\(\int \frac {1}{\sqrt [4]{(2-x) (3-x)}} \, dx\) [168]

Optimal result

Integrand size = 15, antiderivative size = 45 \[ \int \frac {1}{\sqrt [4]{(2-x) (3-x)}} \, dx=-\frac {\sqrt {2} \sqrt [4]{-6+5 x-x^2} E\left (\left .\frac {1}{2} \arcsin (5-2 x)\right |2\right )}{\sqrt [4]{6-5 x+x^2}} \] Output:

(-x^2+5*x-6)^(1/4)*EllipticE(sin(1/2*arcsin(-5+2*x)),2^(1/2))*2^(1/2)/(x^2 
-5*x+6)^(1/4)
                                                                                    
                                                                                    
 

Mathematica [C] (verified)

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

Time = 0.01 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.22 \[ \int \frac {1}{\sqrt [4]{(2-x) (3-x)}} \, dx=-\frac {(5-2 x) \sqrt [4]{-6+5 x-x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {3}{2},(5-2 x)^2\right )}{\sqrt {2} \sqrt [4]{6-5 x+x^2}} \] Input:

Integrate[((2 - x)*(3 - x))^(-1/4),x]
 

Output:

-(((5 - 2*x)*(-6 + 5*x - x^2)^(1/4)*Hypergeometric2F1[1/4, 1/2, 3/2, (5 - 
2*x)^2])/(Sqrt[2]*(6 - 5*x + x^2)^(1/4)))
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(273\) vs. \(2(45)=90\).

Time = 0.49 (sec) , antiderivative size = 273, normalized size of antiderivative = 6.07, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2048, 1094, 834, 761, 1510}

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.

\(\displaystyle \int \frac {1}{\sqrt [4]{(2-x) (3-x)}} \, dx\)

\(\Big \downarrow \) 2048

\(\displaystyle \int \frac {1}{\sqrt [4]{x^2-5 x+6}}dx\)

\(\Big \downarrow \) 1094

\(\displaystyle -\frac {4 \sqrt {(2 x-5)^2} \int \frac {\sqrt {x^2-5 x+6}}{\sqrt {4 \left (x^2-5 x+6\right )+1}}d\sqrt [4]{x^2-5 x+6}}{5-2 x}\)

\(\Big \downarrow \) 834

\(\displaystyle -\frac {4 \sqrt {(2 x-5)^2} \left (\frac {1}{2} \int \frac {1}{\sqrt {4 \left (x^2-5 x+6\right )+1}}d\sqrt [4]{x^2-5 x+6}-\frac {1}{2} \int \frac {1-2 \sqrt {x^2-5 x+6}}{\sqrt {4 \left (x^2-5 x+6\right )+1}}d\sqrt [4]{x^2-5 x+6}\right )}{5-2 x}\)

\(\Big \downarrow \) 761

\(\displaystyle -\frac {4 \sqrt {(2 x-5)^2} \left (\frac {\left (2 \sqrt {x^2-5 x+6}+1\right ) \sqrt {\frac {4 \left (x^2-5 x+6\right )+1}{\left (2 \sqrt {x^2-5 x+6}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {2} \sqrt [4]{x^2-5 x+6}\right ),\frac {1}{2}\right )}{4 \sqrt {2} \sqrt {4 \left (x^2-5 x+6\right )+1}}-\frac {1}{2} \int \frac {1-2 \sqrt {x^2-5 x+6}}{\sqrt {4 \left (x^2-5 x+6\right )+1}}d\sqrt [4]{x^2-5 x+6}\right )}{5-2 x}\)

\(\Big \downarrow \) 1510

\(\displaystyle -\frac {4 \sqrt {(2 x-5)^2} \left (\frac {\left (2 \sqrt {x^2-5 x+6}+1\right ) \sqrt {\frac {4 \left (x^2-5 x+6\right )+1}{\left (2 \sqrt {x^2-5 x+6}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {2} \sqrt [4]{x^2-5 x+6}\right ),\frac {1}{2}\right )}{4 \sqrt {2} \sqrt {4 \left (x^2-5 x+6\right )+1}}+\frac {1}{2} \left (\frac {\sqrt [4]{x^2-5 x+6} \sqrt {4 \left (x^2-5 x+6\right )+1}}{2 \sqrt {x^2-5 x+6}+1}-\frac {\left (2 \sqrt {x^2-5 x+6}+1\right ) \sqrt {\frac {4 \left (x^2-5 x+6\right )+1}{\left (2 \sqrt {x^2-5 x+6}+1\right )^2}} E\left (2 \arctan \left (\sqrt {2} \sqrt [4]{x^2-5 x+6}\right )|\frac {1}{2}\right )}{\sqrt {2} \sqrt {4 \left (x^2-5 x+6\right )+1}}\right )\right )}{5-2 x}\)

