Integrand size = 33, antiderivative size = 315 \[ \int \frac {1}{x^4 \sqrt {a+b x^2+\frac {\left (b d e-a e^2\right ) x^4}{d^2}}} \, dx=-\frac {\sqrt {a+b x^2+\frac {e (b d-a e) x^4}{d^2}}}{3 a x^3}+\frac {2 b d \sqrt {a+b x^2+\frac {e (b d-a e) x^4}{d^2}}}{3 a^2 x \left (d+e x^2\right )}+\frac {2 b \sqrt {e} \left (d+e x^2\right ) \sqrt {\frac {a d+(b d-a e) x^2}{a \left (d+e x^2\right )}} E\left (\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )|2-\frac {b d}{a e}\right )}{3 a d^{3/2} \sqrt {a+b x^2+\frac {e (b d-a e) x^4}{d^2}}}-\frac {\sqrt {e} (b d-a e) \left (d+e x^2\right ) \sqrt {\frac {a d+(b d-a e) x^2}{a \left (d+e x^2\right )}} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),2-\frac {b d}{a e}\right )}{3 a d^{5/2} \sqrt {a+b x^2+\frac {e (b d-a e) x^4}{d^2}}} \] Output:
-1/3*(a+b*x^2+e*(-a*e+b*d)*x^4/d^2)^(1/2)/a/x^3+2/3*b*d*(a+b*x^2+e*(-a*e+b *d)*x^4/d^2)^(1/2)/a^2/x/(e*x^2+d)+2/3*b*e^(1/2)*(e*x^2+d)*((a*d+(-a*e+b*d )*x^2)/a/(e*x^2+d))^(1/2)*EllipticE(e^(1/2)*x/d^(1/2)/(1+e*x^2/d)^(1/2),(2 -b*d/a/e)^(1/2))/a/d^(3/2)/(a+b*x^2+e*(-a*e+b*d)*x^4/d^2)^(1/2)-1/3*e^(1/2 )*(-a*e+b*d)*(e*x^2+d)*((a*d+(-a*e+b*d)*x^2)/a/(e*x^2+d))^(1/2)*InverseJac obiAM(arctan(e^(1/2)*x/d^(1/2)),(2-b*d/a/e)^(1/2))/a/d^(5/2)/(a+b*x^2+e*(- a*e+b*d)*x^4/d^2)^(1/2)
Result contains complex when optimal does not.
Time = 3.37 (sec) , antiderivative size = 256, normalized size of antiderivative = 0.81 \[ \int \frac {1}{x^4 \sqrt {a+b x^2+\frac {\left (b d e-a e^2\right ) x^4}{d^2}}} \, dx=\frac {\sqrt {\frac {e}{d}} \left (-a+2 b x^2\right ) \left (d+e x^2\right ) \left (b d x^2+a \left (d-e x^2\right )\right )+2 i a b d e x^3 \sqrt {1+\frac {b x^2}{a}-\frac {e x^2}{d}} \sqrt {1+\frac {e x^2}{d}} E\left (i \text {arcsinh}\left (\sqrt {\frac {e}{d}} x\right )|-1+\frac {b d}{a e}\right )-i a e (b d+a e) x^3 \sqrt {1+\frac {b x^2}{a}-\frac {e x^2}{d}} \sqrt {1+\frac {e x^2}{d}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {e}{d}} x\right ),-1+\frac {b d}{a e}\right )}{3 a^2 d^2 \sqrt {\frac {e}{d}} x^3 \sqrt {\frac {\left (d+e x^2\right ) \left (b d x^2+a \left (d-e x^2\right )\right )}{d^2}}} \] Input:
Integrate[1/(x^4*Sqrt[a + b*x^2 + ((b*d*e - a*e^2)*x^4)/d^2]),x]
Output:
(Sqrt[e/d]*(-a + 2*b*x^2)*(d + e*x^2)*(b*d*x^2 + a*(d - e*x^2)) + (2*I)*a* b*d*e*x^3*Sqrt[1 + (b*x^2)/a - (e*x^2)/d]*Sqrt[1 + (e*x^2)/d]*EllipticE[I* ArcSinh[Sqrt[e/d]*x], -1 + (b*d)/(a*e)] - I*a*e*(b*d + a*e)*x^3*Sqrt[1 + ( b*x^2)/a - (e*x^2)/d]*Sqrt[1 + (e*x^2)/d]*EllipticF[I*ArcSinh[Sqrt[e/d]*x] , -1 + (b*d)/(a*e)])/(3*a^2*d^2*Sqrt[e/d]*x^3*Sqrt[((d + e*x^2)*(b*d*x^2 + a*(d - e*x^2)))/d^2])
Leaf count is larger than twice the leaf count of optimal. \(639\) vs. \(2(315)=630\).
