Integrand size = 72, antiderivative size = 198 \[ \int \frac {1+\frac {\sqrt {b} \left (a+b-\sqrt {b} \sqrt {a+b}\right ) x^2}{(a+b) \left (-\sqrt {b}+\sqrt {a+b}\right )}}{\left (1-x^2\right ) \sqrt {a+b+b x^4}} \, dx=\frac {\left (1+\frac {\sqrt {b}}{\sqrt {a+b}}\right ) \text {arctanh}\left (\frac {\sqrt {a+2 b} x}{\sqrt {a+b+b x^4}}\right )}{2 \sqrt {a+2 b}}+\frac {\left (1-\frac {\sqrt {b}}{\sqrt {a+b}}\right ) \left (\sqrt [4]{a+b}+\frac {\sqrt {b} x^2}{\sqrt [4]{a+b}}\right ) \sqrt {\frac {a+b+b x^4}{\left (\sqrt {a+b}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {b}+\sqrt {a+b}\right )^2}{4 \sqrt {b} \sqrt {a+b}},2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{b} \sqrt {a+b+b x^4}} \] 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)^(1/4)+b^(1/2)*x^2/(a+b)^(1/4 ))*((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)/(b*x^4+a+b)^(1/2)
Result contains complex when optimal does not.
Time = 12.68 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.05 \[ \int \frac {1+\frac {\sqrt {b} \left (a+b-\sqrt {b} \sqrt {a+b}\right ) x^2}{(a+b) \left (-\sqrt {b}+\sqrt {a+b}\right )}}{\left (1-x^2\right ) \sqrt {a+b+b x^4}} \, dx=\frac {\left (\frac {a+b+b x^4}{a+b}\right )^{3/2} \left (-i b^{3/4} \left (a+b-\sqrt {b} \sqrt {a+b}\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {b}}{\sqrt {-a-b}}} x\right ),-1\right )+\sqrt [4]{-1} a \sqrt {-\frac {\sqrt {b}}{\sqrt {-a-b}}} (a+b)^{3/4} \operatorname {EllipticPi}\left (\frac {i \sqrt {a+b}}{\sqrt {b}},\arcsin \left (\frac {(-1)^{3/4} \sqrt [4]{b} x}{\sqrt [4]{a+b}}\right ),-1\right )\right )}{\sqrt {-\frac {\sqrt {b}}{\sqrt {-a-b}}} \sqrt [4]{b} \left (\sqrt {b}-\sqrt {a+b}\right ) \left (a+b+b x^4\right )^{3/2}} \] Input:
Integrate[(1 + (Sqrt[b]*(a + b - Sqrt[b]*Sqrt[a + b])*x^2)/((a + b)*(-Sqrt [b] + Sqrt[a + b])))/((1 - x^2)*Sqrt[a + b + b*x^4]),x]
Output:
(((a + b + b*x^4)/(a + b))^(3/2)*((-I)*b^(3/4)*(a + b - Sqrt[b]*Sqrt[a + b ])*EllipticF[I*ArcSinh[Sqrt[-(Sqrt[b]/Sqrt[-a - b])]*x], -1] + (-1)^(1/4)* a*Sqrt[-(Sqrt[b]/Sqrt[-a - b])]*(a + b)^(3/4)*EllipticPi[(I*Sqrt[a + b])/S qrt[b], ArcSin[((-1)^(3/4)*b^(1/4)*x)/(a + b)^(1/4)], -1]))/(Sqrt[-(Sqrt[b ]/Sqrt[-a - b])]*b^(1/4)*(Sqrt[b] - Sqrt[a + b])*(a + b + b*x^4)^(3/2))
Leaf count is larger than twice the leaf count of optimal. \(615\) vs. \(2(198)=396\).
