Integrand size = 28, antiderivative size = 641 \[ \int \frac {A+B x^2}{\left (d+e x^2\right )^2 \sqrt {a+c x^4}} \, dx=\frac {\sqrt {c} (B d-A e) x \sqrt {a+c x^4}}{2 d \left (c d^2+a e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {e (B d-A e) x \sqrt {a+c x^4}}{2 d \left (c d^2+a e^2\right ) \left (d+e x^2\right )}-\frac {\left (B c d^3-3 A c d^2 e-a B d e^2-a A e^3\right ) \arctan \left (\frac {\sqrt {c d^2+a e^2} x}{\sqrt {d} \sqrt {e} \sqrt {a+c x^4}}\right )}{4 d^{3/2} \sqrt {e} \left (c d^2+a e^2\right )^{3/2}}-\frac {\sqrt [4]{a} \sqrt [4]{c} (B d-A e) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 d \left (c d^2+a e^2\right ) \sqrt {a+c x^4}}+\frac {A \sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} d \left (\sqrt {c} d-\sqrt {a} e\right ) \sqrt {a+c x^4}}+\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) \left (B c d^3-3 A c d^2 e-a B d e^2-a A e^3\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2}{4 \sqrt {a} \sqrt {c} d e},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{8 \sqrt [4]{a} \sqrt [4]{c} d^2 e \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right ) \sqrt {a+c x^4}} \]
-1/4*(-A*a*e^3-3*A*c*d^2*e-B*a*d*e^2+B*c*d^3)*arctan(x*(a*e^2+c*d^2)^(1/2) /d^(1/2)/e^(1/2)/(c*x^4+a)^(1/2))/d^(3/2)/(a*e^2+c*d^2)^(3/2)/e^(1/2)-1/2* e*(-A*e+B*d)*x*(c*x^4+a)^(1/2)/d/(a*e^2+c*d^2)/(e*x^2+d)+1/2*(-A*e+B*d)*x* c^(1/2)*(c*x^4+a)^(1/2)/d/(a*e^2+c*d^2)/(a^(1/2)+x^2*c^(1/2))-1/2*a^(1/4)* c^(1/4)*(-A*e+B*d)*(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2*arctan (c^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),1/2*2^(1/2 ))*(a^(1/2)+x^2*c^(1/2))*((c*x^4+a)/(a^(1/2)+x^2*c^(1/2))^2)^(1/2)/d/(a*e^ 2+c*d^2)/(c*x^4+a)^(1/2)+1/2*A*c^(1/4)*(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2 )^(1/2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/ a^(1/4))),1/2*2^(1/2))*(a^(1/2)+x^2*c^(1/2))*((c*x^4+a)/(a^(1/2)+x^2*c^(1/ 2))^2)^(1/2)/a^(1/4)/d/(-e*a^(1/2)+d*c^(1/2))/(c*x^4+a)^(1/2)+1/8*(-A*a*e^ 3-3*A*c*d^2*e-B*a*d*e^2+B*c*d^3)*(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)^(1/2 )/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticPi(sin(2*arctan(c^(1/4)*x/a^(1/ 4))),-1/4*(-e*a^(1/2)+d*c^(1/2))^2/d/e/a^(1/2)/c^(1/2),1/2*2^(1/2))*(e*a^( 1/2)+d*c^(1/2))*(a^(1/2)+x^2*c^(1/2))*((c*x^4+a)/(a^(1/2)+x^2*c^(1/2))^2)^ (1/2)/a^(1/4)/c^(1/4)/d^2/e/(a*e^2+c*d^2)/(-e*a^(1/2)+d*c^(1/2))/(c*x^4+a) ^(1/2)
Result contains complex when optimal does not.
