\(\int \frac {a+b+\sqrt {b} \sqrt {a+b}-(b+\sqrt {b} \sqrt {a+b}) x^2}{(1-x^2) \sqrt {a+b-2 b x^2+b x^4}} \, dx\) [189]

Optimal result

Integrand size = 64, antiderivative size = 397 \[ \int \frac {a+b+\sqrt {b} \sqrt {a+b}-\left (b+\sqrt {b} \sqrt {a+b}\right ) x^2}{\left (1-x^2\right ) \sqrt {a+b-2 b x^2+b x^4}} \, dx=\frac {1}{2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} x}{\sqrt {a+b-2 b x^2+b x^4}}\right )+\frac {\sqrt [4]{b} \sqrt [4]{a+b} \left (\sqrt {b}+\sqrt {a+b}\right ) \left (a+b+\sqrt {b} \sqrt {a+b}\right ) \left (1+\frac {\sqrt {b} x^2}{\sqrt {a+b}}\right ) \sqrt {\frac {a+b-2 b x^2+b x^4}{(a+b) \left (1+\frac {\sqrt {b} x^2}{\sqrt {a+b}}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b}}\right ),\frac {1}{2} \left (1+\frac {\sqrt {b}}{\sqrt {a+b}}\right )\right )}{\left (a+2 \left (b+\sqrt {b} \sqrt {a+b}\right )\right ) \sqrt {a+b-2 b x^2+b x^4}}+\frac {a^2 \left (\sqrt {a+b}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b-2 b x^2+b x^4}{\left (\sqrt {a+b}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {b}+\sqrt {a+b}\right )^2}{4 \sqrt {b} \sqrt {a+b}},2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b}}\right ),\frac {1}{2} \left (1+\frac {\sqrt {b}}{\sqrt {a+b}}\right )\right )}{4 \sqrt [4]{b} \sqrt [4]{a+b} \left (a+2 \left (b+\sqrt {b} \sqrt {a+b}\right )\right ) \sqrt {a+b-2 b x^2+b x^4}} \] Output:

1/2*a^(1/2)*arctanh(a^(1/2)*x/(b*x^4-2*b*x^2+a+b)^(1/2))+b^(1/4)*(a+b)^(1/ 
4)*(b^(1/2)+(a+b)^(1/2))*(a+b+b^(1/2)*(a+b)^(1/2))*(1+b^(1/2)*x^2/(a+b)^(1 
/2))*((b*x^4-2*b*x^2+a+b)/(a+b)/(1+b^(1/2)*x^2/(a+b)^(1/2))^2)^(1/2)*Inver 
seJacobiAM(2*arctan(b^(1/4)*x/(a+b)^(1/4)),1/2*(2+2*b^(1/2)/(a+b)^(1/2))^( 
1/2))/(a+2*b+2*b^(1/2)*(a+b)^(1/2))/(b*x^4-2*b*x^2+a+b)^(1/2)+1/4*a^2*((a+ 
b)^(1/2)+b^(1/2)*x^2)*((b*x^4-2*b*x^2+a+b)/((a+b)^(1/2)+b^(1/2)*x^2)^2)^(1 
/2)*EllipticPi(sin(2*arctan(b^(1/4)*x/(a+b)^(1/4))),1/4*(b^(1/2)+(a+b)^(1/ 
2))^2/b^(1/2)/(a+b)^(1/2),1/2*(2+2*b^(1/2)/(a+b)^(1/2))^(1/2))/b^(1/4)/(a+ 
b)^(1/4)/(a+2*b+2*b^(1/2)*(a+b)^(1/2))/(b*x^4-2*b*x^2+a+b)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 18.72 (sec) , antiderivative size = 3113, normalized size of antiderivative = 7.84 \[ \int \frac {a+b+\sqrt {b} \sqrt {a+b}-\left (b+\sqrt {b} \sqrt {a+b}\right ) x^2}{\left (1-x^2\right ) \sqrt {a+b-2 b x^2+b x^4}} \, dx=\text {Result too large to show} \] Input:

Integrate[(a + b + Sqrt[b]*Sqrt[a + b] - (b + Sqrt[b]*Sqrt[a + b])*x^2)/(( 
1 - x^2)*Sqrt[a + b - 2*b*x^2 + b*x^4]),x]
 

Output:

