\(\int \frac {1}{x^2 \sqrt {a-b x^4}} \, dx\) [222]

Optimal result

Integrand size = 16, antiderivative size = 128 \[ \int \frac {1}{x^2 \sqrt {a-b x^4}} \, dx=-\frac {\sqrt {a-b x^4}}{a x}-\frac {\sqrt [4]{b} \sqrt {1-\frac {b x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{a} \sqrt {a-b x^4}}+\frac {\sqrt [4]{b} \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{a} \sqrt {a-b x^4}} \] Output:

-(-b*x^4+a)^(1/2)/a/x-b^(1/4)*(1-b*x^4/a)^(1/2)*EllipticE(b^(1/4)*x/a^(1/4 
),I)/a^(1/4)/(-b*x^4+a)^(1/2)+b^(1/4)*(1-b*x^4/a)^(1/2)*EllipticF(b^(1/4)* 
x/a^(1/4),I)/a^(1/4)/(-b*x^4+a)^(1/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.01 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.39 \[ \int \frac {1}{x^2 \sqrt {a-b x^4}} \, dx=-\frac {\sqrt {1-\frac {b x^4}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},\frac {b x^4}{a}\right )}{x \sqrt {a-b x^4}} \] Input:

Integrate[1/(x^2*Sqrt[a - b*x^4]),x]
 

Output:

-((Sqrt[1 - (b*x^4)/a]*Hypergeometric2F1[-1/4, 1/2, 3/4, (b*x^4)/a])/(x*Sq 
rt[a - b*x^4]))
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.05, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {847, 836, 27, 765, 762, 1390, 1389, 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.

\(\displaystyle \int \frac {1}{x^2 \sqrt {a-b x^4}} \, dx\)

\(\Big \downarrow \) 847

\(\displaystyle -\frac {b \int \frac {x^2}{\sqrt {a-b x^4}}dx}{a}-\frac {\sqrt {a-b x^4}}{a x}\)

\(\Big \downarrow \) 836

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 765

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

\(\Big \downarrow \) 762

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

\(\Big \downarrow \) 1390

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

\(\Big \downarrow \) 1389

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

\(\Big \downarrow \) 327

\(\displaystyle -\frac {b \left (\frac {a^{3/4} \sqrt {1-\frac {b x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{b^{3/4} \sqrt {a-b x^4}}-\frac {a^{3/4} \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{b^{3/4} \sqrt {a-b x^4}}\right )}{a}-\frac {\sqrt {a-b x^4}}{a x}\)

Input:

Int[1/(x^2*Sqrt[a - b*x^4]),x]
 

Output:

-(Sqrt[a - b*x^4]/(a*x)) - (b*((a^(3/4)*Sqrt[1 - (b*x^4)/a]*EllipticE[ArcS 
in[(b^(1/4)*x)/a^(1/4)], -1])/(b^(3/4)*Sqrt[a - b*x^4]) - (a^(3/4)*Sqrt[1 
- (b*x^4)/a]*EllipticF[ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/(b^(3/4)*Sqrt[a - 
 b*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 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 762

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) 
)*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] 
 && GtQ[a, 0]
 

rule 765

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[Sqrt[1 + b*(x^4/a)]/Sqrt 
[a + b*x^4]   Int[1/Sqrt[1 + b*(x^4/a)], x], x] /; FreeQ[{a, b}, x] && NegQ 
[b/a] &&  !GtQ[a, 0]
 

rule 836

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

rule 847

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x 
)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + n*(p + 1) 
+ 1)/(a*c^n*(m + 1)))   Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a 
, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p 
, x]
 

rule 1389

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[d/Sq 
rt[a]   Int[Sqrt[1 + e*(x^2/d)]/Sqrt[1 - e*(x^2/d)], x], x] /; FreeQ[{a, c, 
 d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] && GtQ[a, 0]
 

rule 1390

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[Sqrt 
[1 + c*(x^4/a)]/Sqrt[a + c*x^4]   Int[(d + e*x^2)/Sqrt[1 + c*(x^4/a)], x], 
x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] &&  !GtQ 
[a, 0] &&  !(LtQ[a, 0] && GtQ[c, 0])
 