Time = 1.36 (sec) , antiderivative size = 639, normalized size of antiderivative = 2.03, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.303, Rules used = {1443, 25, 27, 1604, 25, 27, 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.
| \(\displaystyle \int \frac {1}{x^4 \sqrt {\frac {x^4 \left (b d e-a e^2\right )}{d^2}+a+b x^2}} \, dx\) |
| \(\Big \downarrow \) 1443 |
| \(\displaystyle \frac {\int -\frac {2 b d^2+e (b d-a e) x^2}{d^2 x^2 \sqrt {\frac {e (b d-a e) x^4}{d^2}+b x^2+a}}dx}{3 a}-\frac {\sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}{3 a x^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int \frac {2 b d^2+e (b d-a e) x^2}{d^2 x^2 \sqrt {\frac {e (b d-a e) x^4}{d^2}+b x^2+a}}dx}{3 a}-\frac {\sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}{3 a x^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {2 b d^2+e (b d-a e) x^2}{x^2 \sqrt {\frac {e (b d-a e) x^4}{d^2}+b x^2+a}}dx}{3 a d^2}-\frac {\sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}{3 a x^3}\) |
| \(\Big \downarrow \) 1604 |
| \(\displaystyle -\frac {-\frac {\int -\frac {e \left (2 b (b d-a e) x^2+a (b d-a e)\right )}{\sqrt {\frac {e (b d-a e) x^4}{d^2}+b x^2+a}}dx}{a}-\frac {2 b d^2 \sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}{a x}}{3 a d^2}-\frac {\sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}{3 a x^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\int \frac {e (b d-a e) \left (2 b x^2+a\right )}{\sqrt {\frac {e (b d-a e) x^4}{d^2}+b x^2+a}}dx}{a}-\frac {2 b d^2 \sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}{a x}}{3 a d^2}-\frac {\sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}{3 a x^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {e (b d-a e) \int \frac {2 b x^2+a}{\sqrt {\frac {e (b d-a e) x^4}{d^2}+b x^2+a}}dx}{a}-\frac {2 b d^2 \sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}{a x}}{3 a d^2}-\frac {\sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}{3 a x^3}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle -\frac {\frac {e (b d-a e) \left (\sqrt {a} \left (\frac {2 b d}{\sqrt {e} \sqrt {b d-a e}}+\sqrt {a}\right ) \int \frac {1}{\sqrt {\frac {e (b d-a e) x^4}{d^2}+b x^2+a}}dx-\frac {2 \sqrt {a} b d \int \frac {\sqrt {a} d-\sqrt {e} \sqrt {b d-a e} x^2}{\sqrt {a} d \sqrt {\frac {e (b d-a e) x^4}{d^2}+b x^2+a}}dx}{\sqrt {e} \sqrt {b d-a e}}\right )}{a}-\frac {2 b d^2 \sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}{a x}}{3 a d^2}-\frac {\sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}{3 