\(\int \frac {1}{(a^2+2 a b x^2+b^2 x^4)^{2/3}} \, dx\) [670]

Optimal result

Integrand size = 22, antiderivative size = 609 \[ \int \frac {1}{\left (a^2+2 a b x^2+b^2 x^4\right )^{2/3}} \, dx=\frac {3 x \left (a+b x^2\right )}{2 a \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3}}+\frac {3 x \left (1+\frac {b x^2}{a}\right )^{4/3}}{2 \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3} \left (1-\sqrt {3}-\sqrt [3]{1+\frac {b x^2}{a}}\right )}-\frac {3 \sqrt [4]{3} \sqrt {2+\sqrt {3}} a \left (1+\frac {b x^2}{a}\right )^{4/3} \left (1-\sqrt [3]{1+\frac {b x^2}{a}}\right ) \sqrt {\frac {1+\sqrt [3]{1+\frac {b x^2}{a}}+\left (1+\frac {b x^2}{a}\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{1+\frac {b x^2}{a}}\right )^2}} E\left (\arcsin \left (\frac {1+\sqrt {3}-\sqrt [3]{1+\frac {b x^2}{a}}}{1-\sqrt {3}-\sqrt [3]{1+\frac {b x^2}{a}}}\right )|-7+4 \sqrt {3}\right )}{4 b x \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3} \sqrt {-\frac {1-\sqrt [3]{1+\frac {b x^2}{a}}}{\left (1-\sqrt {3}-\sqrt [3]{1+\frac {b x^2}{a}}\right )^2}}}+\frac {3^{3/4} a \left (1+\frac {b x^2}{a}\right )^{4/3} \left (1-\sqrt [3]{1+\frac {b x^2}{a}}\right ) \sqrt {\frac {1+\sqrt [3]{1+\frac {b x^2}{a}}+\left (1+\frac {b x^2}{a}\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{1+\frac {b x^2}{a}}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-\sqrt [3]{1+\frac {b x^2}{a}}}{1-\sqrt {3}-\sqrt [3]{1+\frac {b x^2}{a}}}\right ),-7+4 \sqrt {3}\right )}{\sqrt {2} b x \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3} \sqrt {-\frac {1-\sqrt [3]{1+\frac {b x^2}{a}}}{\left (1-\sqrt {3}-\sqrt [3]{1+\frac {b x^2}{a}}\right )^2}}} \] Output:

3/2*x*(b*x^2+a)/a/(b^2*x^4+2*a*b*x^2+a^2)^(2/3)+3/2*x*(1+b*x^2/a)^(4/3)/(b 
^2*x^4+2*a*b*x^2+a^2)^(2/3)/(1-3^(1/2)-(1+b*x^2/a)^(1/3))-3/4*3^(1/4)*(1/2 
*6^(1/2)+1/2*2^(1/2))*a*(1+b*x^2/a)^(4/3)*(1-(1+b*x^2/a)^(1/3))*((1+(1+b*x 
^2/a)^(1/3)+(1+b*x^2/a)^(2/3))/(1-3^(1/2)-(1+b*x^2/a)^(1/3))^2)^(1/2)*Elli 
pticE((1+3^(1/2)-(1+b*x^2/a)^(1/3))/(1-3^(1/2)-(1+b*x^2/a)^(1/3)),2*I-I*3^ 
(1/2))/b/x/(b^2*x^4+2*a*b*x^2+a^2)^(2/3)/(-(1-(1+b*x^2/a)^(1/3))/(1-3^(1/2 
)-(1+b*x^2/a)^(1/3))^2)^(1/2)+1/2*3^(3/4)*a*(1+b*x^2/a)^(4/3)*(1-(1+b*x^2/ 
a)^(1/3))*((1+(1+b*x^2/a)^(1/3)+(1+b*x^2/a)^(2/3))/(1-3^(1/2)-(1+b*x^2/a)^ 
(1/3))^2)^(1/2)*EllipticF((1+3^(1/2)-(1+b*x^2/a)^(1/3))/(1-3^(1/2)-(1+b*x^ 
2/a)^(1/3)),2*I-I*3^(1/2))*2^(1/2)/b/x/(b^2*x^4+2*a*b*x^2+a^2)^(2/3)/(-(1- 
(1+b*x^2/a)^(1/3))/(1-3^(1/2)-(1+b*x^2/a)^(1/3))^2)^(1/2)
 