((-I)*b*Sqrt[1 - (Sqrt[b]*x^2)/((-I)*Sqrt[a] + Sqrt[b])]*Sqrt[1 - (Sqrt[b] 
*x^2)/(I*Sqrt[a] + Sqrt[b])]*EllipticF[I*ArcSinh[Sqrt[-(Sqrt[b]/((-I)*Sqrt 
[a] + Sqrt[b]))]*x], ((-I)*Sqrt[a] + Sqrt[b])/(I*Sqrt[a] + Sqrt[b])])/(Sqr 
t[-(Sqrt[b]/((-I)*Sqrt[a] + Sqrt[b]))]*Sqrt[a + b*(-1 + x^2)^2]) - (I*Sqrt 
[b]*Sqrt[a + b]*Sqrt[1 - (Sqrt[b]*x^2)/((-I)*Sqrt[a] + Sqrt[b])]*Sqrt[1 - 
(Sqrt[b]*x^2)/(I*Sqrt[a] + Sqrt[b])]*EllipticF[I*ArcSinh[Sqrt[-(Sqrt[b]/(( 
-I)*Sqrt[a] + Sqrt[b]))]*x], ((-I)*Sqrt[a] + Sqrt[b])/(I*Sqrt[a] + Sqrt[b] 
)])/(Sqrt[-(Sqrt[b]/((-I)*Sqrt[a] + Sqrt[b]))]*Sqrt[a + b*(-1 + x^2)^2]) - 
 (a*(Sqrt[1 - (I*Sqrt[a])/Sqrt[b]] + Sqrt[1 + (I*Sqrt[a])/Sqrt[b]])*(-Sqrt 
[1 - (I*Sqrt[a])/Sqrt[b]] + x)^2*Sqrt[(Sqrt[((-I)*Sqrt[a] + Sqrt[b])/Sqrt[ 
b]]*(-Sqrt[1 + (I*Sqrt[a])/Sqrt[b]] + x))/((Sqrt[1 - (I*Sqrt[a])/Sqrt[b]] 
+ Sqrt[1 + (I*Sqrt[a])/Sqrt[b]])*(-Sqrt[1 - (I*Sqrt[a])/Sqrt[b]] + x))]*Sq 
rt[(Sqrt[((-I)*Sqrt[a] + Sqrt[b])/Sqrt[b]]*(Sqrt[1 + (I*Sqrt[a])/Sqrt[b]] 
+ x))/((Sqrt[1 - (I*Sqrt[a])/Sqrt[b]] - Sqrt[1 + (I*Sqrt[a])/Sqrt[b]])*(-S 
qrt[1 - (I*Sqrt[a])/Sqrt[b]] + x))]*Sqrt[((Sqrt[((-I)*Sqrt[a] + Sqrt[b])/S 
qrt[b]] - Sqrt[(I*Sqrt[a] + Sqrt[b])/Sqrt[b]])*(Sqrt[((-I)*Sqrt[a] + Sqrt[ 
b])/Sqrt[b]] + x))/((Sqrt[((-I)*Sqrt[a] + Sqrt[b])/Sqrt[b]] + Sqrt[(I*Sqrt 
[a] + Sqrt[b])/Sqrt[b]])*(Sqrt[((-I)*Sqrt[a] + Sqrt[b])/Sqrt[b]] - x))]*(( 
1 + Sqrt[1 - (I*Sqrt[a])/Sqrt[b]])*EllipticF[ArcSin[Sqrt[((Sqrt[((-I)*Sqrt 
[a] + Sqrt[b])/Sqrt[b]] - Sqrt[(I*Sqrt[a] + Sqrt[b])/Sqrt[b]])*(Sqrt[((...
 

