (\sqrt {a b}+x)} \, dx\)

Optimal. Leaf size=91 \[ \frac {2 \tan ^{-1}\left (\frac {(b+x) \left (\frac {x}{\sqrt {2 \sqrt {a b}-a-b}}+\frac {a}{\sqrt {2 \sqrt {a b}-a-b}}\right )}{\sqrt {x^2 (a+b)+a b x+x^3}}\right )}{\sqrt {2 \sqrt {a b}-a-b}} \]

________________________________________________________________________________________

Rubi [C]  time = 0.87, antiderivative size = 146, normalized size of antiderivative = 1.60, number of steps used = 8, number of rules used = 7, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6718, 1607, 169, 538, 537, 117, 116} \begin {gather*} \frac {2 \sqrt {-a} \sqrt {x} \sqrt {\frac {x}{a}+1} \sqrt {\frac {x}{b}+1} F\left (\sin ^{-1}\left (\frac {\sqrt {x}}{\sqrt {-a}}\right )|\frac {a}{b}\right )}{\sqrt {x (a+x) (b+x)}}-\frac {4 \sqrt {-a} \sqrt {x} \sqrt {\frac {x}{a}+1} \sqrt {\frac {x}{b}+1} \Pi \left (\frac {a}{\sqrt {a b}};\sin ^{-1}\left (\frac {\sqrt {x}}{\sqrt {-a}}\right )|\frac {a}{b}\right )}{\sqrt {x (a+x) (b+x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-Sqrt[a*b] + x)/(Sqrt[x*(a + x)*(b + x)]*(Sqrt[a*b] + x)),x]

[Out]

(2*Sqrt[-a]*Sqrt[x]*Sqrt[1 + x/a]*Sqrt[1 + x/b]*EllipticF[ArcSin[Sqrt[x]/Sqrt[-a]], a/b])/Sqrt[x*(a + x)*(b +
x)] - (4*Sqrt[-a]*Sqrt[x]*Sqrt[1 + x/a]*Sqrt[1 + x/b]*EllipticPi[a/Sqrt[a*b], ArcSin[Sqrt[x]/Sqrt[-a]], a/b])/
Sqrt[x*(a + x)*(b + x)]

Rule 116

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[(2*Rt[-(b/d), 2]*E
llipticF[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[-(b/d), 2])], (c*f)/(d*e)])/(b*Sqrt[e]), x] /; FreeQ[{b, c, d, e, f}, x]
&& GtQ[c, 0] && GtQ[e, 0] && (PosQ[-(b/d)] || NegQ[-(b/f)])

Rule 117

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Dist[(Sqrt[1 + (d*x)/c]
*Sqrt[1 + (f*x)/e])/(Sqrt[c + d*x]*Sqrt[e + f*x]), Int[1/(Sqrt[b*x]*Sqrt[1 + (d*x)/c]*Sqrt[1 + (f*x)/e]), x],
x] /; FreeQ[{b, c, d, e, f}, x] &&  !(GtQ[c, 0] && GtQ[e, 0])

Rule 169

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Sym
bol] :> Dist[-2, Subst[Int[1/(Simp[b*c - a*d - b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + (f*x^2)/d, x]]*Sqrt[Simp[(d
*g - c*h)/d + (h*x^2)/d, x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] &&  !SimplerQ[e
+ f*x, c + d*x] &&  !SimplerQ[g + h*x, c + d*x]

Rule 537

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1*Ellipt
icPi[(b*c)/(a*d), ArcSin[Rt[-(d/c), 2]*x], (c*f)/(d*e)])/(a*Sqrt[c]*Sqrt[e]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b,
c, d, e, f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && SimplerSqrtQ[-(f/e), -(d/c)
])

Rule 538

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[1 +
(d*x^2)/c]/Sqrt[c + d*x^2], Int[1/((a + b*x^2)*Sqrt[1 + (d*x^2)/c]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c,
d, e, f}, x] &&  !GtQ[c, 0]

