\(\int \frac {1}{\sqrt {d+e x^2} \sqrt {a-c x^4}} \, dx\) [454]

Optimal result

Integrand size = 24, antiderivative size = 137 \[ \int \frac {1}{\sqrt {d+e x^2} \sqrt {a-c x^4}} \, dx=\frac {\sqrt {c} \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{\sqrt {a} \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:

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

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.98, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {1562, 1799, 512, 511, 321}

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.

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

\(\Big \downarrow \) 1562

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

\(\Big \downarrow \) 1799

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

\(\Big \downarrow \) 512

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

\(\Big \downarrow \) 511

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

\(\Big \downarrow \) 321

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

Input:

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

Output:

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

Defintions of rubi rules used

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 511

Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Wit 
h[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[q*((c + d*x)/(d + c*q))]/(Sqrt[a]*q*Sqrt 
[c + d*x]))   Subst[Int[1/(Sqrt[1 - 2*d*(x^2/(d + c*q))]*Sqrt[1 - x^2]), x] 
, x, Sqrt[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[ 
a, 0]
 

rule 512

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

rule 1562

Int[1/(Sqrt[(d_) + (e_.)*(x_)^2]*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> S 
imp[x^3*Sqrt[e + d/x^2]*(Sqrt[c + a/x^4]/(Sqrt[d + e*x^2]*Sqrt[a + c*x^4])) 
   Int[1/(x^3*Sqrt[e + d/x^2]*Sqrt[c + a/x^4]), x], x] /; FreeQ[{a, c, d, e 
}, x] && NeQ[c*d^2 + a*e^2, 0]
 

rule 1799

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q 
_.), x_Symbol] :> Simp[1/n   Subst[Int[(d + e*x)^q*(a + c*x^2)^p, x], x, x^ 
n], x] /; FreeQ[{a, c, d, e, m, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[Simplif 
y[m - n + 1], 0]
 

Maple [F]

\[\int \frac {1}{\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}d x\]

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(-sqrt(-c*x^4 + a)*sqrt(e*x^2 + d)/(c*e*x^6 + c*d*x^4 - a*e*x^2 - 
a*d), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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