\(\int \frac {(a \sqrt {a+b}+b \sqrt {a+b}+\sqrt {b} (a+b)) (1-\frac {\sqrt {b} x^2}{\sqrt {a+b}})}{(1-x^2) \sqrt {a+b+b x^4}} \, dx\) [89]

Optimal result

Integrand size = 69, antiderivative size = 301 \[ \int \frac {\left (a \sqrt {a+b}+b \sqrt {a+b}+\sqrt {b} (a+b)\right ) \left (1-\frac {\sqrt {b} x^2}{\sqrt {a+b}}\right )}{\left (1-x^2\right ) \sqrt {a+b+b x^4}} \, dx=\frac {a \sqrt {a+b} \text {arctanh}\left (\frac {\sqrt {a+2 b} x}{\sqrt {a+b+b x^4}}\right )}{2 \sqrt {a+2 b}}+\frac {\sqrt [4]{b} (a+b)^{5/4} \left (1+\frac {\sqrt {b} x^2}{\sqrt {a+b}}\right ) \sqrt {\frac {a+b+b x^4}{(a+b) \left (1+\frac {\sqrt {b} x^2}{\sqrt {a+b}}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b}}\right ),\frac {1}{2}\right )}{\sqrt {a+b+b x^4}}+\frac {(a+b)^{3/4} \left (\sqrt {b}-\sqrt {a+b}\right )^2 \left (1+\frac {\sqrt {b} x^2}{\sqrt {a+b}}\right ) \sqrt {\frac {a+b+b x^4}{(a+b) \left (1+\frac {\sqrt {b} x^2}{\sqrt {a+b}}\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {b}+\sqrt {a+b}\right )^2}{4 \sqrt {b} \sqrt {a+b}},2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{b} \sqrt {a+b+b x^4}} \] Output:

1/2*a*(a+b)^(1/2)*arctanh((a+2*b)^(1/2)*x/(b*x^4+a+b)^(1/2))/(a+2*b)^(1/2) 
+b^(1/4)*(a+b)^(5/4)*(1+b^(1/2)*x^2/(a+b)^(1/2))*((b*x^4+a+b)/(a+b)/(1+b^( 
1/2)*x^2/(a+b)^(1/2))^2)^(1/2)*InverseJacobiAM(2*arctan(b^(1/4)*x/(a+b)^(1 
/4)),1/2*2^(1/2))/(b*x^4+a+b)^(1/2)+1/4*(a+b)^(3/4)*(b^(1/2)-(a+b)^(1/2))^ 
2*(1+b^(1/2)*x^2/(a+b)^(1/2))*((b*x^4+a+b)/(a+b)/(1+b^(1/2)*x^2/(a+b)^(1/2 
))^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^(1/2))/b^(1/4)/(b*x^4+a+b)^(1/2)
                                                                                    
                                                                                    
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.59 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.69 \[ \int \frac {\left (a \sqrt {a+b}+b \sqrt {a+b}+\sqrt {b} (a+b)\right ) \left (1-\frac {\sqrt {b} x^2}{\sqrt {a+b}}\right )}{\left (1-x^2\right ) \sqrt {a+b+b x^4}} \, dx=\frac {\sqrt {a+b} \left (\sqrt {b}+\sqrt {a+b}\right ) \sqrt {\frac {a+b+b x^4}{a+b}} \left (-i b^{3/4} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {b}}{\sqrt {-a-b}}} x\right ),-1\right )+\sqrt [4]{-1} \sqrt {-\frac {\sqrt {b}}{\sqrt {-a-b}}} \sqrt [4]{a+b} \left (\sqrt {b}-\sqrt {a+b}\right ) \operatorname {EllipticPi}\left (\frac {i \sqrt {a+b}}{\sqrt {b}},\arcsin \left (\frac {(-1)^{3/4} \sqrt [4]{b} x}{\sqrt [4]{a+b}}\right ),-1\right )\right )}{\sqrt {-\frac {\sqrt {b}}{\sqrt {-a-b}}} \sqrt [4]{b} \sqrt {a+b+b x^4}} \] Input:

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

Output:

(Sqrt[a + b]*(Sqrt[b] + Sqrt[a + b])*Sqrt[(a + b + b*x^4)/(a + b)]*((-I)*b 
^(3/4)*EllipticF[I*ArcSinh[Sqrt[-(Sqrt[b]/Sqrt[-a - b])]*x], -1] + (-1)^(1 
/4)*Sqrt[-(Sqrt[b]/Sqrt[-a - b])]*(a + b)^(1/4)*(Sqrt[b] - Sqrt[a + b])*El 
lipticPi[(I*Sqrt[a + b])/Sqrt[b], ArcSin[((-1)^(3/4)*b^(1/4)*x)/(a + b)^(1 
/4)], -1]))/(Sqrt[-(Sqrt[b]/Sqrt[-a - b])]*b^(1/4)*Sqrt[a + b + b*x^4])
 

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 379, normalized size of antiderivative = 1.26, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.058, Rules used = {27, 2225, 761, 2223}

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.