Maple [A] (verified)

Time = 0.60 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.83

method	result	size
default	\(-\frac {\sqrt {-b \,x^{4}+a}}{a x}+\frac {\sqrt {b}\, \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )\right )}{\sqrt {a}\, \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\)	\(106\)
risch	\(-\frac {\sqrt {-b \,x^{4}+a}}{a x}+\frac {\sqrt {b}\, \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )\right )}{\sqrt {a}\, \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\)	\(106\)
elliptic	\(-\frac {\sqrt {-b \,x^{4}+a}}{a x}+\frac {\sqrt {b}\, \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )\right )}{\sqrt {a}\, \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\)	\(106\)

Input:

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

Output:

-(-b*x^4+a)^(1/2)/a/x+b^(1/2)/a^(1/2)/(1/a^(1/2)*b^(1/2))^(1/2)*(1-b^(1/2) 
*x^2/a^(1/2))^(1/2)*(1+b^(1/2)*x^2/a^(1/2))^(1/2)/(-b*x^4+a)^(1/2)*(Ellipt 
icF(x*(1/a^(1/2)*b^(1/2))^(1/2),I)-EllipticE(x*(1/a^(1/2)*b^(1/2))^(1/2),I 
))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.53 \[ \int \frac {1}{x^2 \sqrt {a-b x^4}} \, dx=-\frac {\sqrt {a} x \left (\frac {b}{a}\right )^{\frac {3}{4}} E(\arcsin \left (x \left (\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) - \sqrt {a} x \left (\frac {b}{a}\right )^{\frac {3}{4}} F(\arcsin \left (x \left (\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) + \sqrt {-b x^{4} + a}}{a x} \] Input:

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

Output:

-(sqrt(a)*x*(b/a)^(3/4)*elliptic_e(arcsin(x*(b/a)^(1/4)), -1) - sqrt(a)*x* 
(b/a)^(3/4)*elliptic_f(arcsin(x*(b/a)^(1/4)), -1) + sqrt(-b*x^4 + a))/(a*x 
)
 

Sympy [A] (verification not implemented)

Time = 0.50 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.32 \[ \int \frac {1}{x^2 \sqrt {a-b x^4}} \, dx=\frac {\Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {1}{2} \\ \frac {3}{4} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt {a} x \Gamma \left (\frac {3}{4}\right )} \] Input:

integrate(1/x**2/(-b*x**4+a)**(1/2),x)
 

Output:

gamma(-1/4)*hyper((-1/4, 1/2), (3/4,), b*x**4*exp_polar(2*I*pi)/a)/(4*sqrt 
(a)*x*gamma(3/4))
 

Maxima [F]

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

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

Output:

integrate(1/(sqrt(-b*x^4 + a)*x^2), x)
 

Giac [F]

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

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

Output:

integrate(1/(sqrt(-b*x^4 + a)*x^2), x)
 

Mupad [B] (verification not implemented)

Time = 0.47 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.32 \[ \int \frac {1}{x^2 \sqrt {a-b x^4}} \, dx=-\frac {\sqrt {1-\frac {a}{b\,x^4}}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {3}{4};\ \frac {7}{4};\ \frac {a}{b\,x^4}\right )}{3\,x\,\sqrt {a-b\,x^4}} \] Input:

int(1/(x^2*(a - b*x^4)^(1/2)),x)
 

Output:

-((1 - a/(b*x^4))^(1/2)*hypergeom([1/2, 3/4], 7/4, a/(b*x^4)))/(3*x*(a - b 
*x^4)^(1/2))
 

Reduce [F]

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

int(1/x^2/(-b*x^4+a)^(1/2),x)
 

Output:

int(sqrt(a - b*x**4)/(a*x**2 - b*x**6),x)