Rubi [A] (warning: unable to verify)

Time = 1.08 (sec) , antiderivative size = 593, normalized size of antiderivative = 1.49, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.047, Rules used = {2224, 1416, 2222}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {-\left (x^2 \left (\sqrt {b} \sqrt {a+b}+b\right )\right )+\sqrt {b} \sqrt {a+b}+a+b}{\left (1-x^2\right ) \sqrt {a+b x^4-2 b x^2+b}} \, dx\)

\(\Big \downarrow \) 2224

\(\displaystyle \frac {2 \sqrt {b} \left (\sqrt {a+b}+\sqrt {b}\right ) \left (\sqrt {b} \sqrt {a+b}+a+b\right ) \int \frac {1}{\sqrt {b x^4-2 b x^2+a+b}}dx}{2 \left (\sqrt {b} \sqrt {a+b}+b\right )+a}+\frac {a \int \frac {\sqrt {b} \left (\sqrt {b}+\sqrt {a+b}\right ) x^2+a+b+\sqrt {b} \sqrt {a+b}}{\left (1-x^2\right ) \sqrt {b x^4-2 b x^2+a+b}}dx}{2 \left (\sqrt {b} \sqrt {a+b}+b\right )+a}\)

\(\Big \downarrow \) 1416

\(\displaystyle \frac {a \int \frac {\sqrt {b} \left (\sqrt {b}+\sqrt {a+b}\right ) x^2+a+b+\sqrt {b} \sqrt {a+b}}{\left (1-x^2\right ) \sqrt {b x^4-2 b x^2+a+b}}dx}{2 \left (\sqrt {b} \sqrt {a+b}+b\right )+a}+\frac {\sqrt [4]{b} \sqrt [4]{a+b} \left (\sqrt {a+b}+\sqrt {b}\right ) \left (\sqrt {b} \sqrt {a+b}+a+b\right ) \left (\frac {\sqrt {b} x^2}{\sqrt {a+b}}+1\right ) \sqrt {\frac {a+b x^4-2 b x^2+b}{(a+b) \left (\frac {\sqrt {b} x^2}{\sqrt {a+b}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b}}\right ),\frac {1}{2} \left (\frac {\sqrt {b}}{\sqrt {a+b}}+1\right )\right )}{\left (2 \left (\sqrt {b} \sqrt {a+b}+b\right )+a\right ) \sqrt {a+b x^4-2 b x^2+b}}\)

\(\Big \downarrow \) 2222

\(\displaystyle \frac {a \left (\frac {a \sqrt {\sqrt {b} \sqrt {a+b}+a+b} \left (\frac {\sqrt {b} x^2 \left (\sqrt {a+b}+\sqrt {b}\right )}{\sqrt {b} \sqrt {a+b}+a+b}+1\right ) \sqrt {\frac {a+b x^4-2 b x^2+b}{(a+b) \left (\frac {\sqrt {b} x^2 \left (\sqrt {a+b}+\sqrt {b}\right )}{\sqrt {b} \sqrt {a+b}+a+b}+1\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (a+2 \left (b+\sqrt {a+b} \sqrt {b}\right )\right )^2}{4 \sqrt {b} \left (\sqrt {b}+\sqrt {a+b}\right ) \left (a+b+\sqrt {b} \sqrt {a+b}\right )},2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {\sqrt {b}+\sqrt {a+b}} x}{\sqrt {a+b+\sqrt {b} \sqrt {a+b}}}\right ),\frac {1}{2} \left (\frac {\sqrt {b} \left (a+b+\sqrt {b} \sqrt {a+b}\right )}{(a+b) \left (\sqrt {b}+\sqrt {a+b}\right )}+1\right )\right )}{4 \sqrt [4]{b} \sqrt {\sqrt {a+b}+\sqrt {b}} \sqrt {a+b x^4-2 b x^2+b}}+\frac {\left (2 \left (\sqrt {b} \sqrt {a+b}+b\right )+a\right ) \text {arctanh}\left (\frac {\sqrt {a} x}{\sqrt {a+b x^4-2 b x^2+b}}\right )}{2 \sqrt {a}}\right )}{2 \left (\sqrt {b} \sqrt {a+b}+b\right )+a}+\frac {\sqrt [4]{b} \sqrt [4]{a+b} \left (\sqrt {a+b}+\sqrt {b}\right ) \left (\sqrt {b} \sqrt {a+b}+a+b\right ) \left (\frac {\sqrt {b} x^2}{\sqrt {a+b}}+1\right ) \sqrt {\frac {a+b x^4-2 b x^2+b}{(a+b) \left (\frac {\sqrt {b} x^2}{\sqrt {a+b}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b}}\right ),\frac {1}{2} \left (\frac {\sqrt {b}}{\sqrt {a+b}}+1\right )\right )}{\left (2 \left (\sqrt {b} \sqrt {a+b}+b\right )+a\right ) \sqrt {a+b x^4-2 b x^2+b}}\)

