\(\int \frac {1}{x^2 \sqrt {a+\frac {(c d^2+a e^2) x^2}{d e}+c x^4}} \, dx\) [712]

Optimal result

Integrand size = 36, antiderivative size = 153 \[ \int \frac {1}{x^2 \sqrt {a+\frac {\left (c d^2+a e^2\right ) x^2}{d e}+c x^4}} \, dx=-\frac {d \sqrt {a+\left (\frac {c d}{e}+\frac {a e}{d}\right ) x^2+c x^4}}{a x \left (d+e x^2\right )}-\frac {\sqrt {e} \sqrt {\frac {d \left (a e+c d x^2\right )}{a e \left (d+e x^2\right )}} \left (d+e x^2\right ) E\left (\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )|1-\frac {c d^2}{a e^2}\right )}{d^{3/2} \sqrt {a+\left (\frac {c d}{e}+\frac {a e}{d}\right ) x^2+c x^4}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 2.38 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.18 \[ \int \frac {1}{x^2 \sqrt {a+\frac {\left (c d^2+a e^2\right ) x^2}{d e}+c x^4}} \, dx=\frac {-\frac {\left (a e+c d x^2\right ) \left (d+e x^2\right )}{x}-i a d \sqrt {\frac {c d}{a e}} e \sqrt {1+\frac {c d x^2}{a e}} \sqrt {1+\frac {e x^2}{d}} \left (E\left (i \text {arcsinh}\left (\sqrt {\frac {c d}{a e}} x\right )|\frac {a e^2}{c d^2}\right )-\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {c d}{a e}} x\right ),\frac {a e^2}{c d^2}\right )\right )}{a d e \sqrt {\frac {\left (a e+c d x^2\right ) \left (d+e x^2\right )}{d e}}} \] Input:

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

Output:

(-(((a*e + c*d*x^2)*(d + e*x^2))/x) - I*a*d*Sqrt[(c*d)/(a*e)]*e*Sqrt[1 + ( 
c*d*x^2)/(a*e)]*Sqrt[1 + (e*x^2)/d]*(EllipticE[I*ArcSinh[Sqrt[(c*d)/(a*e)] 
*x], (a*e^2)/(c*d^2)] - EllipticF[I*ArcSinh[Sqrt[(c*d)/(a*e)]*x], (a*e^2)/ 
(c*d^2)]))/(a*d*e*Sqrt[((a*e + c*d*x^2)*(d + e*x^2))/(d*e)])
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(396\) vs. \(2(153)=306\).

Time = 0.85 (sec) , antiderivative size = 396, normalized size of antiderivative = 2.59, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1443, 27, 1459, 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.

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

\(\Big \downarrow \) 1443

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1459

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1416

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

\(\Big \downarrow \) 1509

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

Input:

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

Output:

-(Sqrt[a + ((c*d)/e + (a*e)/d)*x^2 + c*x^4]/(a*x)) + (c*(-((-((x*Sqrt[a + 
((c*d)/e + (a*e)/d)*x^2 + c*x^4])/(Sqrt[a] + Sqrt[c]*x^2)) + (a^(1/4)*(Sqr 
t[a] + Sqrt[c]*x^2)*Sqrt[(a + ((c*d)/e + (a*e)/d)*x^2 + c*x^4)/(Sqrt[a] + 
Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - ((c*d)/e + ( 
a*e)/d)/(Sqrt[a]*Sqrt[c]))/4])/(c^(1/4)*Sqrt[a + ((c*d)/e + (a*e)/d)*x^2 + 
 c*x^4]))/Sqrt[c]) + (a^(1/4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + ((c*d)/e + 
 (a*e)/d)*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1 
/4)*x)/a^(1/4)], (2 - ((c*d)/e + (a*e)/d)/(Sqrt[a]*Sqrt[c]))/4])/(2*c^(3/4 
)*Sqrt[a + ((c*d)/e + (a*e)/d)*x^2 + c*x^4])))/a
 

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 1416

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]
 

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 1459

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

rule 1509

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]
 

Maple [A] (verified)

