\(\int \frac {1}{(6-5 x+x^2)^{5/4}} \, dx\) [77]

Optimal result

Integrand size = 12, antiderivative size = 65 \[ \int \frac {1}{\left (6-5 x+x^2\right )^{5/4}} \, dx=\frac {4 (5-2 x)}{\sqrt [4]{6-5 x+x^2}}-\frac {4 \sqrt {2} \sqrt [4]{-6+5 x-x^2} E\left (\left .\frac {1}{2} \arcsin (5-2 x)\right |2\right )}{\sqrt [4]{6-5 x+x^2}} \] Output:

4*(5-2*x)/(x^2-5*x+6)^(1/4)+4*(-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 = 6.92 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.89 \[ \int \frac {1}{\left (6-5 x+x^2\right )^{5/4}} \, dx=\frac {2 (-5+2 x) \left (-2+\sqrt {2} \sqrt [4]{-6+5 x-x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {3}{2},(5-2 x)^2\right )\right )}{\sqrt [4]{6-5 x+x^2}} \] Input:

Integrate[(6 - 5*x + x^2)^(-5/4),x]
 

Output:

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

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(293\) vs. \(2(65)=130\).

Time = 0.51 (sec) , antiderivative size = 293, normalized size of antiderivative = 4.51, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {1089, 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.

\(\displaystyle \int \frac {1}{\left (x^2-5 x+6\right )^{5/4}} \, dx\)

\(\Big \downarrow \) 1089

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

\(\Big \downarrow \) 1094

\(\displaystyle \frac {4 (5-2 x)}{\sqrt [4]{x^2-5 x+6}}-\frac {16 \sqrt {(2 x-5)^2} \int \frac {\sqrt {x^2-5 x+6}}{\sqrt {4 \left (x^2-5 x+6\right )+1}}d\sqrt [4]{x^2-5 x+6}}{5-2 x}\)

\(\Big \downarrow \) 834

\(\displaystyle \frac {4 (5-2 x)}{\sqrt [4]{x^2-5 x+6}}-\frac {16 \sqrt {(2 x-5)^2} \left (\frac {1}{2} \int \frac {1}{\sqrt {4 \left (x^2-5 x+6\right )+1}}d\sqrt [4]{x^2-5 x+6}-\frac {1}{2} \int \frac {1-2 \sqrt {x^2-5 x+6}}{\sqrt {4 \left (x^2-5 x+6\right )+1}}d\sqrt [4]{x^2-5 x+6}\right )}{5-2 x}\)

\(\Big \downarrow \) 761

\(\displaystyle \frac {4 (5-2 x)}{\sqrt [4]{x^2-5 x+6}}-\frac {16 \sqrt {(2 x-5)^2} \left (\frac {\left (2 \sqrt {x^2-5 x+6}+1\right ) \sqrt {\frac {4 \left (x^2-5 x+6\right )+1}{\left (2 \sqrt {x^2-5 x+6}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {2} \sqrt [4]{x^2-5 x+6}\right ),\frac {1}{2}\right )}{4 \sqrt {2} \sqrt {4 \left (x^2-5 x+6\right )+1}}-\frac {1}{2} \int \frac {1-2 \sqrt {x^2-5 x+6}}{\sqrt {4 \left (x^2-5 x+6\right )+1}}d\sqrt [4]{x^2-5 x+6}\right )}{5-2 x}\)

\(\Big \downarrow \) 1510

\(\displaystyle \frac {4 (5-2 x)}{\sqrt [4]{x^2-5 x+6}}-\frac {16 \sqrt {(2 x-5)^2} \left (\frac {\left (2 \sqrt {x^2-5 x+6}+1\right ) \sqrt {\frac {4 \left (x^2-5 x+6\right )+1}{\left (2 \sqrt {x^2-5 x+6}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {2} \sqrt [4]{x^2-5 x+6}\right ),\frac {1}{2}\right )}{4 \sqrt {2} \sqrt {4 \left (x^2-5 x+6\right )+1}}+\frac {1}{2} \left (\frac {\sqrt [4]{x^2-5 x+6} \sqrt {4 \left (x^2-5 x+6\right )+1}}{2 \sqrt {x^2-5 x+6}+1}-\frac {\left (2 \sqrt {x^2-5 x+6}+1\right ) \sqrt {\frac {4 \left (x^2-5 x+6\right )+1}{\left (2 \sqrt {x^2-5 x+6}+1\right )^2}} E\left (2 \arctan \left (\sqrt {2} \sqrt [4]{x^2-5 x+6}\right )|\frac {1}{2}\right )}{\sqrt {2} \sqrt {4 \left (x^2-5 x+6\right )+1}}\right )\right )}{5-2 x}\)

Input:

Int[(6 - 5*x + x^2)^(-5/4),x]
 

Output:

(4*(5 - 2*x))/(6 - 5*x + x^2)^(1/4) - (16*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[Sq 
rt[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 1089

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

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]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

integrate(1/(x^2-5*x+6)^(5/4),x, algorithm="fricas")
 

Output:

integral((x^2 - 5*x + 6)^(3/4)/(x^4 - 10*x^3 + 37*x^2 - 60*x + 36), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate(1/(x^2-5*x+6)^(5/4),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate(1/(x^2-5*x+6)^(5/4),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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