Input:

Int[(a + b + Sqrt[b]*Sqrt[a + b] - (b + Sqrt[b]*Sqrt[a + b])*x^2)/((1 - x^ 
2)*Sqrt[a + b - 2*b*x^2 + b*x^4]),x]
 

Output:

(b^(1/4)*(a + b)^(1/4)*(Sqrt[b] + Sqrt[a + b])*(a + b + Sqrt[b]*Sqrt[a + b 
])*(1 + (Sqrt[b]*x^2)/Sqrt[a + b])*Sqrt[(a + b - 2*b*x^2 + b*x^4)/((a + b) 
*(1 + (Sqrt[b]*x^2)/Sqrt[a + b])^2)]*EllipticF[2*ArcTan[(b^(1/4)*x)/(a + b 
)^(1/4)], (1 + Sqrt[b]/Sqrt[a + b])/2])/((a + 2*(b + Sqrt[b]*Sqrt[a + b])) 
*Sqrt[a + b - 2*b*x^2 + b*x^4]) + (a*(((a + 2*(b + Sqrt[b]*Sqrt[a + b]))*A 
rcTanh[(Sqrt[a]*x)/Sqrt[a + b - 2*b*x^2 + b*x^4]])/(2*Sqrt[a]) + (a*Sqrt[a 
 + b + Sqrt[b]*Sqrt[a + b]]*(1 + (Sqrt[b]*(Sqrt[b] + Sqrt[a + b])*x^2)/(a 
+ b + Sqrt[b]*Sqrt[a + b]))*Sqrt[(a + b - 2*b*x^2 + b*x^4)/((a + b)*(1 + ( 
Sqrt[b]*(Sqrt[b] + Sqrt[a + b])*x^2)/(a + b + Sqrt[b]*Sqrt[a + b]))^2)]*El 
lipticPi[(a + 2*(b + Sqrt[b]*Sqrt[a + b]))^2/(4*Sqrt[b]*(Sqrt[b] + Sqrt[a 
+ b])*(a + b + Sqrt[b]*Sqrt[a + b])), 2*ArcTan[(b^(1/4)*Sqrt[Sqrt[b] + Sqr 
t[a + b]]*x)/Sqrt[a + b + Sqrt[b]*Sqrt[a + b]]], (1 + (Sqrt[b]*(a + b + Sq 
rt[b]*Sqrt[a + b]))/((a + b)*(Sqrt[b] + Sqrt[a + b])))/2])/(4*b^(1/4)*Sqrt 
[Sqrt[b] + Sqrt[a + b]]*Sqrt[a + b - 2*b*x^2 + b*x^4])))/(a + 2*(b + Sqrt[ 
b]*Sqrt[a + b]))
 

Defintions of rubi rules used

rule 1416

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c 
/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ 
(2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) 
], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
 

rule 2222

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + 
 (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[B/A, 2]}, Simp[(-(B*d - A*e))*(A 
rcTanh[Rt[b - c*(d/e) - a*(e/d), 2]*(x/Sqrt[a + b*x^2 + c*x^4])]/(2*d*e*Rt[ 
b - c*(d/e) - a*(e/d), 2])), x] + Simp[(B*d + A*e)*(1 + q^2*x^2)*(Sqrt[(a + 
 b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/(4*d*e*q*Sqrt[a + b*x^2 + c*x^4]))*Ell 
ipticPi[-(e - d*q^2)^2/(4*d*e*q^2), 2*ArcTan[q*x], 1/2 - b/(4*a*q^2)], x]] 
/; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] && 
 EqQ[c*A^2 - a*B^2, 0] && PosQ[B/A] && NegQ[-b + c*(d/e) + a*(e/d)]
 

rule 2224

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + 
 (c_.)*(x_)^4]), x_Symbol] :> Simp[2*A*(B/(B*d + A*e))   Int[1/Sqrt[a + b*x 
^2 + c*x^4], x], x] - Simp[(B*d - A*e)/(B*d + A*e)   Int[(A - B*x^2)/((d + 
e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x] /; FreeQ[{a, b, c, d, e, A, B}, x] 
&& NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] && EqQ[c*A^2 - a*B^2, 0] && NegQ[B/A]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 11.76 (sec) , antiderivative size = 674, normalized size of antiderivative = 1.70

