Integrand size = 20, antiderivative size = 384 \[ \int \frac {x^6}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=-\frac {x \left (a b+\left (b^2-2 a c\right ) x^2\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}+\frac {2 \left (b^2-3 a c\right ) x \sqrt {a+b x^2+c x^4}}{c^{3/2} \left (b^2-4 a c\right ) \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {2 \sqrt [4]{a} \left (b^2-3 a c\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{c^{7/4} \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}+\frac {\sqrt [4]{a} \left (2 b^2+\sqrt {a} b \sqrt {c}-6 a c\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 c^{7/4} \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}} \] Output:
-x*(a*b+(-2*a*c+b^2)*x^2)/c/(-4*a*c+b^2)/(c*x^4+b*x^2+a)^(1/2)+2*(-3*a*c+b ^2)*x*(c*x^4+b*x^2+a)^(1/2)/c^(3/2)/(-4*a*c+b^2)/(a^(1/2)+c^(1/2)*x^2)-2*a ^(1/4)*(-3*a*c+b^2)*(a^(1/2)+c^(1/2)*x^2)*((c*x^4+b*x^2+a)/(a^(1/2)+c^(1/2 )*x^2)^2)^(1/2)*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),1/2*(2-b/a^(1/2 )/c^(1/2))^(1/2))/c^(7/4)/(-4*a*c+b^2)/(c*x^4+b*x^2+a)^(1/2)+1/2*a^(1/4)*( 2*b^2+a^(1/2)*b*c^(1/2)-6*a*c)*(a^(1/2)+c^(1/2)*x^2)*((c*x^4+b*x^2+a)/(a^( 1/2)+c^(1/2)*x^2)^2)^(1/2)*InverseJacobiAM(2*arctan(c^(1/4)*x/a^(1/4)),1/2 *(2-b/a^(1/2)/c^(1/2))^(1/2))/c^(7/4)/(-4*a*c+b^2)/(c*x^4+b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 10.93 (sec) , antiderivative size = 489, normalized size of antiderivative = 1.27 \[ \int \frac {x^6}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\frac {2 c \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x \left (b^2 x^2+a \left (b-2 c x^2\right )\right )-i \left (b^2-3 a c\right ) \left (-b+\sqrt {b^2-4 a c}\right ) \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+4 c x^2}{b-\sqrt {b^2-4 a c}}} E\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )+i \left (-b^3+4 a b c+b^2 \sqrt {b^2-4 a c}-3 a c \sqrt {b^2-4 a c}\right ) \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+4 c x^2}{b-\sqrt {b^2-4 a c}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{2 c^2 \left (-b^2+4 a c\right ) \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \sqrt {a+b x^2+c x^4}} \] Input:
Integrate[x^6/(a + b*x^2 + c*x^4)^(3/2),x]
Output:
(2*c*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x*(b^2*x^2 + a*(b - 2*c*x^2)) - I*(b^ 2 - 3*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])]*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^3 + 4*a *b*c + b^2*Sqrt[b^2 - 4*a*c] - 3*a*c*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])])/(2*c^2*(-b^2 + 4*a*c)*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[a + b*x^2 + c*x^4])
Time = 0.86 (sec) , antiderivative size = 386, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {1440, 27, 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.