Time = 10.79 (sec) , antiderivative size = 297, normalized size of antiderivative = 0.46 \[ \int \frac {A+B x^2}{\left (d+e x^2\right )^2 \sqrt {a+c x^4}} \, dx=\frac {\frac {d e (-B d+A e) x \left (a+c x^4\right )}{\left (c d^2+a e^2\right ) \left (d+e x^2\right )}-\frac {i \sqrt {1+\frac {c x^4}{a}} \left (i \sqrt {a} \sqrt {c} d e (B d-A e) E\left (\left .i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )+\sqrt {c} d \left (\sqrt {c} d-i \sqrt {a} e\right ) (B d-A e) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right ),-1\right )+\left (-B c d^3+3 A c d^2 e+a B d e^2+a A e^3\right ) \operatorname {EllipticPi}\left (-\frac {i \sqrt {a} e}{\sqrt {c} d},i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right ),-1\right )\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} \left (c d^2 e+a e^3\right )}}{2 d^2 \sqrt {a+c x^4}} \]
((d*e*(-(B*d) + A*e)*x*(a + c*x^4))/((c*d^2 + a*e^2)*(d + e*x^2)) - (I*Sqr t[1 + (c*x^4)/a]*(I*Sqrt[a]*Sqrt[c]*d*e*(B*d - A*e)*EllipticE[I*ArcSinh[Sq rt[(I*Sqrt[c])/Sqrt[a]]*x], -1] + Sqrt[c]*d*(Sqrt[c]*d - I*Sqrt[a]*e)*(B*d - A*e)*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] + (-(B*c*d^3 ) + 3*A*c*d^2*e + a*B*d*e^2 + a*A*e^3)*EllipticPi[((-I)*Sqrt[a]*e)/(Sqrt[c ]*d), I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1]))/(Sqrt[(I*Sqrt[c])/Sqrt [a]]*(c*d^2*e + a*e^3)))/(2*d^2*Sqrt[a + c*x^4])
Time = 1.29 (sec) , antiderivative size = 600, normalized size of antiderivative = 0.94, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {2211, 25, 2233, 27, 1510, 2227, 27, 761, 2221}
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.
| \(\displaystyle \int \frac {A+B x^2}{\sqrt {a+c x^4} \left (d+e x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 2211 |
| \(\displaystyle -\frac {\int -\frac {c e (B d-A e) x^4+2 c d (B d-A e) x^2+2 A c d^2+a A e^2+a B d e}{\left (e x^2+d\right ) \sqrt {c x^4+a}}dx}{2 d \left (a e^2+c d^2\right )}-\frac {e x \sqrt {a+c x^4} (B d-A e)}{2 d \left (d+e x^2\right ) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {c e (B d-A e) x^4+2 c d (B d-A e) x^2+2 A c d^2+a A e^2+a B d e}{\left (e x^2+d\right ) \sqrt {c x^4+a}}dx}{2 d \left (a e^2+c d^2\right )}-\frac {e x \sqrt {a+c x^4} (B d-A e)}{2 d \left (d+e x^2\right ) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 2233 |
| \(\displaystyle \frac {\frac {\int \frac {c e \left (2 A c d^2+\sqrt {a} \sqrt {c} (B d-A e) d+\sqrt {c} \left (\sqrt {c} d+\sqrt {a} e\right ) (B d-A e) x^2+a e (B d+A e)\right )}{\left (e x^2+d\right ) \sqrt {c x^4+a}}dx}{c e}-\sqrt {a} \sqrt {c} (B d-A e) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {a} \sqrt {c x^4+a}}dx}{2 d \left (a e^2+c d^2\right )}-\frac {e x \sqrt {a+c x^4} (B d-A e)}{2 d \left (d+e x^2\right ) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {2 A c d^2+\sqrt {a} \sqrt {c} (B d-A e) d+\sqrt {c} \left (\sqrt {c} d+\sqrt {a} e\right ) (B d-A e) x^2+a e (B d+A e)}{\left (e x^2+d\right ) \sqrt {c x^4+a}}dx-\sqrt {c} (B d-A e) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+a}}dx}{2 d \left (a e^2+c d^2\right )}-\frac {e x \sqrt {a+c x^4} (B d-A e)}{2 d \left (d+e x^2\right ) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle \frac {\int \frac {2 A c d^2+\sqrt {a} \sqrt {c} (B d-A e) d+\sqrt {c} \left (\sqrt {c} d+\sqrt {a} e\right ) (B d-A e) x^2+a e (B d+A e)}{\left (e x^2+d\right ) \sqrt {c x^4+a}}dx-\sqrt {c} (B d-A e) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^4}}-\frac {x \sqrt {a+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{2 d \left (a e^2+c d^2\right )}-\frac {e x \sqrt {a+c x^4} (B d-A e)}{2 d \left (d+e x^2\right ) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 2227 |