Rule 1607

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.)*((g_.) + (h_.)*(x_)
)^(q_.), x_Symbol] :> Dist[PolynomialRemainder[Px, a + b*x, x], Int[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*(g + h
*x)^q, x], x] + Int[PolynomialQuotient[Px, a + b*x, x]*(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*(g + h*x)^q,
x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n, p, q}, x] && PolyQ[Px, x] && EqQ[m, -1]

Rule 6718

Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.)*(z_)^(q_.))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a*v^m*w^n*z^q)^FracP
art[p])/(v^(m*FracPart[p])*w^(n*FracPart[p])*z^(q*FracPart[p])), Int[u*v^(m*p)*w^(n*p)*z^(p*q), x], x] /; Free
Q[{a, m, n, p, q}, x] &&  !IntegerQ[p] &&  !FreeQ[v, x] &&  !FreeQ[w, x] &&  !FreeQ[z, x]

Rubi steps

\begin {align*} \int \frac {-\sqrt {a b}+x}{\sqrt {x (a+x) (b+x)} \left (\sqrt {a b}+x\right )} \, dx &=\frac {\left (\sqrt {x} \sqrt {a+x} \sqrt {b+x}\right ) \int \frac {-\sqrt {a b}+x}{\sqrt {x} \sqrt {a+x} \sqrt {b+x} \left (\sqrt {a b}+x\right )} \, dx}{\sqrt {x (a+x) (b+x)}}\\ &=\frac {\left (\sqrt {x} \sqrt {a+x} \sqrt {b+x}\right ) \int \frac {1}{\sqrt {x} \sqrt {a+x} \sqrt {b+x}} \, dx}{\sqrt {x (a+x) (b+x)}}-\frac {\left (2 \sqrt {a b} \sqrt {x} \sqrt {a+x} \sqrt {b+x}\right ) \int \frac {1}{\sqrt {x} \sqrt {a+x} \sqrt {b+x} \left (\sqrt {a b}+x\right )} \, dx}{\sqrt {x (a+x) (b+x)}}\\ &=\frac {\left (4 \sqrt {a b} \sqrt {x} \sqrt {a+x} \sqrt {b+x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt {a b}-x^2\right ) \sqrt {a+x^2} \sqrt {b+x^2}} \, dx,x,\sqrt {x}\right )}{\sqrt {x (a+x) (b+x)}}+\frac {\left (\sqrt {x} \sqrt {1+\frac {x}{a}} \sqrt {1+\frac {x}{b}}\right ) \int \frac {1}{\sqrt {x} \sqrt {1+\frac {x}{a}} \sqrt {1+\frac {x}{b}}} \, dx}{\sqrt {x (a+x) (b+x)}}\\ &=\frac {2 \sqrt {-a} \sqrt {x} \sqrt {1+\frac {x}{a}} \sqrt {1+\frac {x}{b}} F\left (\sin ^{-1}\left (\frac {\sqrt {x}}{\sqrt {-a}}\right )|\frac {a}{b}\right )}{\sqrt {x (a+x) (b+x)}}+\frac {\left (4 \sqrt {a b} \sqrt {x} \sqrt {b+x} \sqrt {1+\frac {x}{a}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt {a b}-x^2\right ) \sqrt {b+x^2} \sqrt {1+\frac {x^2}{a}}} \, dx,x,\sqrt {x}\right )}{\sqrt {x (a+x) (b+x)}}\\ &=\frac {2 \sqrt {-a} \sqrt {x} \sqrt {1+\frac {x}{a}} \sqrt {1+\frac {x}{b}} F\left (\sin ^{-1}\left (\frac {\sqrt {x}}{\sqrt {-a}}\right )|\frac {a}{b}\right )}{\sqrt {x (a+x) (b+x)}}+\frac {\left (4 \sqrt {a b} \sqrt {x} \sqrt {1+\frac {x}{a}} \sqrt {1+\frac {x}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt {a b}-x^2\right ) \sqrt {1+\frac {x^2}{a}} \sqrt {1+\frac {x^2}{b}}} \, dx,x,\sqrt {x}\right )}{\sqrt {x (a+x) (b+x)}}\\ &=\frac {2 \sqrt {-a} \sqrt {x} \sqrt {1+\frac {x}{a}} \sqrt {1+\frac {x}{b}} F\left (\sin ^{-1}\left (\frac {\sqrt {x}}{\sqrt {-a}}\right )|\frac {a}{b}\right )}{\sqrt {x (a+x) (b+x)}}-\frac {4 \sqrt {-a} \sqrt {x} \sqrt {1+\frac {x}{a}} \sqrt {1+\frac {x}{b}} \Pi \left (\frac {a}{\sqrt {a b}};\sin ^{-1}\left (\frac {\sqrt {x}}{\sqrt {-a}}\right )|\frac {a}{b}\right )}{\sqrt {x (a+x) (b+x)}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.58, size = 101, normalized size = 1.11 \begin {gather*} -\frac {2 a x^{3/2} \sqrt {\frac {a}{x}+1} \sqrt {\frac {b}{x}+1} \left (F\left (\sin ^{-1}\left (\frac {\sqrt {-a}}{\sqrt {x}}\right )|\frac {b}{a}\right )-2 \Pi \left (\frac {b}{\sqrt {a b}};\sin ^{-1}\left (\frac {\sqrt {-a}}{\sqrt {x}}\right )|\frac {b}{a}\right )\right )}{(-a)^{3/2} \sqrt {x (a+x) (b+x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-Sqrt[a*b] + x)/(Sqrt[x*(a + x)*(b + x)]*(Sqrt[a*b] + x)),x]