method	result	size
default	\(\frac {b \sqrt {1-\frac {\left (i \sqrt {b}\, \sqrt {a}+b \right ) x^{2}}{a +b}}\, \sqrt {1+\frac {\left (i \sqrt {b}\, \sqrt {a}-b \right ) x^{2}}{a +b}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}\, \sqrt {a}+b}{a +b}}, \sqrt {-1-\frac {2 \left (i \sqrt {b}\, \sqrt {a}-b \right )}{a +b}}\right )}{\sqrt {\frac {i \sqrt {b}\, \sqrt {a}+b}{a +b}}\, \sqrt {b \,x^{4}-2 b \,x^{2}+a +b}}+\frac {\sqrt {b}\, \sqrt {a +b}\, \sqrt {1-\frac {\left (i \sqrt {b}\, \sqrt {a}+b \right ) x^{2}}{a +b}}\, \sqrt {1+\frac {\left (i \sqrt {b}\, \sqrt {a}-b \right ) x^{2}}{a +b}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}\, \sqrt {a}+b}{a +b}}, \sqrt {-1-\frac {2 \left (i \sqrt {b}\, \sqrt {a}-b \right )}{a +b}}\right )}{\sqrt {\frac {i \sqrt {b}\, \sqrt {a}+b}{a +b}}\, \sqrt {b \,x^{4}-2 b \,x^{2}+a +b}}-\frac {a \left (-\frac {\operatorname {arctanh}\left (\frac {\sqrt {a}}{\sqrt {b \,x^{4}-2 b \,x^{2}+a +b}}\right )}{2 \sqrt {a}}-\frac {\sqrt {1-\frac {\left (i \sqrt {b}\, \sqrt {a}+b \right ) x^{2}}{a +b}}\, \sqrt {1+\frac {\left (i \sqrt {b}\, \sqrt {a}-b \right ) x^{2}}{a +b}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {b}\, \sqrt {a}+b}{a +b}}, \frac {a +b}{i \sqrt {b}\, \sqrt {a}+b}, \frac {\sqrt {-\frac {i \sqrt {b}\, \sqrt {a}-b}{a +b}}}{\sqrt {\frac {i \sqrt {b}\, \sqrt {a}+b}{a +b}}}\right )}{\sqrt {\frac {i \sqrt {b}\, \sqrt {a}+b}{a +b}}\, \sqrt {b \,x^{4}-2 b \,x^{2}+a +b}}\right )}{2}+\frac {a \left (-\frac {\operatorname {arctanh}\left (\frac {\sqrt {a}}{\sqrt {b \,x^{4}-2 b \,x^{2}+a +b}}\right )}{2 \sqrt {a}}+\frac {\sqrt {1-\frac {\left (i \sqrt {b}\, \sqrt {a}+b \right ) x^{2}}{a +b}}\, \sqrt {1+\frac {\left (i \sqrt {b}\, \sqrt {a}-b \right ) x^{2}}{a +b}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {b}\, \sqrt {a}+b}{a +b}}, \frac {a +b}{i \sqrt {b}\, \sqrt {a}+b}, \frac {\sqrt {-\frac {i \sqrt {b}\, \sqrt {a}-b}{a +b}}}{\sqrt {\frac {i \sqrt {b}\, \sqrt {a}+b}{a +b}}}\right )}{\sqrt {\frac {i \sqrt {b}\, \sqrt {a}+b}{a +b}}\, \sqrt {b \,x^{4}-2 b \,x^{2}+a +b}}\right )}{2}\)	\(674\)
