Integrand size = 68, antiderivative size = 192 \[ \int \frac {(a+b) \left (-\sqrt {b}+\sqrt {a+b}\right )+\sqrt {b} \left (a+b-\sqrt {b} \sqrt {a+b}\right ) x^2}{\left (1-x^2\right ) \sqrt {a+b+b x^4}} \, dx=\frac {a \sqrt {a+b} \text {arctanh}\left (\frac {\sqrt {a+2 b} x}{\sqrt {a+b+b x^4}}\right )}{2 \sqrt {a+2 b}}+\frac {\sqrt [4]{a+b} \left (\sqrt {b}-\sqrt {a+b}\right )^2 \left (\sqrt {a+b}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b+b x^4}{\left (\sqrt {a+b}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {b}+\sqrt {a+b}\right )^2}{4 \sqrt {b} \sqrt {a+b}},2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{b} \sqrt {a+b+b x^4}} \] 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) +1/4*(a+b)^(1/4)*(b^(1/2)-(a+b)^(1/2))^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)/(b*x^4+a+b)^(1/2)
Result contains complex when optimal does not.
Time = 10.80 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.99 \[ \int \frac {(a+b) \left (-\sqrt {b}+\sqrt {a+b}\right )+\sqrt {b} \left (a+b-\sqrt {b} \sqrt {a+b}\right ) x^2}{\left (1-x^2\right ) \sqrt {a+b+b x^4}} \, dx=\frac {i \sqrt {\frac {a+b+b x^4}{a+b}} \left (b^{3/4} \left (a+b-\sqrt {b} \sqrt {a+b}\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {b}}{\sqrt {-a-b}}} x\right ),-1\right )+(-1)^{3/4} a \sqrt {-\frac {\sqrt {b}}{\sqrt {-a-b}}} (a+b)^{3/4} \operatorname {EllipticPi}\left (\frac {i \sqrt {a+b}}{\sqrt {b}},\arcsin \left (\frac {(-1)^{3/4} \sqrt [4]{b} x}{\sqrt [4]{a+b}}\right ),-1\right )\right )}{\sqrt {-\frac {\sqrt {b}}{\sqrt {-a-b}}} \sqrt [4]{b} \sqrt {a+b+b x^4}} \] Input:
Integrate[((a + b)*(-Sqrt[b] + Sqrt[a + b]) + Sqrt[b]*(a + b - Sqrt[b]*Sqr t[a + b])*x^2)/((1 - x^2)*Sqrt[a + b + b*x^4]),x]
Output:
(I*Sqrt[(a + b + b*x^4)/(a + b)]*(b^(3/4)*(a + b - Sqrt[b]*Sqrt[a + b])*El lipticF[I*ArcSinh[Sqrt[-(Sqrt[b]/Sqrt[-a - b])]*x], -1] + (-1)^(3/4)*a*Sqr t[-(Sqrt[b]/Sqrt[-a - b])]*(a + b)^(3/4)*EllipticPi[(I*Sqrt[a + b])/Sqrt[b ], ArcSin[((-1)^(3/4)*b^(1/4)*x)/(a + b)^(1/4)], -1]))/(Sqrt[-(Sqrt[b]/Sqr t[-a - b])]*b^(1/4)*Sqrt[a + b + b*x^4])
Leaf count is larger than twice the leaf count of optimal. \(616\) vs. \(2(192)=384\).