[Out]

(-2*a*Sqrt[1 + a/x]*Sqrt[1 + b/x]*x^(3/2)*(EllipticF[ArcSin[Sqrt[-a]/Sqrt[x]], b/a] - 2*EllipticPi[b/Sqrt[a*b]
, ArcSin[Sqrt[-a]/Sqrt[x]], b/a]))/((-a)^(3/2)*Sqrt[x*(a + x)*(b + x)])

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.78, size = 91, normalized size = 1.00 \begin {gather*} \frac {2 \tan ^{-1}\left (\frac {(b+x) \left (\frac {a}{\sqrt {-a-b+2 \sqrt {a b}}}+\frac {x}{\sqrt {-a-b+2 \sqrt {a b}}}\right )}{\sqrt {a b x+(a+b) x^2+x^3}}\right )}{\sqrt {-a-b+2 \sqrt {a b}}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-Sqrt[a*b] + x)/(Sqrt[x*(a + x)*(b + x)]*(Sqrt[a*b] + x)),x]

[Out]

(2*ArcTan[((b + x)*(a/Sqrt[-a - b + 2*Sqrt[a*b]] + x/Sqrt[-a - b + 2*Sqrt[a*b]]))/Sqrt[a*b*x + (a + b)*x^2 + x
^3]])/Sqrt[-a - b + 2*Sqrt[a*b]]