Time = 1.31 (sec) , antiderivative size = 615, normalized size of antiderivative = 3.11, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {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.
| \(\displaystyle \int \frac {\frac {\sqrt {b} x^2 \left (-\sqrt {b} \sqrt {a+b}+a+b\right )}{(a+b) \left (\sqrt {a+b}-\sqrt {b}\right )}+1}{\left (1-x^2\right ) \sqrt {a+b x^4+b}} \, dx\) |
| \(\Big \downarrow \) 2225 |
| \(\displaystyle \frac {2 \sqrt {b} \left (-\sqrt {b} \sqrt {a+b}+a+b\right ) \int \frac {1}{\sqrt {b x^4+a+b}}dx}{-2 b \sqrt {a+b}+2 a \sqrt {b}-a \sqrt {a+b}+2 b^{3/2}}-\frac {a \sqrt {a+b} \int \frac {\frac {\sqrt {b} \left (a+b-\sqrt {b} \sqrt {a+b}\right ) x^2}{(a+b) \left (\sqrt {b}-\sqrt {a+b}\right )}+1}{\left (1-x^2\right ) \sqrt {b x^4+a+b}}dx}{-2 b \sqrt {a+b}+2 a \sqrt {b}-a \sqrt {a+b}+2 b^{3/2}}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {\sqrt [4]{b} \sqrt [4]{a+b} \left (-\sqrt {b} \sqrt {a+b}+a+b\right ) \left (\frac {\sqrt {b} x^2}{\sqrt {a+b}}+1\right ) \sqrt {\frac {a+b x^4+b}{(a+b) \left (\frac {\sqrt {b} x^2}{\sqrt {a+b}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b}}\right ),\frac {1}{2}\right )}{\left (-2 b \sqrt {a+b}+2 a \sqrt {b}-a \sqrt {a+b}+2 b^{3/2}\right ) \sqrt {a+b x^4+b}}-\frac {a \sqrt {a+b} \int \frac {\frac {\sqrt {b} \left (a+b-\sqrt {b} \sqrt {a+b}\right ) x^2}{(a+b) \left (\sqrt {b}-\sqrt {a+b}\right )}+1}{\left (1-x^2\right ) \sqrt {b x^4+a+b}}dx}{-2 b \sqrt {a+b}+2 a \sqrt {b}-a \sqrt {a+b}+2 b^{3/2}}\) |
| \(\Big \downarrow \) 2223 |
| \(\displaystyle \frac {\sqrt [4]{b} \sqrt [4]{a+b} \left (-\sqrt {b} \sqrt {a+b}+a+b\right ) \left (\frac {\sqrt {b} x^2}{\sqrt {a+b}}+1\right ) \sqrt {\frac {a+b x^4+b}{(a+b) \left (\frac {\sqrt {b} x^2}{\sqrt {a+b}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b}}\right ),\frac {1}{2}\right )}{\left (-2 b \sqrt {a+b}+2 a \sqrt {b}-a \sqrt {a+b}+2 b^{3/2}\right ) \sqrt {a+b x^4+b}}-\frac {a \sqrt {a+b} \left (\frac {\left (\frac {\sqrt {b} \left (-\sqrt {b} \sqrt {a+b}+a+b\right )}{(a+b) \left (\sqrt {b}-\sqrt {a+b}\right )}+1\right ) \text {arctanh}\left (\frac {x \sqrt {a+2 b}}{\sqrt {a+b x^4+b}}\right )}{2 \sqrt {a+2 b}}-\frac {a \left (\frac {\sqrt {b} x^2 \left (-\sqrt {b} \sqrt {a+b}+a+b\right )}{(a+b) \left (\sqrt {b}-\sqrt {a+b}\right )}+1\right ) \sqrt {\frac {a+b x^4+b}{(a+b) \left (\frac {\sqrt {b} x^2 \left (-\sqrt {b} \sqrt {a+b}+a+b\right )}{(a+b) \left (\sqrt {b}-\sqrt {a+b}\right )}+1\right )^2}} \operatorname {EllipticPi}\left (\frac {a^2+8 b a+8 b^2-\sqrt {b} \sqrt {a+b} (4 a+8 b)}{4 \sqrt {b} \left (2 b^{3/2}-2 \sqrt {a+b} b+2 a \sqrt {b}-a \sqrt {a+b}\right )},2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {a+b-\sqrt {b} \sqrt {a+b}} x}{\sqrt {a+b} \sqrt {\sqrt {b}-\sqrt {a+b}}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{b} \sqrt {\sqrt {b}-\sqrt {a+b}} \sqrt {-\sqrt {b} \sqrt {a+b}+a+b} \sqrt {a+b x^4+b}}\right )}{-2 b \sqrt {a+b}+2 a \sqrt {b}-a \sqrt {a+b}+2 b^{3/2}}\) |
Input:
Int[(1 + (Sqrt[b]*(a + b - Sqrt[b]*Sqrt[a + b])*x^2)/((a + b)*(-Sqrt[b] + Sqrt[a + b])))/((1 - x^2)*Sqrt[a + b + b*x^4]),x]
Output:
(b^(1/4)*(a + b)^(1/4)*(a + b - Sqrt[b]*Sqrt[a + b])*(1 + (Sqrt[b]*x^2)/Sq rt[a + b])*Sqrt[(a + b + 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)], 1/2])/((2*a*Sqrt[b] + 2* b^(3/2) - a*Sqrt[a + b] - 2*b*Sqrt[a + b])*Sqrt[a + b + b*x^4]) - (a*Sqrt[ a + b]*(((1 + (Sqrt[b]*(a + b - Sqrt[b]*Sqrt[a + b]))/((a + b)*(Sqrt[b] - Sqrt[a + b])))*ArcTanh[(Sqrt[a + 2*b]*x)/Sqrt[a + b + b*x^4]])/(2*Sqrt[a + 2*b]) - (a*(1 + (Sqrt[b]*(a + b - Sqrt[b]*Sqrt[a + b])*x^2)/((a + b)*(Sqr t[b] - Sqrt[a + b])))*Sqrt[(a + b + b*x^4)/((a + b)*(1 + (Sqrt[b]*(a + b - Sqrt[b]*Sqrt[a + b])*x^2)/((a + b)*(Sqrt[b] - Sqrt[a + b])))^2)]*Elliptic Pi[(a^2 + 8*a*b + 8*b^2 - Sqrt[b]*Sqrt[a + b]*(4*a + 8*b))/(4*Sqrt[b]*(2*a *Sqrt[b] + 2*b^(3/2) - a*Sqrt[a + b] - 2*b*Sqrt[a + b])), 2*ArcTan[(b^(1/4 )*Sqrt[a + b - Sqrt[b]*Sqrt[a + b]]*x)/(Sqrt[a + b]*Sqrt[Sqrt[b] - Sqrt[a + b]])], 1/2])/(4*b^(1/4)*Sqrt[Sqrt[b] - Sqrt[a + b]]*Sqrt[a + b - Sqrt[b] *Sqrt[a + b]]*Sqrt[a + b + b*x^4])))/(2*a*Sqrt[b] + 2*b^(3/2) - a*Sqrt[a + b] - 2*b*Sqrt[a + b])
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]
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)]
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]
Result contains complex when optimal does not.