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2225

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

\(\Big \downarrow \) 761

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

\(\Big \downarrow \) 2223

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

Input:

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

Output:

Sqrt[a + b]*(a + b + Sqrt[b]*Sqrt[a + b])*((b^(1/4)*(a + b)^(1/4)*(1 + (Sq 
rt[b]*x^2)/Sqrt[a + b])*Sqrt[(a + b + b*x^4)/((a + b)*(1 + (Sqrt[b]*x^2)/S 
qrt[a + b])^2)]*EllipticF[2*ArcTan[(b^(1/4)*x)/(a + b)^(1/4)], 1/2])/((Sqr 
t[b] + Sqrt[a + b])*Sqrt[a + b + b*x^4]) - ((Sqrt[b] - Sqrt[a + b])*(((1 + 
 Sqrt[b]/Sqrt[a + b])*ArcTanh[(Sqrt[a + 2*b]*x)/Sqrt[a + b + b*x^4]])/(2*S 
qrt[a + 2*b]) + ((a + b)^(1/4)*(1 - Sqrt[b]/Sqrt[a + b])*(1 + (Sqrt[b]*x^2 
)/Sqrt[a + b])*Sqrt[(a + b + b*x^4)/((a + b)*(1 + (Sqrt[b]*x^2)/Sqrt[a + b 
])^2)]*EllipticPi[(Sqrt[b] + Sqrt[a + b])^2/(4*Sqrt[b]*Sqrt[a + b]), 2*Arc 
Tan[(b^(1/4)*x)/(a + b)^(1/4)], 1/2])/(4*b^(1/4)*Sqrt[a + b + b*x^4])))/(S 
qrt[b] + Sqrt[a + b]))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 761

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

rule 2223

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]) 
, x_Symbol] :> With[{q = Rt[B/A, 2]}, Simp[(-(B*d - A*e))*(ArcTanh[Rt[(-c)* 
(d/e) - a*(e/d), 2]*(x/Sqrt[a + c*x^4])]/(2*d*e*Rt[(-c)*(d/e) - a*(e/d), 2] 
)), x] + Simp[(B*d + A*e)*(1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^ 
2)]/(4*d*e*q*Sqrt[a + c*x^4]))*EllipticPi[-(e - d*q^2)^2/(4*d*e*q^2), 2*Arc 
Tan[q*x], 1/2], x]] /; FreeQ[{a, 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[c*(d/e) + a*(e 
/d)]
 

rule 2225

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]) 
, x_Symbol] :> Simp[2*A*(B/(B*d + A*e))   Int[1/Sqrt[a + c*x^4], x], x] - S 
imp[(B*d - A*e)/(B*d + A*e)   Int[(A - B*x^2)/((d + e*x^2)*Sqrt[a + c*x^4]) 
, x], x] /; FreeQ[{a, 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 = 4.26 (sec) , antiderivative size = 456, normalized size of antiderivative = 1.51

method	result	size
default	\(-\frac {\left (a \sqrt {a +b}+b \sqrt {a +b}+\sqrt {b}\, \left (a +b \right )\right ) \left (-\frac {\sqrt {b}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a +b}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a +b}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a +b}}}, i\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a +b}}}\, \sqrt {b \,x^{4}+a +b}}+\left (-\frac {\sqrt {b}}{2}+\frac {\sqrt {a +b}}{2}\right ) \left (-\frac {\operatorname {arctanh}\left (\frac {2 b \,x^{2}+2 a +2 b}{2 \sqrt {a +2 b}\, \sqrt {b \,x^{4}+a +b}}\right )}{2 \sqrt {a +2 b}}-\frac {\sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a +b}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a +b}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a +b}}}, -\frac {i \sqrt {a +b}}{\sqrt {b}}, \frac {\sqrt {-\frac {i \sqrt {b}}{\sqrt {a +b}}}}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a +b}}}}\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a +b}}}\, \sqrt {b \,x^{4}+a +b}}\right )+\left (\frac {\sqrt {b}}{2}-\frac {\sqrt {a +b}}{2}\right ) \left (-\frac {\operatorname {arctanh}\left (\frac {2 b \,x^{2}+2 a +2 b}{2 \sqrt {a +2 b}\, \sqrt {b \,x^{4}+a +b}}\right )}{2 \sqrt {a +2 b}}+\frac {\sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a +b}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a +b}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a +b}}}, -\frac {i \sqrt {a +b}}{\sqrt {b}}, \frac {\sqrt {-\frac {i \sqrt {b}}{\sqrt {a +b}}}}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a +b}}}}\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a +b}}}\, \sqrt {b \,x^{4}+a +b}}\right )\right )}{\sqrt {a +b}}\)	\(456\)
elliptic	\(\text {Expression too large to display}\)	\(2104\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

