\(\int \frac {1}{x^2 \sqrt {\frac {-c d^2+b d e}{e^2}+b x^2+c x^4}} \, dx\) [726]

Optimal result

Integrand size = 34, antiderivative size = 178 \[ \int \frac {1}{x^2 \sqrt {\frac {-c d^2+b d e}{e^2}+b x^2+c x^4}} \, dx=\frac {e^2 \sqrt {-\frac {d (c d-b e)}{e^2}+b x^2+c x^4}}{(c d-b e) x \left (d+e x^2\right )}-\frac {\sqrt {e} \left (d+e x^2\right ) \sqrt {\frac {d \left (c d-b e-c e x^2\right )}{(c d-b e) \left (d+e x^2\right )}} E\left (\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )|\frac {2 c d-b e}{c d-b e}\right )}{d^{3/2} \sqrt {-\frac {d (c d-b e)}{e^2}+b x^2+c x^4}} \] Output:

e^2*(-d*(-b*e+c*d)/e^2+b*x^2+c*x^4)^(1/2)/(-b*e+c*d)/x/(e*x^2+d)-e^(1/2)*( 
e*x^2+d)*(d*(-c*e*x^2-b*e+c*d)/(-b*e+c*d)/(e*x^2+d))^(1/2)*EllipticE(e^(1/ 
2)*x/d^(1/2)/(1+e*x^2/d)^(1/2),((-b*e+2*c*d)/(-b*e+c*d))^(1/2))/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 = 3.44 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.47 \[ \int \frac {1}{x^2 \sqrt {\frac {-c d^2+b d e}{e^2}+b x^2+c x^4}} \, dx=\frac {\sqrt {\frac {e}{d}} \left (\sqrt {\frac {e}{d}} \left (d+e x^2\right ) \left (c d-b e-c e x^2\right )+i e (c d-b e) x \sqrt {\frac {-c d+b e+c e x^2}{-c d+b e}} \sqrt {1+\frac {e x^2}{d}} E\left (i \text {arcsinh}\left (\sqrt {\frac {e}{d}} x\right )|\frac {c d}{-c d+b e}\right )-i e (c d-b e) x \sqrt {\frac {-c d+b e+c e x^2}{-c d+b e}} \sqrt {1+\frac {e x^2}{d}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {e}{d}} x\right ),\frac {c d}{-c d+b e}\right )\right )}{e (-c d+b e) x \sqrt {\frac {\left (d+e x^2\right ) \left (-c d+b e+c e x^2\right )}{e^2}}} \] Input:

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

Output:

(Sqrt[e/d]*(Sqrt[e/d]*(d + e*x^2)*(c*d - b*e - c*e*x^2) + I*e*(c*d - b*e)* 
x*Sqrt[(-(c*d) + b*e + c*e*x^2)/(-(c*d) + b*e)]*Sqrt[1 + (e*x^2)/d]*Ellipt 
icE[I*ArcSinh[Sqrt[e/d]*x], (c*d)/(-(c*d) + b*e)] - I*e*(c*d - b*e)*x*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)]))/(e*(-(c*d) + b*e)*x*Sqrt[((d 
 + e*x^2)*(-(c*d) + b*e + c*e*x^2))/e^2])
 

Rubi [A] (verified)

Time = 0.78 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.46, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {1443, 27, 1460, 389, 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.

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

\(\Big \downarrow \) 1443

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1460

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

\(\Big \downarrow \) 389

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

\(\Big \downarrow \) 321

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

\(\Big \downarrow \) 327

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

Input:

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

Defintions of rubi rules used

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 389

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] 
 :> Simp[1/b   Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] - Simp[a/b   Int 
[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] && N 
eQ[b*c - a*d, 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 1460

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> 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[x^2/(Sqrt[1 + 2*c*(x^2/(b - q))]*Sq 
rt[1 + 2*c*(x^2/(b + q))]), x], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a 
*c, 0] && NegQ[c/a]
 

Maple [A] (verified)