elliptic	\(\frac {\sqrt {\left (b \,x^{4}-2 b \,x^{2}+a +b \right ) b \left (a +b \right )}\, \left (\sqrt {a +b}\, \sqrt {b}\, x^{2}+b \,x^{2}-\sqrt {b}\, \sqrt {a +b}-a -b \right ) \left (\frac {b \left (a +b \right ) \sqrt {1-\frac {\left (i b^{\frac {3}{2}} \sqrt {a}+b^{2}\right ) x^{2}}{b \left (a +b \right )}}\, \sqrt {1+\frac {\left (i b^{\frac {3}{2}} \sqrt {a}-b^{2}\right ) x^{2}}{b \left (a +b \right )}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i b^{\frac {3}{2}} \sqrt {a}+b^{2}}{b \left (a +b \right )}}, \sqrt {-1+\frac {\left (-2 b^{2} a -2 b^{3}\right ) \left (i b^{\frac {3}{2}} \sqrt {a}-b^{2}\right )}{b \left (a +b \right ) \left (b^{2} a +b^{3}\right )}}\right )}{\sqrt {\frac {i b^{\frac {3}{2}} \sqrt {a}+b^{2}}{b \left (a +b \right )}}\, \sqrt {a \,b^{2} x^{4}+b^{3} x^{4}-2 a \,b^{2} x^{2}-2 b^{3} x^{2}+b \,a^{2}+2 b^{2} a +b^{3}}}+\frac {b \sqrt {1-\frac {i x^{2} \sqrt {b}\, \sqrt {a}}{a +b}-\frac {b \,x^{2}}{a +b}}\, \sqrt {1+\frac {i x^{2} \sqrt {b}\, \sqrt {a}}{a +b}-\frac {b \,x^{2}}{a +b}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}\, \sqrt {a}+b}{a +b}}, \sqrt {-1-\frac {2 \left (i \sqrt {b}\, \sqrt {a}-b \right )}{a +b}}\right )}{\sqrt {\frac {i \sqrt {b}\, \sqrt {a}}{a +b}+\frac {b}{a +b}}\, \sqrt {b \,x^{4}-2 b \,x^{2}+a +b}}+\frac {a \sqrt {1-\frac {i x^{2} \sqrt {b}\, \sqrt {a}}{a +b}-\frac {b \,x^{2}}{a +b}}\, \sqrt {1+\frac {i x^{2} \sqrt {b}\, \sqrt {a}}{a +b}-\frac {b \,x^{2}}{a +b}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {b}\, \sqrt {a}+b}{a +b}}, \frac {a +b}{i \sqrt {b}\, \sqrt {a}+b}, \frac {\sqrt {-\frac {i \sqrt {b}\, \sqrt {a}-b}{a +b}}}{\sqrt {\frac {i \sqrt {b}\, \sqrt {a}+b}{a +b}}}\right )}{\sqrt {\frac {i \sqrt {b}\, \sqrt {a}}{a +b}+\frac {b}{a +b}}\, \sqrt {b \,x^{4}-2 b \,x^{2}+a +b}}\right )}{\sqrt {b \,x^{4}-2 b \,x^{2}+a +b}\, a b \,x^{2}+\sqrt {b \,x^{4}-2 b \,x^{2}+a +b}\, b^{2} x^{2}+\sqrt {\left (b \,x^{4}-2 b \,x^{2}+a +b \right ) b \left (a +b \right )}\, b \,x^{2}-\sqrt {b \,x^{4}-2 b \,x^{2}+a +b}\, a b -\sqrt {b \,x^{4}-2 b \,x^{2}+a +b}\, b^{2}-\sqrt {\left (b \,x^{4}-2 b \,x^{2}+a +b \right ) b \left (a +b \right )}\, a -\sqrt {\left (b \,x^{4}-2 b \,x^{2}+a +b \right ) b \left (a +b \right )}\, b}\)	\(777\)