| \(\displaystyle \frac {\frac {\sqrt {a} \left (-a A e^3-a B d e^2-3 A c d^2 e+B c d^3\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a} \left (e x^2+d\right ) \sqrt {c x^4+a}}dx}{\sqrt {c} d-\sqrt {a} e}+\frac {2 A \sqrt {c} \left (a e^2+c d^2\right ) \int \frac {1}{\sqrt {c x^4+a}}dx}{\sqrt {c} d-\sqrt {a} e}-\sqrt {c} (B d-A e) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^4}}-\frac {x \sqrt {a+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{2 d \left (a e^2+c d^2\right )}-\frac {e x \sqrt {a+c x^4} (B d-A e)}{2 d \left (d+e x^2\right ) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\left (-a A e^3-a B d e^2-3 A c d^2 e+B c d^3\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (e x^2+d\right ) \sqrt {c x^4+a}}dx}{\sqrt {c} d-\sqrt {a} e}+\frac {2 A \sqrt {c} \left (a e^2+c d^2\right ) \int \frac {1}{\sqrt {c x^4+a}}dx}{\sqrt {c} d-\sqrt {a} e}-\sqrt {c} (B d-A e) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^4}}-\frac {x \sqrt {a+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{2 d \left (a e^2+c d^2\right )}-\frac {e x \sqrt {a+c x^4} (B d-A e)}{2 d \left (d+e x^2\right ) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {\frac {\left (-a A e^3-a B d e^2-3 A c d^2 e+B c d^3\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (e x^2+d\right ) \sqrt {c x^4+a}}dx}{\sqrt {c} d-\sqrt {a} e}-\sqrt {c} (B d-A e) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^4}}-\frac {x \sqrt {a+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )+\frac {A \sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (a e^2+c d^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{\sqrt [4]{a} \sqrt {a+c x^4} \left (\sqrt {c} d-\sqrt {a} e\right )}}{2 d \left (a e^2+c d^2\right )}-\frac {e x \sqrt {a+c x^4} (B d-A e)}{2 d \left (d+e x^2\right ) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 2221 |
| \(\displaystyle \frac {\frac {\left (-a A e^3-a B d e^2-3 A c d^2 e+B c d^3\right ) \left (\frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (\sqrt {a} e+\sqrt {c} d\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \left (\frac {\sqrt {c} d}{\sqrt {a}}-e\right )^2}{4 \sqrt {c} d e},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{c} d e \sqrt {a+c x^4}}-\frac {\left (\sqrt {c} d-\sqrt {a} e\right ) \arctan \left (\frac {x \sqrt {a e^2+c d^2}}{\sqrt {d} \sqrt {e} \sqrt {a+c x^4}}\right )}{2 \sqrt {d} \sqrt {e} \sqrt {a e^2+c d^2}}\right )}{\sqrt {c} d-\sqrt {a} e}-\sqrt {c} (B d-A e) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^4}}-\frac {x \sqrt {a+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )+\frac {A \sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (a e^2+c d^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{\sqrt [4]{a} \sqrt {a+c x^4} \left (\sqrt {c} d-\sqrt {a} e\right )}}{2 d \left (a e^2+c d^2\right )}-\frac {e x \sqrt {a+c x^4} (B d-A e)}{2 d \left (d+e x^2\right ) \left (a e^2+c d^2\right )}\) |