Time = 4.62 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.38

method	result	size
default	\(-\frac {e^{2} \sqrt {c \,x^{4}+b \,x^{2}+\frac {b d}{e}-\frac {c \,d^{2}}{e^{2}}}}{d \left (b e -c d \right ) x}-\frac {2 e^{2} c \left (\frac {b d}{e}-\frac {c \,d^{2}}{e^{2}}\right ) \sqrt {1+\frac {c e \,x^{2}}{b e -c d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {c e}{b e -c d}}, \sqrt {-1+\frac {b e}{c d}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {c e}{b e -c d}}, \sqrt {-1+\frac {b e}{c d}}\right )\right )}{d \left (b e -c d \right ) \sqrt {-\frac {c e}{b e -c d}}\, \sqrt {c \,x^{4}+b \,x^{2}+\frac {b d}{e}-\frac {c \,d^{2}}{e^{2}}}\, \left (b +\frac {b e -2 c d}{e}\right )}\)	\(245\)
elliptic	\(-\frac {e^{2} \sqrt {c \,x^{4}+b \,x^{2}+\frac {b d}{e}-\frac {c \,d^{2}}{e^{2}}}}{d \left (b e -c d \right ) x}-\frac {2 e^{2} c \left (\frac {b d}{e}-\frac {c \,d^{2}}{e^{2}}\right ) \sqrt {1+\frac {c e \,x^{2}}{b e -c d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {c e}{b e -c d}}, \sqrt {-1+\frac {b e}{c d}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {c e}{b e -c d}}, \sqrt {-1+\frac {b e}{c d}}\right )\right )}{d \left (b e -c d \right ) \sqrt {-\frac {c e}{b e -c d}}\, \sqrt {c \,x^{4}+b \,x^{2}+\frac {b d}{e}-\frac {c \,d^{2}}{e^{2}}}\, \left (b +\frac {b e -2 c d}{e}\right )}\)	\(245\)
risch	\(-\frac {\left (e \,x^{2}+d \right ) \left (c e \,x^{2}+b e -c d \right )}{d \left (b e -c d \right ) x \sqrt {\frac {\left (e \,x^{2}+d \right ) \left (c e \,x^{2}+b e -c d \right )}{e^{2}}}}-\frac {2 e^{2} c \left (b d e -c \,d^{2}\right ) \sqrt {1+\frac {c e \,x^{2}}{b e -c d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {c e}{b e -c d}}, \sqrt {-1+\frac {b e}{c d}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {c e}{b e -c d}}, \sqrt {-1+\frac {b e}{c d}}\right )\right ) \sqrt {\left (e \,x^{2}+d \right ) \left (c e \,x^{2}+b e -c d \right )}}{d \left (b e -c d \right ) \sqrt {-\frac {c e}{b e -c d}}\, \sqrt {c \,x^{4} e^{2}+b \,x^{2} e^{2}+b d e -c \,d^{2}}\, \left (b \,e^{2}+e \left (b e -2 c d \right )\right ) \sqrt {\frac {\left (e \,x^{2}+d \right ) \left (c e \,x^{2}+b e -c d \right )}{e^{2}}}}\)	\(311\)

Input:

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

Output:

-1/d/(b*e-c*d)*e^2*(c*x^4+b*x^2+b*d/e-c*d^2/e^2)^(1/2)/x-2/d/(b*e-c*d)*e^2 
*c*(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)*(Ell 
ipticF(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 = 0.08 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.22 \[ \int \frac {1}{x^2 \sqrt {\frac {-c d^2+b d e}{e^2}+b x^2+c x^4}} \, dx=\frac {\sqrt {-c d^{2} + b d e} c \sqrt {\frac {c e}{c d - b e}} e^{2} x E(\arcsin \left (\sqrt {\frac {c e}{c d - b e}} x\right )\,|\,-\frac {c d - b e}{c d}) - \sqrt {-c d^{2} + b d e} c \sqrt {\frac {c e}{c d - b e}} e^{2} x F(\arcsin \left (\sqrt {\frac {c e}{c d - b e}} x\right )\,|\,-\frac {c d - b e}{c d}) + {\left (c d e^{2} - b e^{3}\right )} \sqrt {\frac {c e^{2} x^{4} + b e^{2} x^{2} - c d^{2} + b d e}{e^{2}}}}{{\left (c^{2} d^{3} - 2 \, b c d^{2} e + b^{2} d e^{2}\right )} x} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate(1/x^2/((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^2), x)
 

Giac [F]

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

integrate(1/x^2/((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^2), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

int(1/x^2/((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**2 + b*e**2*x** 
4 - c*d**2*x**2 + c*e**2*x**6),x)*e