Time = 1.12 (sec) , antiderivative size = 616, normalized size of antiderivative = 3.21, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {2225, 25, 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.
| \(\displaystyle \int \frac {\sqrt {b} x^2 \left (-\sqrt {b} \sqrt {a+b}+a+b\right )+(a+b) \left (\sqrt {a+b}-\sqrt {b}\right )}{\left (1-x^2\right ) \sqrt {a+b x^4+b}} \, dx\) |
| \(\Big \downarrow \) 2225 |
| \(\displaystyle -\frac {2 \sqrt {b} (a+b) \left (\sqrt {b}-\sqrt {a+b}\right ) \left (-\sqrt {b} \sqrt {a+b}+a+b\right ) \int \frac {1}{\sqrt {b x^4+a+b}}dx}{-2 b \sqrt {a+b}+2 a \sqrt {b}-a \sqrt {a+b}+2 b^{3/2}}-\frac {a \sqrt {a+b} \int -\frac {\sqrt {b} \left (a+b-\sqrt {b} \sqrt {a+b}\right ) x^2+(a+b) \left (\sqrt {b}-\sqrt {a+b}\right )}{\left (1-x^2\right ) \sqrt {b x^4+a+b}}dx}{-2 b \sqrt {a+b}+2 a \sqrt {b}-a \sqrt {a+b}+2 b^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {a \sqrt {a+b} \int \frac {\sqrt {b} \left (a+b-\sqrt {b} \sqrt {a+b}\right ) x^2+(a+b) \left (\sqrt {b}-\sqrt {a+b}\right )}{\left (1-x^2\right ) \sqrt {b x^4+a+b}}dx}{-2 b \sqrt {a+b}+2 a \sqrt {b}-a \sqrt {a+b}+2 b^{3/2}}-\frac {2 \sqrt {b} (a+b) \left (\sqrt {b}-\sqrt {a+b}\right ) \left (-\sqrt {b} \sqrt {a+b}+a+b\right ) \int \frac {1}{\sqrt {b x^4+a+b}}dx}{-2 b \sqrt {a+b}+2 a \sqrt {b}-a \sqrt {a+b}+2 b^{3/2}}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {a \sqrt {a+b} \int \frac {\sqrt {b} \left (a+b-\sqrt {b} \sqrt {a+b}\right ) x^2+(a+b) \left (\sqrt {b}-\sqrt {a+b}\right )}{\left (1-x^2\right ) \sqrt {b x^4+a+b}}dx}{-2 b \sqrt {a+b}+2 a \sqrt {b}-a \sqrt {a+b}+2 b^{3/2}}-\frac {\sqrt [4]{b} (a+b)^{5/4} \left (\sqrt {b}-\sqrt {a+b}\right ) \left (-\sqrt {b} \sqrt {a+b}+a+b\right ) \left (\frac {\sqrt {b} x^2}{\sqrt {a+b}}+1\right ) \sqrt {\frac {a+b x^4+b}{(a+b) \left (\frac {\sqrt {b} x^2}{\sqrt {a+b}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b}}\right ),\frac {1}{2}\right )}{\left (-2 b \sqrt {a+b}+2 a \sqrt {b}-a \sqrt {a+b}+2 b^{3/2}\right ) \sqrt {a+b x^4+b}}\) |
| \(\Big \downarrow \) 2223 |
| \(\displaystyle \frac {a \sqrt {a+b} \left (\frac {\left (2 a \sqrt {b}-\sqrt {a+b} (a+2 b)+2 b^{3/2}\right ) \text {arctanh}\left (\frac {x \sqrt {a+2 b}}{\sqrt {a+b x^4+b}}\right )}{2 \sqrt {a+2 b}}-\frac {a (a+b) \sqrt {\sqrt {b}-\sqrt {a+b}} \left (\frac {\sqrt {b} x^2 \left (-\sqrt {b} \sqrt {a+b}+a+b\right )}{(a+b) \left (\sqrt {b}-\sqrt {a+b}\right )}+1\right ) \sqrt {\frac {a+b