________________________________________________________________________________________

fricas [A]  time = 0.70, size = 791, normalized size = 8.69 \begin {gather*} \left [-\frac {\sqrt {a + b + 2 \, \sqrt {a b}} \log \left (-\frac {a^{5} b^{4} - a^{4} b^{5} + {\left (a - b\right )} x^{8} + 8 \, {\left (a^{2} - b^{2}\right )} x^{7} + 4 \, {\left (2 \, a^{3} + 17 \, a^{2} b - 17 \, a b^{2} - 2 \, b^{3}\right )} x^{6} + 120 \, {\left (a^{3} b - a b^{3}\right )} x^{5} + 2 \, {\left (24 \, a^{4} b + 91 \, a^{3} b^{2} - 91 \, a^{2} b^{3} - 24 \, a b^{4}\right )} x^{4} + 120 \, {\left (a^{4} b^{2} - a^{2} b^{4}\right )} x^{3} + 4 \, {\left (2 \, a^{5} b^{2} + 17 \, a^{4} b^{3} - 17 \, a^{3} b^{4} - 2 \, a^{2} b^{5}\right )} x^{2} + 4 \, {\left (a^{4} b^{3} + a^{3} b^{4} + {\left (a + b\right )} x^{6} + 2 \, {\left (a^{2} + 8 \, a b + b^{2}\right )} x^{5} + 31 \, {\left (a^{2} b + a b^{2}\right )} x^{4} + 4 \, {\left (3 \, a^{3} b + 16 \, a^{2} b^{2} + 3 \, a b^{3}\right )} x^{3} + 31 \, {\left (a^{3} b^{2} + a^{2} b^{3}\right )} x^{2} + 2 \, {\left (a^{4} b^{2} + 8 \, a^{3} b^{3} + a^{2} b^{4}\right )} x - 2 \, {\left (a^{3} b^{3} + 5 \, {\left (a + b\right )} x^{5} + x^{6} + {\left (4 \, a^{2} + 23 \, a b + 4 \, b^{2}\right )} x^{4} + 22 \, {\left (a^{2} b + a b^{2}\right )} x^{3} + {\left (4 \, a^{3} b + 23 \, a^{2} b^{2} + 4 \, a b^{3}\right )} x^{2} + 5 \, {\left (a^{3} b^{2} + a^{2} b^{3}\right )} x\right )} \sqrt {a b}\right )} \sqrt {a b x + {\left (a + b\right )} x^{2} + x^{3}} \sqrt {a + b + 2 \, \sqrt {a b}} + 8 \, {\left (a^{5} b^{3} - a^{3} b^{5}\right )} x - 16 \, {\left ({\left (a - b\right )} x^{7} + 3 \, {\left (a^{2} - b^{2}\right )} x^{6} + {\left (2 \, a^{3} + 9 \, a^{2} b - 9 \, a b^{2} - 2 \, b^{3}\right )} x^{5} + 10 \, {\left (a^{3} b - a b^{3}\right )} x^{4} + {\left (2 \, a^{4} b + 9 \, a^{3} b^{2} - 9 \, a^{2} b^{3} - 2 \, a b^{4}\right )} x^{3} + 3 \, {\left (a^{4} b^{2} - a^{2} b^{4}\right )} x^{2} + {\left (a^{4} b^{3} - a^{3} b^{4}\right )} x\right )} \sqrt {a b}}{a^{4} b^{4} - 4 \, a^{3} b^{3} x^{2} + 6 \, a^{2} b^{2} x^{4} - 4 \, a b x^{6} + x^{8}}\right )}{2 \, {\left (a - b\right )}}, \frac {\sqrt {-a - b - 2 \, \sqrt {a b}} \arctan \left (\frac {\sqrt {a b x + {\left (a + b\right )} x^{2} + x^{3}} {\left (a b + 2 \, {\left (a + b\right )} x + x^{2} - 2 \, \sqrt {a b} x\right )} \sqrt {-a - b - 2 \, \sqrt {a b}}}{2 \, {\left ({\left (a - b\right )} x^{3} + {\left (a^{2} - b^{2}\right )} x^{2} + {\left (a^{2} b - a b^{2}\right )} x\right )}}\right )}{a - b}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-(a*b)^(1/2)+x)/(x*(a+x)*(b+x))^(1/2)/((a*b)^(1/2)+x),x, algorithm="fricas")

[Out]

