Integrand size = 17, antiderivative size = 405 \[ \int \frac {1}{\left (\text {a1}+\text {a2}+b x^2+c x^4\right )^{3/2}} \, dx=\frac {x \left (b^2-2 (\text {a1}+\text {a2}) c+b c x^2\right )}{(\text {a1}+\text {a2}) \left (b^2-4 (\text {a1}+\text {a2}) c\right ) \sqrt {\text {a1}+\text {a2}+b x^2+c x^4}}-\frac {b \sqrt {c} x \sqrt {\text {a1}+\text {a2}+b x^2+c x^4}}{(\text {a1}+\text {a2}) \left (b^2-4 (\text {a1}+\text {a2}) c\right ) \left (\sqrt {\text {a1}+\text {a2}}+\sqrt {c} x^2\right )}+\frac {b \sqrt [4]{c} \left (\sqrt {\text {a1}+\text {a2}}+\sqrt {c} x^2\right ) \sqrt {\frac {\text {a1}+\text {a2}+b x^2+c x^4}{\left (\sqrt {\text {a1}+\text {a2}}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{\text {a1}+\text {a2}}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {\text {a1}+\text {a2}} \sqrt {c}}\right )\right )}{(\text {a1}+\text {a2})^{3/4} \left (b^2-4 (\text {a1}+\text {a2}) c\right ) \sqrt {\text {a1}+\text {a2}+b x^2+c x^4}}-\frac {\left (b+2 \sqrt {\text {a1}+\text {a2}} \sqrt {c}\right ) \sqrt [4]{c} \left (\sqrt {\text {a1}+\text {a2}}+\sqrt {c} x^2\right ) \sqrt {\frac {\text {a1}+\text {a2}+b x^2+c x^4}{\left (\sqrt {\text {a1}+\text {a2}}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{\text {a1}+\text {a2}}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {\text {a1}+\text {a2}} \sqrt {c}}\right )\right )}{2 (\text {a1}+\text {a2})^{3/4} \left (b^2-4 (\text {a1}+\text {a2}) c\right ) \sqrt {\text {a1}+\text {a2}+b x^2+c x^4}} \] Output:
x*(b^2-2*(a1+a2)*c+b*c*x^2)/(a1+a2)/(b^2-4*(a1+a2)*c)/(c*x^4+b*x^2+a1+a2)^ (1/2)-b*c^(1/2)*x*(c*x^4+b*x^2+a1+a2)^(1/2)/(a1+a2)/(b^2-4*(a1+a2)*c)/(c^( 1/2)*x^2+(a1+a2)^(1/2))+b*c^(1/4)*(c^(1/2)*x^2+(a1+a2)^(1/2))*((c*x^4+b*x^ 2+a1+a2)/(c^(1/2)*x^2+(a1+a2)^(1/2))^2)^(1/2)*EllipticE(sin(2*arctan(c^(1/ 4)*x/(a1+a2)^(1/4))),1/2*(2-b/(a1+a2)^(1/2)/c^(1/2))^(1/2))/(a1+a2)^(3/4)/ (b^2-4*(a1+a2)*c)/(c*x^4+b*x^2+a1+a2)^(1/2)-1/2*(b+2*(a1+a2)^(1/2)*c^(1/2) )*c^(1/4)*(c^(1/2)*x^2+(a1+a2)^(1/2))*((c*x^4+b*x^2+a1+a2)/(c^(1/2)*x^2+(a 1+a2)^(1/2))^2)^(1/2)*InverseJacobiAM(2*arctan(c^(1/4)*x/(a1+a2)^(1/4)),1/ 2*(2-b/(a1+a2)^(1/2)/c^(1/2))^(1/2))/(a1+a2)^(3/4)/(b^2-4*(a1+a2)*c)/(c*x^ 4+b*x^2+a1+a2)^(1/2)
Result contains complex when optimal does not.
