Integrand size = 27, antiderivative size = 403 \[ \int \frac {x^4 \left (A+B x^2\right )}{\sqrt {a+b x^2+c x^4}} \, dx=-\frac {(4 b B-5 A c) x \sqrt {a+b x^2+c x^4}}{15 c^2}+\frac {B x^3 \sqrt {a+b x^2+c x^4}}{5 c}+\frac {\left (8 b^2 B-10 A b c-9 a B c\right ) x \sqrt {a+b x^2+c x^4}}{15 c^{5/2} \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {\sqrt [4]{a} \left (8 b^2 B-10 A b c-9 a B c\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{15 c^{11/4} \sqrt {a+b x^2+c x^4}}+\frac {\sqrt [4]{a} \left (8 b^2 B-10 A b c-9 a B c+\sqrt {a} \sqrt {c} (4 b B-5 A c)\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{30 c^{11/4} \sqrt {a+b x^2+c x^4}} \]
-1/15*(-5*A*c+4*B*b)*x*(c*x^4+b*x^2+a)^(1/2)/c^2+1/5*B*x^3*(c*x^4+b*x^2+a) ^(1/2)/c+1/15*(-10*A*b*c-9*B*a*c+8*B*b^2)*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*b*c-9*B*a*c+8*B*b^2)*(cos(2*arct an(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))*(8*B*b^2-10*A*b *c-9*B*a*c+(-5*A*c+4*B*b)*a^(1/2)*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)
Result contains complex when optimal does not.
Time = 11.51 (sec) , antiderivative size = 532, normalized size of antiderivative = 1.32 \[ \int \frac {x^4 \left (A+B x^2\right )}{\sqrt {a+b x^2+c x^4}} \, dx=\frac {4 c \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x \left (-4 b B+5 A c+3 B c x^2\right ) \left (a+b x^2+c x^4\right )+i \left (8 b^2 B-10 A b c-9 a B c\right ) \left (-b+\sqrt {b^2-4 a c}\right ) \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+4 c x^2}{b-\sqrt {b^2-4 a c}}} E\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )-i \left (-8 b^3 B+b c \left (17 a B-10 A \sqrt {b^2-4 a c}\right )+2 b^2 \left (5 A c+4 B \sqrt {b^2-4 a c}\right )-a c \left (10 A c+9 B \sqrt {b^2-4 a c}\right )\right ) \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+4 c x^2}{b-\sqrt {b^2-4 a c}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{60 c^3 \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \sqrt {a+b x^2+c x^4}} \]
(4*c*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x*(-4*b*B + 5*A*c + 3*B*c*x^2)*(a + b *x^2 + c*x^4) + I*(8*b^2*B - 10*A*b*c - 9*a*B*c)*(-b + Sqrt[b^2 - 4*a*c])* 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])]*EllipticE[I*ArcS inh[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 + b*c*(17*a*B - 10*A*Sqrt[b^2 - 4*a*c ]) + 2*b^2*(5*A*c + 4*B*Sqrt[b^2 - 4*a*c]) - a*c*(10*A*c + 9*B*Sqrt[b^2 - 4*a*c]))*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*S qrt[(2*b - 2*Sqrt[b^2 - 4*a*c] + 4*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*Ellipti cF[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])])/(60*c^3*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*S qrt[a + b*x^2 + c*x^4])
Time = 0.56 (sec) , antiderivative size = 400, normalized size of antiderivative = 0.99, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1602, 1602, 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.
