Integrand size = 18, antiderivative size = 407 \[ \int \frac {1}{\left (a-b x^2-c x^4\right )^{3/2}} \, dx=\frac {x \left (b^2+2 a c+b c x^2\right )}{a \left (b^2+4 a c\right ) \sqrt {a-b x^2-c x^4}}+\frac {b \sqrt {c} x \sqrt {a-b x^2-c x^4}}{a \left (b^2+4 a c\right ) \left (\sqrt {-a}+\sqrt {c} x^2\right )}-\frac {b \sqrt [4]{c} \left (1+\frac {\sqrt {c} x^2}{\sqrt {-a}}\right ) \sqrt {\frac {a-b x^2-c x^4}{a \left (1+\frac {\sqrt {c} x^2}{\sqrt {-a}}\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{-a}}\right )|\frac {1}{4} \left (2+\frac {a b}{(-a)^{3/2} \sqrt {c}}\right )\right )}{\sqrt [4]{-a} \left (b^2+4 a c\right ) \sqrt {a-b x^2-c x^4}}+\frac {\left (b+2 \sqrt {-a} \sqrt {c}\right ) \sqrt [4]{c} \left (1+\frac {\sqrt {c} x^2}{\sqrt {-a}}\right ) \sqrt {\frac {a-b x^2-c x^4}{a \left (1+\frac {\sqrt {c} x^2}{\sqrt {-a}}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{-a}}\right ),\frac {1}{4} \left (2+\frac {a b}{(-a)^{3/2} \sqrt {c}}\right )\right )}{2 \sqrt [4]{-a} \left (b^2+4 a c\right ) \sqrt {a-b x^2-c x^4}} \] Output:
x*(b*c*x^2+2*a*c+b^2)/a/(4*a*c+b^2)/(-c*x^4-b*x^2+a)^(1/2)+b*c^(1/2)*x*(-c *x^4-b*x^2+a)^(1/2)/a/(4*a*c+b^2)/(c^(1/2)*x^2+(-a)^(1/2))-b*c^(1/4)*(1+c^ (1/2)*x^2/(-a)^(1/2))*((-c*x^4-b*x^2+a)/a/(1+c^(1/2)*x^2/(-a)^(1/2))^2)^(1 /2)*EllipticE(sin(2*arctan(c^(1/4)*x/(-a)^(1/4))),1/2*(2+a*b/(-a)^(3/2)/c^ (1/2))^(1/2))/(-a)^(1/4)/(4*a*c+b^2)/(-c*x^4-b*x^2+a)^(1/2)+1/2*(b+2*(-a)^ (1/2)*c^(1/2))*c^(1/4)*(1+c^(1/2)*x^2/(-a)^(1/2))*((-c*x^4-b*x^2+a)/a/(1+c ^(1/2)*x^2/(-a)^(1/2))^2)^(1/2)*InverseJacobiAM(2*arctan(c^(1/4)*x/(-a)^(1 /4)),1/2*(2+a*b/(-a)^(3/2)/c^(1/2))^(1/2))/(-a)^(1/4)/(4*a*c+b^2)/(-c*x^4- b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 10.67 (sec) , antiderivative size = 462, normalized size of antiderivative = 1.14 \[ \int \frac {1}{\left (a-b x^2-c x^4\right )^{3/2}} \, dx=\frac {4 \sqrt {\frac {c}{b-\sqrt {b^2+4 a c}}} x \left (b^2+2 a c+b c x^2\right )+i b \left (b+\sqrt {b^2+4 a c}\right ) \sqrt {\frac {b+\sqrt {b^2+4 a c}+2 c x^2}{b+\sqrt {b^2+4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2+4 a c}+4 c x^2}{b-\sqrt {b^2+4 a c}}} E\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b-\sqrt {b^2+4 a c}}} x\right )|\frac {b-\sqrt {b^2+4 a c}}{b+\sqrt {b^2+4 a c}}\right )-i \left (b^2+4 a c+b \sqrt {b^2+4 a c}\right ) \sqrt {\frac {b+\sqrt {b^2+4 a c}+2 c x^2}{b+\sqrt {b^2+4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2+4 a c}+4 c x^2}{b-\sqrt {b^2+4 a c}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b-\sqrt {b^2+4 a c}}} x\right ),\frac {b-\sqrt {b^2+4 a c}}{b+\sqrt {b^2+4 a c}}\right )}{4 a \left (b^2+4 a c\right ) \sqrt {\frac {c}{b-\sqrt {b^2+4 a c}}} \sqrt {a-x^2 \left (b+c x^2\right )}} \] Input:
Integrate[(a - b*x^2 - c*x^4)^(-3/2),x]
Output:
(4*Sqrt[c/(b - Sqrt[b^2 + 4*a*c])]*x*(b^2 + 2*a*c + b*c*x^2) + I*b*(b + Sq rt[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*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*(b^2 + 4*a*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])]*EllipticF[ 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])])/(4*a*(b^2 + 4*a*c)*Sqrt[c/(b - Sqrt[b^2 + 4* a*c])]*Sqrt[a - x^2*(b + c*x^2)])
Time = 1.46 (sec) , antiderivative size = 662, normalized size of antiderivative = 1.63, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {1405, 25, 27, 1514, 406, 320, 388, 313}
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.