x^4+b}{(a+b) \left (\frac {\sqrt {b} x^2 \left (-\sqrt {b} \sqrt {a+b}+a+b\right )}{(a+b) \left (\sqrt {b}-\sqrt {a+b}\right )}+1\right )^2}} \operatorname {EllipticPi}\left (\frac {a^2+8 b a+8 b^2-\sqrt {b} \sqrt {a+b} (4 a+8 b)}{4 \sqrt {b} \left (2 b^{3/2}-2 \sqrt {a+b} b+2 a \sqrt {b}-a \sqrt {a+b}\right )},2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {a+b-\sqrt {b} \sqrt {a+b}} x}{\sqrt {a+b} \sqrt {\sqrt {b}-\sqrt {a+b}}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{b} \sqrt {-\sqrt {b} \sqrt {a+b}+a+b} \sqrt {a+b x^4+b}}\right )}{-2 b \sqrt {a+b}+2 a \sqrt {b}-a \sqrt {a+b}+2 b^{3/2}}-\frac {\sqrt [4]{b} (a+b)^{5/4} \left (\sqrt {b}-\sqrt {a+b}\right ) \left (-\sqrt {b} \sqrt {a+b}+a+b\right ) \left (\frac {\sqrt {b} x^2}{\sqrt {a+b}}+1\right ) \sqrt {\frac {a+b x^4+b}{(a+b) \left (\frac {\sqrt {b} x^2}{\sqrt {a+b}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b}}\right ),\frac {1}{2}\right )}{\left (-2 b \sqrt {a+b}+2 a \sqrt {b}-a \sqrt {a+b}+2 b^{3/2}\right ) \sqrt {a+b x^4+b}}\) |
Input:
Int[((a + b)*(-Sqrt[b] + Sqrt[a + b]) + Sqrt[b]*(a + b - Sqrt[b]*Sqrt[a + b])*x^2)/((1 - x^2)*Sqrt[a + b + b*x^4]),x]
Output:
-((b^(1/4)*(a + b)^(5/4)*(Sqrt[b] - Sqrt[a + b])*(a + b - Sqrt[b]*Sqrt[a + b])*(1 + (Sqrt[b]*x^2)/Sqrt[a + b])*Sqrt[(a + b + b*x^4)/((a + b)*(1 + (S qrt[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]*(((2*a*Sqrt[b] + 2*b^(3/2) - Sqrt[a + b] *(a + 2*b))*ArcTanh[(Sqrt[a + 2*b]*x)/Sqrt[a + b + b*x^4]])/(2*Sqrt[a + 2* b]) - (a*(a + b)*Sqrt[Sqrt[b] - Sqrt[a + b]]*(1 + (Sqrt[b]*(a + b - Sqrt[b ]*Sqrt[a + b])*x^2)/((a + b)*(Sqrt[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)]*EllipticPi[(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)/(S qrt[a + b]*Sqrt[Sqrt[b] - Sqrt[a + b]])], 1/2])/(4*b^(1/4)*Sqrt[a + b - Sq rt[b]*Sqrt[a + b]]*Sqrt[a + b + b*x^4])))/(2*a*Sqrt[b] + 2*b^(3/2) - a*Sqr t[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 = 3.45 (sec) , antiderivative size = 582, normalized size of antiderivative = 3.03
| method | result | size |
| default | \(-\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}\) | \(582\) |
| elliptic | \(\text {Expression too large to display}\) | \(994\) |
Input:
int(((a+b)*(-b^(1/2)+(a+b)^(1/2))+b^(1/2)*(a+b-b^(1/2)*(a+b)^(1/2))*x^2)/( -x^2+1)/(b*x^4+a+b)^(1/2),x,method=_RETURNVERBOSE)
Output:
-b^(3/2)/(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)-b^(1/2)*a/(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)+b*(a+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*a*(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*a*(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)*Ellipti cPi(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)))
\[ \int \frac {(a+b) \left (-\sqrt {b}+\sqrt {a+b}\right )+\sqrt {b} \left (a+b-\sqrt {b} \sqrt {a+b}\right ) x^2}{\left (1-x^2\right ) \sqrt {a+b+b x^4}} \, dx=\int { -\frac {{\left (a - \sqrt {a + b} \sqrt {b} + b\right )} \sqrt {b} x^{2} + {\left (a + b\right )} {\left (\sqrt {a + b} - \sqrt {b}\right )}}{\sqrt {b x^{4} + a + b} {\left (x^{2} - 1\right )}} \,d x } \] Input:
integrate(((a+b)*(-b^(1/2)+(a+b)^(1/2))+b^(1/2)*(a+b-b^(1/2)*(a+b)^(1/2))* x^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)
\[ \int \frac {(a+b) \left (-\sqrt {b}+\sqrt {a+b}\right )+\sqrt {b} \left (a+b-\sqrt {b} \sqrt {a+b}\right ) x^2}{\left (1-x^2\right ) \sqrt {a+b+b x^4}} \, dx=- \int \left (- \frac {b^{\frac {3}{2}}}{x^{2} \sqrt {a + b x^{4} + b} - \sqrt {a + b x^{4} + b}}\right )\, dx - \int \left (- \frac {a \sqrt {b}}{x^{2} \sqrt {a + b x^{4} + b} - \sqrt {a + b x^{4} + b}}\right )\, dx - \int \frac {a \sqrt {a + b}}{x^{2} \sqrt {a + b x^{4} + b} - \sqrt {a + b x^{4} + b}}\, dx - \int \frac {b \sqrt {a + b}}{x^{2} \sqrt {a + b x^{4} + b} - \sqrt {a + b x^{4} + b}}\, dx - \int \frac {b^{\frac {3}{2}} x^{2}}{x^{2} \sqrt {a + b x^{4} + b} - \sqrt {a + b x^{4} + b}}\, dx - \int \frac {a \sqrt {b} x^{2}}{x^{2} \sqrt {a + b x^{4} + b} - \sqrt {a + b x^{4} + b}}\, dx - \int \left (- \frac {b x^{2} \sqrt {a + b}}{x^{2} \sqrt {a + b x^{4} + b} - \sqrt {a + b x^{4} + b}}\right )\, dx \] Input:
integrate(((a+b)*(-b**(1/2)+(a+b)**(1/2))+b**(1/2)*(a+b-b**(1/2)*(a+b)**(1 /2))*x**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) - sqrt (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)
\[ \int \frac {(a+b) \left (-\sqrt {b}+\sqrt {a+b}\right )+\sqrt {b} \left (a+b-\sqrt {b} \sqrt {a+b}\right ) x^2}{\left (1-x^2\right ) \sqrt {a+b+b x^4}} \, dx=\int { -\frac {{\left (a - \sqrt {a + b} \sqrt {b} + b\right )} \sqrt {b} x^{2} + {\left (a + b\right )} {\left (\sqrt {a + b} - \sqrt {b}\right )}}{\sqrt {b x^{4} + a + b} {\left (x^{2} - 1\right )}} \,d x } \] Input:
integrate(((a+b)*(-b^(1/2)+(a+b)^(1/2))+b^(1/2)*(a+b-b^(1/2)*(a+b)^(1/2))* x^2)/(-x^2+1)/(b*x^4+a+b)^(1/2),x, algorithm="maxima")
Output:
-integrate(((a - sqrt(a + b)*sqrt(b) + b)*sqrt(b)*x^2 + (a + b)*(sqrt(a + b) - sqrt(b)))/(sqrt(b*x^4 + a + b)*(x^2 - 1)), x)