| \(\displaystyle \int \frac {x^6}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1440 |
| \(\displaystyle \frac {x^3 \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {\int \frac {3 x^2 \left (b x^2+2 a\right )}{\sqrt {c x^4+b x^2+a}}dx}{b^2-4 a c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^3 \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {3 \int \frac {x^2 \left (b x^2+2 a\right )}{\sqrt {c x^4+b x^2+a}}dx}{b^2-4 a c}\) |
| \(\Big \downarrow \) 1602 |
| \(\displaystyle \frac {x^3 \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {3 \left (\frac {b x \sqrt {a+b x^2+c x^4}}{3 c}-\frac {\int \frac {2 \left (b^2-3 a c\right ) x^2+a b}{\sqrt {c x^4+b x^2+a}}dx}{3 c}\right )}{b^2-4 a c}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {x^3 \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {3 \left (\frac {b x \sqrt {a+b x^2+c x^4}}{3 c}-\frac {\sqrt {a} \left (\frac {2 \left (b^2-3 a c\right )}{\sqrt {c}}+\sqrt {a} b\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx-\frac {2 \sqrt {a} \left (b^2-3 a c\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {a} \sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}}{3 c}\right )}{b^2-4 a c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^3 \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {3 \left (\frac {b x \sqrt {a+b x^2+c x^4}}{3 c}-\frac {\sqrt {a} \left (\frac {2 \left (b^2-3 a c\right )}{\sqrt {c}}+\sqrt {a} b\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx-\frac {2 \left (b^2-3 a c\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}}{3 c}\right )}{b^2-4 a c}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {x^3 \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {3 \left (\frac {b x \sqrt {a+b x^2+c x^4}}{3 c}-\frac {\frac {\sqrt [4]{a} \left (\frac {2 \left (b^2-3 a c\right )}{\sqrt {c}}+\sqrt {a} b\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {2 \left (b^2-3 a c\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}}{3 c}\right )}{b^2-4 a c}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {x^3 \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {3 \left (\frac {b x \sqrt {a+b x^2+c x^4}}{3 c}-\frac {\frac {\sqrt [4]{a} \left (\frac {2 \left (b^2-3 a c\right )}{\sqrt {c}}+\sqrt {a} b\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {2 \left (b^2-3 a c\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {c}}}{3 c}\right )}{b^2-4 a c}\) |
Input:
Int[x^6/(a + b*x^2 + c*x^4)^(3/2),x]
Output:
(x^3*(2*a + b*x^2))/((b^2 - 4*a*c)*Sqrt[a + b*x^2 + c*x^4]) - (3*((b*x*Sqr t[a + b*x^2 + c*x^4])/(3*c) - ((-2*(b^2 - 3*a*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)*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]) ))/Sqrt[c] + (a^(1/4)*(Sqrt[a]*b + (2*(b^2 - 3*a*c))/Sqrt[c])*(Sqrt[a] + S qrt[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^(1/4)*Sq rt[a + b*x^2 + c*x^4]))/(3*c)))/(b^2 - 4*a*c)
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_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(-d^3)*(d*x)^(m - 3)*(2*a + b*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2* (p + 1)*(b^2 - 4*a*c))), x] + Simp[d^4/(2*(p + 1)*(b^2 - 4*a*c)) Int[(d*x )^(m - 4)*(2*a*(m - 3) + b*(m + 4*p + 3)*x^2)*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && Gt Q[m, 3] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])
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 = 1.64 (sec) , antiderivative size = 482, normalized size of antiderivative = 1.26
| method | result | size |