Time = 2.46 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.46

method	result	size
default	\(-\frac {\sqrt {a +\frac {x^{2} e a}{d}+\frac {x^{2} d c}{e}+c \,x^{4}}}{a x}-\frac {2 c \sqrt {1+\frac {x^{2} d c}{a e}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {c d}{a e}}, \sqrt {-1+\frac {\left (\frac {c d}{e}+\frac {a e}{d}\right ) e}{d c}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {c d}{a e}}, \sqrt {-1+\frac {\left (\frac {c d}{e}+\frac {a e}{d}\right ) e}{d c}}\right )\right )}{\sqrt {-\frac {c d}{a e}}\, \sqrt {a +\frac {x^{2} e a}{d}+\frac {x^{2} d c}{e}+c \,x^{4}}\, \left (\frac {c d}{e}+\frac {a e}{d}+\frac {a \,e^{2}-c \,d^{2}}{d e}\right )}\)	\(224\)
elliptic	\(-\frac {\sqrt {a +\frac {x^{2} e a}{d}+\frac {x^{2} d c}{e}+c \,x^{4}}}{a x}-\frac {2 c \sqrt {1+\frac {x^{2} d c}{a e}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {c d}{a e}}, \sqrt {-1+\frac {\left (\frac {c d}{e}+\frac {a e}{d}\right ) e}{d c}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {c d}{a e}}, \sqrt {-1+\frac {\left (\frac {c d}{e}+\frac {a e}{d}\right ) e}{d c}}\right )\right )}{\sqrt {-\frac {c d}{a e}}\, \sqrt {a +\frac {x^{2} e a}{d}+\frac {x^{2} d c}{e}+c \,x^{4}}\, \left (\frac {c d}{e}+\frac {a e}{d}+\frac {a \,e^{2}-c \,d^{2}}{d e}\right )}\)	\(224\)
risch	\(-\frac {\left (e \,x^{2}+d \right ) \left (c d \,x^{2}+a e \right )}{e d a x \sqrt {\frac {\left (e \,x^{2}+d \right ) \left (c d \,x^{2}+a e \right )}{d e}}}-\frac {2 c \,e^{2} d^{2} \sqrt {1+\frac {x^{2} d c}{a e}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {c d}{a e}}, \sqrt {-1+\frac {a d \,e^{3}+c \,d^{3} e}{e \,d^{3} c}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {c d}{a e}}, \sqrt {-1+\frac {a d \,e^{3}+c \,d^{3} e}{e \,d^{3} c}}\right )\right ) \sqrt {\left (e \,x^{2}+d \right ) \left (c d \,x^{2}+a e \right ) d e}}{\sqrt {-\frac {c d}{a e}}\, \sqrt {c \,d^{2} e^{2} x^{4}+a \,e^{3} x^{2} d +c e \,x^{2} d^{3}+a \,d^{2} e^{2}}\, \left (a d \,e^{3}+c \,d^{3} e +d e \left (a \,e^{2}-c \,d^{2}\right )\right ) \sqrt {\frac {\left (e \,x^{2}+d \right ) \left (c d \,x^{2}+a e \right )}{d e}}}\)	\(313\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.94 \[ \int \frac {1}{x^2 \sqrt {a+\frac {\left (c d^2+a e^2\right ) x^2}{d e}+c x^4}} \, dx=\frac {\sqrt {a} c d x \sqrt {-\frac {c d}{a e}} E(\arcsin \left (x \sqrt {-\frac {c d}{a e}}\right )\,|\,\frac {a e^{2}}{c d^{2}}) - \sqrt {a} c d x \sqrt {-\frac {c d}{a e}} F(\arcsin \left (x \sqrt {-\frac {c d}{a e}}\right )\,|\,\frac {a e^{2}}{c d^{2}}) - a e \sqrt {\frac {c d e x^{4} + a d e + {\left (c d^{2} + a e^{2}\right )} x^{2}}{d e}}}{a^{2} e x} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

sqrt(e)*sqrt(d)*int((sqrt(d + e*x**2)*sqrt(a*e + c*d*x**2))/(a*d*e*x**2 + 
a*e**2*x**4 + c*d**2*x**4 + c*d*e*x**6),x)