Input:

int((a+b+b^(1/2)*(a+b)^(1/2)-(b+b^(1/2)*(a+b)^(1/2))*x^2)/(-x^2+1)/(b*x^4- 
2*b*x^2+a+b)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

b/((I*b^(1/2)*a^(1/2)+b)/(a+b))^(1/2)*(1-(I*b^(1/2)*a^(1/2)+b)/(a+b)*x^2)^ 
(1/2)*(1+(I*b^(1/2)*a^(1/2)-b)/(a+b)*x^2)^(1/2)/(b*x^4-2*b*x^2+a+b)^(1/2)* 
EllipticF(x*((I*b^(1/2)*a^(1/2)+b)/(a+b))^(1/2),(-1-2*(I*b^(1/2)*a^(1/2)-b 
)/(a+b))^(1/2))+b^(1/2)*(a+b)^(1/2)/((I*b^(1/2)*a^(1/2)+b)/(a+b))^(1/2)*(1 
-(I*b^(1/2)*a^(1/2)+b)/(a+b)*x^2)^(1/2)*(1+(I*b^(1/2)*a^(1/2)-b)/(a+b)*x^2 
)^(1/2)/(b*x^4-2*b*x^2+a+b)^(1/2)*EllipticF(x*((I*b^(1/2)*a^(1/2)+b)/(a+b) 
)^(1/2),(-1-2*(I*b^(1/2)*a^(1/2)-b)/(a+b))^(1/2))-1/2*a*(-1/2/a^(1/2)*arct 
anh(a^(1/2)/(b*x^4-2*b*x^2+a+b)^(1/2))-1/((I*b^(1/2)*a^(1/2)+b)/(a+b))^(1/ 
2)*(1-(I*b^(1/2)*a^(1/2)+b)/(a+b)*x^2)^(1/2)*(1+(I*b^(1/2)*a^(1/2)-b)/(a+b 
)*x^2)^(1/2)/(b*x^4-2*b*x^2+a+b)^(1/2)*EllipticPi(x*((I*b^(1/2)*a^(1/2)+b) 
/(a+b))^(1/2),1/(I*b^(1/2)*a^(1/2)+b)*(a+b),(-(I*b^(1/2)*a^(1/2)-b)/(a+b)) 
^(1/2)/((I*b^(1/2)*a^(1/2)+b)/(a+b))^(1/2)))+1/2*a*(-1/2/a^(1/2)*arctanh(a 
^(1/2)/(b*x^4-2*b*x^2+a+b)^(1/2))+1/((I*b^(1/2)*a^(1/2)+b)/(a+b))^(1/2)*(1 
-(I*b^(1/2)*a^(1/2)+b)/(a+b)*x^2)^(1/2)*(1+(I*b^(1/2)*a^(1/2)-b)/(a+b)*x^2 
)^(1/2)/(b*x^4-2*b*x^2+a+b)^(1/2)*EllipticPi(x*((I*b^(1/2)*a^(1/2)+b)/(a+b 
))^(1/2),1/(I*b^(1/2)*a^(1/2)+b)*(a+b),(-(I*b^(1/2)*a^(1/2)-b)/(a+b))^(1/2 
)/((I*b^(1/2)*a^(1/2)+b)/(a+b))^(1/2)))
 

Fricas [F]

\[ \int \frac {a+b+\sqrt {b} \sqrt {a+b}-\left (b+\sqrt {b} \sqrt {a+b}\right ) x^2}{\left (1-x^2\right ) \sqrt {a+b-2 b x^2+b x^4}} \, dx=\int { \frac {{\left (\sqrt {a + b} \sqrt {b} + b\right )} x^{2} - a - \sqrt {a + b} \sqrt {b} - b}{\sqrt {b x^{4} - 2 \, b x^{2} + a + b} {\left (x^{2} - 1\right )}} \,d x } \] Input:

integrate((a+b+b^(1/2)*(a+b)^(1/2)-(b+b^(1/2)*(a+b)^(1/2))*x^2)/(-x^2+1)/( 
b*x^4-2*b*x^2+a+b)^(1/2),x, algorithm="fricas")
 

Output:

integral(sqrt(b*x^4 - 2*b*x^2 + a + b)*(b*x^2 + (x^2 - 1)*sqrt(a + b)*sqrt 
(b) - a - b)/(b*x^6 - 3*b*x^4 + (a + 3*b)*x^2 - a - b), x)
 

Sympy [F]

\[ \int \frac {a+b+\sqrt {b} \sqrt {a+b}-\left (b+\sqrt {b} \sqrt {a+b}\right ) x^2}{\left (1-x^2\right ) \sqrt {a+b-2 b x^2+b x^4}} \, dx=\int \frac {- a + \sqrt {b} x^{2} \sqrt {a + b} - \sqrt {b} \sqrt {a + b} + b x^{2} - b}{\left (x - 1\right ) \left (x + 1\right ) \sqrt {a + b x^{4} - 2 b x^{2} + b}}\, dx \] Input:

integrate((a+b+b**(1/2)*(a+b)**(1/2)-(b+b**(1/2)*(a+b)**(1/2))*x**2)/(-x** 
2+1)/(b*x**4-2*b*x**2+a+b)**(1/2),x)
 

Output:

Integral((-a + sqrt(b)*x**2*sqrt(a + b) - sqrt(b)*sqrt(a + b) + b*x**2 - b 
)/((x - 1)*(x + 1)*sqrt(a + b*x**4 - 2*b*x**2 + b)), x)
 