| default | \(-\frac {2 c \left (\frac {\left (2 a c -b^{2}\right ) x^{3}}{2 c^{2} \left (4 a c -b^{2}\right )}-\frac {a b x}{2 c^{2} \left (4 a c -b^{2}\right )}\right )}{\sqrt {\left (x^{4}+\frac {b \,x^{2}}{c}+\frac {a}{c}\right ) c}}-\frac {a b \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 c \left (4 a c -b^{2}\right ) \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {\left (\frac {1}{c}+\frac {2 a c -b^{2}}{c \left (4 a c -b^{2}\right )}\right ) a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{2 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}\) | \(482\) |
| elliptic | \(-\frac {2 c \left (\frac {\left (2 a c -b^{2}\right ) x^{3}}{2 c^{2} \left (4 a c -b^{2}\right )}-\frac {a b x}{2 c^{2} \left (4 a c -b^{2}\right )}\right )}{\sqrt {\left (x^{4}+\frac {b \,x^{2}}{c}+\frac {a}{c}\right ) c}}-\frac {a b \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 c \left (4 a c -b^{2}\right ) \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {\left (\frac {1}{c}+\frac {2 a c -b^{2}}{c \left (4 a c -b^{2}\right )}\right ) a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{2 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}\) | \(482\) |
Input:
int(x^6/(c*x^4+b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
-2*c*(1/2/c^2*(2*a*c-b^2)/(4*a*c-b^2)*x^3-1/2*a*b/c^2/(4*a*c-b^2)*x)/((x^4 +1/c*b*x^2+1/c*a)*c)^(1/2)-1/4*a*b/c/(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*(1/c+(2*a*c-b^2)/c/(4*a*c-b^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))*(Elliptic F(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 = 0.08 (sec) , antiderivative size = 616, normalized size of antiderivative = 1.60 \[ \int \frac {x^6}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=-\frac {2 \, \sqrt {\frac {1}{2}} {\left ({\left (b^{3} c - 3 \, a b c^{2}\right )} x^{5} + {\left (b^{4} - 3 \, a b^{2} c\right )} x^{3} + {\left (a b^{3} - 3 \, a^{2} b c\right )} x - {\left ({\left (b^{2} c^{2} - 3 \, a c^{3}\right )} x^{5} + {\left (b^{3} c - 3 \, a b c^{2}\right )} x^{3} + {\left (a b^{2} c - 3 \, a^{2} c^{2}\right )} x\right )} \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}}\right )} \sqrt {c} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}} E(\arcsin \left (\frac {\sqrt {\frac {1}{2}} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}}}{x}\right )\,|\,\frac {b c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} + b^{2} - 2 \, a c}{2 \, a c}) - \sqrt {\frac {1}{2}} {\left ({\left (2 \, b^{3} c - {\left (6 \, a b - b^{2}\right )} c^{2}\right )} x^{5} + {\left (2 \, b^{4} - {\left (6 \, a b^{2} - b^{3}\right )} c\right )} x^{3} + {\left (2 \, a b^{3} - {\left (6 \, a^{2} b - a b^{2}\right )} c\right )} x - {\left ({\left (2 \, b^{2} c^{2} - {\left (6 \, a + b\right )} c^{3}\right )} x^{5} + {\left (2 \, b^{3} c - {\left (6 \, a b + b^{2}\right )} c^{2}\right )} x^{3} + {\left (2 \, a b^{2} c - {\left (6 \, a^{2} + a b\right )} c^{2}\right )} x\right )} \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}}\right )} \sqrt {c} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}} F(\arcsin \left (\frac {\sqrt {\frac {1}{2}} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}}}{x}\right )\,|\,\frac {b c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} + b^{2} - 2 \, a c}{2 \, a c}) - 2 \, {\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{4} + 2 \, a b^{2} c - 6 \, a^{2} c^{2} + {\left (2 \, b^{3} c - 7 \, a b c^{2}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2} + a}}{2 \, {\left ({\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x^{5} + {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} x^{3} + {\left (a b^{2} c^{3} - 4 \, a^{2} c^{4}\right )} x\right )}} \] Input:
integrate(x^6/(c*x^4+b*x^2+a)^(3/2),x, algorithm="fricas")
Output:
-1/2*(2*sqrt(1/2)*((b^3*c - 3*a*b*c^2)*x^5 + (b^4 - 3*a*b^2*c)*x^3 + (a*b^ 3 - 3*a^2*b*c)*x - ((b^2*c^2 - 3*a*c^3)*x^5 + (b^3*c - 3*a*b*c^2)*x^3 + (a *b^2*c - 3*a^2*c^2)*x)*sqrt((b^2 - 4*a*c)/c^2))*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)*((2*b^3*c - (6*a*b - b^2)*c^2)*x^5 + (2*b^4 - (6*a*b^2 - b^3) *c)*x^3 + (2*a*b^3 - (6*a^2*b - a*b^2)*c)*x - ((2*b^2*c^2 - (6*a + b)*c^3) *x^5 + (2*b^3*c - (6*a*b + b^2)*c^2)*x^3 + (2*a*b^2*c - (6*a^2 + a*b)*c^2) *x)*sqrt((b^2 - 4*a*c)/c^2))*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*((b^2*c^2 - 4* a*c^3)*x^4 + 2*a*b^2*c - 6*a^2*c^2 + (2*b^3*c - 7*a*b*c^2)*x^2)*sqrt(c*x^4 + b*x^2 + a))/((b^2*c^4 - 4*a*c^5)*x^5 + (b^3*c^3 - 4*a*b*c^4)*x^3 + (a*b ^2*c^3 - 4*a^2*c^4)*x)
\[ \int \frac {x^6}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int \frac {x^{6}}{\left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(x**6/(c*x**4+b*x**2+a)**(3/2),x)
Output:
Integral(x**6/(a + b*x**2 + c*x**4)**(3/2), x)
\[ \int \frac {x^6}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int { \frac {x^{6}}{{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(x^6/(c*x^4+b*x^2+a)^(3/2),x, algorithm="maxima")
Output:
integrate(x^6/(c*x^4 + b*x^2 + a)^(3/2), x)
\[ \int \frac {x^6}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int { \frac {x^{6}}{{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(x^6/(c*x^4+b*x^2+a)^(3/2),x, algorithm="giac")
Output:
integrate(x^6/(c*x^4 + b*x^2 + a)^(3/2), x)
Timed out. \[ \int \frac {x^6}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int \frac {x^6}{{\left (c\,x^4+b\,x^2+a\right )}^{3/2}} \,d x \] Input:
int(x^6/(a + b*x^2 + c*x^4)^(3/2),x)
Output:
int(x^6/(a + b*x^2 + c*x^4)^(3/2), x)
\[ \int \frac {x^6}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\frac {2 \sqrt {c \,x^{4}+b \,x^{2}+a}\, b x +\sqrt {c \,x^{4}+b \,x^{2}+a}\, c \,x^{3}-2 \left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{c^{2} x^{8}+2 b c \,x^{6}+2 a c \,x^{4}+b^{2} x^{4}+2 a b \,x^{2}+a^{2}}d x \right ) a^{2} b -2 \left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{c^{2} x^{8}+2 b c \,x^{6}+2 a c \,x^{4}+b^{2} x^{4}+2 a b \,x^{2}+a^{2}}d x \right ) a \,b^{2} x^{2}-2 \left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{c^{2} x^{8}+2 b c \,x^{6}+2 a c \,x^{4}+b^{2} x^{4}+2 a b \,x^{2}+a^{2}}d x \right ) a b c \,x^{4}-3 \left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{c^{2} x^{8}+2 b c \,x^{6}+2 a c \,x^{4}+b^{2} x^{4}+2 a b \,x^{2}+a^{2}}d x \right ) a^{2} c -3 \left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{c^{2} x^{8}+2 b c \,x^{6}+2 a c \,x^{4}+b^{2} x^{4}+2 a b \,x^{2}+a^{2}}d x \right ) a b c \,x^{2}-3 \left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{c^{2} x^{8}+2 b c \,x^{6}+2 a c \,x^{4}+b^{2} x^{4}+2 a b \,x^{2}+a^{2}}d x \right ) a \,c^{2} x^{4}}{c^{2} \left (c \,x^{4}+b \,x^{2}+a \right )} \] Input:
int(x^6/(c*x^4+b*x^2+a)^(3/2),x)
Output:
(2*sqrt(a + b*x**2 + c*x**4)*b*x + sqrt(a + b*x**2 + c*x**4)*c*x**3 - 2*in t(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)*a**2*b - 2*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)*a*b**2* x**2 - 2*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)*a*b*c*x**4 - 3*int((sqrt(a + b*x**2 + c*x**4)*x**2)/(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)*a**2*c - 3*int((sqrt(a + b*x**2 + c*x**4)*x**2)/(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)*a*b*c*x**2 - 3*int((sqrt(a + b*x**2 + c*x**4)*x**2)/(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)*a*c**2*x**4)/(c**2*(a + b*x**2 + c* x**4))