\(\int \frac {1}{x^4 \sqrt {\frac {-c d^2+b d e}{e^2}+b x^2+c x^4}} \, dx\) [727]

Optimal result

Integrand size = 34, antiderivative size = 378 \[ \int \frac {1}{x^4 \sqrt {\frac {-c d^2+b d e}{e^2}+b x^2+c x^4}} \, dx=\frac {e^2 \sqrt {-\frac {d (c d-b e)}{e^2}+b x^2+c x^4}}{3 d (c d-b e) x^3}+\frac {2 b e^4 \sqrt {-\frac {d (c d-b e)}{e^2}+b x^2+c x^4}}{3 d^2 (c d-b e)^2 x}-\frac {2 b \sqrt {c} e^{5/2} \sqrt {1+\frac {e x^2}{d}} \sqrt {1-\frac {c e x^2}{c d-b e}} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {e} x}{\sqrt {c d-b e}}\right )|-1+\frac {b e}{c d}\right )}{3 d (c d-b e)^{3/2} \sqrt {-\frac {d (c d-b e)}{e^2}+b x^2+c x^4}}+\frac {\sqrt {c} e^{3/2} (c d+b e) \sqrt {1+\frac {e x^2}{d}} \sqrt {1-\frac {c e x^2}{c d-b e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {e} x}{\sqrt {c d-b e}}\right ),-1+\frac {b e}{c d}\right )}{3 d (c d-b e)^{3/2} \sqrt {-\frac {d (c d-b e)}{e^2}+b x^2+c x^4}} \] Output:

1/3*e^2*(-d*(-b*e+c*d)/e^2+b*x^2+c*x^4)^(1/2)/d/(-b*e+c*d)/x^3+2/3*b*e^4*( 
-d*(-b*e+c*d)/e^2+b*x^2+c*x^4)^(1/2)/d^2/(-b*e+c*d)^2/x-2/3*b*c^(1/2)*e^(5 
/2)*(1+e*x^2/d)^(1/2)*(1-c*e*x^2/(-b*e+c*d))^(1/2)*EllipticE(c^(1/2)*e^(1/ 
2)*x/(-b*e+c*d)^(1/2),(-1+b*e/c/d)^(1/2))/d/(-b*e+c*d)^(3/2)/(-d*(-b*e+c*d 
)/e^2+b*x^2+c*x^4)^(1/2)+1/3*c^(1/2)*e^(3/2)*(b*e+c*d)*(1+e*x^2/d)^(1/2)*( 
1-c*e*x^2/(-b*e+c*d))^(1/2)*EllipticF(c^(1/2)*e^(1/2)*x/(-b*e+c*d)^(1/2),( 
-1+b*e/c/d)^(1/2))/d/(-b*e+c*d)^(3/2)/(-d*(-b*e+c*d)/e^2+b*x^2+c*x^4)^(1/2 
)
                                                                                    
                                                                                    
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 4.02 (sec) , antiderivative size = 330, normalized size of antiderivative = 0.87 \[ \int \frac {1}{x^4 \sqrt {\frac {-c d^2+b d e}{e^2}+b x^2+c x^4}} \, dx=\frac {-\sqrt {\frac {e}{d}} \left (d+e x^2\right ) \left (b^2 e^2 \left (d-2 e x^2\right )+c^2 d^2 \left (d-e x^2\right )+b c e \left (-2 d^2+3 d e x^2-2 e^2 x^4\right )\right )+2 i b e^3 (-c d+b e) x^3 \sqrt {\frac {-c d+b e+c e x^2}{-c d+b e}} \sqrt {1+\frac {e x^2}{d}} E\left (i \text {arcsinh}\left (\sqrt {\frac {e}{d}} x\right )|\frac {c d}{-c d+b e}\right )-i e^2 \left (c^2 d^2-3 b c d e+2 b^2 e^2\right ) x^3 \sqrt {\frac {-c d+b e+c e x^2}{-c d+b e}} \sqrt {1+\frac {e x^2}{d}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {e}{d}} x\right ),\frac {c d}{-c d+b e}\right )}{3 d^2 \sqrt {\frac {e}{d}} (c d-b e)^2 x^3 \sqrt {\frac {\left (d+e x^2\right ) \left (-c d+b e+c e x^2\right )}{e^2}}} \] Input:

Integrate[1/(x^4*Sqrt[(-(c*d^2) + b*d*e)/e^2 + b*x^2 + c*x^4]),x]
 

Output:

(-(Sqrt[e/d]*(d + e*x^2)*(b^2*e^2*(d - 2*e*x^2) + c^2*d^2*(d - e*x^2) + b* 
c*e*(-2*d^2 + 3*d*e*x^2 - 2*e^2*x^4))) + (2*I)*b*e^3*(-(c*d) + b*e)*x^3*Sq 
rt[(-(c*d) + b*e + c*e*x^2)/(-(c*d) + b*e)]*Sqrt[1 + (e*x^2)/d]*EllipticE[ 
I*ArcSinh[Sqrt[e/d]*x], (c*d)/(-(c*d) + b*e)] - I*e^2*(c^2*d^2 - 3*b*c*d*e 
 + 2*b^2*e^2)*x^3*Sqrt[(-(c*d) + b*e + c*e*x^2)/(-(c*d) + b*e)]*Sqrt[1 + ( 
e*x^2)/d]*EllipticF[I*ArcSinh[Sqrt[e/d]*x], (c*d)/(-(c*d) + b*e)])/(3*d^2* 
Sqrt[e/d]*(c*d - b*e)^2*x^3*Sqrt[((d + e*x^2)*(-(c*d) + b*e + c*e*x^2))/e^ 
2])
 

Rubi [A] (verified)

Time = 1.00 (sec) , antiderivative size = 338, normalized size of antiderivative = 0.89, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {1443, 25, 1604, 27, 1514, 399, 321, 327}

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.

\(\displaystyle \int \frac {1}{x^4 \sqrt {\frac {b d e-c d^2}{e^2}+b x^2+c x^4}} \, dx\)

\(\Big \downarrow \) 1443

\(\displaystyle \frac {e^2 \sqrt {-\frac {d (c d-b e)}{e^2}+b x^2+c x^4}}{3 d x^3 (c d-b e)}-\frac {e^2 \int -\frac {c x^2+2 b}{x^2 \sqrt {c x^4+b x^2-\frac {d (c d-b e)}{e^2}}}dx}{3 d (c d-b e)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {e^2 \int \frac {c x^2+2 b}{x^2 \sqrt {c x^4+b x^2-\frac {d (c d-b e)}{e^2}}}dx}{3 d (c d-b e)}+\frac {e^2 \sqrt {-\frac {d (c d-b e)}{e^2}+b x^2+c x^4}}{3 d x^3 (c d-b e)}\)

\(\Big \downarrow \) 1604

\(\displaystyle \frac {e^2 \left (\frac {e^2 \int \frac {c \left (d (c d-b e)-2 b e^2 x^2\right )}{e^2 \sqrt {c x^4+b x^2-\frac {d (c d-b e)}{e^2}}}dx}{d (c d-b e)}+\frac {2 b e^2 \sqrt {-\frac {d (c d-b e)}{e^2}+b x^2+c x^4}}{d x (c d-b e)}\right )}{3 d (c d-b e)}+\frac {e^2 \sqrt {-\frac {d (c d-b e)}{e^2}+b x^2+c x^4}}{3 d x^3 (c d-b e)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {e^2 \left (\frac {c \int \frac {d (c d-b e)-2 b e^2 x^2}{\sqrt {c x^4+b x^2-\frac {d (c d-b e)}{e^2}}}dx}{d (c d-b e)}+\frac {2 b e^2 \sqrt {-\frac {d (c d-b e)}{e^2}+b x^2+c x^4}}{d x (c d-b e)}\right )}{3 d (c d-b e)}+\frac {e^2 \sqrt {-\frac {d (c d-b e)}{e^2}+b x^2+c x^4}}{3 d x^3 (c d-b e)}\)

\(\Big \downarrow \) 1514

\(\displaystyle \frac {e^2 \left (\frac {c \sqrt {\frac {e x^2}{d}+1} \sqrt {1-\frac {c e x^2}{c d-b e}} \int \frac {d (c d-b e)-2 b e^2 x^2}{\sqrt {\frac {e x^2}{d}+1} \sqrt {1-\frac {c e x^2}{c d-b e}}}dx}{d (c d-b e) \sqrt {-\frac {d (c d-b e)}{e^2}+b x^2+c x^4}}+\frac {2 b e^2 \sqrt {-\frac {d (c d-b e)}{e^2}+b x^2+c x^4}}{d x (c d-b e)}\right )}{3 d (c d-b e)}+\frac {e^2 \sqrt {-\frac {d (c d-b e)}{e^2}+b x^2+c x^4}}{3 d x^3 (c d-b e)}\)