| \(\displaystyle \int \frac {x^4 \left (A+B x^2\right )}{\sqrt {a+b x^2+c x^4}} \, dx\) |
| \(\Big \downarrow \) 1602 |
| \(\displaystyle \frac {B x^3 \sqrt {a+b x^2+c x^4}}{5 c}-\frac {\int \frac {x^2 \left ((4 b B-5 A c) x^2+3 a B\right )}{\sqrt {c x^4+b x^2+a}}dx}{5 c}\) |
| \(\Big \downarrow \) 1602 |
| \(\displaystyle \frac {B x^3 \sqrt {a+b x^2+c x^4}}{5 c}-\frac {\frac {x \sqrt {a+b x^2+c x^4} (4 b B-5 A c)}{3 c}-\frac {\int \frac {\left (8 B b^2-10 A c b-9 a B c\right ) x^2+a (4 b B-5 A c)}{\sqrt {c x^4+b x^2+a}}dx}{3 c}}{5 c}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {B x^3 \sqrt {a+b x^2+c x^4}}{5 c}-\frac {\frac {x \sqrt {a+b x^2+c x^4} (4 b B-5 A c)}{3 c}-\frac {\frac {\sqrt {a} \left (\sqrt {a} \sqrt {c} (4 b B-5 A c)-9 a B c-10 A b c+8 b^2 B\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}-\frac {\sqrt {a} \left (-9 a B c-10 A b c+8 b^2 B\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {a} \sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}}{3 c}}{5 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {B x^3 \sqrt {a+b x^2+c x^4}}{5 c}-\frac {\frac {x \sqrt {a+b x^2+c x^4} (4 b B-5 A c)}{3 c}-\frac {\frac {\sqrt {a} \left (\sqrt {a} \sqrt {c} (4 b B-5 A c)-9 a B c-10 A b c+8 b^2 B\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}-\frac {\left (-9 a B c-10 A b c+8 b^2 B\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}}{3 c}}{5 c}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {B x^3 \sqrt {a+b x^2+c x^4}}{5 c}-\frac {\frac {x \sqrt {a+b x^2+c x^4} (4 b B-5 A c)}{3 c}-\frac {\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (\sqrt {a} \sqrt {c} (4 b B-5 A c)-9 a B c-10 A b c+8 b^2 B\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 c^{3/4} \sqrt {a+b x^2+c x^4}}-\frac {\left (-9 a B c-10 A b c+8 b^2 B\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}}{3 c}}{5 c}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {B x^3 \sqrt {a+b x^2+c x^4}}{5 c}-\frac {\frac {x \sqrt {a+b x^2+c x^4} (4 b B-5 A c)}{3 c}-\frac {\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (\sqrt {a} \sqrt {c} (4 b B-5 A c)-9 a B c-10 A b c+8 b^2 B\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 c^{3/4} \sqrt {a+b x^2+c x^4}}-\frac {\left (-9 a B c-10 A b c+8 b^2 B\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {c}}}{3 c}}{5 c}\) |
(B*x^3*Sqrt[a + b*x^2 + c*x^4])/(5*c) - (((4*b*B - 5*A*c)*x*Sqrt[a + b*x^2 + c*x^4])/(3*c) - (-(((8*b^2*B - 10*A*b*c - 9*a*B*c)*(-((x*Sqrt[a + b*x^2 + c*x^4])/(Sqrt[a] + Sqrt[c]*x^2)) + (a^(1/4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqr t[(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])))/Sqrt[c]) + (a^(1/4)*(8*b^2*B - 10*A*b*c - 9*a*B*c + Sqrt[a]*Sqrt[c ]*(4*b*B - 5*A*c))*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[ a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt [a]*Sqrt[c]))/4])/(2*c^(3/4)*Sqrt[a + b*x^2 + c*x^4]))/(3*c))/(5*c)
3.2.76.3.1 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[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*( x_)^4)^(p_), x_Symbol] :> Simp[e*f*(f*x)^(m - 1)*((a + b*x^2 + c*x^4)^(p + 1)/(c*(m + 4*p + 3))), x] - Simp[f^2/(c*(m + 4*p + 3)) Int[(f*x)^(m - 2)* (a + b*x^2 + c*x^4)^p*Simp[a*e*(m - 1) + (b*e*(m + 2*p + 1) - c*d*(m + 4*p + 3))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && NeQ[b^2 - 4*a*c , 0] && GtQ[m, 1] && NeQ[m + 4*p + 3, 0] && IntegerQ[2*p] && (IntegerQ[p] | | IntegerQ[m])
Time = 5.78 (sec) , antiderivative size = 448, normalized size of antiderivative = 1.11
| method | result | size |
| elliptic | \(\frac {B \,x^{3} \sqrt {c \,x^{4}+b \,x^{2}+a}}{5 c}+\frac {\left (A -\frac {4 B b}{5 c}\right ) x \sqrt {c \,x^{4}+b \,x^{2}+a}}{3 c}-\frac {a \left (A -\frac {4 B b}{5 c}\right ) \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, F\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{12 c \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {\left (-\frac {3 a B}{5 c}-\frac {2 b \left (A -\frac {4 B b}{5 c}\right )}{3 c}\right ) a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (F\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-E\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{2 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}\) | \(448\) |