Mathematica [C] (verified)

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

Time = 4.86 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.11 \[ \int \frac {1}{\left (a^2+2 a b x^2+b^2 x^4\right )^{2/3}} \, dx=-\frac {x \left (a+b x^2\right ) \left (-3+\sqrt [3]{1+\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {3}{2},-\frac {b x^2}{a}\right )\right )}{2 a \left (\left (a+b x^2\right )^2\right )^{2/3}} \] Input:

Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(-2/3),x]
 

Output:

-1/2*(x*(a + b*x^2)*(-3 + (1 + (b*x^2)/a)^(1/3)*Hypergeometric2F1[1/3, 1/2 
, 3/2, -((b*x^2)/a)]))/(a*((a + b*x^2)^2)^(2/3))
 

Rubi [A] (warning: unable to verify)

Time = 0.76 (sec) , antiderivative size = 576, normalized size of antiderivative = 0.95, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {1385, 215, 233, 833, 760, 2418}

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.

\(\displaystyle \int \frac {1}{\left (a^2+2 a b x^2+b^2 x^4\right )^{2/3}} \, dx\)

\(\Big \downarrow \) 1385

\(\displaystyle \frac {\left (\frac {b x^2}{a}+1\right )^{4/3} \int \frac {1}{\left (\frac {b x^2}{a}+1\right )^{4/3}}dx}{\left (a^2+2 a b x^2+b^2 x^4\right )^{2/3}}\)

\(\Big \downarrow \) 215

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

\(\Big \downarrow \) 233

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

\(\Big \downarrow \) 833

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

\(\Big \downarrow \) 760

\(\displaystyle \frac {\left (\frac {b x^2}{a}+1\right )^{4/3} \left (\frac {3 x}{2 \sqrt [3]{\frac {b x^2}{a}+1}}-\frac {3 a \sqrt {\frac {b x^2}{a}} \left (-\int \frac {-\sqrt [3]{\frac {b x^2}{a}+1}+\sqrt {3}+1}{\sqrt {\frac {b x^2}{a}}}d\sqrt [3]{\frac {b x^2}{a}+1}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (1-\sqrt [3]{\frac {b x^2}{a}+1}\right ) \sqrt {\frac {\left (\frac {b x^2}{a}+1\right )^{2/3}+\sqrt [3]{\frac {b x^2}{a}+1}+1}{\left (-\sqrt [3]{\frac {b x^2}{a}+1}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{\frac {b x^2}{a}+1}+\sqrt {3}+1}{-\sqrt [3]{\frac {b x^2}{a}+1}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {b x^2}{a}} \sqrt {-\frac {1-\sqrt [3]{\frac {b x^2}{a}+1}}{\left (-\sqrt [3]{\frac {b x^2}{a}+1}-\sqrt {3}+1\right )^2}}}\right )}{4 b x}\right )}{\left (a^2+2 a b x^2+b^2 x^4\right )^{2/3}}\)

\(\Big \downarrow \) 2418

\(\displaystyle \frac {\left (\frac {b x^2}{a}+1\right )^{4/3} \left (\frac {3 x}{2 \sqrt [3]{\frac {b x^2}{a}+1}}-\frac {3 a \sqrt {\frac {b x^2}{a}} \left (-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (1-\sqrt [3]{\frac {b x^2}{a}+1}\right ) \sqrt {\frac {\left (\frac {b x^2}{a}+1\right )^{2/3}+\sqrt [3]{\frac {b x^2}{a}+1}+1}{\left (-\sqrt [3]{\frac {b x^2}{a}+1}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{\frac {b x^2}{a}+1}+\sqrt {3}+1}{-\sqrt [3]{\frac {b x^2}{a}+1}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {b x^2}{a}} \sqrt {-\frac {1-\sqrt [3]{\frac {b x^2}{a}+1}}{\left (-\sqrt [3]{\frac {b x^2}{a}+1}-\sqrt {3}+1\right )^2}}}+\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{\frac {b x^2}{a}+1}\right ) \sqrt {\frac {\left (\frac {b x^2}{a}+1\right )^{2/3}+\sqrt [3]{\frac {b x^2}{a}+1}+1}{\left (-\sqrt [3]{\frac {b x^2}{a}+1}-\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {-\sqrt [3]{\frac {b x^2}{a}+1}+\sqrt {3}+1}{-\sqrt [3]{\frac {b x^2}{a}+1}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{\sqrt {\frac {b x^2}{a}} \sqrt {-\frac {1-\sqrt [3]{\frac {b x^2}{a}+1}}{\left (-\sqrt [3]{\frac {b x^2}{a}+1}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {\frac {b x^2}{a}}}{-\sqrt [3]{\frac {b x^2}{a}+1}-\sqrt {3}+1}\right )}{4 b x}\right )}{\left (a^2+2 a b x^2+b^2 x^4\right )^{2/3}}\)