\(\displaystyle \int \frac {1}{\left (a-b x^2-c x^4\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 1405 |
\(\displaystyle \frac {x \left (2 a c+b^2+b c x^2\right )}{a \left (4 a c+b^2\right ) \sqrt {a-b x^2-c x^4}}-\frac {\int -\frac {c \left (2 a-b x^2\right )}{\sqrt {-c x^4-b x^2+a}}dx}{a \left (4 a c+b^2\right )}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \frac {c \left (2 a-b x^2\right )}{\sqrt {-c x^4-b x^2+a}}dx}{a \left (4 a c+b^2\right )}+\frac {x \left (2 a c+b^2+b c x^2\right )}{a \left (4 a c+b^2\right ) \sqrt {a-b x^2-c x^4}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {c \int \frac {2 a-b x^2}{\sqrt {-c x^4-b x^2+a}}dx}{a \left (4 a c+b^2\right )}+\frac {x \left (2 a c+b^2+b c x^2\right )}{a \left (4 a c+b^2\right ) \sqrt {a-b x^2-c x^4}}\) |
\(\Big \downarrow \) 1514 |
\(\displaystyle \frac {c \sqrt {\frac {2 c x^2}{b-\sqrt {4 a c+b^2}}+1} \sqrt {\frac {2 c x^2}{\sqrt {4 a c+b^2}+b}+1} \int \frac {2 a-b x^2}{\sqrt {\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}+1} \sqrt {\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}+1}}dx}{a \left (4 a c+b^2\right ) \sqrt {a-b x^2-c x^4}}+\frac {x \left (2 a c+b^2+b c x^2\right )}{a \left (4 a c+b^2\right ) \sqrt {a-b x^2-c x^4}}\) |
\(\Big \downarrow \) 406 |
\(\displaystyle \frac {c \sqrt {\frac {2 c x^2}{b-\sqrt {4 a c+b^2}}+1} \sqrt {\frac {2 c x^2}{\sqrt {4 a c+b^2}+b}+1} \left (2 a \int \frac {1}{\sqrt {\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}+1} \sqrt {\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}+1}}dx-b \int \frac {x^2}{\sqrt {\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}+1} \sqrt {\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}+1}}dx\right )}{a \left (4 a c+b^2\right ) \sqrt {a-b x^2-c x^4}}+\frac {x \left (2 a c+b^2+b c x^2\right )}{a \left (4 a c+b^2\right ) \sqrt {a-b x^2-c x^4}}\) |
\(\Big \downarrow \) 320 |
\(\displaystyle \frac {c \sqrt {\frac {2 c x^2}{b-\sqrt {4 a c+b^2}}+1} \sqrt {\frac {2 c x^2}{\sqrt {4 a c+b^2}+b}+1} \left (\frac {\sqrt {2} a \sqrt {\sqrt {4 a c+b^2}+b} \sqrt {\frac {2 c x^2}{b-\sqrt {4 a c+b^2}}+1} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2+4 a c}}}\right ),-\frac {2 \sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{\sqrt {c} \sqrt {\frac {\frac {2 c x^2}{b-\sqrt {4 a c+b^2}}+1}{\frac {2 c x^2}{\sqrt {4 a c+b^2}+b}+1}} \sqrt {\frac {2 c x^2}{\sqrt {4 a c+b^2}+b}+1}}-b \int \frac {x^2}{\sqrt {\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}+1} \sqrt {\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}+1}}dx\right )}{a \left (4 a c+b^2\right ) \sqrt {a-b x^2-c x^4}}+\frac {x \left (2 a c+b^2+b c x^2\right )}{a \left (4 a c+b^2\right ) \sqrt {a-b x^2-c x^4}}\) |
\(\Big \downarrow \) 388 |