\(\Big \downarrow \) 399

\(\displaystyle \frac {e^2 \left (\frac {c \sqrt {\frac {e x^2}{d}+1} \sqrt {1-\frac {c e x^2}{c d-b e}} \left (d (b e+c d) \int \frac {1}{\sqrt {\frac {e x^2}{d}+1} \sqrt {1-\frac {c e x^2}{c d-b e}}}dx-2 b d e \int \frac {\sqrt {\frac {e x^2}{d}+1}}{\sqrt {1-\frac {c e x^2}{c d-b e}}}dx\right )}{d (c d-b e) \sqrt {-\frac {d (c d-b e)}{e^2}+b x^2+c x^4}}+\frac {2 b e^2 \sqrt {-\frac {d (c d-b e)}{e^2}+b x^2+c x^4}}{d x (c d-b e)}\right )}{3 d (c d-b e)}+\frac {e^2 \sqrt {-\frac {d (c d-b e)}{e^2}+b x^2+c x^4}}{3 d x^3 (c d-b e)}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {e^2 \left (\frac {c \sqrt {\frac {e x^2}{d}+1} \sqrt {1-\frac {c e x^2}{c d-b e}} \left (\frac {d \sqrt {c d-b e} (b e+c d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {e} x}{\sqrt {c d-b e}}\right ),\frac {b e}{c d}-1\right )}{\sqrt {c} \sqrt {e}}-2 b d e \int \frac {\sqrt {\frac {e x^2}{d}+1}}{\sqrt {1-\frac {c e x^2}{c d-b e}}}dx\right )}{d (c d-b e) \sqrt {-\frac {d (c d-b e)}{e^2}+b x^2+c x^4}}+\frac {2 b e^2 \sqrt {-\frac {d (c d-b e)}{e^2}+b x^2+c x^4}}{d x (c d-b e)}\right )}{3 d (c d-b e)}+\frac {e^2 \sqrt {-\frac {d (c d-b e)}{e^2}+b x^2+c x^4}}{3 d x^3 (c d-b e)}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {e^2 \left (\frac {c \sqrt {\frac {e x^2}{d}+1} \sqrt {1-\frac {c e x^2}{c d-b e}} \left (\frac {d \sqrt {c d-b e} (b e+c d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {e} x}{\sqrt {c d-b e}}\right ),\frac {b e}{c d}-1\right )}{\sqrt {c} \sqrt {e}}-\frac {2 b d \sqrt {e} \sqrt {c d-b e} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {e} x}{\sqrt {c d-b e}}\right )|\frac {b e}{c d}-1\right )}{\sqrt {c}}\right )}{d (c d-b e) \sqrt {-\frac {d (c d-b e)}{e^2}+b x^2+c x^4}}+\frac {2 b e^2 \sqrt {-\frac {d (c d-b e)}{e^2}+b x^2+c x^4}}{d x (c d-b e)}\right )}{3 d (c d-b e)}+\frac {e^2 \sqrt {-\frac {d (c d-b e)}{e^2}+b x^2+c x^4}}{3 d x^3 (c d-b e)}\)

Input:

Int[1/(x^4*Sqrt[(-(c*d^2) + b*d*e)/e^2 + b*x^2 + c*x^4]),x]
 

Output:

(e^2*Sqrt[-((d*(c*d - b*e))/e^2) + b*x^2 + c*x^4])/(3*d*(c*d - b*e)*x^3) + 
 (e^2*((2*b*e^2*Sqrt[-((d*(c*d - b*e))/e^2) + b*x^2 + c*x^4])/(d*(c*d - b* 
e)*x) + (c*Sqrt[1 + (e*x^2)/d]*Sqrt[1 - (c*e*x^2)/(c*d - b*e)]*((-2*b*d*Sq 
rt[e]*Sqrt[c*d - b*e]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[e]*x)/Sqrt[c*d - b*e] 
], -1 + (b*e)/(c*d)])/Sqrt[c] + (d*Sqrt[c*d - b*e]*(c*d + b*e)*EllipticF[A 
rcSin[(Sqrt[c]*Sqrt[e]*x)/Sqrt[c*d - b*e]], -1 + (b*e)/(c*d)])/(Sqrt[c]*Sq 
rt[e])))/(d*(c*d - b*e)*Sqrt[-((d*(c*d - b*e))/e^2) + b*x^2 + c*x^4])))/(3 
*d*(c*d - b*e))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c 
/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 
0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 399

Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_) 
^2]), x_Symbol] :> Simp[f/b   Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] + 
 Simp[(b*e - a*f)/b   Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; Fr 
eeQ[{a, b, c, d, e, f}, x] &&  !((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && 
(PosQ[d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c])))))
 

rule 1443

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])
 

rule 1514

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]
 

rule 1604

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])
 

Maple [A] (verified)

Time = 16.56 (sec) , antiderivative size = 423, normalized size of antiderivative = 1.12

method	result	size
default	\(-\frac {e^{2} \sqrt {c \,x^{4}+b \,x^{2}+\frac {b d}{e}-\frac {c \,d^{2}}{e^{2}}}}{3 d \left (b e -c d \right ) x^{3}}+\frac {2 e^{4} b \sqrt {c \,x^{4}+b \,x^{2}+\frac {b d}{e}-\frac {c \,d^{2}}{e^{2}}}}{3 d^{2} \left (b e -c d \right )^{2} x}-\frac {e^{2} c \sqrt {1+\frac {c e \,x^{2}}{b e -c d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {c e}{b e -c d}}, \sqrt {-1+\frac {b e}{c d}}\right )}{3 d \left (b e -c d \right ) \sqrt {-\frac {c e}{b e -c d}}\, \sqrt {c \,x^{4}+b \,x^{2}+\frac {b d}{e}-\frac {c \,d^{2}}{e^{2}}}}+\frac {4 e^{4} b c \left (\frac {b d}{e}-\frac {c \,d^{2}}{e^{2}}\right ) \sqrt {1+\frac {c e \,x^{2}}{b e -c d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {c e}{b e -c d}}, \sqrt {-1+\frac {b e}{c d}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {c e}{b e -c d}}, \sqrt {-1+\frac {b e}{c d}}\right )\right )}{3 d^{2} \left (b e -c d \right )^{2} \sqrt {-\frac {c e}{b e -c d}}\, \sqrt {c \,x^{4}+b \,x^{2}+\frac {b d}{e}-\frac {c \,d^{2}}{e^{2}}}\, \left (b +\frac {b e -2 c d}{e}\right )}\)	\(423\)
elliptic	\(-\frac {e^{2} \sqrt {c \,x^{4}+b \,x^{2}+\frac {b d}{e}-\frac {c \,d^{2}}{e^{2}}}}{3 d \left (b e -c d \right ) x^{3}}+\frac {2 e^{4} b \sqrt {c \,x^{4}+b \,x^{2}+\frac {b d}{e}-\frac {c \,d^{2}}{e^{2}}}}{3 d^{2} \left (b e -c d \right )^{2} x}-\frac {e^{2} c \sqrt {1+\frac {c e \,x^{2}}{b e -c d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {c e}{b e -c d}}, \sqrt {-1+\frac {b e}{c d}}\right )}{3 d \left (b e -c d \right ) \sqrt {-\frac {c e}{b e -c d}}\, \sqrt {c \,x^{4}+b \,x^{2}+\frac {b d}{e}-\frac {c \,d^{2}}{e^{2}}}}+\frac {4 e^{4} b c \left (\frac {b d}{e}-\frac {c \,d^{2}}{e^{2}}\right ) \sqrt {1+\frac {c e \,x^{2}}{b e -c d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {c e}{b e -c d}}, \sqrt {-1+\frac {b e}{c d}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {c e}{b e -c d}}, \sqrt {-1+\frac {b e}{c d}}\right )\right )}{3 d^{2} \left (b e -c d \right )^{2} \sqrt {-\frac {c e}{b e -c d}}\, \sqrt {c \,x^{4}+b \,x^{2}+\frac {b d}{e}-\frac {c \,d^{2}}{e^{2}}}\, \left (b +\frac {b e -2 c d}{e}\right )}\)	\(423\)
risch	\(-\frac {\left (e \,x^{2}+d \right ) \left (c e \,x^{2}+b e -c d \right ) \left (-2 b \,x^{2} e^{2}+b d e -c \,d^{2}\right )}{3 d^{2} \left (b e -c d \right )^{2} x^{3} \sqrt {\frac {\left (e \,x^{2}+d \right ) \left (c e \,x^{2}+b e -c d \right )}{e^{2}}}}-\frac {e^{2} c \left (\frac {b d e \sqrt {1+\frac {c e \,x^{2}}{b e -c d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {c e}{b e -c d}}, \sqrt {-1+\frac {b e}{c d}}\right )}{\sqrt {-\frac {c e}{b e -c d}}\, \sqrt {c \,x^{4} e^{2}+b \,x^{2} e^{2}+b d e -c \,d^{2}}}-\frac {c \,d^{2} \sqrt {1+\frac {c e \,x^{2}}{b e -c d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {c e}{b e -c d}}, \sqrt {-1+\frac {b e}{c d}}\right )}{\sqrt {-\frac {c e}{b e -c d}}\, \sqrt {c \,x^{4} e^{2}+b \,x^{2} e^{2}+b d e -c \,d^{2}}}-\frac {4 b \,e^{2} \left (b d e -c \,d^{2}\right ) \sqrt {1+\frac {c e \,x^{2}}{b e -c d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {c e}{b e -c d}}, \sqrt {-1+\frac {b e}{c d}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {c e}{b e -c d}}, \sqrt {-1+\frac {b e}{c d}}\right )\right )}{\sqrt {-\frac {c e}{b e -c d}}\, \sqrt {c \,x^{4} e^{2}+b \,x^{2} e^{2}+b d e -c \,d^{2}}\, \left (b \,e^{2}+e \left (b e -2 c d \right )\right )}\right ) \sqrt {\left (e \,x^{2}+d \right ) \left (c e \,x^{2}+b e -c d \right )}}{3 d^{2} \left (b e -c d \right )^{2} \sqrt {\frac {\left (e \,x^{2}+d \right ) \left (c e \,x^{2}+b e -c d \right )}{e^{2}}}}\)	\(566\)