Input:

Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^(-2/3),x]
 

Output:

((1 + (b*x^2)/a)^(4/3)*((3*x)/(2*(1 + (b*x^2)/a)^(1/3)) - (3*a*Sqrt[(b*x^2 
)/a]*((-2*Sqrt[(b*x^2)/a])/(1 - Sqrt[3] - (1 + (b*x^2)/a)^(1/3)) + (3^(1/4 
)*Sqrt[2 + Sqrt[3]]*(1 - (1 + (b*x^2)/a)^(1/3))*Sqrt[(1 + (1 + (b*x^2)/a)^ 
(1/3) + (1 + (b*x^2)/a)^(2/3))/(1 - Sqrt[3] - (1 + (b*x^2)/a)^(1/3))^2]*El 
lipticE[ArcSin[(1 + Sqrt[3] - (1 + (b*x^2)/a)^(1/3))/(1 - Sqrt[3] - (1 + ( 
b*x^2)/a)^(1/3))], -7 + 4*Sqrt[3]])/(Sqrt[(b*x^2)/a]*Sqrt[-((1 - (1 + (b*x 
^2)/a)^(1/3))/(1 - Sqrt[3] - (1 + (b*x^2)/a)^(1/3))^2)]) - (2*Sqrt[2 - Sqr 
t[3]]*(1 + Sqrt[3])*(1 - (1 + (b*x^2)/a)^(1/3))*Sqrt[(1 + (1 + (b*x^2)/a)^ 
(1/3) + (1 + (b*x^2)/a)^(2/3))/(1 - Sqrt[3] - (1 + (b*x^2)/a)^(1/3))^2]*El 
lipticF[ArcSin[(1 + Sqrt[3] - (1 + (b*x^2)/a)^(1/3))/(1 - Sqrt[3] - (1 + ( 
b*x^2)/a)^(1/3))], -7 + 4*Sqrt[3]])/(3^(1/4)*Sqrt[(b*x^2)/a]*Sqrt[-((1 - ( 
1 + (b*x^2)/a)^(1/3))/(1 - Sqrt[3] - (1 + (b*x^2)/a)^(1/3))^2)])))/(4*b*x) 
))/(a^2 + 2*a*b*x^2 + b^2*x^4)^(2/3)
 

Defintions of rubi rules used

rule 215

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) 
/(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1))   Int[(a + b*x^2)^(p + 1 
), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 
*p])
 

rule 233

Int[((a_) + (b_.)*(x_)^2)^(-1/3), x_Symbol] :> Simp[3*(Sqrt[b*x^2]/(2*b*x)) 
   Subst[Int[x/Sqrt[-a + x^3], x], x, (a + b*x^2)^(1/3)], x] /; FreeQ[{a, b 
}, x]
 

rule 760

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], 
s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt[2 - Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s 
*x + r^2*x^2)/((1 - Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[(- 
s)*((s + r*x)/((1 - Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 + Sqrt[3]) 
*s + r*x)/((1 - Sqrt[3])*s + r*x)], -7 + 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x 
] && NegQ[a]
 

rule 833

Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3] 
], s = Denom[Rt[b/a, 3]]}, Simp[(-(1 + Sqrt[3]))*(s/r)   Int[1/Sqrt[a + b*x 
^3], x], x] + Simp[1/r   Int[((1 + Sqrt[3])*s + r*x)/Sqrt[a + b*x^3], x], x 
]] /; FreeQ[{a, b}, x] && NegQ[a]
 

