Integrand size = 33, antiderivative size = 528 \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^2}{\sqrt {a+b x^2+c x^4}} \, dx=\frac {e (10 B c d-4 b B e+5 A c e) x \sqrt {a+b x^2+c x^4}}{15 c^2}+\frac {B e^2 x^3 \sqrt {a+b x^2+c x^4}}{5 c}+\frac {\left (10 A c e (3 c d-b e)+B \left (15 c^2 d^2+8 b^2 e^2-c e (20 b d+9 a e)\right )\right ) x \sqrt {a+b x^2+c x^4}}{15 c^{5/2} \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {\sqrt [4]{a} \left (10 A c e (3 c d-b e)+B \left (15 c^2 d^2+8 b^2 e^2-c e (20 b d+9 a e)\right )\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{15 c^{11/4} \sqrt {a+b x^2+c x^4}}+\frac {\sqrt [4]{a} \left (10 A c e (3 c d-b e)+B \left (15 c^2 d^2+8 b^2 e^2-c e (20 b d+9 a e)\right )-\frac {\sqrt {c} \left (2 a B e (5 c d-2 b e)-5 A c \left (3 c d^2-a e^2\right )\right )}{\sqrt {a}}\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{30 c^{11/4} \sqrt {a+b x^2+c x^4}} \]
1/15*e*(5*A*c*e-4*B*b*e+10*B*c*d)*x*(c*x^4+b*x^2+a)^(1/2)/c^2+1/5*B*e^2*x^ 3*(c*x^4+b*x^2+a)^(1/2)/c+1/15*(10*A*c*e*(-b*e+3*c*d)+B*(15*c^2*d^2+8*b^2* e^2-c*e*(9*a*e+20*b*d)))*x*(c*x^4+b*x^2+a)^(1/2)/c^(5/2)/(a^(1/2)+x^2*c^(1 /2))-1/15*a^(1/4)*(10*A*c*e*(-b*e+3*c*d)+B*(15*c^2*d^2+8*b^2*e^2-c*e*(9*a* e+20*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-b/a^(1/2)/ c^(1/2))^(1/2))*(a^(1/2)+x^2*c^(1/2))*((c*x^4+b*x^2+a)/(a^(1/2)+x^2*c^(1/2 ))^2)^(1/2)/c^(11/4)/(c*x^4+b*x^2+a)^(1/2)+1/30*a^(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-b/a^(1/2)/c^(1/2))^(1/2))*(a^(1/2)+x^2*c ^(1/2))*(10*A*c*e*(-b*e+3*c*d)+B*(15*c^2*d^2+8*b^2*e^2-c*e*(9*a*e+20*b*d)) -(2*a*B*e*(-2*b*e+5*c*d)-5*A*c*(-a*e^2+3*c*d^2))*c^(1/2)/a^(1/2))*((c*x^4+ b*x^2+a)/(a^(1/2)+x^2*c^(1/2))^2)^(1/2)/c^(11/4)/(c*x^4+b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 12.81 (sec) , antiderivative size = 674, normalized size of antiderivative = 1.28 \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^2}{\sqrt {a+b x^2+c x^4}} \, dx=\frac {4 c \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} e x \left (a+b x^2+c x^4\right ) \left (5 A c e+B \left (10 c d-4 b e+3 c e x^2\right )\right )+i \left (-b+\sqrt {b^2-4 a c}\right ) \left (10 A c e (3 c d-b e)+B \left (15 c^2 d^2+8 b^2 e^2-c e (20 b d+9 a e)\right )\right ) \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+4 c x^2}{b-\sqrt {b^2-4 a c}}} E\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )-i \left (-8 b^3 B e^2+2 b^2 e \left (10 B c d+5 A c e+4 B \sqrt {b^2-4 a c} e\right )-b c \left (15 B c d^2+B e \left (20 \sqrt {b^2-4 a c} d-17 a e\right )+10 A e \left (3 c d+\sqrt {b^2-4 a c} e\right )\right )+c \left (B \left (-9 a \sqrt {b^2-4 a c} e^2+5 c d \left (3 \sqrt {b^2-4 a c} d-4 a e\right )\right )+10 A c \left (3 c d^2+e \left (3 \sqrt {b^2-4 a c} d-a e\right )\right )\right )\right ) \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+4 c x^2}{b-\sqrt {b^2-4 a c}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{60 c^3 \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \sqrt {a+b x^2+c x^4}} \]