Input:

int(1/x^4/((b*d*e-c*d^2)/e^2+b*x^2+c*x^4)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/3/d/(b*e-c*d)*e^2*(c*x^4+b*x^2+b*d/e-c*d^2/e^2)^(1/2)/x^3+2/3*e^4*b/d^2 
/(b*e-c*d)^2*(c*x^4+b*x^2+b*d/e-c*d^2/e^2)^(1/2)/x-1/3/d/(b*e-c*d)*e^2*c/( 
-c*e/(b*e-c*d))^(1/2)*(1+c*e/(b*e-c*d)*x^2)^(1/2)*(1+e*x^2/d)^(1/2)/(c*x^4 
+b*x^2+b*d/e-c*d^2/e^2)^(1/2)*EllipticF(x*(-c*e/(b*e-c*d))^(1/2),(-1+b*e/c 
/d)^(1/2))+4/3*e^4*b*c/d^2/(b*e-c*d)^2*(b*d/e-c*d^2/e^2)/(-c*e/(b*e-c*d))^ 
(1/2)*(1+c*e/(b*e-c*d)*x^2)^(1/2)*(1+e*x^2/d)^(1/2)/(c*x^4+b*x^2+b*d/e-c*d 
^2/e^2)^(1/2)/(b+(b*e-2*c*d)/e)*(EllipticF(x*(-c*e/(b*e-c*d))^(1/2),(-1+b* 
e/c/d)^(1/2))-EllipticE(x*(-c*e/(b*e-c*d))^(1/2),(-1+b*e/c/d)^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 305, normalized size of antiderivative = 0.81 \[ \int \frac {1}{x^4 \sqrt {\frac {-c d^2+b d e}{e^2}+b x^2+c x^4}} \, dx=\frac {2 \, \sqrt {-c d^{2} + b d e} b c \sqrt {\frac {c e}{c d - b e}} e^{4} x^{3} E(\arcsin \left (\sqrt {\frac {c e}{c d - b e}} x\right )\,|\,-\frac {c d - b e}{c d}) - {\left (c^{2} d^{2} e^{2} - 2 \, b c d e^{3} + {\left (b^{2} + 2 \, b c\right )} e^{4}\right )} \sqrt {-c d^{2} + b d e} \sqrt {\frac {c e}{c d - b e}} x^{3} F(\arcsin \left (\sqrt {\frac {c e}{c d - b e}} x\right )\,|\,-\frac {c d - b e}{c d}) + {\left (c^{2} d^{3} e^{2} - 2 \, b c d^{2} e^{3} + b^{2} d e^{4} + 2 \, {\left (b c d e^{4} - b^{2} e^{5}\right )} x^{2}\right )} \sqrt {\frac {c e^{2} x^{4} + b e^{2} x^{2} - c d^{2} + b d e}{e^{2}}}}{3 \, {\left (c^{3} d^{5} - 3 \, b c^{2} d^{4} e + 3 \, b^{2} c d^{3} e^{2} - b^{3} d^{2} e^{3}\right )} x^{3}} \] Input:

integrate(1/x^4/((b*d*e-c*d^2)/e^2+b*x^2+c*x^4)^(1/2),x, algorithm="fricas 
")
 