Time = 11.07 (sec) , antiderivative size = 509, normalized size of antiderivative = 1.26 \[ \int \frac {1}{\left (\text {a1}+\text {a2}+b x^2+c x^4\right )^{3/2}} \, dx=-\frac {-4 \sqrt {\frac {c}{b+\sqrt {b^2-4 (\text {a1}+\text {a2}) c}}} x \left (b^2-2 (\text {a1}+\text {a2}) c+b c x^2\right )+i \sqrt {2} b \left (-b+\sqrt {b^2-4 (\text {a1}+\text {a2}) c}\right ) \sqrt {\frac {-b+\sqrt {b^2-4 (\text {a1}+\text {a2}) c}-2 c x^2}{-b+\sqrt {b^2-4 (\text {a1}+\text {a2}) c}}} \sqrt {\frac {b+\sqrt {b^2-4 (\text {a1}+\text {a2}) c}+2 c x^2}{b+\sqrt {b^2-4 (\text {a1}+\text {a2}) c}}} E\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 (\text {a1}+\text {a2}) c}}} x\right )|-\frac {b+\sqrt {b^2-4 (\text {a1}+\text {a2}) c}}{-b+\sqrt {b^2-4 (\text {a1}+\text {a2}) c}}\right )-i \sqrt {2} \left (-b^2+4 (\text {a1}+\text {a2}) c+b \sqrt {b^2-4 (\text {a1}+\text {a2}) c}\right ) \sqrt {\frac {-b+\sqrt {b^2-4 (\text {a1}+\text {a2}) c}-2 c x^2}{-b+\sqrt {b^2-4 (\text {a1}+\text {a2}) c}}} \sqrt {\frac {b+\sqrt {b^2-4 (\text {a1}+\text {a2}) c}+2 c x^2}{b+\sqrt {b^2-4 (\text {a1}+\text {a2}) c}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 (\text {a1}+\text {a2}) c}}} x\right ),-\frac {b+\sqrt {b^2-4 (\text {a1}+\text {a2}) c}}{-b+\sqrt {b^2-4 (\text {a1}+\text {a2}) c}}\right )}{4 (\text {a1}+\text {a2}) \left (b^2-4 (\text {a1}+\text {a2}) c\right ) \sqrt {\frac {c}{b+\sqrt {b^2-4 (\text {a1}+\text {a2}) c}}} \sqrt {\text {a1}+\text {a2}+b x^2+c x^4}} \] Input:
Integrate[(a1 + a2 + b*x^2 + c*x^4)^(-3/2),x]
Output:
-1/4*(-4*Sqrt[c/(b + Sqrt[b^2 - 4*(a1 + a2)*c])]*x*(b^2 - 2*(a1 + a2)*c + b*c*x^2) + I*Sqrt[2]*b*(-b + Sqrt[b^2 - 4*(a1 + a2)*c])*Sqrt[(-b + Sqrt[b^ 2 - 4*(a1 + a2)*c] - 2*c*x^2)/(-b + Sqrt[b^2 - 4*(a1 + a2)*c])]*Sqrt[(b + Sqrt[b^2 - 4*(a1 + a2)*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*(a1 + a2)*c])]*Elli pticE[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*(a1 + a2)*c])]*x], -((b + Sqrt[b^2 - 4*(a1 + a2)*c])/(-b + Sqrt[b^2 - 4*(a1 + a2)*c]))] - I*Sqrt[2] *(-b^2 + 4*(a1 + a2)*c + b*Sqrt[b^2 - 4*(a1 + a2)*c])*Sqrt[(-b + Sqrt[b^2 - 4*(a1 + a2)*c] - 2*c*x^2)/(-b + Sqrt[b^2 - 4*(a1 + a2)*c])]*Sqrt[(b + Sq rt[b^2 - 4*(a1 + a2)*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*(a1 + a2)*c])]*Ellipt icF[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*(a1 + a2)*c])]*x], -((b + S qrt[b^2 - 4*(a1 + a2)*c])/(-b + Sqrt[b^2 - 4*(a1 + a2)*c]))])/((a1 + a2)*( b^2 - 4*(a1 + a2)*c)*Sqrt[c/(b + Sqrt[b^2 - 4*(a1 + a2)*c])]*Sqrt[a1 + a2 + b*x^2 + c*x^4])
Time = 0.96 (sec) , antiderivative size = 415, normalized size of antiderivative = 1.02, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {1405, 27, 1511, 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.