Maxima [F]

\[ \int \frac {a+b+\sqrt {b} \sqrt {a+b}-\left (b+\sqrt {b} \sqrt {a+b}\right ) x^2}{\left (1-x^2\right ) \sqrt {a+b-2 b x^2+b x^4}} \, dx=\int { \frac {{\left (\sqrt {a + b} \sqrt {b} + b\right )} x^{2} - a - \sqrt {a + b} \sqrt {b} - b}{\sqrt {b x^{4} - 2 \, b x^{2} + a + b} {\left (x^{2} - 1\right )}} \,d x } \] Input:

integrate((a+b+b^(1/2)*(a+b)^(1/2)-(b+b^(1/2)*(a+b)^(1/2))*x^2)/(-x^2+1)/( 
b*x^4-2*b*x^2+a+b)^(1/2),x, algorithm="maxima")
 

Output:

integrate(((sqrt(a + b)*sqrt(b) + b)*x^2 - a - sqrt(a + b)*sqrt(b) - b)/(s 
qrt(b*x^4 - 2*b*x^2 + a + b)*(x^2 - 1)), x)
 

Giac [F]

\[ \int \frac {a+b+\sqrt {b} \sqrt {a+b}-\left (b+\sqrt {b} \sqrt {a+b}\right ) x^2}{\left (1-x^2\right ) \sqrt {a+b-2 b x^2+b x^4}} \, dx=\int { \frac {{\left (\sqrt {a + b} \sqrt {b} + b\right )} x^{2} - a - \sqrt {a + b} \sqrt {b} - b}{\sqrt {b x^{4} - 2 \, b x^{2} + a + b} {\left (x^{2} - 1\right )}} \,d x } \] Input:

integrate((a+b+b^(1/2)*(a+b)^(1/2)-(b+b^(1/2)*(a+b)^(1/2))*x^2)/(-x^2+1)/( 
b*x^4-2*b*x^2+a+b)^(1/2),x, algorithm="giac")
 

Output:

integrate(((sqrt(a + b)*sqrt(b) + b)*x^2 - a - sqrt(a + b)*sqrt(b) - b)/(s 
qrt(b*x^4 - 2*b*x^2 + a + b)*(x^2 - 1)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b+\sqrt {b} \sqrt {a+b}-\left (b+\sqrt {b} \sqrt {a+b}\right ) x^2}{\left (1-x^2\right ) \sqrt {a+b-2 b x^2+b x^4}} \, dx=\int -\frac {a+b-x^2\,\left (b+\sqrt {b}\,\sqrt {a+b}\right )+\sqrt {b}\,\sqrt {a+b}}{\left (x^2-1\right )\,\sqrt {b\,x^4-2\,b\,x^2+a+b}} \,d x \] Input:

int(-(a + b - x^2*(b + b^(1/2)*(a + b)^(1/2)) + b^(1/2)*(a + b)^(1/2))/((x 
^2 - 1)*(a + b - 2*b*x^2 + b*x^4)^(1/2)),x)
 

Output:

int(-(a + b - x^2*(b + b^(1/2)*(a + b)^(1/2)) + b^(1/2)*(a + b)^(1/2))/((x 
^2 - 1)*(a + b - 2*b*x^2 + b*x^4)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {a+b+\sqrt {b} \sqrt {a+b}-\left (b+\sqrt {b} \sqrt {a+b}\right ) x^2}{\left (1-x^2\right ) \sqrt {a+b-2 b x^2+b x^4}} \, dx=\int \frac {a +b +\sqrt {b}\, \sqrt {a +b}-\left (b +\sqrt {b}\, \sqrt {a +b}\right ) x^{2}}{\left (-x^{2}+1\right ) \sqrt {b \,x^{4}-2 b \,x^{2}+a +b}}d x \] Input:

int((a+b+b^(1/2)*(a+b)^(1/2)-(b+b^(1/2)*(a+b)^(1/2))*x^2)/(-x^2+1)/(b*x^4- 
2*b*x^2+a+b)^(1/2),x)
 

Output:

int((a+b+b^(1/2)*(a+b)^(1/2)-(b+b^(1/2)*(a+b)^(1/2))*x^2)/(-x^2+1)/(b*x^4- 
2*b*x^2+a+b)^(1/2),x)