-1/2*(e*(B*d - A*e)*x*Sqrt[a + c*x^4])/(d*(c*d^2 + a*e^2)*(d + e*x^2)) + ( -(Sqrt[c]*(B*d - A*e)*(-((x*Sqrt[a + c*x^4])/(Sqrt[a] + Sqrt[c]*x^2)) + (a ^(1/4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2] *EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(c^(1/4)*Sqrt[a + c*x^4])) ) + (A*c^(1/4)*(c*d^2 + a*e^2)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(S qrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(a ^(1/4)*(Sqrt[c]*d - Sqrt[a]*e)*Sqrt[a + c*x^4]) + ((B*c*d^3 - 3*A*c*d^2*e - a*B*d*e^2 - a*A*e^3)*(-1/2*((Sqrt[c]*d - Sqrt[a]*e)*ArcTan[(Sqrt[c*d^2 + a*e^2]*x)/(Sqrt[d]*Sqrt[e]*Sqrt[a + c*x^4])])/(Sqrt[d]*Sqrt[e]*Sqrt[c*d^2 + a*e^2]) + ((Sqrt[c]*d + Sqrt[a]*e)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c* x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticPi[-1/4*(Sqrt[a]*((Sqrt[c]*d)/Sqrt [a] - e)^2)/(Sqrt[c]*d*e), 2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(4*a^(1/4) *c^(1/4)*d*e*Sqrt[a + c*x^4])))/(Sqrt[c]*d - Sqrt[a]*e))/(2*d*(c*d^2 + a*e ^2))
3.1.6.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d* (1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q*Sqrt[a + c*x^4]))*E llipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e }, x] && PosQ[c/a]
Int[((P4x_)*((d_) + (e_.)*(x_)^2)^(q_))/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol ] :> With[{A = Coeff[P4x, x, 0], B = Coeff[P4x, x, 2], C = Coeff[P4x, x, 4] }, Simp[(-(C*d^2 - B*d*e + A*e^2))*x*(d + e*x^2)^(q + 1)*(Sqrt[a + c*x^4]/( 2*d*(q + 1)*(c*d^2 + a*e^2))), x] + Simp[1/(2*d*(q + 1)*(c*d^2 + a*e^2)) Int[((d + e*x^2)^(q + 1)/Sqrt[a + c*x^4])*Simp[a*d*(C*d - B*e) + A*(a*e^2*( 2*q + 3) + 2*c*d^2*(q + 1)) + 2*d*(B*c*d - A*c*e + a*C*e)*(q + 1)*x^2 + c*( C*d^2 - B*d*e + A*e^2)*(2*q + 5)*x^4, x], x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[P4x, x^2] && LeQ[Expon[P4x, x], 4] && ILtQ[q, -1]
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))*(ArcTan[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*ArcTan[q*x ], 1/2], x]] /; FreeQ[{a, c, d, e, A, B}, x] && NeQ[c*d^2 - a*e^2, 0] && Po sQ[c/a] && EqQ[c*A^2 - a*B^2, 0] && PosQ[B/A] && PosQ[c*(d/e) + a*(e/d)]
Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]) , x_Symbol] :> With[{q = Rt[c/a, 2]}, Simp[(A*(c*d + a*e*q) - a*B*(e + d*q) )/(c*d^2 - a*e^2) Int[1/Sqrt[a + c*x^4], x], x] + Simp[a*(B*d - A*e)*((e + d*q)/(c*d^2 - a*e^2)) Int[(1 + q*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] && NeQ[c*A^2 - a*B^2, 0]
Int[(P4x_)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[c/a, 2], A = Coeff[P4x, x, 0], B = Coeff[P4x, x, 2], C = Coeff [P4x, x, 4]}, Simp[-C/(e*q) Int[(1 - q*x^2)/Sqrt[a + c*x^4], x], x] + Sim p[1/(c*e) Int[(A*c*e + a*C*d*q + (B*c*e - C*(c*d - a*e*q))*x^2)/((d + e*x ^2)*Sqrt[a + c*x^4]), x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[P4x, x^2, 2] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a]
Result contains complex when optimal does not.
Time = 1.42 (sec) , antiderivative size = 679, normalized size of antiderivative = 1.06
| method | result | size |
| default | \(\frac {B \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \Pi \left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, \frac {i \sqrt {a}\, e}{\sqrt {c}\, d}, \frac {\sqrt {-\frac {i \sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}\right )}{e d \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {\left (A e -B d \right ) \left (\frac {e^{2} x \sqrt {c \,x^{4}+a}}{2 d \left (a \,e^{2}+c \,d^{2}\right ) \left (e \,x^{2}+d \right )}-\frac {c \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{2 \left (a \,e^{2}+c \,d^{2}\right ) \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}-\frac {i \sqrt {c}\, e \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{2 d \left (a \,e^{2}+c \,d^{2}\right ) \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {i \sqrt {c}\, e \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, E\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{2 d \left (a \,e^{2}+c \,d^{2}\right ) \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {e^{2} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \Pi \left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, \frac {i \sqrt {a}\, e}{\sqrt {c}\, d}, \frac {\sqrt {-\frac {i \sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}\right ) a}{2 d^{2} \left (a \,e^{2}+c \,d^{2}\right ) \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {3 \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \Pi \left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, \frac {i \sqrt {a}\, e}{\sqrt {c}\, d}, \frac {\sqrt {-\frac {i \sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}\right ) c}{2 \left (a \,e^{2}+c \,d^{2}\right ) \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\right )}{e}\) | \(679\) |
| elliptic | \(\text {Expression too large to display}\) | \(1084\) |
B/e/d/(I/a^(1/2)*c^(1/2))^(1/2)*(1-I/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+I/a^(1/ 2)*c^(1/2)*x^2)^(1/2)/(c*x^4+a)^(1/2)*EllipticPi(x*(I/a^(1/2)*c^(1/2))^(1/ 2),I*a^(1/2)/c^(1/2)*e/d,(-I/a^(1/2)*c^(1/2))^(1/2)/(I/a^(1/2)*c^(1/2))^(1 /2))+(A*e-B*d)/e*(1/2*e^2*x*(c*x^4+a)^(1/2)/d/(a*e^2+c*d^2)/(e*x^2+d)-1/2* c/(a*e^2+c*d^2)/(I/a^(1/2)*c^(1/2))^(1/2)*(1-I/a^(1/2)*c^(1/2)*x^2)^(1/2)* (1+I/a^(1/2)*c^(1/2)*x^2)^(1/2)/(c*x^4+a)^(1/2)*EllipticF(x*(I/a^(1/2)*c^( 1/2))^(1/2),I)-1/2*I*c^(1/2)*e/d/(a*e^2+c*d^2)*a^(1/2)/(I/a^(1/2)*c^(1/2)) ^(1/2)*(1-I/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*c^(1/2)*x^2)^(1/2)/(c* x^4+a)^(1/2)*EllipticF(x*(I/a^(1/2)*c^(1/2))^(1/2),I)+1/2*I*c^(1/2)*e/d/(a *e^2+c*d^2)*a^(1/2)/(I/a^(1/2)*c^(1/2))^(1/2)*(1-I/a^(1/2)*c^(1/2)*x^2)^(1 /2)*(1+I/a^(1/2)*c^(1/2)*x^2)^(1/2)/(c*x^4+a)^(1/2)*EllipticE(x*(I/a^(1/2) *c^(1/2))^(1/2),I)+1/2/d^2/(a*e^2+c*d^2)*e^2/(I/a^(1/2)*c^(1/2))^(1/2)*(1- I/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*c^(1/2)*x^2)^(1/2)/(c*x^4+a)^(1/ 2)*EllipticPi(x*(I/a^(1/2)*c^(1/2))^(1/2),I*a^(1/2)/c^(1/2)*e/d,(-I/a^(1/2 )*c^(1/2))^(1/2)/(I/a^(1/2)*c^(1/2))^(1/2))*a+3/2/(a*e^2+c*d^2)/(I/a^(1/2) *c^(1/2))^(1/2)*(1-I/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*c^(1/2)*x^2)^ (1/2)/(c*x^4+a)^(1/2)*EllipticPi(x*(I/a^(1/2)*c^(1/2))^(1/2),I*a^(1/2)/c^( 1/2)*e/d,(-I/a^(1/2)*c^(1/2))^(1/2)/(I/a^(1/2)*c^(1/2))^(1/2))*c)
Timed out. \[ \int \frac {A+B x^2}{\left (d+e x^2\right )^2 \sqrt {a+c x^4}} \, dx=\text {Timed out} \]
\[ \int \frac {A+B x^2}{\left (d+e x^2\right )^2 \sqrt {a+c x^4}} \, dx=\int \frac {A + B x^{2}}{\sqrt {a + c x^{4}} \left (d + e x^{2}\right )^{2}}\, dx \]
\[ \int \frac {A+B x^2}{\left (d+e x^2\right )^2 \sqrt {a+c x^4}} \, dx=\int { \frac {B x^{2} + A}{\sqrt {c x^{4} + a} {\left (e x^{2} + d\right )}^{2}} \,d x } \]
\[ \int \frac {A+B x^2}{\left (d+e x^2\right )^2 \sqrt {a+c x^4}} \, dx=\int { \frac {B x^{2} + A}{\sqrt {c x^{4} + a} {\left (e x^{2} + d\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {A+B x^2}{\left (d+e x^2\right )^2 \sqrt {a+c x^4}} \, dx=\int \frac {B\,x^2+A}{\sqrt {c\,x^4+a}\,{\left (e\,x^2+d\right )}^2} \,d x \]