a x^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {e (b d-a e) \left (\sqrt {a} \left (\frac {2 b d}{\sqrt {e} \sqrt {b d-a e}}+\sqrt {a}\right ) \int \frac {1}{\sqrt {\frac {e (b d-a e) x^4}{d^2}+b x^2+a}}dx-\frac {2 b \int \frac {\sqrt {a} d-\sqrt {e} \sqrt {b d-a e} x^2}{\sqrt {\frac {e (b d-a e) x^4}{d^2}+b x^2+a}}dx}{\sqrt {e} \sqrt {b d-a e}}\right )}{a}-\frac {2 b d^2 \sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}{a x}}{3 a d^2}-\frac {\sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}{3 a x^3}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle -\frac {\frac {e (b d-a e) \left (\frac {\sqrt [4]{a} \left (\frac {2 b d}{\sqrt {e} \sqrt {b d-a e}}+\sqrt {a}\right ) \left (\sqrt {e} x^2 \sqrt {b d-a e}+\sqrt {a} d\right ) \sqrt {\frac {d^2 \left (\frac {e x^4 (b d-a e)}{d^2}+a+b x^2\right )}{\left (\sqrt {e} x^2 \sqrt {b d-a e}+\sqrt {a} d\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{e} \sqrt [4]{b d-a e} x}{\sqrt [4]{a} \sqrt {d}}\right ),\frac {1}{4} \left (2-\frac {b d}{\sqrt {a} \sqrt {e} \sqrt {b d-a e}}\right )\right )}{2 \sqrt {d} \sqrt [4]{e} \sqrt [4]{b d-a e} \sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}-\frac {2 b \int \frac {\sqrt {a} d-\sqrt {e} \sqrt {b d-a e} x^2}{\sqrt {\frac {e (b d-a e) x^4}{d^2}+b x^2+a}}dx}{\sqrt {e} \sqrt {b d-a e}}\right )}{a}-\frac {2 b d^2 \sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}{a x}}{3 a d^2}-\frac {\sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}{3 a x^3}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle -\frac {\frac {e (b d-a e) \left (\frac {\sqrt [4]{a} \left (\frac {2 b d}{\sqrt {e} \sqrt {b d-a e}}+\sqrt {a}\right ) \left (\sqrt {e} x^2 \sqrt {b d-a e}+\sqrt {a} d\right ) \sqrt {\frac {d^2 \left (\frac {e x^4 (b d-a e)}{d^2}+a+b x^2\right )}{\left (\sqrt {e} x^2 \sqrt {b d-a e}+\sqrt {a} d\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{e} \sqrt [4]{b d-a e} x}{\sqrt [4]{a} \sqrt {d}}\right ),\frac {1}{4} \left (2-\frac {b d}{\sqrt {a} \sqrt {e} \sqrt {b d-a e}}\right )\right )}{2 \sqrt {d} \sqrt [4]{e} \sqrt [4]{b d-a e} \sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}-\frac {2 b \left (\frac {\sqrt [4]{a} \sqrt {d} \left (\sqrt {e} x^2 \sqrt {b d-a e}+\sqrt {a} d\right ) \sqrt {\frac {d^2 \left (\frac {e x^4 (b d-a e)}{d^2}+a+b x^2\right )}{\left (\sqrt {e} x^2 \sqrt {b d-a e}+\sqrt {a} d\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{e} \sqrt [4]{b d-a e} x}{\sqrt [4]{a} \sqrt {d}}\right )|\frac {1}{4} \left (2-\frac {b d}{\sqrt {a} \sqrt {e} \sqrt {b d-a e}}\right )\right )}{\sqrt [4]{e} \sqrt [4]{b d-a e} \sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}-\frac {d^2 x \sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}{\sqrt {e} x^2 \sqrt {b d-a e}+\sqrt {a} d}\right )}{\sqrt {e} \sqrt {b d-a e}}\right )}{a}-\frac {2 b d^2 \sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}{a x}}{3 a d^2}-\frac {\sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}{3 a x^3}\) |
Input:
Int[1/(x^4*Sqrt[a + b*x^2 + ((b*d*e - a*e^2)*x^4)/d^2]),x]
Output:
-1/3*Sqrt[a + b*x^2 + (e*(b*d - a*e)*x^4)/d^2]/(a*x^3) - ((-2*b*d^2*Sqrt[a + b*x^2 + (e*(b*d - a*e)*x^4)/d^2])/(a*x) + (e*(b*d - a*e)*((-2*b*(-((d^2 *x*Sqrt[a + b*x^2 + (e*(b*d - a*e)*x^4)/d^2])/(Sqrt[a]*d + Sqrt[e]*Sqrt[b* d - a*e]*x^2)) + (a^(1/4)*Sqrt[d]*(Sqrt[a]*d + Sqrt[e]*Sqrt[b*d - a*e]*x^2 )*Sqrt[(d^2*(a + b*x^2 + (e*(b*d - a*e)*x^4)/d^2))/(Sqrt[a]*d + Sqrt[e]*Sq rt[b*d - a*e]*x^2)^2]*EllipticE[2*ArcTan[(e^(1/4)*(b*d - a*e)^(1/4)*x)/(a^ (1/4)*Sqrt[d])], (2 - (b*d)/(Sqrt[a]*Sqrt[e]*Sqrt[b*d - a*e]))/4])/(e^(1/4 )*(b*d - a*e)^(1/4)*Sqrt[a + b*x^2 + (e*(b*d - a*e)*x^4)/d^2])))/(Sqrt[e]* Sqrt[b*d - a*e]) + (a^(1/4)*(Sqrt[a] + (2*b*d)/(Sqrt[e]*Sqrt[b*d - a*e]))* (Sqrt[a]*d + Sqrt[e]*Sqrt[b*d - a*e]*x^2)*Sqrt[(d^2*(a + b*x^2 + (e*(b*d - a*e)*x^4)/d^2))/(Sqrt[a]*d + Sqrt[e]*Sqrt[b*d - a*e]*x^2)^2]*EllipticF[2* ArcTan[(e^(1/4)*(b*d - a*e)^(1/4)*x)/(a^(1/4)*Sqrt[d])], (2 - (b*d)/(Sqrt[ a]*Sqrt[e]*Sqrt[b*d - a*e]))/4])/(2*Sqrt[d]*e^(1/4)*(b*d - a*e)^(1/4)*Sqrt [a + b*x^2 + (e*(b*d - a*e)*x^4)/d^2])))/a)/(3*a*d^2)
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*x)^(m + 1)*((a + b*x^2 + c*x^4)^(p + 1)/(a*d*(m + 1))), x] - Sim p[1/(a*d^2*(m + 1)) Int[(d*x)^(m + 2)*(b*(m + 2*p + 3) + c*(m + 4*p + 5)* x^2)*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[m, -1] && 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[d*(f*x)^(m + 1)*((a + b*x^2 + c*x^4)^(p + 1) /(a*f*(m + 1))), x] + Simp[1/(a*f^2*(m + 1)) Int[(f*x)^(m + 2)*(a + b*x^2 + c*x^4)^p*Simp[a*e*(m + 1) - b*d*(m + 2*p + 3) - c*d*(m + 4*p + 5)*x^2, x ], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[ m, -1] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])
Time = 4.49 (sec) , antiderivative size = 442, normalized size of antiderivative = 1.40
| method | result | size |