Time = 3.96 (sec) , antiderivative size = 588, normalized size of antiderivative = 2.97
| method | result | size |
| elliptic | \(\frac {\left (1+\frac {\sqrt {b}\, \left (\sqrt {b}\, \sqrt {a +b}-a -b \right ) x^{2}}{\left (a +b \right ) \left (\sqrt {b}-\sqrt {a +b}\right )}\right ) \sqrt {\left (b \,x^{4}+a +b \right ) b \left (a +b \right )}\, \left (-\frac {b \sqrt {1-\frac {i \sqrt {b}\, \left (a +b \right )^{\frac {3}{2}} x^{2}}{a^{2}+2 b a +b^{2}}}\, \sqrt {1+\frac {i \sqrt {b}\, \left (a +b \right )^{\frac {3}{2}} x^{2}}{a^{2}+2 b a +b^{2}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}\, \left (a +b \right )^{\frac {3}{2}}}{a^{2}+2 b a +b^{2}}}, i\right )}{\sqrt {\frac {i \sqrt {b}\, \left (a +b \right )^{\frac {3}{2}}}{a^{2}+2 b a +b^{2}}}\, \sqrt {b^{2} a \,x^{4}+b^{3} x^{4}+b \,a^{2}+2 b^{2} a +b^{3}}}+\frac {b \sqrt {1-\frac {i \sqrt {b}\, \left (a +b \right )^{\frac {3}{2}} x^{2}}{a^{2}+2 b a +b^{2}}}\, \sqrt {1+\frac {i \sqrt {b}\, \left (a +b \right )^{\frac {3}{2}} x^{2}}{a^{2}+2 b a +b^{2}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {b}\, \left (a +b \right )^{\frac {3}{2}}}{a^{2}+2 b a +b^{2}}}, -\frac {i \left (a^{2}+2 b a +b^{2}\right )}{\sqrt {b}\, \left (a +b \right )^{\frac {3}{2}}}, \frac {\sqrt {-\frac {i \sqrt {b}\, \left (a +b \right )^{\frac {3}{2}}}{a^{2}+2 b a +b^{2}}}}{\sqrt {\frac {i \sqrt {b}\, \left (a +b \right )^{\frac {3}{2}}}{a^{2}+2 b a +b^{2}}}}\right )}{\sqrt {\frac {i \sqrt {b}\, \left (a +b \right )^{\frac {3}{2}}}{a^{2}+2 b a +b^{2}}}\, \sqrt {b^{2} a \,x^{4}+b^{3} x^{4}+b \,a^{2}+2 b^{2} a +b^{3}}}+\frac {\sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a +b}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a +b}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a +b}}}, -\frac {i \sqrt {a +b}}{\sqrt {b}}, \frac {\sqrt {-\frac {i \sqrt {b}}{\sqrt {a +b}}}}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a +b}}}}\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a +b}}}\, \sqrt {b \,x^{4}+a +b}}\right )}{b \,x^{2} \sqrt {b \,x^{4}+a +b}+\sqrt {\left (b \,x^{4}+a +b \right ) b \left (a +b \right )}}\) | \(588\) |
| default | \(\frac {\frac {b^{\frac {3}{2}} \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}}+\frac {\sqrt {b}\, a \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}}-\frac {b \sqrt {a +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}}+\frac {a \sqrt {a +b}\, \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 )}{2}-\frac {a \sqrt {a +b}\, \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 )}{2}}{\left (a +b \right ) \left (\sqrt {b}-\sqrt {a +b}\right )}\) | \(600\) |
Input:
int((1+b^(1/2)*(a+b-b^(1/2)*(a+b)^(1/2))*x^2/(a+b)/(-b^(1/2)+(a+b)^(1/2))) /(-x^2+1)/(b*x^4+a+b)^(1/2),x,method=_RETURNVERBOSE)
Output:
(1+b^(1/2)*(b^(1/2)*(a+b)^(1/2)-a-b)*x^2/(a+b)/(b^(1/2)-(a+b)^(1/2)))/(b*x ^2*(b*x^4+a+b)^(1/2)+((b*x^4+a+b)*b*(a+b))^(1/2))*((b*x^4+a+b)*b*(a+b))^(1 /2)*(-b/(I*b^(1/2)*(a+b)^(3/2)/(a^2+2*a*b+b^2))^(1/2)*(1-I*b^(1/2)*(a+b)^( 3/2)/(a^2+2*a*b+b^2)*x^2)^(1/2)*(1+I*b^(1/2)*(a+b)^(3/2)/(a^2+2*a*b+b^2)*x ^2)^(1/2)/(a*b^2*x^4+b^3*x^4+a^2*b+2*a*b^2+b^3)^(1/2)*EllipticF(x*(I*b^(1/ 2)*(a+b)^(3/2)/(a^2+2*a*b+b^2))^(1/2),I)+b/(I*b^(1/2)*(a+b)^(3/2)/(a^2+2*a *b+b^2))^(1/2)*(1-I*b^(1/2)*(a+b)^(3/2)/(a^2+2*a*b+b^2)*x^2)^(1/2)*(1+I*b^ (1/2)*(a+b)^(3/2)/(a^2+2*a*b+b^2)*x^2)^(1/2)/(a*b^2*x^4+b^3*x^4+a^2*b+2*a* b^2+b^3)^(1/2)*EllipticPi(x*(I*b^(1/2)*(a+b)^(3/2)/(a^2+2*a*b+b^2))^(1/2), -I/b^(1/2)/(a+b)^(3/2)*(a^2+2*a*b+b^2),(-I*b^(1/2)*(a+b)^(3/2)/(a^2+2*a*b+ b^2))^(1/2)/(I*b^(1/2)*(a+b)^(3/2)/(a^2+2*a*b+b^2))^(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)))
Timed out. \[ \int \frac {1+\frac {\sqrt {b} \left (a+b-\sqrt {b} \sqrt {a+b}\right ) x^2}{(a+b) \left (-\sqrt {b}+\sqrt {a+b}\right )}}{\left (1-x^2\right ) \sqrt {a+b+b x^4}} \, dx=\text {Timed out} \] Input:
integrate((1+b^(1/2)*(a+b-b^(1/2)*(a+b)^(1/2))*x^2/(a+b)/(-b^(1/2)+(a+b)^( 1/2)))/(-x^2+1)/(b*x^4+a+b)^(1/2),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {1+\frac {\sqrt {b} \left (a+b-\sqrt {b} \sqrt {a+b}\right ) x^2}{(a+b) \left (-\sqrt {b}+\sqrt {a+b}\right )}}{\left (1-x^2\right ) \sqrt {a+b+b x^4}} \, dx=- \frac {\int \left (- \frac {b^{\frac {3}{2}}}{x^{2} \sqrt {a + b x^{4} + b} - \sqrt {a + b x^{4} + b}}\right )\, dx + \int \left (- \frac {a \sqrt {b}}{x^{2} \sqrt {a + b x^{4} + b} - \sqrt {a + b x^{4} + b}}\right )\, dx + \int \frac {a \sqrt {a + b}}{x^{2} \sqrt {a + b x^{4} + b} - \sqrt {a + b x^{4} + b}}\, dx + \int \frac {b \sqrt {a + b}}{x^{2} \sqrt {a + b x^{4} + b} - \sqrt {a + b x^{4} + b}}\, dx + \int \frac {b^{\frac {3}{2}} x^{2}}{x^{2} \sqrt {a + b x^{4} + b} - \sqrt {a + b x^{4} + b}}\, dx + \int \frac {a \sqrt {b} x^{2}}{x^{2} \sqrt {a + b x^{4} + b} - \sqrt {a + b x^{4} + b}}\, dx + \int \left (- \frac {b x^{2} \sqrt {a + b}}{x^{2} \sqrt {a + b x^{4} + b} - \sqrt {a + b x^{4} + b}}\right )\, dx}{\left (a + b\right ) \left (- \sqrt {b} + \sqrt {a + b}\right )} \] Input:
integrate((1+b**(1/2)*(a+b-b**(1/2)*(a+b)**(1/2))*x**2/(a+b)/(-b**(1/2)+(a +b)**(1/2)))/(-x**2+1)/(b*x**4+a+b)**(1/2),x)
Output:
-(Integral(-b**(3/2)/(x**2*sqrt(a + b*x**4 + b) - sqrt(a + b*x**4 + b)), x ) + Integral(-a*sqrt(b)/(x**2*sqrt(a + b*x**4 + b) - sqrt(a + b*x**4 + b)) , x) + Integral(a*sqrt(a + b)/(x**2*sqrt(a + b*x**4 + b) - sqrt(a + b*x**4 + b)), x) + Integral(b*sqrt(a + b)/(x**2*sqrt(a + b*x**4 + b) - sqrt(a + b*x**4 + b)), x) + Integral(b**(3/2)*x**2/(x**2*sqrt(a + b*x**4 + b) - sqr t(a + b*x**4 + b)), x) + Integral(a*sqrt(b)*x**2/(x**2*sqrt(a + b*x**4 + b ) - sqrt(a + b*x**4 + b)), x) + Integral(-b*x**2*sqrt(a + b)/(x**2*sqrt(a + b*x**4 + b) - sqrt(a + b*x**4 + b)), x))/((a + b)*(-sqrt(b) + sqrt(a + b )))
Exception generated. \[ \int \frac {1+\frac {\sqrt {b} \left (a+b-\sqrt {b} \sqrt {a+b}\right ) x^2}{(a+b) \left (-\sqrt {b}+\sqrt {a+b}\right )}}{\left (1-x^2\right ) \sqrt {a+b+b x^4}} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((1+b^(1/2)*(a+b-b^(1/2)*(a+b)^(1/2))*x^2/(a+b)/(-b^(1/2)+(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.