(4*c*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*e*x*(a + b*x^2 + c*x^4)*(5*A*c*e + B* (10*c*d - 4*b*e + 3*c*e*x^2)) + I*(-b + Sqrt[b^2 - 4*a*c])*(10*A*c*e*(3*c* d - b*e) + B*(15*c^2*d^2 + 8*b^2*e^2 - c*e*(20*b*d + 9*a*e)))*Sqrt[(b + Sq rt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[(2*b - 2*Sqrt[b^2 - 4*a*c] + 4*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*EllipticE[I*ArcSinh[Sqrt[2]* Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] - I*(-8*b^3*B*e^2 + 2*b^2*e*(10*B*c*d + 5*A*c*e + 4*B*Sqrt[b^2 - 4*a*c]*e) - b*c*(15*B*c*d^2 + B*e*(20*Sqrt[b^2 - 4*a*c]*d - 17*a*e) + 10 *A*e*(3*c*d + Sqrt[b^2 - 4*a*c]*e)) + c*(B*(-9*a*Sqrt[b^2 - 4*a*c]*e^2 + 5 *c*d*(3*Sqrt[b^2 - 4*a*c]*d - 4*a*e)) + 10*A*c*(3*c*d^2 + e*(3*Sqrt[b^2 - 4*a*c]*d - a*e))))*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[(2*b - 2*Sqrt[b^2 - 4*a*c] + 4*c*x^2)/(b - Sqrt[b^2 - 4*a*c] )]*EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sq rt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])])/(60*c^3*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[a + b*x^2 + c*x^4])
Time = 0.89 (sec) , antiderivative size = 498, normalized size of antiderivative = 0.94, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {2207, 2207, 25, 1511, 27, 1416, 1509}
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.
| \(\displaystyle \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^2}{\sqrt {a+b x^2+c x^4}} \, dx\) |
| \(\Big \downarrow \) 2207 |
| \(\displaystyle \frac {\int \frac {e (10 B c d-4 b B e+5 A c e) x^4+\left (5 B c d^2+10 A c e d-3 a B e^2\right ) x^2+5 A c d^2}{\sqrt {c x^4+b x^2+a}}dx}{5 c}+\frac {B e^2 x^3 \sqrt {a+b x^2+c x^4}}{5 c}\) |
| \(\Big \downarrow \) 2207 |
| \(\displaystyle \frac {\frac {\int -\frac {-\left (\left (10 A c e (3 c d-b e)+B \left (15 c^2 d^2+8 b^2 e^2-c e (20 b d+9 a e)\right )\right ) x^2\right )+2 a B e (5 c d-2 b e)-5 A c \left (3 c d^2-a e^2\right )}{\sqrt {c x^4+b x^2+a}}dx}{3 c}+\frac {e x \sqrt {a+b x^2+c x^4} (5 A c e-4 b B e+10 B c d)}{3 c}}{5 c}+\frac {B e^2 x^3 \sqrt {a+b x^2+c x^4}}{5 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {e x \sqrt {a+b x^2+c x^4} (5 A c e-4 b B e+10 B c d)}{3 c}-\frac {\int \frac {-\left (\left (10 A c e (3 c d-b e)+B \left (15 c^2 d^2+8 b^2 e^2-c e (20 b d+9 a e)\right )\right ) x^2\right )+2 a B e (5 c d-2 b e)-5 A c \left (3 c d^2-a e^2\right )}{\sqrt {c x^4+b x^2+a}}dx}{3 c}}{5 c}+\frac {B e^2 x^3 \sqrt {a+b x^2+c x^4}}{5 c}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {\frac {e x \sqrt {a+b x^2+c x^4} (5 A c e-4 b B e+10 B c d)}{3 c}-\frac {\frac {\sqrt {a} \left (B \left (-c e (9 a e+20 b d)+8 b^2 e^2+15 c^2 d^2\right )+10 A c e (3 c d-b e)\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {a} \sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}-\frac {\left (-9 a^{3/2} B c e^2+\sqrt {a} \left (10 A c e (3 c d-b e)+B \left (8 b^2 e^2-20 b c d e+15 c^2 d^2\right )\right )-a \sqrt {c} e (5 A c e-4 b B e+10 B c d)+15 A c^{5/2} d^2\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}}{3 c}}{5 c}+\frac {B e^2 x^3 \sqrt {a+b x^2+c x^4}}{5 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {e x \sqrt {a+b x^2+c x^4} (5 A c e-4 b B e+10 B c d)}{3 c}-\frac {\frac {\left (B \left (-c e (9 a e+20 b d)+8 b^2 e^2+15 c^2 d^2\right )+10 A c e (3 c d-b e)\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}-\frac {\left (-9 a^{3/2} B c e^2+\sqrt {a} \left (10 A c e (3 c d-b e)+B \left (8 b^2 e^2-20 b c d e+15 c^2 d^2\right )\right )-a \sqrt {c} e (5 A c e-4 b B e+10 B c d)+15 A c^{5/2} d^2\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}}{3 c}}{5 c}+\frac {B e^2 x^3 \sqrt {a+b x^2+c x^4}}{5 c}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {\frac {e x \sqrt {a+b x^2+c x^4} (5 A c e-4 b B e+10 B c d)}{3 c}-\frac {\frac {\left (B \left (-c e (9 a e+20 b d)+8 b^2 e^2+15 c^2 d^2\right )+10 A c e (3 c d-b e)\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}-\frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) \left (-9 a^{3/2} B c e^2+\sqrt {a} \left (10 A c e (3 c d-b e)+B \left (8 b^2 e^2-20 b c d e+15 c^2 d^2\right )\right )-a \sqrt {c} e (5 A c e-4 b B e+10 B c d)+15 A c^{5/2} d^2\right )}{2 \sqrt [4]{a} c^{3/4} \sqrt {a+b x^2+c x^4}}}{3 c}}{5 c}+\frac {B e^2 x^3 \sqrt {a+b x^2+c x^4}}{5 c}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {\frac {e x \sqrt {a+b x^2+c x^4} (5 A c e-4 b B e+10 B c d)}{3 c}-\frac {\frac {\left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right ) \left (B \left (-c e (9 a e+20 b d)+8 b^2 e^2+15 c^2 d^2\right )+10 A c e (3 c d-b e)\right )}{\sqrt {c}}-\frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) \left (-9 a^{3/2} B c e^2+\sqrt {a} \left (10 A c e (3 c d-b e)+B \left (8 b^2 e^2-20 b c d e+15 c^2 d^2\right )\right )-a \sqrt {c} e (5 A c e-4 b B e+10 B c d)+15 A c^{5/2} d^2\right )}{2 \sqrt [4]{a} c^{3/4} \sqrt {a+b x^2+c x^4}}}{3 c}}{5 c}+\frac {B e^2 x^3 \sqrt {a+b x^2+c x^4}}{5 c}\) |
(B*e^2*x^3*Sqrt[a + b*x^2 + c*x^4])/(5*c) + ((e*(10*B*c*d - 4*b*B*e + 5*A* c*e)*x*Sqrt[a + b*x^2 + c*x^4])/(3*c) - (((10*A*c*e*(3*c*d - b*e) + B*(15* c^2*d^2 + 8*b^2*e^2 - c*e*(20*b*d + 9*a*e)))*(-((x*Sqrt[a + b*x^2 + c*x^4] )/(Sqrt[a] + Sqrt[c]*x^2)) + (a^(1/4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b* x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^( 1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(c^(1/4)*Sqrt[a + b*x^2 + c*x^4])))/S qrt[c] - ((15*A*c^(5/2)*d^2 - 9*a^(3/2)*B*c*e^2 - a*Sqrt[c]*e*(10*B*c*d - 4*b*B*e + 5*A*c*e) + Sqrt[a]*(10*A*c*e*(3*c*d - b*e) + B*(15*c^2*d^2 - 20* b*c*d*e + 8*b^2*e^2)))*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(S qrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/( Sqrt[a]*Sqrt[c]))/4])/(2*a^(1/4)*c^(3/4)*Sqrt[a + b*x^2 + c*x^4]))/(3*c))/ (5*c)
3.1.18.3.1 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_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c /a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ (2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) ], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q ^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2* x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2 /(4*c))], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q Int[1/Sqrt[a + b*x^2 + c*x^ 4], x], x] - Simp[e/q Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] /; NeQ[e + d*q, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && Pos Q[c/a]
Int[(Px_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{n = Expon[Px, x^2], e = Coeff[Px, x^2, Expon[Px, x^2]]}, Simp[e*x^(2*n - 3)*(( a + b*x^2 + c*x^4)^(p + 1)/(c*(2*n + 4*p + 1))), x] + Simp[1/(c*(2*n + 4*p + 1)) Int[(a + b*x^2 + c*x^4)^p*ExpandToSum[c*(2*n + 4*p + 1)*Px - a*e*(2 *n - 3)*x^(2*n - 4) - b*e*(2*n + 2*p - 1)*x^(2*n - 2) - c*e*(2*n + 4*p + 1) *x^(2*n), x], x], x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Px, x^2] && Expon[ Px, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] && !LtQ[p, -1]
Time = 4.28 (sec) , antiderivative size = 508, normalized size of antiderivative = 0.96
| method | result | size |