| default | \(-\frac {\sqrt {-\frac {x^{4} a \,e^{2}}{d^{2}}+\frac {e \,x^{4} b}{d}+b \,x^{2}+a}}{3 a \,x^{3}}+\frac {2 b \sqrt {-\frac {x^{4} a \,e^{2}}{d^{2}}+\frac {e \,x^{4} b}{d}+b \,x^{2}+a}}{3 a^{2} x}+\frac {e \left (a e -b d \right ) \sqrt {1-\frac {\left (a e -b d \right ) x^{2}}{a d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {a e -b d}{a d}}, \sqrt {-1+\frac {b e}{d \left (-\frac {a \,e^{2}}{d^{2}}+\frac {e b}{d}\right )}}\right )}{3 a \,d^{2} \sqrt {\frac {a e -b d}{a d}}\, \sqrt {-\frac {x^{4} a \,e^{2}}{d^{2}}+\frac {e \,x^{4} b}{d}+b \,x^{2}+a}}-\frac {4 b e \left (a e -b d \right ) \sqrt {1-\frac {\left (a e -b d \right ) x^{2}}{a d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {a e -b d}{a d}}, \sqrt {-1+\frac {b e}{d \left (-\frac {a \,e^{2}}{d^{2}}+\frac {e b}{d}\right )}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {a e -b d}{a d}}, \sqrt {-1+\frac {b e}{d \left (-\frac {a \,e^{2}}{d^{2}}+\frac {e b}{d}\right )}}\right )\right )}{3 a \,d^{2} \sqrt {\frac {a e -b d}{a d}}\, \sqrt {-\frac {x^{4} a \,e^{2}}{d^{2}}+\frac {e \,x^{4} b}{d}+b \,x^{2}+a}\, \left (b +\frac {2 a e -b d}{d}\right )}\) | \(442\) |
| elliptic | \(-\frac {\sqrt {-\frac {x^{4} a \,e^{2}}{d^{2}}+\frac {e \,x^{4} b}{d}+b \,x^{2}+a}}{3 a \,x^{3}}+\frac {2 b \sqrt {-\frac {x^{4} a \,e^{2}}{d^{2}}+\frac {e \,x^{4} b}{d}+b \,x^{2}+a}}{3 a^{2} x}+\frac {e \left (a e -b d \right ) \sqrt {1-\frac {\left (a e -b d \right ) x^{2}}{a d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {a e -b d}{a d}}, \sqrt {-1+\frac {b e}{d \left (-\frac {a \,e^{2}}{d^{2}}+\frac {e b}{d}\right )}}\right )}{3 a \,d^{2} \sqrt {\frac {a e -b d}{a d}}\, \sqrt {-\frac {x^{4} a \,e^{2}}{d^{2}}+\frac {e \,x^{4} b}{d}+b \,x^{2}+a}}-\frac {4 b e \left (a e -b d \right ) \sqrt {1-\frac {\left (a e -b d \right ) x^{2}}{a d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {a e -b d}{a d}}, \sqrt {-1+\frac {b e}{d \left (-\frac {a \,e^{2}}{d^{2}}+\frac {e b}{d}\right )}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {a e -b d}{a d}}, \sqrt {-1+\frac {b e}{d \left (-\frac {a \,e^{2}}{d^{2}}+\frac {e b}{d}\right )}}\right )\right )}{3 a \,d^{2} \sqrt {\frac {a e -b d}{a d}}\, \sqrt {-\frac {x^{4} a \,e^{2}}{d^{2}}+\frac {e \,x^{4} b}{d}+b \,x^{2}+a}\, \left (b +\frac {2 a e -b d}{d}\right )}\) | \(442\) |
| risch | \(-\frac {\left (e \,x^{2}+d \right ) \left (-a e \,x^{2}+b d \,x^{2}+a d \right ) \left (-2 b \,x^{2}+a \right ) \sqrt {\left (e \,x^{2}+d \right ) \left (-a e \,x^{2}+b d \,x^{2}+a d \right )}}{3 a^{2} d^{2} x^{3} \sqrt {-\left (e \,x^{2}+d \right ) \left (a e \,x^{2}-b d \,x^{2}-a d \right )}\, \sqrt {\frac {\left (e \,x^{2}+d \right ) \left (-a e \,x^{2}+b d \,x^{2}+a d \right )}{d^{2}}}}+\frac {e \left (\frac {a^{2} e \sqrt {1-\frac {\left (a e -b d \right ) x^{2}}{a d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {a e -b d}{a d}}, \sqrt {-1+\frac {b d e}{-a \,e^{2}+b d e}}\right )}{\sqrt {\frac {a e -b d}{a d}}\, \sqrt {-a \,e^{2} x^{4}+b d e \,x^{4}+b \,d^{2} x^{2}+a \,d^{2}}}-\frac {4 b \left (a e -b d \right ) a \,d^{2} \sqrt {1-\frac {\left (a e -b d \right ) x^{2}}{a d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {a e -b d}{a d}}, \sqrt {-1+\frac {b d e}{-a \,e^{2}+b d e}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {a e -b d}{a d}}, \sqrt {-1+\frac {b d e}{-a \,e^{2}+b d e}}\right )\right )}{\sqrt {\frac {a e -b d}{a d}}\, \sqrt {-a \,e^{2} x^{4}+b d e \,x^{4}+b \,d^{2} x^{2}+a \,d^{2}}\, \left (b \,d^{2}+d \left (2 a e -b d \right )\right )}-\frac {a b d \sqrt {1-\frac {\left (a e -b d \right ) x^{2}}{a d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {a e -b d}{a d}}, \sqrt {-1+\frac {b d e}{-a \,e^{2}+b d e}}\right )}{\sqrt {\frac {a e -b d}{a d}}\, \sqrt {-a \,e^{2} x^{4}+b d e \,x^{4}+b \,d^{2} x^{2}+a \,d^{2}}}\right ) \sqrt {\left (e \,x^{2}+d \right ) \left (-a e \,x^{2}+b d \,x^{2}+a d \right )}}{3 a^{2} d^{2} \sqrt {\frac {\left (e \,x^{2}+d \right ) \left (-a e \,x^{2}+b d \,x^{2}+a d \right )}{d^{2}}}}\) | \(661\) |
Input:
int(1/x^4/(a+b*x^2+(-a*e^2+b*d*e)*x^4/d^2)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/3/a*(-x^4*a/d^2*e^2+e/d*x^4*b+b*x^2+a)^(1/2)/x^3+2/3*b/a^2*(-x^4*a/d^2* e^2+e/d*x^4*b+b*x^2+a)^(1/2)/x+1/3*e/a*(a*e-b*d)/d^2/((a*e-b*d)/a/d)^(1/2) *(1-(a*e-b*d)/a/d*x^2)^(1/2)*(1+e*x^2/d)^(1/2)/(-x^4*a/d^2*e^2+e/d*x^4*b+b *x^2+a)^(1/2)*EllipticF(x*((a*e-b*d)/a/d)^(1/2),(-1+b*e/d/(-a/d^2*e^2+e/d* b))^(1/2))-4/3*b*e*(a*e-b*d)/a/d^2/((a*e-b*d)/a/d)^(1/2)*(1-(a*e-b*d)/a/d* x^2)^(1/2)*(1+e*x^2/d)^(1/2)/(-x^4*a/d^2*e^2+e/d*x^4*b+b*x^2+a)^(1/2)/(b+( 2*a*e-b*d)/d)*(EllipticF(x*((a*e-b*d)/a/d)^(1/2),(-1+b*e/d/(-a/d^2*e^2+e/d *b))^(1/2))-EllipticE(x*((a*e-b*d)/a/d)^(1/2),(-1+b*e/d/(-a/d^2*e^2+e/d*b) )^(1/2)))
Time = 0.09 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.72 \[ \int \frac {1}{x^4 \sqrt {a+b x^2+\frac {\left (b d e-a e^2\right ) x^4}{d^2}}} \, dx=-\frac {2 \, {\left (b^{2} d - a b e\right )} \sqrt {a d^{2}} x^{3} \sqrt {-\frac {b d - a e}{a d}} E(\arcsin \left (x \sqrt {-\frac {b d - a e}{a d}}\right )\,|\,\frac {a e}{b d - a e}) - {\left (2 \, b^{2} d + {\left (a^{2} - 2 \, a b\right )} e\right )} \sqrt {a d^{2}} x^{3} \sqrt {-\frac {b d - a e}{a d}} F(\arcsin \left (x \sqrt {-\frac {b d - a e}{a d}}\right )\,|\,\frac {a e}{b d - a e}) - {\left (2 \, a b d^{2} x^{2} - a^{2} d^{2}\right )} \sqrt {\frac {b d^{2} x^{2} + {\left (b d e - a e^{2}\right )} x^{4} + a d^{2}}{d^{2}}}}{3 \, a^{3} d^{2} x^{3}} \] Input:
integrate(1/x^4/(a+b*x^2+(-a*e^2+b*d*e)*x^4/d^2)^(1/2),x, algorithm="frica s")
Output:
-1/3*(2*(b^2*d - a*b*e)*sqrt(a*d^2)*x^3*sqrt(-(b*d - a*e)/(a*d))*elliptic_ e(arcsin(x*sqrt(-(b*d - a*e)/(a*d))), a*e/(b*d - a*e)) - (2*b^2*d + (a^2 - 2*a*b)*e)*sqrt(a*d^2)*x^3*sqrt(-(b*d - a*e)/(a*d))*elliptic_f(arcsin(x*sq rt(-(b*d - a*e)/(a*d))), a*e/(b*d - a*e)) - (2*a*b*d^2*x^2 - a^2*d^2)*sqrt ((b*d^2*x^2 + (b*d*e - a*e^2)*x^4 + a*d^2)/d^2))/(a^3*d^2*x^3)
\[ \int \frac {1}{x^4 \sqrt {a+b x^2+\frac {\left (b d e-a e^2\right ) x^4}{d^2}}} \, dx=\int \frac {1}{x^{4} \sqrt {- \left (1 + \frac {e x^{2}}{d}\right ) \left (- a + \frac {a e x^{2}}{d} - b x^{2}\right )}}\, dx \] Input:
integrate(1/x**4/(a+b*x**2+(-a*e**2+b*d*e)*x**4/d**2)**(1/2),x)
Output:
Integral(1/(x**4*sqrt(-(1 + e*x**2/d)*(-a + a*e*x**2/d - b*x**2))), x)
\[ \int \frac {1}{x^4 \sqrt {a+b x^2+\frac {\left (b d e-a e^2\right ) x^4}{d^2}}} \, dx=\int { \frac {1}{\sqrt {b x^{2} + \frac {{\left (b d e - a e^{2}\right )} x^{4}}{d^{2}} + a} x^{4}} \,d x } \] Input:
integrate(1/x^4/(a+b*x^2+(-a*e^2+b*d*e)*x^4/d^2)^(1/2),x, algorithm="maxim a")
Output:
integrate(1/(sqrt(b*x^2 + (b*d*e - a*e^2)*x^4/d^2 + a)*x^4), x)
\[ \int \frac {1}{x^4 \sqrt {a+b x^2+\frac {\left (b d e-a e^2\right ) x^4}{d^2}}} \, dx=\int { \frac {1}{\sqrt {b x^{2} + \frac {{\left (b d e - a e^{2}\right )} x^{4}}{d^{2}} + a} x^{4}} \,d x } \] Input:
integrate(1/x^4/(a+b*x^2+(-a*e^2+b*d*e)*x^4/d^2)^(1/2),x, algorithm="giac" )
Output:
integrate(1/(sqrt(b*x^2 + (b*d*e - a*e^2)*x^4/d^2 + a)*x^4), x)
Timed out. \[ \int \frac {1}{x^4 \sqrt {a+b x^2+\frac {\left (b d e-a e^2\right ) x^4}{d^2}}} \, dx=\int \frac {1}{x^4\,\sqrt {a+b\,x^2-\frac {x^4\,\left (a\,e^2-b\,d\,e\right )}{d^2}}} \,d x \] Input:
int(1/(x^4*(a + b*x^2 - (x^4*(a*e^2 - b*d*e))/d^2)^(1/2)),x)
Output:
int(1/(x^4*(a + b*x^2 - (x^4*(a*e^2 - b*d*e))/d^2)^(1/2)), x)
\[ \int \frac {1}{x^4 \sqrt {a+b x^2+\frac {\left (b d e-a e^2\right ) x^4}{d^2}}} \, dx=\left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-a e \,x^{2}+b d \,x^{2}+a d}}{-a \,e^{2} x^{8}+b d e \,x^{8}+b \,d^{2} x^{6}+a \,d^{2} x^{4}}d x \right ) d \] Input:
int(1/x^4/(a+b*x^2+(-a*e^2+b*d*e)*x^4/d^2)^(1/2),x)
Output:
int((sqrt(d + e*x**2)*sqrt(a*d - a*e*x**2 + b*d*x**2))/(a*d**2*x**4 - a*e* *2*x**8 + b*d**2*x**6 + b*d*e*x**8),x)*d