[-1/2*sqrt(a + b + 2*sqrt(a*b))*log(-(a^5*b^4 - a^4*b^5 + (a - b)*x^8 + 8*(a^2 - b^2)*x^7 + 4*(2*a^3 + 17*a^2*
b - 17*a*b^2 - 2*b^3)*x^6 + 120*(a^3*b - a*b^3)*x^5 + 2*(24*a^4*b + 91*a^3*b^2 - 91*a^2*b^3 - 24*a*b^4)*x^4 +
120*(a^4*b^2 - a^2*b^4)*x^3 + 4*(2*a^5*b^2 + 17*a^4*b^3 - 17*a^3*b^4 - 2*a^2*b^5)*x^2 + 4*(a^4*b^3 + a^3*b^4 +
(a + b)*x^6 + 2*(a^2 + 8*a*b + b^2)*x^5 + 31*(a^2*b + a*b^2)*x^4 + 4*(3*a^3*b + 16*a^2*b^2 + 3*a*b^3)*x^3 + 3
1*(a^3*b^2 + a^2*b^3)*x^2 + 2*(a^4*b^2 + 8*a^3*b^3 + a^2*b^4)*x - 2*(a^3*b^3 + 5*(a + b)*x^5 + x^6 + (4*a^2 +
23*a*b + 4*b^2)*x^4 + 22*(a^2*b + a*b^2)*x^3 + (4*a^3*b + 23*a^2*b^2 + 4*a*b^3)*x^2 + 5*(a^3*b^2 + a^2*b^3)*x)
*sqrt(a*b))*sqrt(a*b*x + (a + b)*x^2 + x^3)*sqrt(a + b + 2*sqrt(a*b)) + 8*(a^5*b^3 - a^3*b^5)*x - 16*((a - b)*
x^7 + 3*(a^2 - b^2)*x^6 + (2*a^3 + 9*a^2*b - 9*a*b^2 - 2*b^3)*x^5 + 10*(a^3*b - a*b^3)*x^4 + (2*a^4*b + 9*a^3*
b^2 - 9*a^2*b^3 - 2*a*b^4)*x^3 + 3*(a^4*b^2 - a^2*b^4)*x^2 + (a^4*b^3 - a^3*b^4)*x)*sqrt(a*b))/(a^4*b^4 - 4*a^
3*b^3*x^2 + 6*a^2*b^2*x^4 - 4*a*b*x^6 + x^8))/(a - b), sqrt(-a - b - 2*sqrt(a*b))*arctan(1/2*sqrt(a*b*x + (a +
b)*x^2 + x^3)*(a*b + 2*(a + b)*x + x^2 - 2*sqrt(a*b)*x)*sqrt(-a - b - 2*sqrt(a*b))/((a - b)*x^3 + (a^2 - b^2)
*x^2 + (a^2*b - a*b^2)*x))/(a - b)]

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x - \sqrt {a b}}{\sqrt {{\left (a + x\right )} {\left (b + x\right )} x} {\left (x + \sqrt {a b}\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-(a*b)^(1/2)+x)/(x*(a+x)*(b+x))^(1/2)/((a*b)^(1/2)+x),x, algorithm="giac")

[Out]

integrate((x - sqrt(a*b))/(sqrt((a + x)*(b + x)*x)*(x + sqrt(a*b))), x)