| \(\displaystyle \int \frac {1}{\left (\text {a1}+\text {a2}+b x^2+c x^4\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1405 |
| \(\displaystyle \frac {x \left (-2 c (\text {a1}+\text {a2})+b^2+b c x^2\right )}{(\text {a1}+\text {a2}) \left (b^2-4 c (\text {a1}+\text {a2})\right ) \sqrt {\text {a1}+\text {a2}+b x^2+c x^4}}-\frac {\int \frac {c \left (b x^2+2 (\text {a1}+\text {a2})\right )}{\sqrt {c x^4+b x^2+\text {a1}+\text {a2}}}dx}{(\text {a1}+\text {a2}) \left (b^2-4 c (\text {a1}+\text {a2})\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x \left (-2 c (\text {a1}+\text {a2})+b^2+b c x^2\right )}{(\text {a1}+\text {a2}) \left (b^2-4 c (\text {a1}+\text {a2})\right ) \sqrt {\text {a1}+\text {a2}+b x^2+c x^4}}-\frac {c \int \frac {b x^2+2 (\text {a1}+\text {a2})}{\sqrt {c x^4+b x^2+\text {a1}+\text {a2}}}dx}{(\text {a1}+\text {a2}) \left (b^2-4 c (\text {a1}+\text {a2})\right )}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {x \left (-2 c (\text {a1}+\text {a2})+b^2+b c x^2\right )}{(\text {a1}+\text {a2}) \left (b^2-4 c (\text {a1}+\text {a2})\right ) \sqrt {\text {a1}+\text {a2}+b x^2+c x^4}}-\frac {c \left (\sqrt {\text {a1}+\text {a2}} \left (2 \sqrt {\text {a1}+\text {a2}}+\frac {b}{\sqrt {c}}\right ) \int \frac {1}{\sqrt {c x^4+b x^2+\text {a1}+\text {a2}}}dx-\frac {b \sqrt {\text {a1}+\text {a2}} \int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {\text {a1}+\text {a2}}}}{\sqrt {c x^4+b x^2+\text {a1}+\text {a2}}}dx}{\sqrt {c}}\right )}{(\text {a1}+\text {a2}) \left (b^2-4 c (\text {a1}+\text {a2})\right )}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {x \left (-2 c (\text {a1}+\text {a2})+b^2+b c x^2\right )}{(\text {a1}+\text {a2}) \left (b^2-4 c (\text {a1}+\text {a2})\right ) \sqrt {\text {a1}+\text {a2}+b x^2+c x^4}}-\frac {c \left (\frac {(\text {a1}+\text {a2})^{3/4} \left (2 \sqrt {\text {a1}+\text {a2}}+\frac {b}{\sqrt {c}}\right ) \left (\frac {\sqrt {c} x^2}{\sqrt {\text {a1}+\text {a2}}}+1\right ) \sqrt {\frac {\text {a1}+\text {a2}+b x^2+c x^4}{(\text {a1}+\text {a2}) \left (\frac {\sqrt {c} x^2}{\sqrt {\text {a1}+\text {a2}}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{\text {a1}+\text {a2}}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {\text {a1}+\text {a2}} \sqrt {c}}\right )\right )}{2 \sqrt [4]{c} \sqrt {\text {a1}+\text {a2}+b x^2+c x^4}}-\frac {b \sqrt {\text {a1}+\text {a2}} \int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {\text {a1}+\text {a2}}}}{\sqrt {c x^4+b x^2+\text {a1}+\text {a2}}}dx}{\sqrt {c}}\right )}{(\text {a1}+\text {a2}) \left (b^2-4 c (\text {a1}+\text {a2})\right )}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {x \left (-2 c (\text {a1}+\text {a2})+b^2+b