| elliptic | \(\frac {B \,e^{2} x^{3} \sqrt {c \,x^{4}+b \,x^{2}+a}}{5 c}+\frac {\left (A \,e^{2}+2 B e d -\frac {4 b B \,e^{2}}{5 c}\right ) x \sqrt {c \,x^{4}+b \,x^{2}+a}}{3 c}+\frac {\left (d^{2} A -\frac {a \left (A \,e^{2}+2 B e d -\frac {4 b B \,e^{2}}{5 c}\right )}{3 c}\right ) \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, F\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {\left (2 A e d +B \,d^{2}-\frac {3 a B \,e^{2}}{5 c}-\frac {2 b \left (A \,e^{2}+2 B e d -\frac {4 b B \,e^{2}}{5 c}\right )}{3 c}\right ) a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (F\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-E\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{2 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}\) | \(508\) |
| risch | \(\frac {e x \left (3 B e \,x^{2} c +5 A c e -4 B b e +10 B c d \right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{15 c^{2}}-\frac {-\frac {15 A \,c^{2} d^{2} \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, F\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {5 A a c \,e^{2} \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, F\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {a b B \,e^{2} \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, F\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {5 B a c d e \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, F\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{2 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {\left (10 A b c \,e^{2}-30 A \,c^{2} d e +9 B a c \,e^{2}-8 B \,b^{2} e^{2}+20 B b c d e -15 B \,c^{2} d^{2}\right ) a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (F\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-E\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{2 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}}{15 c^{2}}\) | \(914\) |
| default | \(\text {Expression too large to display}\) | \(1201\) |
1/5*B*e^2*x^3*(c*x^4+b*x^2+a)^(1/2)/c+1/3*(A*e^2+2*B*e*d-4/5*b/c*B*e^2)/c* x*(c*x^4+b*x^2+a)^(1/2)+1/4*(d^2*A-1/3*a/c*(A*e^2+2*B*e*d-4/5*b/c*B*e^2))* 2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*(4-2*(-b+(-4*a*c+b^2)^(1/2))/a*x ^2)^(1/2)*(4+2*(b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)/(c*x^4+b*x^2+a)^(1/2)*E llipticF(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4+2*b*(b+(- 4*a*c+b^2)^(1/2))/a/c)^(1/2))-1/2*(2*A*e*d+B*d^2-3/5*a/c*B*e^2-2/3*b/c*(A* e^2+2*B*e*d-4/5*b/c*B*e^2))*a*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*(4 -2*(-b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)*(4+2*(b+(-4*a*c+b^2)^(1/2))/a*x^2) ^(1/2)/(c*x^4+b*x^2+a)^(1/2)/(b+(-4*a*c+b^2)^(1/2))*(EllipticF(1/2*x*2^(1/ 2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/ c)^(1/2))-EllipticE(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(- 4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2)))
Time = 0.11 (sec) , antiderivative size = 681, normalized size of antiderivative = 1.29 \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^2}{\sqrt {a+b x^2+c x^4}} \, dx=\frac {\sqrt {\frac {1}{2}} {\left ({\left (15 \, B a c^{3} d^{2} - 10 \, {\left (2 \, B a b c^{2} - 3 \, A a c^{3}\right )} d e + {\left (8 \, B a b^{2} c - {\left (9 \, B a^{2} + 10 \, A a b\right )} c^{2}\right )} e^{2}\right )} x \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - {\left (15 \, B a b c^{2} d^{2} - 10 \, {\left (2 \, B a b^{2} c - 3 \, A a b c^{2}\right )} d e + {\left (8 \, B a b^{3} - {\left (9 \, B a^{2} b + 10 \, A a b^{2}\right )} c\right )} e^{2}\right )} x\right )} \sqrt {c} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}} E(\arcsin \left (\frac {\sqrt {\frac {1}{2}} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}}}{x}\right )\,|\,\frac {b c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} + b^{2} - 2 \, a c}{2 \, a c}) - \sqrt {\frac {1}{2}} {\left ({\left (15 \, {\left (B a c^{3} - A c^{4}\right )} d^{2} - 10 \, {\left (2 \, B a b c^{2} - {\left (3 \, A + B\right )} a c^{3}\right )} d e + {\left (8 \, B a b^{2} c + 5 \, A a c^{3} - {\left (9 \, B a^{2} + 2 \, {\left (5 \, A + 2 \, B\right )} a b\right )} c^{2}\right )} e^{2}\right )} x \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - {\left (15 \, {\left (B a b c^{2} + A b c^{3}\right )} d^{2} - 10 \, {\left (2 \, B a b^{2} c - {\left (3 \, A - B\right )} a b c^{2}\right )} d e + {\left (8 \, B a b^{3} - 5 \, A a b c^{2} - {\left (9 \, B a^{2} b + 2 \, {\left (5 \, A - 2 \, B\right )} a b^{2}\right )} c\right )} e^{2}\right )} x\right )} \sqrt {c} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}} F(\arcsin \left (\frac {\sqrt {\frac {1}{2}} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}}}{x}\right )\,|\,\frac {b c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} + b^{2} - 2 \, a c}{2 \, a c}) + 2 \, {\left (3 \, B a c^{3} e^{2} x^{4} + 15 \, B a c^{3} d^{2} - 10 \, {\left (2 \, B a b c^{2} - 3 \, A a c^{3}\right )} d e + {\left (8 \, B a b^{2} c - {\left (9 \, B a^{2} + 10 \, A a b\right )} c^{2}\right )} e^{2} + {\left (10 \, B a c^{3} d e - {\left (4 \, B a b c^{2} - 5 \, A a c^{3}\right )} e^{2}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2} + a}}{30 \, a c^{4} x} \]
1/30*(sqrt(1/2)*((15*B*a*c^3*d^2 - 10*(2*B*a*b*c^2 - 3*A*a*c^3)*d*e + (8*B *a*b^2*c - (9*B*a^2 + 10*A*a*b)*c^2)*e^2)*x*sqrt((b^2 - 4*a*c)/c^2) - (15* B*a*b*c^2*d^2 - 10*(2*B*a*b^2*c - 3*A*a*b*c^2)*d*e + (8*B*a*b^3 - (9*B*a^2 *b + 10*A*a*b^2)*c)*e^2)*x)*sqrt(c)*sqrt((c*sqrt((b^2 - 4*a*c)/c^2) - b)/c )*elliptic_e(arcsin(sqrt(1/2)*sqrt((c*sqrt((b^2 - 4*a*c)/c^2) - b)/c)/x), 1/2*(b*c*sqrt((b^2 - 4*a*c)/c^2) + b^2 - 2*a*c)/(a*c)) - sqrt(1/2)*((15*(B *a*c^3 - A*c^4)*d^2 - 10*(2*B*a*b*c^2 - (3*A + B)*a*c^3)*d*e + (8*B*a*b^2* c + 5*A*a*c^3 - (9*B*a^2 + 2*(5*A + 2*B)*a*b)*c^2)*e^2)*x*sqrt((b^2 - 4*a* c)/c^2) - (15*(B*a*b*c^2 + A*b*c^3)*d^2 - 10*(2*B*a*b^2*c - (3*A - B)*a*b* c^2)*d*e + (8*B*a*b^3 - 5*A*a*b*c^2 - (9*B*a^2*b + 2*(5*A - 2*B)*a*b^2)*c) *e^2)*x)*sqrt(c)*sqrt((c*sqrt((b^2 - 4*a*c)/c^2) - b)/c)*elliptic_f(arcsin (sqrt(1/2)*sqrt((c*sqrt((b^2 - 4*a*c)/c^2) - b)/c)/x), 1/2*(b*c*sqrt((b^2 - 4*a*c)/c^2) + b^2 - 2*a*c)/(a*c)) + 2*(3*B*a*c^3*e^2*x^4 + 15*B*a*c^3*d^ 2 - 10*(2*B*a*b*c^2 - 3*A*a*c^3)*d*e + (8*B*a*b^2*c - (9*B*a^2 + 10*A*a*b) *c^2)*e^2 + (10*B*a*c^3*d*e - (4*B*a*b*c^2 - 5*A*a*c^3)*e^2)*x^2)*sqrt(c*x ^4 + b*x^2 + a))/(a*c^4*x)
\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^2}{\sqrt {a+b x^2+c x^4}} \, dx=\int \frac {\left (A + B x^{2}\right ) \left (d + e x^{2}\right )^{2}}{\sqrt {a + b x^{2} + c x^{4}}}\, dx \]
\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^2}{\sqrt {a+b x^2+c x^4}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{2}}{\sqrt {c x^{4} + b x^{2} + a}} \,d x } \]
\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^2}{\sqrt {a+b x^2+c x^4}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{2}}{\sqrt {c x^{4} + b x^{2} + a}} \,d x } \]
Timed out. \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^2}{\sqrt {a+b x^2+c x^4}} \, dx=\int \frac {\left (B\,x^2+A\right )\,{\left (e\,x^2+d\right )}^2}{\sqrt {c\,x^4+b\,x^2+a}} \,d x \]