________________________________________________________________________________________

maple [C]  time = 0.20, size = 564, normalized size = 6.20

method	result	size
default	\(\frac {2 b \sqrt {\frac {b +x}{b}}\, \sqrt {\frac {a +x}{a -b}}\, \sqrt {-\frac {x}{b}}\, \EllipticF \left (\sqrt {\frac {b +x}{b}}, \sqrt {-\frac {b}{a -b}}\right )}{\sqrt {a b x +a \,x^{2}+b \,x^{2}+x^{3}}}-2 \sqrt {a b}\, \left (\frac {b \sqrt {1+\frac {x}{b}}\, \sqrt {\frac {a}{a -b}+\frac {x}{a -b}}\, \sqrt {-\frac {x}{b}}\, \EllipticPi \left (\sqrt {\frac {b +x}{b}}, -\frac {b}{-b -\sqrt {a b}}, \sqrt {-\frac {b}{a -b}}\right )}{\sqrt {a b x +a \,x^{2}+b \,x^{2}+x^{3}}\, \left (-b -\sqrt {a b}\right )}+\frac {b \sqrt {1+\frac {x}{b}}\, \sqrt {\frac {a}{a -b}+\frac {x}{a -b}}\, \sqrt {-\frac {x}{b}}\, \EllipticPi \left (\sqrt {\frac {b +x}{b}}, -\frac {b}{-b +\sqrt {a b}}, \sqrt {-\frac {b}{a -b}}\right )}{\sqrt {a b x +a \,x^{2}+b \,x^{2}+x^{3}}\, \left (-b +\sqrt {a b}\right )}-\frac {a \,b^{2} \sqrt {1+\frac {x}{b}}\, \sqrt {\frac {a}{a -b}+\frac {x}{a -b}}\, \sqrt {-\frac {x}{b}}\, \EllipticPi \left (\sqrt {\frac {b +x}{b}}, -\frac {b}{-b -\sqrt {a b}}, \sqrt {-\frac {b}{a -b}}\right )}{\sqrt {a b}\, \sqrt {a^{2} b^{2} x +a^{2} b \,x^{2}+a \,b^{2} x^{2}+a b \,x^{3}}\, \left (-b -\sqrt {a b}\right )}+\frac {a \,b^{2} \sqrt {1+\frac {x}{b}}\, \sqrt {\frac {a}{a -b}+\frac {x}{a -b}}\, \sqrt {-\frac {x}{b}}\, \EllipticPi \left (\sqrt {\frac {b +x}{b}}, -\frac {b}{-b +\sqrt {a b}}, \sqrt {-\frac {b}{a -b}}\right )}{\sqrt {a b}\, \sqrt {a^{2} b^{2} x +a^{2} b \,x^{2}+a \,b^{2} x^{2}+a b \,x^{3}}\, \left (-b +\sqrt {a b}\right )}\right )\)	\(564\)
elliptic	\(\frac {2 b \sqrt {1+\frac {x}{b}}\, \sqrt {\frac {a}{a -b}+\frac {x}{a -b}}\, \sqrt {-\frac {x}{b}}\, \EllipticF \left (\sqrt {\frac {b +x}{b}}, \sqrt {-\frac {b}{a -b}}\right )}{\sqrt {a b x +a \,x^{2}+b \,x^{2}+x^{3}}}+\frac {2 a \,b^{2} \sqrt {1+\frac {x}{b}}\, \sqrt {\frac {a}{a -b}+\frac {x}{a -b}}\, \sqrt {-\frac {x}{b}}\, \EllipticPi \left (\sqrt {\frac {b +x}{b}}, -\frac {b}{-b -\sqrt {a b}}, \sqrt {-\frac {b}{a -b}}\right )}{\sqrt {a b}\, \sqrt {a b x +a \,x^{2}+b \,x^{2}+x^{3}}\, \left (-b -\sqrt {a b}\right )}-\frac {2 a \,b^{2} \sqrt {1+\frac {x}{b}}\, \sqrt {\frac {a}{a -b}+\frac {x}{a -b}}\, \sqrt {-\frac {x}{b}}\, \EllipticPi \left (\sqrt {\frac {b +x}{b}}, -\frac {b}{-b +\sqrt {a b}}, \sqrt {-\frac {b}{a -b}}\right )}{\sqrt {a b}\, \sqrt {a b x +a \,x^{2}+b \,x^{2}+x^{3}}\, \left (-b +\sqrt {a b}\right )}-\frac {2 a \,b^{2} \sqrt {1+\frac {x}{b}}\, \sqrt {\frac {a}{a -b}+\frac {x}{a -b}}\, \sqrt {-\frac {x}{b}}\, \EllipticPi \left (\sqrt {\frac {b +x}{b}}, -\frac {b}{-b -\sqrt {a b}}, \sqrt {-\frac {b}{a -b}}\right )}{\sqrt {a^{2} b^{2} x +a^{2} b \,x^{2}+a \,b^{2} x^{2}+a b \,x^{3}}\, \left (-b -\sqrt {a b}\right )}-\frac {2 a \,b^{2} \sqrt {1+\frac {x}{b}}\, \sqrt {\frac {a}{a -b}+\frac {x}{a -b}}\, \sqrt {-\frac {x}{b}}\, \EllipticPi \left (\sqrt {\frac {b +x}{b}}, -\frac {b}{-b +\sqrt {a b}}, \sqrt {-\frac {b}{a -b}}\right )}{\sqrt {a^{2} b^{2} x +a^{2} b \,x^{2}+a \,b^{2} x^{2}+a b \,x^{3}}\, \left (-b +\sqrt {a b}\right )}\)	\(573\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-(a*b)^(1/2)+x)/(x*(a+x)*(b+x))^(1/2)/((a*b)^(1/2)+x),x,method=_RETURNVERBOSE)