c x^2\right )}{(\text {a1}+\text {a2}) \left (b^2-4 c (\text {a1}+\text {a2})\right ) \sqrt {\text {a1}+\text {a2}+b x^2+c x^4}}-\frac {c \left (\frac {(\text {a1}+\text {a2})^{3/4} \left (2 \sqrt {\text {a1}+\text {a2}}+\frac {b}{\sqrt {c}}\right ) \left (\frac {\sqrt {c} x^2}{\sqrt {\text {a1}+\text {a2}}}+1\right ) \sqrt {\frac {\text {a1}+\text {a2}+b x^2+c x^4}{(\text {a1}+\text {a2}) \left (\frac {\sqrt {c} x^2}{\sqrt {\text {a1}+\text {a2}}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{\text {a1}+\text {a2}}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {\text {a1}+\text {a2}} \sqrt {c}}\right )\right )}{2 \sqrt [4]{c} \sqrt {\text {a1}+\text {a2}+b x^2+c x^4}}-\frac {b \sqrt {\text {a1}+\text {a2}} \left (\frac {\sqrt [4]{\text {a1}+\text {a2}} \left (\frac {\sqrt {c} x^2}{\sqrt {\text {a1}+\text {a2}}}+1\right ) \sqrt {\frac {\text {a1}+\text {a2}+b x^2+c x^4}{(\text {a1}+\text {a2}) \left (\frac {\sqrt {c} x^2}{\sqrt {\text {a1}+\text {a2}}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{\text {a1}+\text {a2}}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {\text {a1}+\text {a2}} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {\text {a1}+\text {a2}+b x^2+c x^4}}-\frac {x \sqrt {\text {a1}+\text {a2}+b x^2+c x^4}}{(\text {a1}+\text {a2}) \left (\frac {\sqrt {c} x^2}{\sqrt {\text {a1}+\text {a2}}}+1\right )}\right )}{\sqrt {c}}\right )}{(\text {a1}+\text {a2}) \left (b^2-4 c (\text {a1}+\text {a2})\right )}\) |
Input:
Int[(a1 + a2 + b*x^2 + c*x^4)^(-3/2),x]
Output:
(x*(b^2 - 2*(a1 + a2)*c + b*c*x^2))/((a1 + a2)*(b^2 - 4*(a1 + a2)*c)*Sqrt[ a1 + a2 + b*x^2 + c*x^4]) - (c*(-((Sqrt[a1 + a2]*b*(-((x*Sqrt[a1 + a2 + b* x^2 + c*x^4])/((a1 + a2)*(1 + (Sqrt[c]*x^2)/Sqrt[a1 + a2]))) + ((a1 + a2)^ (1/4)*(1 + (Sqrt[c]*x^2)/Sqrt[a1 + a2])*Sqrt[(a1 + a2 + b*x^2 + c*x^4)/((a 1 + a2)*(1 + (Sqrt[c]*x^2)/Sqrt[a1 + a2])^2)]*EllipticE[2*ArcTan[(c^(1/4)* x)/(a1 + a2)^(1/4)], (2 - b/(Sqrt[a1 + a2]*Sqrt[c]))/4])/(c^(1/4)*Sqrt[a1 + a2 + b*x^2 + c*x^4])))/Sqrt[c]) + ((a1 + a2)^(3/4)*(2*Sqrt[a1 + a2] + b/ Sqrt[c])*(1 + (Sqrt[c]*x^2)/Sqrt[a1 + a2])*Sqrt[(a1 + a2 + b*x^2 + c*x^4)/ ((a1 + a2)*(1 + (Sqrt[c]*x^2)/Sqrt[a1 + a2])^2)]*EllipticF[2*ArcTan[(c^(1/ 4)*x)/(a1 + a2)^(1/4)], (2 - b/(Sqrt[a1 + a2]*Sqrt[c]))/4])/(2*c^(1/4)*Sqr t[a1 + a2 + b*x^2 + c*x^4])))/((a1 + a2)*(b^2 - 4*(a1 + a2)*c))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(-x)*(b^2 - 2*a*c + b*c*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c)) ), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c)) Int[(b^2 - 2*a*c + 2*(p + 1)*( b^2 - 4*a*c) + b*c*(4*p + 7)*x^2)*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; Fr eeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IntegerQ[2*p]