\(\displaystyle \frac {c \sqrt {\frac {2 c x^2}{b-\sqrt {4 a c+b^2}}+1} \sqrt {\frac {2 c x^2}{\sqrt {4 a c+b^2}+b}+1} \left (\frac {\sqrt {2} a \sqrt {\sqrt {4 a c+b^2}+b} \sqrt {\frac {2 c x^2}{b-\sqrt {4 a c+b^2}}+1} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2+4 a c}}}\right ),-\frac {2 \sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{\sqrt {c} \sqrt {\frac {\frac {2 c x^2}{b-\sqrt {4 a c+b^2}}+1}{\frac {2 c x^2}{\sqrt {4 a c+b^2}+b}+1}} \sqrt {\frac {2 c x^2}{\sqrt {4 a c+b^2}+b}+1}}-b \left (\frac {x \left (b-\sqrt {4 a c+b^2}\right ) \sqrt {\frac {2 c x^2}{b-\sqrt {4 a c+b^2}}+1}}{2 c \sqrt {\frac {2 c x^2}{\sqrt {4 a c+b^2}+b}+1}}-\frac {\left (b-\sqrt {4 a c+b^2}\right ) \int \frac {\sqrt {\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}+1}}{\left (\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}+1\right )^{3/2}}dx}{2 c}\right )\right )}{a \left (4 a c+b^2\right ) \sqrt {a-b x^2-c x^4}}+\frac {x \left (2 a c+b^2+b c x^2\right )}{a \left (4 a c+b^2\right ) \sqrt {a-b x^2-c x^4}}\) |
\(\Big \downarrow \) 313 |
\(\displaystyle \frac {c \sqrt {\frac {2 c x^2}{b-\sqrt {4 a c+b^2}}+1} \sqrt {\frac {2 c x^2}{\sqrt {4 a c+b^2}+b}+1} \left (\frac {\sqrt {2} a \sqrt {\sqrt {4 a c+b^2}+b} \sqrt {\frac {2 c x^2}{b-\sqrt {4 a c+b^2}}+1} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2+4 a c}}}\right ),-\frac {2 \sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{\sqrt {c} \sqrt {\frac {\frac {2 c x^2}{b-\sqrt {4 a c+b^2}}+1}{\frac {2 c x^2}{\sqrt {4 a c+b^2}+b}+1}} \sqrt {\frac {2 c x^2}{\sqrt {4 a c+b^2}+b}+1}}-b \left (\frac {x \left (b-\sqrt {4 a c+b^2}\right ) \sqrt {\frac {2 c x^2}{b-\sqrt {4 a c+b^2}}+1}}{2 c \sqrt {\frac {2 c x^2}{\sqrt {4 a c+b^2}+b}+1}}-\frac {\left (b-\sqrt {4 a c+b^2}\right ) \sqrt {\sqrt {4 a c+b^2}+b} \sqrt {\frac {2 c x^2}{b-\sqrt {4 a c+b^2}}+1} E\left (\arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2+4 a c}}}\right )|-\frac {2 \sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{2 \sqrt {2} c^{3/2} \sqrt {\frac {\frac {2 c x^2}{b-\sqrt {4 a c+b^2}}+1}{\frac {2 c x^2}{\sqrt {4 a c+b^2}+b}+1}} \sqrt {\frac {2 c x^2}{\sqrt {4 a c+b^2}+b}+1}}\right )\right )}{a \left (4 a c+b^2\right ) \sqrt {a-b x^2-c x^4}}+\frac {x \left (2 a c+b^2+b c x^2\right )}{a \left (4 a c+b^2\right ) \sqrt {a-b x^2-c x^4}}\) |
Input:
Int[(a - b*x^2 - c*x^4)^(-3/2),x]
Output:
(x*(b^2 + 2*a*c + b*c*x^2))/(a*(b^2 + 4*a*c)*Sqrt[a - b*x^2 - c*x^4]) + (c *Sqrt[1 + (2*c*x^2)/(b - Sqrt[b^2 + 4*a*c])]*Sqrt[1 + (2*c*x^2)/(b + Sqrt[ b^2 + 4*a*c])]*(-(b*(((b - Sqrt[b^2 + 4*a*c])*x*Sqrt[1 + (2*c*x^2)/(b - Sq rt[b^2 + 4*a*c])])/(2*c*Sqrt[1 + (2*c*x^2)/(b + Sqrt[b^2 + 4*a*c])]) - ((b - Sqrt[b^2 + 4*a*c])*Sqrt[b + Sqrt[b^2 + 4*a*c]]*Sqrt[1 + (2*c*x^2)/(b - Sqrt[b^2 + 4*a*c])]*EllipticE[ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 + 4*a*c]]], (-2*Sqrt[b^2 + 4*a*c])/(b - Sqrt[b^2 + 4*a*c])])/(2*Sqrt[2]*c ^(3/2)*Sqrt[(1 + (2*c*x^2)/(b - Sqrt[b^2 + 4*a*c]))/(1 + (2*c*x^2)/(b + Sq rt[b^2 + 4*a*c]))]*Sqrt[1 + (2*c*x^2)/(b + Sqrt[b^2 + 4*a*c])]))) + (Sqrt[ 2]*a*Sqrt[b + Sqrt[b^2 + 4*a*c]]*Sqrt[1 + (2*c*x^2)/(b - Sqrt[b^2 + 4*a*c] )]*EllipticF[ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 + 4*a*c]]], (-2* Sqrt[b^2 + 4*a*c])/(b - Sqrt[b^2 + 4*a*c])])/(Sqrt[c]*Sqrt[(1 + (2*c*x^2)/ (b - Sqrt[b^2 + 4*a*c]))/(1 + (2*c*x^2)/(b + Sqrt[b^2 + 4*a*c]))]*Sqrt[1 + (2*c*x^2)/(b + Sqrt[b^2 + 4*a*c])])))/(a*(b^2 + 4*a*c)*Sqrt[a - b*x^2 - c *x^4])
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Sim p[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c + d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; FreeQ [{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*( c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; Fre eQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] && !SimplerSqrtQ[b/a, d/c]
Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt[c + d*x^2])), x] - Simp[c/b Int[Sqrt[ a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && PosQ[b/a] && PosQ[d/c] && !SimplerSqrtQ[b/a, d/c]
Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( x_)^2), x_Symbol] :> Simp[e Int[(a + b*x^2)^p*(c + d*x^2)^q, x], x] + Sim p[f Int[x^2*(a + b*x^2)^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e, f, p, q}, 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[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[Sqrt[1 + 2*c*(x^2/(b - q))]*(Sqrt [1 + 2*c*(x^2/(b + q))]/Sqrt[a + b*x^2 + c*x^4]) Int[(d + e*x^2)/(Sqrt[1 + 2*c*(x^2/(b - q))]*Sqrt[1 + 2*c*(x^2/(b + q))]), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NegQ[c/a]
Time = 0.56 (sec) , antiderivative size = 473, normalized size of antiderivative = 1.16
method | result | size |
default | \(\frac {2 c \left (\frac {b \,x^{3}}{2 a \left (4 a c +b^{2}\right )}+\frac {\left (2 a c +b^{2}\right ) x}{2 a \left (4 a c +b^{2}\right ) c}\right )}{\sqrt {-\left (x^{4}+\frac {b \,x^{2}}{c}-\frac {a}{c}\right ) c}}+\frac {\left (\frac {1}{a}-\frac {2 a c +b^{2}}{a \left (4 a c +b^{2}\right )}\right ) \sqrt {2}\, \sqrt {4-\frac {2 \left (b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {b +\sqrt {4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (-b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 \sqrt {\frac {b +\sqrt {4 a c +b^{2}}}{a}}\, \sqrt {-c \,x^{4}-b \,x^{2}+a}}+\frac {b c \sqrt {2}\, \sqrt {4-\frac {2 \left (b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {b +\sqrt {4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (-b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {b +\sqrt {4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (-b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{2 \left (4 a c +b^{2}\right ) \sqrt {\frac {b +\sqrt {4 a c +b^{2}}}{a}}\, \sqrt {-c \,x^{4}-b \,x^{2}+a}\, \left (-b +\sqrt {4 a c +b^{2}}\right )}\) | \(473\) |