Exception generated. \[ \int \frac {1+\frac {\sqrt {b} \left (a+b-\sqrt {b} \sqrt {a+b}\right ) x^2}{(a+b) \left (-\sqrt {b}+\sqrt {a+b}\right )}}{\left (1-x^2\right ) \sqrt {a+b+b x^4}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((1+b^(1/2)*(a+b-b^(1/2)*(a+b)^(1/2))*x^2/(a+b)/(-b^(1/2)+(a+b)^( 1/2)))/(-x^2+1)/(b*x^4+a+b)^(1/2),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Recursive assumption sageVARa>=(-sa geVARb) ignoredRecursive assumption sageVARa>=(-sageVARb) ignoredRecursive assumpti
Timed out. \[ \int \frac {1+\frac {\sqrt {b} \left (a+b-\sqrt {b} \sqrt {a+b}\right ) x^2}{(a+b) \left (-\sqrt {b}+\sqrt {a+b}\right )}}{\left (1-x^2\right ) \sqrt {a+b+b x^4}} \, dx=\int -\frac {\frac {\sqrt {b}\,x^2\,\left (a+b-\sqrt {b}\,\sqrt {a+b}\right )}{\left (\sqrt {a+b}-\sqrt {b}\right )\,\left (a+b\right )}+1}{\left (x^2-1\right )\,\sqrt {b\,x^4+a+b}} \,d x \] Input:
int(-((b^(1/2)*x^2*(a + b - b^(1/2)*(a + b)^(1/2)))/(((a + b)^(1/2) - b^(1 /2))*(a + b)) + 1)/((x^2 - 1)*(a + b + b*x^4)^(1/2)),x)
Output:
int(-((b^(1/2)*x^2*(a + b - b^(1/2)*(a + b)^(1/2)))/(((a + b)^(1/2) - b^(1 /2))*(a + b)) + 1)/((x^2 - 1)*(a + b + b*x^4)^(1/2)), x)
\[ \int \frac {1+\frac {\sqrt {b} \left (a+b-\sqrt {b} \sqrt {a+b}\right ) x^2}{(a+b) \left (-\sqrt {b}+\sqrt {a+b}\right )}}{\left (1-x^2\right ) \sqrt {a+b+b x^4}} \, dx=-\sqrt {b}\, \sqrt {a +b}\, \left (\int \frac {\sqrt {b \,x^{4}+a +b}\, x^{2}}{a b \,x^{6}+b^{2} x^{6}-a b \,x^{4}-b^{2} x^{4}+a^{2} x^{2}+2 a b \,x^{2}+b^{2} x^{2}-a^{2}-2 a b -b^{2}}d x \right )-\left (\int \frac {\sqrt {b \,x^{4}+a +b}}{a b \,x^{6}+b^{2} x^{6}-a b \,x^{4}-b^{2} x^{4}+a^{2} x^{2}+2 a b \,x^{2}+b^{2} x^{2}-a^{2}-2 a b -b^{2}}d x \right ) a -\left (\int \frac {\sqrt {b \,x^{4}+a +b}}{a b \,x^{6}+b^{2} x^{6}-a b \,x^{4}-b^{2} x^{4}+a^{2} x^{2}+2 a b \,x^{2}+b^{2} x^{2}-a^{2}-2 a b -b^{2}}d x \right ) b \] Input:
int((1+b^(1/2)*(a+b-b^(1/2)*(a+b)^(1/2))*x^2/(a+b)/(-b^(1/2)+(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)