\[ \int \frac {\left (a \sqrt {a+b}+b \sqrt {a+b}+\sqrt {b} (a+b)\right ) \left (1-\frac {\sqrt {b} x^2}{\sqrt {a+b}}\right )}{\left (1-x^2\right ) \sqrt {a+b+b x^4}} \, dx=\frac {\left (\int \left (- \frac {\sqrt {a + b}}{x^{2} \sqrt {a + b x^{4} + b} - \sqrt {a + b x^{4} + b}}\right )\, dx + \int \frac {\sqrt {b} x^{2}}{x^{2} \sqrt {a + b x^{4} + b} - \sqrt {a + b x^{4} + b}}\, dx\right ) \left (a \sqrt {b} + a \sqrt {a + b} + b^{\frac {3}{2}} + b \sqrt {a + b}\right )}{\sqrt {a + b}} \] Input:

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

Output:

(Integral(-sqrt(a + b)/(x**2*sqrt(a + b*x**4 + b) - sqrt(a + b*x**4 + b)), 
 x) + Integral(sqrt(b)*x**2/(x**2*sqrt(a + b*x**4 + b) - sqrt(a + b*x**4 + 
 b)), x))*(a*sqrt(b) + a*sqrt(a + b) + b**(3/2) + b*sqrt(a + b))/sqrt(a + 
b)
 

Maxima [F(-2)]

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

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

Output:

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un 
defined.
 

Giac [F(-2)]

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

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

Output:

Exception raised: RuntimeError >> an error occurred running a Giac command 
:INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve 
cteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (a \sqrt {a+b}+b \sqrt {a+b}+\sqrt {b} (a+b)\right ) \left (1-\frac {\sqrt {b} x^2}{\sqrt {a+b}}\right )}{\left (1-x^2\right ) \sqrt {a+b+b x^4}} \, dx=-\sqrt {a +b}\, \left (\int \frac {\sqrt {b \,x^{4}+a +b}}{b \,x^{6}-b \,x^{4}+a \,x^{2}+b \,x^{2}-a -b}d x \right ) a -\sqrt {a +b}\, \left (\int \frac {\sqrt {b \,x^{4}+a +b}}{b \,x^{6}-b \,x^{4}+a \,x^{2}+b \,x^{2}-a -b}d x \right ) b +\sqrt {a +b}\, \left (\int \frac {\sqrt {b \,x^{4}+a +b}\, x^{2}}{b \,x^{6}-b \,x^{4}+a \,x^{2}+b \,x^{2}-a -b}d x \right ) b -\sqrt {b}\, \left (\int \frac {\sqrt {b \,x^{4}+a +b}}{b \,x^{6}-b \,x^{4}+a \,x^{2}+b \,x^{2}-a -b}d x \right ) a -\sqrt {b}\, \left (\int \frac {\sqrt {b \,x^{4}+a +b}}{b \,x^{6}-b \,x^{4}+a \,x^{2}+b \,x^{2}-a -b}d x \right ) b +\sqrt {b}\, \left (\int \frac {\sqrt {b \,x^{4}+a +b}\, x^{2}}{b \,x^{6}-b \,x^{4}+a \,x^{2}+b \,x^{2}-a -b}d x \right ) a +\sqrt {b}\, \left (\int \frac {\sqrt {b \,x^{4}+a +b}\, x^{2}}{b \,x^{6}-b \,x^{4}+a \,x^{2}+b \,x^{2}-a -b}d x \right ) b \] Input:

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

Output:

 - sqrt(a + b)*int(sqrt(a + b*x**4 + b)/(a*x**2 - a + b*x**6 - b*x**4 + b* 
x**2 - b),x)*a - sqrt(a + b)*int(sqrt(a + b*x**4 + b)/(a*x**2 - a + b*x**6 
 - b*x**4 + b*x**2 - b),x)*b + sqrt(a + b)*int((sqrt(a + b*x**4 + b)*x**2) 
/(a*x**2 - a + b*x**6 - b*x**4 + b*x**2 - b),x)*b - sqrt(b)*int(sqrt(a + b 
*x**4 + b)/(a*x**2 - a + b*x**6 - b*x**4 + b*x**2 - b),x)*a - sqrt(b)*int( 
sqrt(a + b*x**4 + b)/(a*x**2 - a + b*x**6 - b*x**4 + b*x**2 - b),x)*b + sq 
rt(b)*int((sqrt(a + b*x**4 + b)*x**2)/(a*x**2 - a + b*x**6 - b*x**4 + b*x* 
*2 - b),x)*a + sqrt(b)*int((sqrt(a + b*x**4 + b)*x**2)/(a*x**2 - a + b*x** 
6 - b*x**4 + b*x**2 - b),x)*b