Output:

1/3*(2*sqrt(-c*d^2 + b*d*e)*b*c*sqrt(c*e/(c*d - b*e))*e^4*x^3*elliptic_e(a 
rcsin(sqrt(c*e/(c*d - b*e))*x), -(c*d - b*e)/(c*d)) - (c^2*d^2*e^2 - 2*b*c 
*d*e^3 + (b^2 + 2*b*c)*e^4)*sqrt(-c*d^2 + b*d*e)*sqrt(c*e/(c*d - b*e))*x^3 
*elliptic_f(arcsin(sqrt(c*e/(c*d - b*e))*x), -(c*d - b*e)/(c*d)) + (c^2*d^ 
3*e^2 - 2*b*c*d^2*e^3 + b^2*d*e^4 + 2*(b*c*d*e^4 - b^2*e^5)*x^2)*sqrt((c*e 
^2*x^4 + b*e^2*x^2 - c*d^2 + b*d*e)/e^2))/((c^3*d^5 - 3*b*c^2*d^4*e + 3*b^ 
2*c*d^3*e^2 - b^3*d^2*e^3)*x^3)
 

Sympy [F]

\[ \int \frac {1}{x^4 \sqrt {\frac {-c d^2+b d e}{e^2}+b x^2+c x^4}} \, dx=\int \frac {1}{x^{4} \sqrt {\left (\frac {d}{e} + x^{2}\right ) \left (b - \frac {c d}{e} + c x^{2}\right )}}\, dx \] Input:

integrate(1/x**4/((b*d*e-c*d**2)/e**2+b*x**2+c*x**4)**(1/2),x)
 

Output:

Integral(1/(x**4*sqrt((d/e + x**2)*(b - c*d/e + c*x**2))), x)
 

Maxima [F]

\[ \int \frac {1}{x^4 \sqrt {\frac {-c d^2+b d e}{e^2}+b x^2+c x^4}} \, dx=\int { \frac {1}{\sqrt {c x^{4} + b x^{2} - \frac {c d^{2} - b d e}{e^{2}}} x^{4}} \,d x } \] Input:

integrate(1/x^4/((b*d*e-c*d^2)/e^2+b*x^2+c*x^4)^(1/2),x, algorithm="maxima 
")
 

Output:

integrate(1/(sqrt(c*x^4 + b*x^2 - (c*d^2 - b*d*e)/e^2)*x^4), x)
 

Giac [F]

\[ \int \frac {1}{x^4 \sqrt {\frac {-c d^2+b d e}{e^2}+b x^2+c x^4}} \, dx=\int { \frac {1}{\sqrt {c x^{4} + b x^{2} - \frac {c d^{2} - b d e}{e^{2}}} x^{4}} \,d x } \] Input:

integrate(1/x^4/((b*d*e-c*d^2)/e^2+b*x^2+c*x^4)^(1/2),x, algorithm="giac")
 

Output:

integrate(1/(sqrt(c*x^4 + b*x^2 - (c*d^2 - b*d*e)/e^2)*x^4), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^4 \sqrt {\frac {-c d^2+b d e}{e^2}+b x^2+c x^4}} \, dx=\int \frac {1}{x^4\,\sqrt {b\,x^2-\frac {c\,d^2-b\,d\,e}{e^2}+c\,x^4}} \,d x \] Input:

int(1/(x^4*(b*x^2 - (c*d^2 - b*d*e)/e^2 + c*x^4)^(1/2)),x)
                                                                                    
                                                                                    
 

Output:

int(1/(x^4*(b*x^2 - (c*d^2 - b*d*e)/e^2 + c*x^4)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {1}{x^4 \sqrt {\frac {-c d^2+b d e}{e^2}+b x^2+c x^4}} \, dx=\left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {c e \,x^{2}+b e -c d}}{c \,e^{2} x^{8}+b \,e^{2} x^{6}+b d e \,x^{4}-c \,d^{2} x^{4}}d x \right ) e \] Input:

int(1/x^4/((b*d*e-c*d^2)/e^2+b*x^2+c*x^4)^(1/2),x)
 

Output:

int((sqrt(d + e*x**2)*sqrt(b*e - c*d + c*e*x**2))/(b*d*e*x**4 + b*e**2*x** 
6 - c*d**2*x**4 + c*e**2*x**8),x)*e