[Out]

2*b*((b+x)/b)^(1/2)*((a+x)/(a-b))^(1/2)*(-x/b)^(1/2)/(a*b*x+a*x^2+b*x^2+x^3)^(1/2)*EllipticF(((b+x)/b)^(1/2),(
-b/(a-b))^(1/2))-2*(a*b)^(1/2)*(b*(1+x/b)^(1/2)*(a/(a-b)+1/(a-b)*x)^(1/2)*(-x/b)^(1/2)/(a*b*x+a*x^2+b*x^2+x^3)
^(1/2)/(-b-(a*b)^(1/2))*EllipticPi(((b+x)/b)^(1/2),-b/(-b-(a*b)^(1/2)),(-b/(a-b))^(1/2))+b*(1+x/b)^(1/2)*(a/(a
-b)+1/(a-b)*x)^(1/2)*(-x/b)^(1/2)/(a*b*x+a*x^2+b*x^2+x^3)^(1/2)/(-b+(a*b)^(1/2))*EllipticPi(((b+x)/b)^(1/2),-b
/(-b+(a*b)^(1/2)),(-b/(a-b))^(1/2))-a*b^2/(a*b)^(1/2)*(1+x/b)^(1/2)*(a/(a-b)+1/(a-b)*x)^(1/2)*(-x/b)^(1/2)/(a^
2*b^2*x+a^2*b*x^2+a*b^2*x^2+a*b*x^3)^(1/2)/(-b-(a*b)^(1/2))*EllipticPi(((b+x)/b)^(1/2),-b/(-b-(a*b)^(1/2)),(-b
/(a-b))^(1/2))+a*b^2/(a*b)^(1/2)*(1+x/b)^(1/2)*(a/(a-b)+1/(a-b)*x)^(1/2)*(-x/b)^(1/2)/(a^2*b^2*x+a^2*b*x^2+a*b
^2*x^2+a*b*x^3)^(1/2)/(-b+(a*b)^(1/2))*EllipticPi(((b+x)/b)^(1/2),-b/(-b+(a*b)^(1/2)),(-b/(a-b))^(1/2)))

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x - \sqrt {a b}}{\sqrt {{\left (a + x\right )} {\left (b + x\right )} x} {\left (x + \sqrt {a b}\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-(a*b)^(1/2)+x)/(x*(a+x)*(b+x))^(1/2)/((a*b)^(1/2)+x),x, algorithm="maxima")

[Out]

integrate((x - sqrt(a*b))/(sqrt((a + x)*(b + x)*x)*(x + sqrt(a*b))), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x-\sqrt {a\,b}}{\left (x+\sqrt {a\,b}\right )\,\sqrt {x\,\left (a+x\right )\,\left (b+x\right )}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-(a*b)**(1/2)+x)/(x*(a+x)*(b+x))**(1/2)/((a*b)**(1/2)+x),x)

[Out]

Timed out

________________________________________________________________________________________