\(\int \frac {1+\frac {\sqrt {b} x^2}{\sqrt {a+b}}}{(1-x^2) \sqrt {a+b+b x^4}} \, dx\) [85]

Optimal result

Integrand size = 40, antiderivative size = 198 \[ \int \frac {1+\frac {\sqrt {b} x^2}{\sqrt {a+b}}}{\left (1-x^2\right ) \sqrt {a+b+b x^4}} \, dx=\frac {\left (1+\frac {\sqrt {b}}{\sqrt {a+b}}\right ) \text {arctanh}\left (\frac {\sqrt {a+2 b} x}{\sqrt {a+b+b x^4}}\right )}{2 \sqrt {a+2 b}}+\frac {\left (1-\frac {\sqrt {b}}{\sqrt {a+b}}\right ) \left (\sqrt {a+b}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b+b x^4}{\left (\sqrt {a+b}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {b}+\sqrt {a+b}\right )^2}{4 \sqrt {b} \sqrt {a+b}},2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{b} \sqrt [4]{a+b} \sqrt {a+b+b x^4}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.97 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.97 \[ \int \frac {1+\frac {\sqrt {b} x^2}{\sqrt {a+b}}}{\left (1-x^2\right ) \sqrt {a+b+b x^4}} \, dx=\frac {i \sqrt {a+b} \left (\frac {a+b+b x^4}{a+b}\right )^{3/2} \left (b^{3/4} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {b}}{\sqrt {-a-b}}} x\right ),-1\right )+(-1)^{3/4} \sqrt {-\frac {\sqrt {b}}{\sqrt {-a-b}}} \sqrt [4]{a+b} \left (\sqrt {b}+\sqrt {a+b}\right ) \operatorname {EllipticPi}\left (\frac {i \sqrt {a+b}}{\sqrt {b}},\arcsin \left (\frac {(-1)^{3/4} \sqrt [4]{b} x}{\sqrt [4]{a+b}}\right ),-1\right )\right )}{\sqrt {-\frac {\sqrt {b}}{\sqrt {-a-b}}} \sqrt [4]{b} \left (a+b+b x^4\right )^{3/2}} \] Input:

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

Output:

(I*Sqrt[a + b]*((a + b + b*x^4)/(a + b))^(3/2)*(b^(3/4)*EllipticF[I*ArcSin 
h[Sqrt[-(Sqrt[b]/Sqrt[-a - b])]*x], -1] + (-1)^(3/4)*Sqrt[-(Sqrt[b]/Sqrt[- 
a - b])]*(a + b)^(1/4)*(Sqrt[b] + Sqrt[a + b])*EllipticPi[(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)*(a + b + b*x^4)^(3/2))
 

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.04, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.025, Rules used = {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.

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

\(\Big \downarrow \) 2223

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

Input:

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

Output:

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

Defintions of rubi rules used

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)]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 3.43 (sec) , antiderivative size = 433, normalized size of antiderivative = 2.19