elliptic | \(\frac {2 c \left (\frac {b \,x^{3}}{2 a \left (4 a c +b^{2}\right )}+\frac {\left (2 a c +b^{2}\right ) x}{2 a \left (4 a c +b^{2}\right ) c}\right )}{\sqrt {-\left (x^{4}+\frac {b \,x^{2}}{c}-\frac {a}{c}\right ) c}}+\frac {\left (\frac {1}{a}-\frac {2 a c +b^{2}}{a \left (4 a c +b^{2}\right )}\right ) \sqrt {2}\, \sqrt {4-\frac {2 \left (b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {b +\sqrt {4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (-b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 \sqrt {\frac {b +\sqrt {4 a c +b^{2}}}{a}}\, \sqrt {-c \,x^{4}-b \,x^{2}+a}}+\frac {b c \sqrt {2}\, \sqrt {4-\frac {2 \left (b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {b +\sqrt {4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (-b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {b +\sqrt {4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (-b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{2 \left (4 a c +b^{2}\right ) \sqrt {\frac {b +\sqrt {4 a c +b^{2}}}{a}}\, \sqrt {-c \,x^{4}-b \,x^{2}+a}\, \left (-b +\sqrt {4 a c +b^{2}}\right )}\) | \(473\) |
Input:
int(1/(-c*x^4-b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
2*c*(1/2*b/a/(4*a*c+b^2)*x^3+1/2*(2*a*c+b^2)/a/(4*a*c+b^2)/c*x)/(-(x^4+1/c *b*x^2-1/c*a)*c)^(1/2)+1/4*(1/a-(2*a*c+b^2)/a/(4*a*c+b^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)*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*b/(4*a*c+b^2)*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)/(-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 = 466, normalized size of antiderivative = 1.14 \[ \int \frac {1}{\left (a-b x^2-c x^4\right )^{3/2}} \, dx=\frac {\sqrt {\frac {1}{2}} {\left ({\left (a b c x^{4} + a b^{2} x^{2} - a^{2} b\right )} \sqrt {a} \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} + {\left (b^{2} c x^{4} + b^{3} x^{2} - a b^{2}\right )} \sqrt {a}\right )} \sqrt {\frac {a \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} + b}{a}} E(\arcsin \left (\sqrt {\frac {1}{2}} x \sqrt {\frac {a \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} + b}{a}}\right )\,|\,\frac {a b \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} - b^{2} - 2 \, a c}{2 \, a c}) - \sqrt {\frac {1}{2}} {\left ({\left ({\left (2 \, a^{2} + a b\right )} c x^{4} - 2 \, a^{3} - a^{2} b + {\left (2 \, a^{2} b + a b^{2}\right )} x^{2}\right )} \sqrt {a} \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} - {\left ({\left (2 \, a b - b^{2}\right )} c x^{4} - 2 \, a^{2} b + a b^{2} + {\left (2 \, a b^{2} - b^{3}\right )} x^{2}\right )} \sqrt {a}\right )} \sqrt {\frac {a \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} + b}{a}} F(\arcsin \left (\sqrt {\frac {1}{2}} x \sqrt {\frac {a \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} + b}{a}}\right )\,|\,\frac {a b \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} - b^{2} - 2 \, a c}{2 \, a c}) + 2 \, {\left (a b c x^{3} + {\left (a b^{2} + 2 \, a^{2} c\right )} x\right )} \sqrt {-c x^{4} - b x^{2} + a}}{2 \, {\left (a^{3} b^{2} + 4 \, a^{4} c - {\left (a^{2} b^{2} c + 4 \, a^{3} c^{2}\right )} x^{4} - {\left (a^{2} b^{3} + 4 \, a^{3} b c\right )} x^{2}\right )}} \] Input:
integrate(1/(-c*x^4-b*x^2+a)^(3/2),x, algorithm="fricas")
Output:
1/2*(sqrt(1/2)*((a*b*c*x^4 + a*b^2*x^2 - a^2*b)*sqrt(a)*sqrt((b^2 + 4*a*c) /a^2) + (b^2*c*x^4 + b^3*x^2 - a*b^2)*sqrt(a))*sqrt((a*sqrt((b^2 + 4*a*c)/ a^2) + b)/a)*elliptic_e(arcsin(sqrt(1/2)*x*sqrt((a*sqrt((b^2 + 4*a*c)/a^2) + b)/a)), 1/2*(a*b*sqrt((b^2 + 4*a*c)/a^2) - b^2 - 2*a*c)/(a*c)) - sqrt(1 /2)*(((2*a^2 + a*b)*c*x^4 - 2*a^3 - a^2*b + (2*a^2*b + a*b^2)*x^2)*sqrt(a) *sqrt((b^2 + 4*a*c)/a^2) - ((2*a*b - b^2)*c*x^4 - 2*a^2*b + a*b^2 + (2*a*b ^2 - b^3)*x^2)*sqrt(a))*sqrt((a*sqrt((b^2 + 4*a*c)/a^2) + b)/a)*elliptic_f (arcsin(sqrt(1/2)*x*sqrt((a*sqrt((b^2 + 4*a*c)/a^2) + b)/a)), 1/2*(a*b*sqr t((b^2 + 4*a*c)/a^2) - b^2 - 2*a*c)/(a*c)) + 2*(a*b*c*x^3 + (a*b^2 + 2*a^2 *c)*x)*sqrt(-c*x^4 - b*x^2 + a))/(a^3*b^2 + 4*a^4*c - (a^2*b^2*c + 4*a^3*c ^2)*x^4 - (a^2*b^3 + 4*a^3*b*c)*x^2)
\[ \int \frac {1}{\left (a-b x^2-c x^4\right )^{3/2}} \, dx=\int \frac {1}{\left (a - b x^{2} - c x^{4}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(1/(-c*x**4-b*x**2+a)**(3/2),x)
Output:
Integral((a - b*x**2 - c*x**4)**(-3/2), x)
\[ \int \frac {1}{\left (a-b x^2-c x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-c x^{4} - b x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(-c*x^4-b*x^2+a)^(3/2),x, algorithm="maxima")
Output:
integrate((-c*x^4 - b*x^2 + a)^(-3/2), x)
\[ \int \frac {1}{\left (a-b x^2-c x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-c x^{4} - b x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(-c*x^4-b*x^2+a)^(3/2),x, algorithm="giac")
Output:
integrate((-c*x^4 - b*x^2 + a)^(-3/2), x)
Timed out. \[ \int \frac {1}{\left (a-b x^2-c x^4\right )^{3/2}} \, dx=\int \frac {1}{{\left (-c\,x^4-b\,x^2+a\right )}^{3/2}} \,d x \] Input:
int(1/(a - b*x^2 - c*x^4)^(3/2),x)
Output:
int(1/(a - b*x^2 - c*x^4)^(3/2), x)
\[ \int \frac {1}{\left (a-b x^2-c x^4\right )^{3/2}} \, dx=\int \frac {\sqrt {-c \,x^{4}-b \,x^{2}+a}}{c^{2} x^{8}+2 b c \,x^{6}-2 a c \,x^{4}+b^{2} x^{4}-2 a b \,x^{2}+a^{2}}d x \] Input:
int(1/(-c*x^4-b*x^2+a)^(3/2),x)
Output:
int(sqrt(a - b*x**2 - c*x**4)/(a**2 - 2*a*b*x**2 - 2*a*c*x**4 + b**2*x**4 + 2*b*c*x**6 + c**2*x**8),x)