| risch | \(\frac {x \left (3 B \,x^{2} c +5 A c -4 B b \right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{15 c^{2}}-\frac {\frac {5 A a c \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, F\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {a b B \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, F\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {\left (10 A b c +9 B a c -8 B \,b^{2}\right ) a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (F\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-E\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{2 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}}{15 c^{2}}\) | \(568\) |
| default | \(B \left (\frac {x^{3} \sqrt {c \,x^{4}+b \,x^{2}+a}}{5 c}-\frac {4 b x \sqrt {c \,x^{4}+b \,x^{2}+a}}{15 c^{2}}+\frac {a b \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 )}{15 c^{2} \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {\left (-\frac {3 a}{5 c}+\frac {8 b^{2}}{15 c^{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 )}\right )+A \left (\frac {x \sqrt {c \,x^{4}+b \,x^{2}+a}}{3 c}-\frac {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}}\, 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 )}{12 c \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {b 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 )}{3 c \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 )}\right )\) | \(815\) |
1/5*B*x^3*(c*x^4+b*x^2+a)^(1/2)/c+1/3*(A-4/5*B*b/c)/c*x*(c*x^4+b*x^2+a)^(1 /2)-1/12*a/c*(A-4/5*B*b/c)*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)*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))-1/2*(-3/5*a/c*B-2 /3*b/c*(A-4/5*B*b/c))*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 = 0.09 (sec) , antiderivative size = 431, normalized size of antiderivative = 1.07 \[ \int \frac {x^4 \left (A+B x^2\right )}{\sqrt {a+b x^2+c x^4}} \, dx=\frac {\sqrt {\frac {1}{2}} {\left ({\left (8 \, B b^{2} c - {\left (9 \, B a + 10 \, A b\right )} c^{2}\right )} x \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - {\left (8 \, B b^{3} - {\left (9 \, B a b + 10 \, A b^{2}\right )} c\right )} x\right )} \sqrt {c} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}} E(\arcsin \left (\frac {\sqrt {\frac {1}{2}} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}}}{x}\right )\,|\,\frac {b c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} + b^{2} - 2 \, a c}{2 \, a c}) - \sqrt {\frac {1}{2}} {\left ({\left (8 \, B b^{2} c + 5 \, A c^{3} - {\left (9 \, B a + 2 \, {\left (5 \, A + 2 \, B\right )} b\right )} c^{2}\right )} x \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - {\left (8 \, B b^{3} - 5 \, A b c^{2} - {\left (9 \, B a b + 2 \, {\left (5 \, A - 2 \, B\right )} b^{2}\right )} c\right )} x\right )} \sqrt {c} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}} F(\arcsin \left (\frac {\sqrt {\frac {1}{2}} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}}}{x}\right )\,|\,\frac {b c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} + b^{2} - 2 \, a c}{2 \, a c}) + 2 \, {\left (3 \, B c^{3} x^{4} + 8 \, B b^{2} c - {\left (9 \, B a + 10 \, A b\right )} c^{2} - {\left (4 \, B b c^{2} - 5 \, A c^{3}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2} + a}}{30 \, c^{4} x} \]
1/30*(sqrt(1/2)*((8*B*b^2*c - (9*B*a + 10*A*b)*c^2)*x*sqrt((b^2 - 4*a*c)/c ^2) - (8*B*b^3 - (9*B*a*b + 10*A*b^2)*c)*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)) - s qrt(1/2)*((8*B*b^2*c + 5*A*c^3 - (9*B*a + 2*(5*A + 2*B)*b)*c^2)*x*sqrt((b^ 2 - 4*a*c)/c^2) - (8*B*b^3 - 5*A*b*c^2 - (9*B*a*b + 2*(5*A - 2*B)*b^2)*c)* 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*c^3*x^4 + 8*B*b^2*c - (9*B*a + 10*A *b)*c^2 - (4*B*b*c^2 - 5*A*c^3)*x^2)*sqrt(c*x^4 + b*x^2 + a))/(c^4*x)
\[ \int \frac {x^4 \left (A+B x^2\right )}{\sqrt {a+b x^2+c x^4}} \, dx=\int \frac {x^{4} \left (A + B x^{2}\right )}{\sqrt {a + b x^{2} + c x^{4}}}\, dx \]
\[ \int \frac {x^4 \left (A+B x^2\right )}{\sqrt {a+b x^2+c x^4}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} x^{4}}{\sqrt {c x^{4} + b x^{2} + a}} \,d x } \]
\[ \int \frac {x^4 \left (A+B x^2\right )}{\sqrt {a+b x^2+c x^4}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} x^{4}}{\sqrt {c x^{4} + b x^{2} + a}} \,d x } \]
Timed out. \[ \int \frac {x^4 \left (A+B x^2\right )}{\sqrt {a+b x^2+c x^4}} \, dx=\int \frac {x^4\,\left (B\,x^2+A\right )}{\sqrt {c\,x^4+b\,x^2+a}} \,d x \]