rule 1385

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> S 
imp[a^IntPart[p]*((a + b*x^n + c*x^(2*n))^FracPart[p]/(1 + 2*c*(x^n/b))^(2* 
FracPart[p]))   Int[u*(1 + 2*c*(x^n/b))^(2*p), x], x] /; FreeQ[{a, b, c, n, 
 p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] &&  !IntegerQ[2*p] && NeQ[u, 
 x^(n - 1)] && NeQ[u, x^(2*n - 1)]
 

rule 2418

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N 
umer[Simplify[(1 + Sqrt[3])*(d/c)]], s = Denom[Simplify[(1 + Sqrt[3])*(d/c) 
]]}, Simp[2*d*s^3*(Sqrt[a + b*x^3]/(a*r^2*((1 - Sqrt[3])*s + r*x))), x] + S 
imp[3^(1/4)*Sqrt[2 + Sqrt[3]]*d*s*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/( 
(1 - Sqrt[3])*s + r*x)^2]/(r^2*Sqrt[a + b*x^3]*Sqrt[(-s)*((s + r*x)/((1 - S 
qrt[3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 + Sqrt[3])*s + r*x)/((1 - Sqrt[ 
3])*s + r*x)], -7 + 4*Sqrt[3]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[a] && 
 EqQ[b*c^3 - 2*(5 + 3*Sqrt[3])*a*d^3, 0]
 

Maple [F]

\[\int \frac {1}{\left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{\frac {2}{3}}}d x\]

Input:

int(1/(b^2*x^4+2*a*b*x^2+a^2)^(2/3),x)
 

Output:

int(1/(b^2*x^4+2*a*b*x^2+a^2)^(2/3),x)
 

Fricas [F]

\[ \int \frac {1}{\left (a^2+2 a b x^2+b^2 x^4\right )^{2/3}} \, dx=\int { \frac {1}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac {2}{3}}} \,d x } \] Input:

integrate(1/(b^2*x^4+2*a*b*x^2+a^2)^(2/3),x, algorithm="fricas")
 

Output:

integral((b^2*x^4 + 2*a*b*x^2 + a^2)^(-2/3), x)
 

Sympy [F]

\[ \int \frac {1}{\left (a^2+2 a b x^2+b^2 x^4\right )^{2/3}} \, dx=\int \frac {1}{\left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac {2}{3}}}\, dx \] Input:

integrate(1/(b**2*x**4+2*a*b*x**2+a**2)**(2/3),x)
 

Output:

Integral((a**2 + 2*a*b*x**2 + b**2*x**4)**(-2/3), x)
 

Maxima [F]

\[ \int \frac {1}{\left (a^2+2 a b x^2+b^2 x^4\right )^{2/3}} \, dx=\int { \frac {1}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac {2}{3}}} \,d x } \] Input:

integrate(1/(b^2*x^4+2*a*b*x^2+a^2)^(2/3),x, algorithm="maxima")
 

Output:

integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(-2/3), x)
 

Giac [F]

\[ \int \frac {1}{\left (a^2+2 a b x^2+b^2 x^4\right )^{2/3}} \, dx=\int { \frac {1}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac {2}{3}}} \,d x } \] Input:

integrate(1/(b^2*x^4+2*a*b*x^2+a^2)^(2/3),x, algorithm="giac")
 

Output:

integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(-2/3), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a^2+2 a b x^2+b^2 x^4\right )^{2/3}} \, dx=\int \frac {1}{{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{2/3}} \,d x \] Input:

int(1/(a^2 + b^2*x^4 + 2*a*b*x^2)^(2/3),x)
 

Output:

int(1/(a^2 + b^2*x^4 + 2*a*b*x^2)^(2/3), x)
 

Reduce [F]

\[ \int \frac {1}{\left (a^2+2 a b x^2+b^2 x^4\right )^{2/3}} \, dx=\int \frac {1}{\left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{\frac {2}{3}}}d x \] Input:

int(1/(b^2*x^4+2*a*b*x^2+a^2)^(2/3),x)
                                                                                    
                                                                                    
 

Output:

int(1/(a**2 + 2*a*b*x**2 + b**2*x**4)**(2/3),x)