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]
Time = 0.57 (sec) , antiderivative size = 593, normalized size of antiderivative = 1.46
| method | result | size |
| default | \(-\frac {2 c \left (\frac {b \,x^{3}}{2 \left (\operatorname {a1} +\operatorname {a2} \right ) \left (4 \operatorname {a1} c +4 c \operatorname {a2} -b^{2}\right )}-\frac {\left (2 \operatorname {a1} c +2 c \operatorname {a2} -b^{2}\right ) x}{2 \left (\operatorname {a1} +\operatorname {a2} \right ) \left (4 \operatorname {a1} c +4 c \operatorname {a2} -b^{2}\right ) c}\right )}{\sqrt {\left (x^{4}+\frac {b \,x^{2}}{c}+\frac {\operatorname {a1} +\operatorname {a2}}{c}\right ) c}}+\frac {\left (\frac {1}{\operatorname {a1} +\operatorname {a2}}-\frac {2 \operatorname {a1} c +2 c \operatorname {a2} -b^{2}}{\left (\operatorname {a1} +\operatorname {a2} \right ) \left (4 \operatorname {a1} c +4 c \operatorname {a2} -b^{2}\right )}\right ) \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 \operatorname {a1} c -4 c \operatorname {a2} +b^{2}}\right ) x^{2}}{\operatorname {a1} +\operatorname {a2}}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 \operatorname {a1} c -4 c \operatorname {a2} +b^{2}}\right ) x^{2}}{\operatorname {a1} +\operatorname {a2}}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 \operatorname {a1} c -4 c \operatorname {a2} +b^{2}}}{\operatorname {a1} +\operatorname {a2}}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 \operatorname {a1} c -4 c \operatorname {a2} +b^{2}}\right )}{\left (\operatorname {a1} +\operatorname {a2} \right ) c}}}{2}\right )}{4 \sqrt {\frac {-b +\sqrt {-4 \operatorname {a1} c -4 c \operatorname {a2} +b^{2}}}{\operatorname {a1} +\operatorname {a2}}}\, \sqrt {c \,x^{4}+b \,x^{2}+\operatorname {a1} +\operatorname {a2}}}-\frac {b c \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 \operatorname {a1} c -4 c \operatorname {a2} +b^{2}}\right ) x^{2}}{\operatorname {a1} +\operatorname {a2}}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 \operatorname {a1} c -4 c \operatorname {a2} +b^{2}}\right ) x^{2}}{\operatorname {a1} +\operatorname {a2}}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 \operatorname {a1} c -4 c \operatorname {a2} +b^{2}}}{\operatorname {a1} +\operatorname {a2}}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 \operatorname {a1} c -4 c \operatorname {a2} +b^{2}}\right )}{\left (\operatorname {a1} +\operatorname {a2} \right ) c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 \operatorname {a1} c -4 c \operatorname {a2} +b^{2}}}{\operatorname {a1} +\operatorname {a2}}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 \operatorname {a1} c -4 c \operatorname {a2} +b^{2}}\right )}{\left (\operatorname {a1} +\operatorname {a2} \right ) c}}}{2}\right )\right )}{2 \left (4 \operatorname {a1} c +4 c \operatorname {a2} -b^{2}\right ) \sqrt {\frac {-b +\sqrt {-4 \operatorname {a1} c -4 c \operatorname {a2} +b^{2}}}{\operatorname {a1} +\operatorname {a2}}}\, \sqrt {c \,x^{4}+b \,x^{2}+\operatorname {a1} +\operatorname {a2}}\, \left (b +\sqrt {-4 \operatorname {a1} c -4 c \operatorname {a2} +b^{2}}\right )}\) | \(593\) |
| elliptic | \(-\frac {2 c \left (\frac {b \,x^{3}}{2 \left (\operatorname {a1} +\operatorname {a2} \right ) \left (4 \operatorname {a1} c +4 c \operatorname {a2} -b^{2}\right )}-\frac {\left (2 \operatorname {a1} c +2 c \operatorname {a2} -b^{2}\right ) x}{2 \left (\operatorname {a1} +\operatorname {a2} \right ) \left (4 \operatorname {a1} c +4 c \operatorname {a2} -b^{2}\right ) c}\right )}{\sqrt {\left (x^{4}+\frac {b \,x^{2}}{c}+\frac {\operatorname {a1} +\operatorname {a2}}{c}\right ) c}}+\frac {\left (\frac {1}{\operatorname {a1} +\operatorname {a2}}-\frac {2 \operatorname {a1} c +2 c \operatorname {a2} -b^{2}}{\left (\operatorname {a1} +\operatorname {a2} \right ) \left (4 \operatorname {a1} c +4 c \operatorname {a2} -b^{2}\right )}\right ) \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 \operatorname {a1} c -4 c \operatorname {a2} +b^{2}}\right ) x^{2}}{\operatorname {a1} +\operatorname {a2}}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 \operatorname {a1} c -4 c \operatorname {a2} +b^{2}}\right ) x^{2}}{\operatorname {a1} +\operatorname {a2}}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 \operatorname {a1} c -4 c \operatorname {a2} +b^{2}}}{\operatorname {a1} +\operatorname {a2}}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 \operatorname {a1} c -4 c \operatorname {a2} +b^{2}}\right )}{\left (\operatorname {a1} +\operatorname {a2} \right ) c}}}{2}\right )}{4 \sqrt {\frac {-b +\sqrt {-4 \operatorname {a1} c -4 c \operatorname {a2} +b^{2}}}{\operatorname {a1} +\operatorname {a2}}}\, \sqrt {c \,x^{4}+b \,x^{2}+\operatorname {a1} +\operatorname {a2}}}-\frac {b c \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 \operatorname {a1} c -4 c \operatorname {a2} +b^{2}}\right ) x^{2}}{\operatorname {a1} +\operatorname {a2}}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 \operatorname {a1} c -4 c \operatorname {a2} +b^{2}}\right ) x^{2}}{\operatorname {a1} +\operatorname {a2}}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 \operatorname {a1} c -4 c \operatorname {a2} +b^{2}}}{\operatorname {a1} +\operatorname {a2}}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 \operatorname {a1} c -4 c \operatorname {a2} +b^{2}}\right )}{\left (\operatorname {a1} +\operatorname {a2} \right ) c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 \operatorname {a1} c -4 c \operatorname {a2} +b^{2}}}{\operatorname {a1} +\operatorname {a2}}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 \operatorname {a1} c -4 c \operatorname {a2} +b^{2}}\right )}{\left (\operatorname {a1} +\operatorname {a2} \right ) c}}}{2}\right )\right )}{2 \left (4 \operatorname {a1} c +4 c \operatorname {a2} -b^{2}\right ) \sqrt {\frac {-b +\sqrt {-4 \operatorname {a1} c -4 c \operatorname {a2} +b^{2}}}{\operatorname {a1} +\operatorname {a2}}}\, \sqrt {c \,x^{4}+b \,x^{2}+\operatorname {a1} +\operatorname {a2}}\, \left (b +\sqrt {-4 \operatorname {a1} c -4 c \operatorname {a2} +b^{2}}\right )}\) | \(593\) |
Input:
int(1/(c*x^4+b*x^2+a1+a2)^(3/2),x,method=_RETURNVERBOSE)
Output:
-2*c*(1/2/(a1+a2)*b/(4*a1*c+4*a2*c-b^2)*x^3-1/2*(2*a1*c+2*a2*c-b^2)/(a1+a2 )/(4*a1*c+4*a2*c-b^2)/c*x)/((x^4+1/c*b*x^2+(a1+a2)/c)*c)^(1/2)+1/4*(1/(a1+ a2)-(2*a1*c+2*a2*c-b^2)/(a1+a2)/(4*a1*c+4*a2*c-b^2))*2^(1/2)/((-b+(-4*a1*c -4*a2*c+b^2)^(1/2))/(a1+a2))^(1/2)*(4-2*(-b+(-4*a1*c-4*a2*c+b^2)^(1/2))/(a 1+a2)*x^2)^(1/2)*(4+2*(b+(-4*a1*c-4*a2*c+b^2)^(1/2))/(a1+a2)*x^2)^(1/2)/(c *x^4+b*x^2+a1+a2)^(1/2)*EllipticF(1/2*x*2^(1/2)*((-b+(-4*a1*c-4*a2*c+b^2)^ (1/2))/(a1+a2))^(1/2),1/2*(-4+2*b*(b+(-4*a1*c-4*a2*c+b^2)^(1/2))/(a1+a2)/c )^(1/2))-1/2*b/(4*a1*c+4*a2*c-b^2)*c*2^(1/2)/((-b+(-4*a1*c-4*a2*c+b^2)^(1/ 2))/(a1+a2))^(1/2)*(4-2*(-b+(-4*a1*c-4*a2*c+b^2)^(1/2))/(a1+a2)*x^2)^(1/2) *(4+2*(b+(-4*a1*c-4*a2*c+b^2)^(1/2))/(a1+a2)*x^2)^(1/2)/(c*x^4+b*x^2+a1+a2 )^(1/2)/(b+(-4*a1*c-4*a2*c+b^2)^(1/2))*(EllipticF(1/2*x*2^(1/2)*((-b+(-4*a 1*c-4*a2*c+b^2)^(1/2))/(a1+a2))^(1/2),1/2*(-4+2*b*(b+(-4*a1*c-4*a2*c+b^2)^ (1/2))/(a1+a2)/c)^(1/2))-EllipticE(1/2*x*2^(1/2)*((-b+(-4*a1*c-4*a2*c+b^2) ^(1/2))/(a1+a2))^(1/2),1/2*(-4+2*b*(b+(-4*a1*c-4*a2*c+b^2)^(1/2))/(a1+a2)/ c)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 755 vs. \(2 (342) = 684\).
Time = 0.09 (sec) , antiderivative size = 755, normalized size of antiderivative = 1.86 \[ \int \frac {1}{\left (\text {a1}+\text {a2}+b x^2+c x^4\right )^{3/2}} \, dx=\text {Too large to display} \] Input:
integrate(1/(c*x^4+b*x^2+a1+a2)^(3/2),x, algorithm="fricas")
Output:
-1/2*(sqrt(1/2)*(b^2*c*x^4 + b^3*x^2 + (a1 + a2)*b^2 - ((a1 + a2)*b*c*x^4 + (a1 + a2)*b^2*x^2 + (a1^2 + 2*a1*a2 + a2^2)*b)*sqrt((b^2 - 4*(a1 + a2)*c )/(a1^2 + 2*a1*a2 + a2^2)))*sqrt(a1 + a2)*sqrt(((a1 + a2)*sqrt((b^2 - 4*(a 1 + a2)*c)/(a1^2 + 2*a1*a2 + a2^2)) - b)/(a1 + a2))*elliptic_e(arcsin(sqrt (1/2)*x*sqrt(((a1 + a2)*sqrt((b^2 - 4*(a1 + a2)*c)/(a1^2 + 2*a1*a2 + a2^2) ) - b)/(a1 + a2))), 1/2*((a1 + a2)*b*sqrt((b^2 - 4*(a1 + a2)*c)/(a1^2 + 2* a1*a2 + a2^2)) + b^2 - 2*(a1 + a2)*c)/((a1 + a2)*c)) - sqrt(1/2)*((2*(a1 + a2)*b + b^2)*c*x^4 + (a1 + a2)*b^2 + (2*(a1 + a2)*b^2 + b^3)*x^2 + 2*(a1^ 2 + 2*a1*a2 + a2^2)*b + ((2*a1^2 + 4*a1*a2 + 2*a2^2 - (a1 + a2)*b)*c*x^4 + 2*a1^3 + 6*a1^2*a2 + 6*a1*a2^2 + 2*a2^3 - ((a1 + a2)*b^2 - 2*(a1^2 + 2*a1 *a2 + a2^2)*b)*x^2 - (a1^2 + 2*a1*a2 + a2^2)*b)*sqrt((b^2 - 4*(a1 + a2)*c) /(a1^2 + 2*a1*a2 + a2^2)))*sqrt(a1 + a2)*sqrt(((a1 + a2)*sqrt((b^2 - 4*(a1 + a2)*c)/(a1^2 + 2*a1*a2 + a2^2)) - b)/(a1 + a2))*elliptic_f(arcsin(sqrt( 1/2)*x*sqrt(((a1 + a2)*sqrt((b^2 - 4*(a1 + a2)*c)/(a1^2 + 2*a1*a2 + a2^2)) - b)/(a1 + a2))), 1/2*((a1 + a2)*b*sqrt((b^2 - 4*(a1 + a2)*c)/(a1^2 + 2*a 1*a2 + a2^2)) + b^2 - 2*(a1 + a2)*c)/((a1 + a2)*c)) - 2*((a1 + a2)*b*c*x^3 + ((a1 + a2)*b^2 - 2*(a1^2 + 2*a1*a2 + a2^2)*c)*x)*sqrt(c*x^4 + b*x^2 + a 1 + a2))/(((a1^2 + 2*a1*a2 + a2^2)*b^2*c - 4*(a1^3 + 3*a1^2*a2 + 3*a1*a2^2 + a2^3)*c^2)*x^4 + (a1^3 + 3*a1^2*a2 + 3*a1*a2^2 + a2^3)*b^2 + ((a1^2 + 2 *a1*a2 + a2^2)*b^3 - 4*(a1^3 + 3*a1^2*a2 + 3*a1*a2^2 + a2^3)*b*c)*x^2 -...
\[ \int \frac {1}{\left (\text {a1}+\text {a2}+b x^2+c x^4\right )^{3/2}} \, dx=\int \frac {1}{\left (a_{1} + a_{2} + b x^{2} + c x^{4}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(1/(c*x**4+b*x**2+a1+a2)**(3/2),x)
Output:
Integral((a1 + a2 + b*x**2 + c*x**4)**(-3/2), x)
\[ \int \frac {1}{\left (\text {a1}+\text {a2}+b x^2+c x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (c x^{4} + b x^{2} + a_{1} + a_{2}\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(c*x^4+b*x^2+a1+a2)^(3/2),x, algorithm="maxima")
Output:
integrate((c*x^4 + b*x^2 + a1 + a2)^(-3/2), x)
\[ \int \frac {1}{\left (\text {a1}+\text {a2}+b x^2+c x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (c x^{4} + b x^{2} + a_{1} + a_{2}\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(c*x^4+b*x^2+a1+a2)^(3/2),x, algorithm="giac")
Output:
integrate((c*x^4 + b*x^2 + a1 + a2)^(-3/2), x)
Timed out. \[ \int \frac {1}{\left (\text {a1}+\text {a2}+b x^2+c x^4\right )^{3/2}} \, dx=\int \frac {1}{{\left (c\,x^4+b\,x^2+a_{1}+a_{2}\right )}^{3/2}} \,d x \] Input:
int(1/(a1 + a2 + b*x^2 + c*x^4)^(3/2),x)
Output:
int(1/(a1 + a2 + b*x^2 + c*x^4)^(3/2), x)
\[ \int \frac {1}{\left (\text {a1}+\text {a2}+b x^2+c x^4\right )^{3/2}} \, dx=\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+\mathit {a1} +\mathit {a2}}}{c^{2} x^{8}+2 b c \,x^{6}+2 \mathit {a1} c \,x^{4}+2 \mathit {a2} c \,x^{4}+b^{2} x^{4}+2 \mathit {a1} b \,x^{2}+2 \mathit {a2} b \,x^{2}+\mathit {a1}^{2}+2 \mathit {a1} \mathit {a2} +\mathit {a2}^{2}}d x \] Input:
int(1/(c*x^4+b*x^2+a1+a2)^(3/2),x)
Output:
int(sqrt(a1 + a2 + b*x**2 + c*x**4)/(a1**2 + 2*a1*a2 + 2*a1*b*x**2 + 2*a1* c*x**4 + a2**2 + 2*a2*b*x**2 + 2*a2*c*x**4 + b**2*x**4 + 2*b*c*x**6 + c**2 *x**8),x)