method	result	size
default	\(-\frac {\frac {\sqrt {b}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a +b}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a +b}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a +b}}}, i\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a +b}}}\, \sqrt {b \,x^{4}+a +b}}+\left (-\frac {\sqrt {b}}{2}-\frac {\sqrt {a +b}}{2}\right ) \left (-\frac {\operatorname {arctanh}\left (\frac {2 b \,x^{2}+2 a +2 b}{2 \sqrt {a +2 b}\, \sqrt {b \,x^{4}+a +b}}\right )}{2 \sqrt {a +2 b}}+\frac {\sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a +b}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a +b}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a +b}}}, -\frac {i \sqrt {a +b}}{\sqrt {b}}, \frac {\sqrt {-\frac {i \sqrt {b}}{\sqrt {a +b}}}}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a +b}}}}\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a +b}}}\, \sqrt {b \,x^{4}+a +b}}\right )+\left (\frac {\sqrt {b}}{2}+\frac {\sqrt {a +b}}{2}\right ) \left (-\frac {\operatorname {arctanh}\left (\frac {2 b \,x^{2}+2 a +2 b}{2 \sqrt {a +2 b}\, \sqrt {b \,x^{4}+a +b}}\right )}{2 \sqrt {a +2 b}}-\frac {\sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a +b}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a +b}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a +b}}}, -\frac {i \sqrt {a +b}}{\sqrt {b}}, \frac {\sqrt {-\frac {i \sqrt {b}}{\sqrt {a +b}}}}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a +b}}}}\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a +b}}}\, \sqrt {b \,x^{4}+a +b}}\right )}{\sqrt {a +b}}\)	\(433\)
elliptic	\(\frac {\sqrt {\left (b \,x^{4}+a +b \right ) b \left (a +b \right )}\, \left (\sqrt {a +b}+\sqrt {b}\, x^{2}\right ) \left (-\frac {b \sqrt {1-\frac {i \sqrt {b}\, \left (a +b \right )^{\frac {3}{2}} x^{2}}{a^{2}+2 b a +b^{2}}}\, \sqrt {1+\frac {i \sqrt {b}\, \left (a +b \right )^{\frac {3}{2}} x^{2}}{a^{2}+2 b a +b^{2}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}\, \left (a +b \right )^{\frac {3}{2}}}{a^{2}+2 b a +b^{2}}}, i\right )}{\sqrt {\frac {i \sqrt {b}\, \left (a +b \right )^{\frac {3}{2}}}{a^{2}+2 b a +b^{2}}}\, \sqrt {b^{2} a \,x^{4}+b^{3} x^{4}+b \,a^{2}+2 b^{2} a +b^{3}}}+\frac {b \sqrt {1-\frac {i \sqrt {b}\, \left (a +b \right )^{\frac {3}{2}} x^{2}}{a^{2}+2 b a +b^{2}}}\, \sqrt {1+\frac {i \sqrt {b}\, \left (a +b \right )^{\frac {3}{2}} x^{2}}{a^{2}+2 b a +b^{2}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {b}\, \left (a +b \right )^{\frac {3}{2}}}{a^{2}+2 b a +b^{2}}}, -\frac {i \left (a^{2}+2 b a +b^{2}\right )}{\sqrt {b}\, \left (a +b \right )^{\frac {3}{2}}}, \frac {\sqrt {-\frac {i \sqrt {b}\, \left (a +b \right )^{\frac {3}{2}}}{a^{2}+2 b a +b^{2}}}}{\sqrt {\frac {i \sqrt {b}\, \left (a +b \right )^{\frac {3}{2}}}{a^{2}+2 b a +b^{2}}}}\right )}{\sqrt {\frac {i \sqrt {b}\, \left (a +b \right )^{\frac {3}{2}}}{a^{2}+2 b a +b^{2}}}\, \sqrt {b^{2} a \,x^{4}+b^{3} x^{4}+b \,a^{2}+2 b^{2} a +b^{3}}}+\frac {\sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a +b}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a +b}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a +b}}}, -\frac {i \sqrt {a +b}}{\sqrt {b}}, \frac {\sqrt {-\frac {i \sqrt {b}}{\sqrt {a +b}}}}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a +b}}}}\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a +b}}}\, \sqrt {b \,x^{4}+a +b}}\right )}{\left (b \,x^{2} \sqrt {b \,x^{4}+a +b}+\sqrt {\left (b \,x^{4}+a +b \right ) b \left (a +b \right )}\right ) \sqrt {a +b}}\)	\(563\)

Input:

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

Output:

-1/(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)*Elli 
pticF(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(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

integrate((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))/sqrt(a + b)
                                                                                    
                                                                                    
 

Maxima [F(-2)]

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

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

Output:

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

Giac [F(-2)]

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

integrate((1+b^(1/2)*x^2/(a+b)^(1/2))/(-x^2+1)/(b*x^4+a+b)^(1/2),x, algori 
thm="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. \[ \int \frac {1+\frac {\sqrt {b} x^2}{\sqrt {a+b}}}{\left (1-x^2\right ) \sqrt {a+b+b x^4}} \, dx=\int -\frac {\frac {\sqrt {b}\,x^2}{\sqrt {a+b}}+1}{\left (x^2-1\right )\,\sqrt {b\,x^4+a+b}} \,d x \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

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