Input:

Int[((2 - x)*(3 - x))^(-1/4),x]
 

Output:

(-4*Sqrt[(-5 + 2*x)^2]*((((6 - 5*x + x^2)^(1/4)*Sqrt[1 + 4*(6 - 5*x + x^2) 
])/(1 + 2*Sqrt[6 - 5*x + x^2]) - ((1 + 2*Sqrt[6 - 5*x + x^2])*Sqrt[(1 + 4* 
(6 - 5*x + x^2))/(1 + 2*Sqrt[6 - 5*x + x^2])^2]*EllipticE[2*ArcTan[Sqrt[2] 
*(6 - 5*x + x^2)^(1/4)], 1/2])/(Sqrt[2]*Sqrt[1 + 4*(6 - 5*x + x^2)]))/2 + 
((1 + 2*Sqrt[6 - 5*x + x^2])*Sqrt[(1 + 4*(6 - 5*x + x^2))/(1 + 2*Sqrt[6 - 
5*x + x^2])^2]*EllipticF[2*ArcTan[Sqrt[2]*(6 - 5*x + x^2)^(1/4)], 1/2])/(4 
*Sqrt[2]*Sqrt[1 + 4*(6 - 5*x + x^2)])))/(5 - 2*x)
 

Defintions of rubi rules used

rule 761

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

rule 834

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

rule 1094

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[4*(Sqrt[(b 
+ 2*c*x)^2]/(b + 2*c*x))   Subst[Int[x^(4*(p + 1) - 1)/Sqrt[b^2 - 4*a*c + 4 
*c*x^4], x], x, (a + b*x + c*x^2)^(1/4)], x] /; FreeQ[{a, b, c}, x] && Inte 
gerQ[4*p]
 

rule 1510

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = 
 Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d* 
(1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q*Sqrt[a + c*x^4]))*E 
llipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e 
}, x] && PosQ[c/a]
 

rule 2048

Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))*((c_) + (d_.)*(x_)^(n_.)))^(p_) 
, x_Symbol] :> Int[u*(a*c*e + (b*c + a*d)*e*x^n + b*d*e*x^(2*n))^p, x] /; F 
reeQ[{a, b, c, d, e, n, p}, x]
 

Maple [F]

\[\int \frac {1}{\left (\left (2-x \right ) \left (3-x \right )\right )^{\frac {1}{4}}}d x\]

Input:

int(1/((2-x)*(3-x))^(1/4),x)
 

Output:

int(1/((2-x)*(3-x))^(1/4),x)
 

Fricas [F]

\[ \int \frac {1}{\sqrt [4]{(2-x) (3-x)}} \, dx=\int { \frac {1}{\left ({\left (x - 2\right )} {\left (x - 3\right )}\right )^{\frac {1}{4}}} \,d x } \] Input:

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

Output:

integral((x^2 - 5*x + 6)^(-1/4), x)
 

Sympy [F]

\[ \int \frac {1}{\sqrt [4]{(2-x) (3-x)}} \, dx=\int \frac {1}{\sqrt [4]{\left (2 - x\right ) \left (3 - x\right )}}\, dx \] Input:

integrate(1/((2-x)*(3-x))**(1/4),x)
 

Output:

Integral(((2 - x)*(3 - x))**(-1/4), x)
 

Maxima [F]

\[ \int \frac {1}{\sqrt [4]{(2-x) (3-x)}} \, dx=\int { \frac {1}{\left ({\left (x - 2\right )} {\left (x - 3\right )}\right )^{\frac {1}{4}}} \,d x } \] Input:

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

Output:

integrate(((x - 2)*(x - 3))^(-1/4), x)
 

Giac [F]

\[ \int \frac {1}{\sqrt [4]{(2-x) (3-x)}} \, dx=\int { \frac {1}{\left ({\left (x - 2\right )} {\left (x - 3\right )}\right )^{\frac {1}{4}}} \,d x } \] Input:

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

Output:

integrate(((x - 2)*(x - 3))^(-1/4), x)
 

Mupad [F(-1)]

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

int(1/((x - 2)*(x - 3))^(1/4),x)
 

Output:

int(1/((x - 2)*(x - 3))^(1/4), x)
 

Reduce [F]

\[ \int \frac {1}{\sqrt [4]{(2-x) (3-x)}} \, dx=\int \frac {1}{\left (x^{2}-5 x +6\right )^{\frac {1}{4}}}d x \] Input:

int(1/((2-x)*(3-x))^(1/4),x)
 

Output:

int(1/(x**2 - 5*x + 6)**(1/4),x)