\[ \int \frac {(a+b) \left (-\sqrt {b}+\sqrt {a+b}\right )+\sqrt {b} \left (a+b-\sqrt {b} \sqrt {a+b}\right ) x^2}{\left (1-x^2\right ) \sqrt {a+b+b x^4}} \, dx=\int { -\frac {{\left (a - \sqrt {a + b} \sqrt {b} + b\right )} \sqrt {b} x^{2} + {\left (a + b\right )} {\left (\sqrt {a + b} - \sqrt {b}\right )}}{\sqrt {b x^{4} + a + b} {\left (x^{2} - 1\right )}} \,d x } \] Input:
integrate(((a+b)*(-b^(1/2)+(a+b)^(1/2))+b^(1/2)*(a+b-b^(1/2)*(a+b)^(1/2))* x^2)/(-x^2+1)/(b*x^4+a+b)^(1/2),x, algorithm="giac")
Output:
integrate(-((a - sqrt(a + b)*sqrt(b) + b)*sqrt(b)*x^2 + (a + b)*(sqrt(a + b) - sqrt(b)))/(sqrt(b*x^4 + a + b)*(x^2 - 1)), x)
Timed out. \[ \int \frac {(a+b) \left (-\sqrt {b}+\sqrt {a+b}\right )+\sqrt {b} \left (a+b-\sqrt {b} \sqrt {a+b}\right ) x^2}{\left (1-x^2\right ) \sqrt {a+b+b x^4}} \, dx=\int -\frac {\left (\sqrt {a+b}-\sqrt {b}\right )\,\left (a+b\right )+\sqrt {b}\,x^2\,\left (a+b-\sqrt {b}\,\sqrt {a+b}\right )}{\left (x^2-1\right )\,\sqrt {b\,x^4+a+b}} \,d x \] Input:
int(-(((a + b)^(1/2) - b^(1/2))*(a + b) + b^(1/2)*x^2*(a + b - b^(1/2)*(a + b)^(1/2)))/((x^2 - 1)*(a + b + b*x^4)^(1/2)),x)
Output:
int(-(((a + b)^(1/2) - b^(1/2))*(a + b) + b^(1/2)*x^2*(a + b - b^(1/2)*(a + b)^(1/2)))/((x^2 - 1)*(a + b + b*x^4)^(1/2)), x)
\[ \int \frac {(a+b) \left (-\sqrt {b}+\sqrt {a+b}\right )+\sqrt {b} \left (a+b-\sqrt {b} \sqrt {a+b}\right ) x^2}{\left (1-x^2\right ) \sqrt {a+b+b x^4}} \, dx=-\sqrt {a +b}\, \left (\int \frac {\sqrt {b \,x^{4}+a +b}}{b \,x^{6}-b \,x^{4}+a \,x^{2}+b \,x^{2}-a -b}d x \right ) a -\sqrt {a +b}\, \left (\int \frac {\sqrt {b \,x^{4}+a +b}}{b \,x^{6}-b \,x^{4}+a \,x^{2}+b \,x^{2}-a -b}d x \right ) b +\sqrt {a +b}\, \left (\int \frac {\sqrt {b \,x^{4}+a +b}\, x^{2}}{b \,x^{6}-b \,x^{4}+a \,x^{2}+b \,x^{2}-a -b}d x \right ) b +\sqrt {b}\, \left (\int \frac {\sqrt {b \,x^{4}+a +b}}{b \,x^{6}-b \,x^{4}+a \,x^{2}+b \,x^{2}-a -b}d x \right ) a +\sqrt {b}\, \left (\int \frac {\sqrt {b \,x^{4}+a +b}}{b \,x^{6}-b \,x^{4}+a \,x^{2}+b \,x^{2}-a -b}d x \right ) b -\sqrt {b}\, \left (\int \frac {\sqrt {b \,x^{4}+a +b}\, x^{2}}{b \,x^{6}-b \,x^{4}+a \,x^{2}+b \,x^{2}-a -b}d x \right ) a -\sqrt {b}\, \left (\int \frac {\sqrt {b \,x^{4}+a +b}\, x^{2}}{b \,x^{6}-b \,x^{4}+a \,x^{2}+b \,x^{2}-a -b}d x \right ) b \] Input:
int(((a+b)*(-b^(1/2)+(a+b)^(1/2))+b^(1/2)*(a+b-b^(1/2)*(a+b)^(1/2))*x^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